Poincaré Against Foundationalists Old and New

Michael Thomas Sands, Jr.
Submitted in partial fulfillment of the requirements
for the degree of Master of Philosophy
at the University of St Andrews.

August 24, 2009

Abstract: The early 20th century witnessed concerted research in foundationalism
in mathematics. Those pursuing a basis for mathematics included Hilbert, Russell,Zermelo, Frege, and Dedekind. They found a vocal opponent in Poincaré, whose
attacks were numerous, vituperative, and often indiscriminant. One of the objections
was the petitio argument that claimed a circularity in foundationalist arguments. Any
derivation of mathematical axioms from a supposedly simpler system would employ
induction, one of the very axioms purportedly derived.
Historically, these attacks became somewhat moot as both Frege and Hilbert had
their programs devastated—Frege’s by Russell’s paradox and Hilbert’s by Gödel’s
incompleteness result. However, the publication of Frege’s Conception of Numbers as Objects [63] by Crispin Wright began the neo-logicist program of reviving
Frege’s project while avoiding Russell’s paradox. The neo-logicist holds that Frege’s
theorem—the derivation of mathematical axioms from Hume’s Principle(HP) andsecond-order logic—combined with the transparency of logic and the analyticity of
HP guarantees knowledge of numbers. Moreover, the neo-logicist conception of language and reality as inextricably intertwined guarantees the objective existence of
numbers. In this context, whether or not a revived version of the petitio objection
can be made against the revived logicist project.
The current project investigates Poincaré’s philosophy of arithmetic—his psychologism, conception of intuition, and understanding of induction, and then evaluates
the effectiveness of his petitio objection against three foundationalist groups: Hilbert’s
early and late programs, the logicists, and the neo-logicists.

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Contents

1 Introduction

4

2 Poincaré

7

2.1

2.2

2.3

2.4

Poincaré’s Psychologism . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.1

Science Advancing Psychology . . . . . . . . . . . . . . . . . .

9

2.1.2

Psychology Advancing Science . . . . . . . . . . . . . . . . . .

10

2.1.3

Psychological Principles as Bedrock . . . . . . . . . . . . . . .

11

2.1.4

Permissibility of Psychological Arguments . . . . . . . . . . .

12Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2.1

Mathematical Ability . . . . . . . . . . . . . . . . . . . . . . .

17

2.2.2

Temporal Progression . . . . . . . . . . . . . . . . . . . . . . .

22

2.2.3

Psychological Necessity . . . . . . . . . . . . . . . . . . . . . .

28

Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.3.1

Induction as Moving from the Particular to the General . . . .

34

2.3.2

Induction and Temporal Progression . . . . . . . . . . . . . .

36

2.3.3

Induction and Mathematical Ability . . . . . . . . . . . . . . .

37

2.3.4

Induction Justified by Psychological Necessity . . . . . . . . .

39

Kant’s Philosophy of Arithmetic . . . . . . . . . . . . . . . . . . . . .

40

2.4.1

Syntheticity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.4.2

Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

421

3 Hilbert

44

3.1

Early Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.1.1

Hilbert’s Foundation of Geometry . . . . . . . . . . . . . . . .

45

3.1.2

Initial Sketches of the Consistency of Arithmetic . . . . . . . .

51

3.1.3

Poincaré’s Petitio against Early Hilbert . . . . . . . . . . . . .

59

The Development of Hilbert’s 1920s Program . . . . . . . . . . . . . .

61

3.2.1

Theory vs. Metatheory: 1922 . . . . . . . . . . . . . . . . . .

62

3.2.2

Finite vs. Transfinite: 1923 . . . . . . . . . . . . . . . . . . .

65

3.2.3

Ideal vs. Concrete: 1925 . . . . . . . . . . . . . . . . . . . . .

67

3.2.4

Formal vs. Contentual . . . . . . . . . . . . . . . . . . . . . .

72

3.2

3.3

3.4

Hilbert’s Reply to Poincaré. . . . . . . . . . . . . . . . . . . . . . .

75

3.3.1

Giaquinto’s Hilbert . . . . . . . . . . . . . . . . . . . . . . . .

77

3.3.2

Detlefsen’s Hilbert . . . . . . . . . . . . . . . . . . . . . . . .

79

Poincaré Redeemed . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4 Poincare against the Logicists

89

4.1

Original Petitio and the Logicist . . . . . . . . . . . . . . . . . . . . .

90

4.2

Sophisticated Petitio . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5 Neo-Logicist
5.1

5.2

5.3

99

Neo-Logicism: Reviving Frege . . . . . . . . . . . . . . . . . . . . . . 100
5.1.1

The Failure of Frege and Basic Law V . . . . . . . . . . . . . 101

5.1.2

The Neo-Logicist Project . . . . . . . . . . . . . . . . . . . . . 106

5.1.3

Proof of the Consistency of HP with Second-Order Logic . . . 111

Respective Goals of Logicism and Neo-Logicism . . . . . . . . . . . . 114
5.2.1

Ontological Thesis . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.2

Strong Epistemic Thesis . . . . . . . . . . . . . . . . . . . . . 118

Poincaréan objections posed to Neo-Logicism . . . . . . . . . . . . . . 121
2

5.3.1The Sophisticated Petitio and the Mathematical Quality ofSecond-Order Logic . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3.2

Consistency Proofs and Bad Company Objections . . . . . . . 126

6 Conclusion

133

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3

Chapter 1
Introduction
The early 20th century witnessed a huge rise in interest in foundationalism in mathematics. Those pursuing a solid foundation for mathematics included Hilbert, Russell,Zermelo, Frege, and Dedekind. They found a vocal opponent in Poincaré, whose attacks were numerous, vituperative, and often indiscriminant. One of the objections
was the petitio argument that claimed a circularity in the foundationalist argument.
Any derivation of mathematical axioms from a supposedly simpler system would employ induction, one of the very axioms purportedly derived.
Historically, these attacks became somewhat moot as both Frege and Hilbert had
their programs devestated—Frege’s by Russell’s paradox and Hilbert’s by Gödel’s
incompleteness result.

However, the publication of Frege’s Conception of Num-bers as Objects [63] by Crispin Wright began the neo-logicist program of reviving
Frege’s project while avoiding Russell’s paradox. The neo-logicist holds that Frege’s
theorem—the derivation of mathematical axioms from Hume’s Principle(HP) andsecond-order logic—combined with the transparency of logic and the analyticity of
HP guarantees knowledge of numbers. Moreover, the neo-logicist conception of language and reality as inextricably intertwined guarantees the objective existence of
numbers. In this context, whether or not a revived version of the petitio objection
4

can be made against the revived logicist project.
The current piece has an ultimate goal of demonstrating that a revived Poincaréanobjection can be posed to the neo-logicist and that the neo-logicist does not currently
offer a solution. Along the way, I hope to achieve a number of historical and interpretive goals.
Chapter 2 has two separate interpretive goals. The first is to explicate Poincaré’s
usage of ‘intuition’ and demonstrate both its origins in Poincaré’s thoroughly psychologistic background and its effect on his conception of induction. Poincaré’s intuition will be separated into three distinct threads—mathematical ability, temporalawareness, and psychological necessity. Secondly, the chapter aims to differentiate
Poincaré’s views on the philosophy of mathematics from those of Kant, whose positions Poincaré largely adopts as his own. The chapter focuses on Poincaré’s philosophyof arithmetic independently of his philosophy of geometry or science.
Chapter 3 divides Hilbert’s work into the early and late periods. For the former, I will demonstrate exactly how Poincaré’s petitio objection is effective; I will
give Hilbert’s proof of the consistency of the Peano axioms and highlight where thisproof relies on induction. The second half of the chapter looks to impose an intepretive structure on Hilbert’s papers of the 1920s. This structure will identify
four contrasts fundamental to Hilbert’s program and help evaluate a reformulated
Poincaréan objection. The chapter also whether either of two other reconstructions,
Michael Detlefesen’s or Marcus Giaquinto’s, fare any better (they do not).
Chapter 4 recapitulates the argument given by Warren Goldfarb [23] that Poincaré’s
objection was not effective against the original logicists because of their unique conception of logic. The goal of this recapitulation is twofold. First, it demonstrates why
a similar defense for the neo-logicists is not possible. Second, it shows how the petitio
could fail, in contrast to its success against Hilbert’s program.
Finally, Chapter 5 spells out the neo-logicist program in detail and lodges two5

revised Poincaréan objections. The chapter also compares these objections to preexisting ones so as to throw those into a Poincaréan light. Ultimately, it does notargue that these objections are knock down arguments, but rather that they require
an explanation from the neo-logicist that does not currently exist.

6

Chapter 2
Poincaré
The current chapter has two distinct and sometimes competing goals. The first is
historical; it seeks to give an accurate account of Poincaré’s views on the mathematical
notions of intuition and induction. Moreover, it looks to contrast these views withKant’s philosophy of arithmetic, which Poincaré is often labeled as following. One of
the main problems with an attempt to accurately represent Poincaré’s position is that
his writings on arithmetic are scant and not always self-coherent. The second goalis therefore to impose an interpretive structure on Poincaré’s writing so as to makehis position a coherent whole. Both goals can be achieved by proposing a structurestrongly supported by textual evidence.
To this end, the chapter will first identify four distinct aspects of Poincaré’s
psychologism. Second, it will identify three notions of mathematical intuition in
Poincaré’s work and trace their roots back to elements of his psychologism. Third, it
will argue that these three underlying conceptions directly inform Poincaré’s views onmathematical induction, which lies not only at the heart of mathematical science forPoincaré, but also as the basis for scientific progress. This conclusion will demonstratethat Poincaré’s petitio objection stems immediately from his conception of induction.
Finally, it will briefly present Kantian notions of intuition and syntheticity in order7

to argue that they differ significantly from Poincaré’s conception of those concepts,
thereby demonstrating in what way Poincaré’s philosophy of arithmetic differs from
Kant’s.
Secondary literature directly pertaining Poincaré’s philosophy of arithmetic issparse, with most commentators focusing on Poincaré’s broader philosophy of science(see Zahar [66] or Ben-Menahem [1] recent takes) or else his philosophy of geometry(Torretti [62]). Dedicated manuscripts primarily about his philosophy of arithmetic
appear from Folina [17], de Lorenzo [10], and Mooij [39], with only the first in English. Much of the scarcity can probably be attributed to the limited quantity ofPoincaré’s writings directly on the subject. Instead, much of his philosophy of arithmetic, intuition, and induction must be gleaned from his objections to Hilbert and the
Logicists. Some of the blame may be due to the fact that while Poincaré’s views on
geometry differed significantly from Kant, his views on arithmetic are seen to ape his
predecessor. While true generally, I will discuss two aspects of Poincaré’s philosophyof arithmetic that significantly differ from Kant—intuition and syntheticity.
One goal not taken up by this chapter is an for the correctness of Poincaré’s views.
There is an inherent plausibility in the formulation of these principles, and this chapter
will attempt to present them in just such a favorable light. However, I will not provide
detailed arguments for why Poincaré is ultimately correct in his views. Ultimately,
the dialectical role of this chapter is to clarify Poincaré’s philosophy of arithmetic,
to expose differences between Poincaré and Kant, and to lay bare the assumptions
brought to bear in Poincaréan objections to the logicists and neo-logicists. 1

1

Page citations of Poincaré are from [45], [46], and [44] where appropriate. Where possible, I
have given the paper citations from the original paper as republished in Ewald [14].

8

2.1

Poincaré’s Psychologism

The lens through which one must view Poincaré is thoroughly psychologistic; his
psychologism is not limited to his belief that logic is psychological because the laws oflogic are the laws of thought, nor is it limited to his belief that logics is psychologicalbecause it is ultimately justified on psychological grounds. Rather, psychology plays afundamental role in all aspects of Poincaré’s philosophy. This role can be categorized
by four principles.
1. Regardless of field, psychological principles are the ultimate bedrock upon which
justifications rest.
2. Arguments from psychological grounds are permissible and convincing.
The relationship between human psychology and the sciences is bilateral:
1. Advances in science can lead to greater understanding of the human psyche.
2. Our psychological experience of these subjects must accord with new theories
and can weigh in favor of a particular theory over another.

2.1.1

Science Advancing Psychology[Mathematical science] reflecting upon itself is reflecting upon the humanmind which has created it; the more so because, of all its creations, mathematics is the one for which it has borrowed the least from the outside[45, p. 36]
By examining the products of the human mind—by reflecting upon the structure the mind places on the world—one can come to know more about the structuresof human psychology, Poincaré argues. These structures cannot help but reflect thestructure of the mind that created them (Compare to the neo-logicist view of language9(§5.1.2). These structures are simply the scientific theories created to explain phenomena. Among these, mathematics holds a privileged place because it is informed
the least by empirical data—it springs forth from the mind most “purely”. Therefore,
it bears most strongly the psychological structures of the human mind, and its study
is best suited to reveal deep-seated psychological rules. Not only is mathematics most
likely to be fruitful on examination, but also most likely to reveal the fundamental
principles of human psychology.
Poincaré writes,
The genesis of the mathematical discovery is a problem which must inspire
the psychologist with the keenest interest. For this is the process in which
the human mind seems to borrow least from the exterior world, in which
it acts, or appears to act, only by itself and on itself, so that by studying
the process of geometric thought we may hope to arrive at what is mostessential in the human mind [45, p.46].

2.1.2

Psychology Advancing Science

This method [of the logicists] is evidently contrary to all healthy psychology. It is certainly not in this manner that the human mind proceeded toconstruct mathematics” [45, pp.144-5].

Poincaré here objects to the logicist project by arguing that the construction of
number from logical principles is not in accordance with the psychological historyof the construction of mathematics. Given Poincaré’s view that mathematics andarithmetic are structures of the human mind, any theory that correctly explains theirfoundations must be analogous to the personal psychological experience of learning
mathematics(that personal experience is itself analogous to the original psychological construction of mathematics). Thus, reflection on psychological experiences atlearning mathematics can and should, according to Poincaré, act as a guide to mathematical theory.
10

Poincaré goes further and argues that not only can rational cognizing on psychological experiences bring about scientific advances, but that psychological experiences
themselves play an integral role in scientific examination, and thus in the science’s
epistemic justification. This position is put forth in an extended discussion of theprocess by which he discovered a particular theorem about Fuschian groups (Cf. [45,p.54]) Poincaré argues that aesthetic sense of the mathematician recognizes the inherent elegance of the correct proof and thus functions as a reliable tool. That themathematician possesses an ability to discern correct proofs by their aesthetic value
reinforces the view that mathematics is a pure product of the human mind. The
mathematician’s appreciation can be seen as an unconscious recognition of the accordance of the mathematical proof with pre-existent psychological structures. That
psychological principles and experiences should be followed to scientific discovery isa consequence of Poincaré’s view that science is a psychological structure.

2.1.3

Psychological Principles as Bedrock

We cannot believe that two quantities which are equal to a third are notequal to one another, and we are thus led to suppose that A is different
from B, and B from C [45, pp.22-3].
Poincaré holds that the only ultimate justification of bedrock principles are psychological in nature. Consider the Principle of Non-Contradiction. Poincaré’s justification of it above is not logical, but rather based on our psychological inability toconceive that it is false. It is a product of psychological necessity. The unavoidable
belief in the principle is what justifies it, not a sense of logical validity. Thus, ifeven the Law of Non-Contradiction has a psychological epistemic basis, than these
psychological rules are bedrock.
It should be noted that these psychologically justified bedrock propositions inform
Poincaré’s definition of the synthetic a priori :11

. . . the term employed by Kant to designate the judgments that can neither be demonstrated analytically, nor reduced to identity, nor establishedexperimentally [46, p.146].
Something that cannot be reduced to identity, established experimentally, nordemonstrated analytically must be something that is self-evident upon reflection,
for these options exhaust all other potential avenues of justification. This notion ofself-evidentness, for Poincaré, is nothing more than psychological inevitability.2 For
Poincaré, psychological necessity is sufficient to constitute epistemic justification.

2.1.4

Permissibility of Psychological Arguments

How does it happen that there are people who do not understand mathematics? If the science invokes only the rules of logic, those accepted by all
well-formed minds, if its evidence is founded on principles that are common to all men, and that none but a madman would attempt to deny, howdoes it happen that there are so many people who are entirely impervious
to it [45, p.47]?
In arguing mathematics being solely logical, Poincaré employs a purely psychological argument, consisting wholly of premises about human understanding and human
psychology, yet making a metaphysical conclusion. This particular argument takes
the following form:
P1 If mathematics were described wholly by the rules of logic, then everyone would
be able to understand it;
P2 Some people do not understand mathematics;
Concl. Therefore, mathematics is not described wholly by the rules of logic.
P2 is true and Concl. follows from P1 and P2. However, Poincaré needs to justify
P1, and here is where Poincaré employs psychological principles as bedrock justification. Accordingly, his argument is outlined thus:
2

Compare this notion of the synthetic priori for Kant (§2.4).

12

P1 If mathematics were described wholly by the rules of logic, then everyone would
be able to understand it.
a) The rules of logic are self-evident upon reflection.
b) The rules of logic are universally valid, across space and time.
c) If the rules of logic are universal, then they are shared by everyone.
d) The rules of logic are shared by everyone. [From b) and c)]e) If the rules of logic are shared by everyone and they are self-evident uponreflection, then everyone has access to them, upon reflection.
Subconcl. Everyone has access to the rules of logic, upon reflection. [from a),d), and e)]
P2 Some people do not understand mathematics, even when it is explained to them.
Concl. Therefore, mathematics is not described wholly by the rules of logic.
Note that a) is a psychological claim. That the rules of logic are self-evident uponreflection is a claim about what the human mind must believe; it claims that therules of logic are forced upon human understanding by the very nature of human
psychology. This rule of human psychology is the basis for P1, and consequently theentire argument. Thus, Poincaré’s admission of psychological argument can be seen
as another aspect of his admission of psychological principles as bedrock.

2.2

Intuition

Poincaré bases much of his views on the foundations of mathematics on intuition.
However, he presents no clear coherent conception of intuition. There is no evolving
notion that comes clear. Nor exists a single thread from which one can draw a single latent principle. Rather, there appear three separate, yet inextricable aspects of13

intuition in Poincaré’s usage that he himself often conflates. These three are mathematical ability, grasp of the succession of moments, and immediate psychologicalacquaintance.
The first we have seen the foundation of in Poincaré’s psychologism. There, heclaim that part of a mathematician’s ability derived from an unconscious aesthetic
appreciation of mathematics. This appreciation is something that can be honed, but
only in the presence of a pre-existing talent. Some people have it and are able toundertake mathematical research, while others do not. Poincaré describes it in thefollowing passage:
We can understand that this feeling, this intuition of mathematical order,
which enables us to guess hidden harmonies and relations, cannot belong
to every one. Some have neither this delicate feeling that is difficult todefine, nor a power of memory and attention above the common, andso they are absolutely incapable of understanding even the first steps ofhigher mathematics. This applies to the majority of people. Other havethe feeling only in a slight degree, but they are gifted with an uncommon
memory and a great capacity for attention. They learn the details one
after the other by heart, they can understand mathematics and sometimes
apply them, but they are not in a condition to create. Lastly, others
possess the special intuition I have spoken of more of less highly developed,
and they can not only understand mathematics, even though their memory
is in no way extraordinary, but they can become creators, and seek to makediscovery with more or less chance of success, according as their intuition
is more or less developed [45, p. 50].
Intuition as mathematical ability allows one to foresee the direction of proofs
and perceive the correct path. As such, it demonstrates that Poincaré adheres to a
Kantian position of mathematics as synthetic. A mathematical proof is not a seriesof deductive steps resulting in a theorem. Instead, every step in the proof requires anintuitive leap made possible by the mathematician’s ability.
The second aspect of Poincaré’s intuition is an immediate, instinctual grasp of
temporal progression. More precisely, an understanding of mathematical notions canonly be achieved against the backdrop of a succession of instants of time. Mathe14matical reasoning is done in a temporal context. This progression of time is not acontinuum, but instead instants following after instants, cascading one after anotherconstantly. It is an instinctual grasp of moments in time being well-ordered, but
discontinuous.
The succession of instants to which we have immediate, instinctual access isanalogous to the rationals embedded in the real line. Rational numbers exist prior to
the irrationals, and hence real numbers. The fact that one learns about the rational
numbers before the reals guarantees this fact for Poincaré (Cf. §2.1.2). In fact, thedefinition of real numbers—and hence human understanding for Poincaré—is built
out of and in reference to the rationals. The two main definitions that in mind
here are Cauchy sequences and Dedekind cuts. They are both comprised of sets ofrational numbers and postulate the existence of a new number from these sets. That
these definitions make reference to rationals reflects for Poincaré the structure of
our psychology. Our understanding of the rationals is prior to our understanding ofirrationals.
Similarly, our grasp of the succession of instants is psychologically (and therefore
epistemically) prior to our grasp of a notion of a continuity of time—our pre-theoretic
notion of “moment succession” is the foundation out of which our theoretical notionof the continuity of time springs. There are many additional parallels to be drawn
between the two relationships.
First, just as there are infinitely many rationals, there are infinitely many instants in time. Moreover, just there are for any given moment in time, infinitely
many moments of time clustered around that moment, so too the case with the rationals. Moreover, between any two rationals, there can be found another rational.
Similarly, between any two moments of time, there can be found an additional moment. Finally, like the rationals, Poincaré’s notion of moments of time is that they
are totally disconnected. That is, there are no non-trivial connected subsets. Each15

moment in time, like each rational, is disconnected in some way from every othermoment in time.
The third and final aspect of Poincaré’s use of ‘intuition’ is perhaps the closest to
traditional uses of the word, both within and outwith philosophy. Poincaré claims that
one can know a proposition by intuition meaning that one has an immediate grasp,
unquestioning readiness to accept, or the proposition is self-evident.3 For example,
recall that Poincaré insists that the Principle of Non-Contradiction can rely uponno further epistemic justification (§2.1.3). Our intuitive grasp (read:psychologicalcertainty) of particular principles outstrips any potential objections to them. This
strand of intuition is simply a repackaging of Poincaré’s prior commitment to the acceptability of psychological facts about human nature(e.g. an inability to disbelievea proposition) as epistemic justification for the truth of a proposition. To say thatsomething is known intuitively on this usage is simply to claim that the psychologically inevitability of that assertion guarantees its truth. Thus, the “intuition” that
underlies “intuitive” propositions is nothing more than psychological force.
These three notions do not interact in clearly delineated ways within Poincaré’s
texts. None is more fundamental than the other two. Instead they are mutually basic
and alternatively reinforcing. There are instances of Poincaré using the notion of temporal continuity to explain the self-evidentness of a proposition; of Poincaré showingthat strong mathematical abilities rely on strong temporal intuitions; and of Poincaréunder-girding our understanding of the continuity of time via self-evidentness. Drawing out these three separate threads is not aimed at performing a reduction on
Poincaré’s arguments. Rather, it is aimed at explicating Poincaré’s unique notions of
intuition and identifying how these three conceptions coalesce in Poincaré’s concep3

Self-evidentness in particular among these construals is controversial insofar as one might beable to give an account of self-evidentness that is logical, rather than psychological in nature. For
Poincaré, however, it would be undoubted that to say that something is self evident would be to
make a psychological claim.

16

tion and justification of mathematical induction. A secondary goal of so explaining
Poincaré’s notions is to witness their character as distinct from Kantian notions ofarithmetic, to be taken up in §{Kant.

2.2.1

Mathematical Ability

We begin with Poincaré’s regard of intuition as mathematical ability. He starts Science and Hypothesis by claiming, “The very possibility of mathematical science seems
an insoluble contradiction”[43, p. 1]. This contradiction springs forth from the belief
that although mathematics might seem purely deductive, a purely deductive mathematical proof would generate no new truths. It would, in essence, simply be a retellingof known facts in a slightly different way, something akin to showing that ‘All bachelors are unmarried’ “proves” that ‘If you are a bachelor, then you are unmarried.
Poincaré’s notion of deduction here is restrictive. He holds that deductive reasoning
is complete obedience to a set of rules. Deduction can only be done in a formalized
system, and because of these rules, deductive systems have all their theorems predetermined. New truths, are then those things that are not predetermined by therules. Poincaré seems to have as a model of deduction a complete, sound system likefirst-order logic. Deductions within the system don’t serve to introduce new truths
because the truth of those theorems was pre-ordained by the formation rules of theformal system.
Were mathematics to be purely deductive, Poincaré argues, all mathematicalprogress would be reducible to mere tautology. Moreover, were mathematics to becomplete–were mathematics to be merely the result of applications of deductive rulesof logic—there could be a mind capable, with one glance, to immediately apprehend all of mathematics.4 That such a mind is unimaginable is proof enough for
4

Recall that the completeness of first-order logic had not been proven by this point. Poincaré’s
position can be construed as a poorly formed argument that mathematics is not part of first-order17

Poincaré that there must be something beyond mere deductive prowess in mathematical proofs.5 This something extra makes real mathematical progress possible, according to Poincaré. He makes the connection between creative ability and mathematicalreasoning: “. . . it must be granted that mathematical reasoning has of itself a kind ofcreative virtue, and is therefore to be distinguished from the syllogism. . . [modes ofreasoning that] retain the analytical character, ipso facto, lose their power”[43, p. 3].
The power being lost is the power to generate new knowledge and progress beyond
the known truths. I will return to explore this seeming contradiction in depth later in§2.3 and argue that the something extra is inductive reasoning. For now, I leave thisnotion to the side to examine thoroughly Poincaré’s detailed investigation of intuitionas mathematical ability.
Poincaré’s discussion of intuition as mathematical ability takes place under theguise of an examination of mathematical education and the psychology behind someone coming to learn mathematics. That Poincaré develops this notion via an exploration of pedagogical principles is emblematic of psychological bases playing afoundational role in his metaphysical arguments (§2.1.4).
However, we must attempt to clarify the type of mathematical ability Poincaré
has in mind. He argues that “The principle aim of mathematical education is todevelop certain faculties of the mind, and among these intuition is not the least
precious”[45, p. 128]. This intuition, which is the aim of mathematical education,
stands in contrast to logic:
Logic teaches us that on such and such a road we are sure of not meetingan obstacle; it does not tell us which is the road that leads to the desiredend. For this it is necessary to see the end from afar, and the faculty
which teaches us to see is intuition. Without it, the geometrician would
logic, and whatever system mathematics is a part of, it is not complete. Of course, Poincaré does
not seem to recognize the difference between first-order and second-order logic in his rants against
the neo-logicist, so this interpretation may be too charitable.
5Note also that this argument is fundamentally psychologistic: the fact that a thing is inconceivable to us is taken as a premise in the argument.

18

be like a writer well up in grammar but destitute of ideas [45, p.130].
Logic appears to Poincaré to be without content—it is merely a set of rulesthat functions regardless of the subject of its inquiry. It is purely mechanical andnon-creative. However, mathematics has substantial content and is fundamentallycreative. Someone who is quite capable in logic might very well determine that a
proof is correct, but for Poincaré, this logician would fail to recognize what the proof
is actually about. This recognition of the substance of mathematics via the creative
spark is the intuition of which Poincaré speaks.
An important feature of this description is a comprehension of unity. It is vitalthat mathematical ability is somehow able to perceive or apprehend the totality of amathematical argument, rather than simply verifying each individual step. This verification process is precisely how Poincaré characterizes logic and deductive reasoning,which checks that one can reason safely from the premisespp⊃q

to the conclusionqMathematical ability is marked by a comprehension of the whole proof and the collection of those individual thoughts and steps into a coherent whole. Those truly giftedwith this ability (intuition) will be able to foresee a proof’s steps and the general
direction without actually verifying each and every one of the deductive steps. They
will be able to sketch a proof and know it is correct without verification of the formal
steps. This ability echoes Poincaré’s notion of the principle of induction—moving
19

from particular inferences to a globally true inference (See §2.3.1).
Intuitive mathematical ability as unifying particulars into a general principle
appears again in the same chapter:
It is the same in mathematics. When the logician has resolved eachdemonstration into a host of elementary operations, all of them correct,
he will not yet be in possession of the whole reality; that indefinable something that constitutes the unity of the demonstration will still escape himcompletely.
What good is it to admire the mason’s work in the edifices erected bygreat architects, if we cannot understand the general plan of the master?
Now pure logic cannot give us this view of the whole; it is to intuition wemust look for it[45, p. 126].
Again, Poincaré draws significant contrast between the specifics of logic and thewider, grander vision of intuition. Poincaré attributes an aesthetic quality to mathematical ability—a psychological experience helping to improve mathematical discovery as in §2.1.2. Someone who possesses such intuition is likened to one with anappreciation of the general plan of a master architect.
Poincaré later develops a description of how this aesthetic intuition combines
subconscious thoughts having been activated by conscious work into solutions hitherto
unknown. The subconscious performs a mental re-arrangement on these agitatedideas to re-combine them into (true) conclusions unattainable via purely conscious
thought. Poincaré then guarantees the accuracy of this subconscious work by a second
appeal to aesthetic appreciation:
More commonly the privileged unconscious phenomena, those that arecapable of becoming conscious, are those which directly or indirectly, mostdeeply affect our sensibility.
It may appear surprising that sensibility should be introduced in connexion with mathematical demonstrations, which, it would seem, can onlyinterest the intellect. But not if we bear in mind the feeling of mathematical beauty, of the harmony of numbers and forms and of geometricelegance. It is a real aesthetic feeling that all true mathematicians recognize, and this is true sensibility.[45, pp.58-9, my emphasis]
20

Poincaré’s notion of intuition as mathematical ability extends beyond intellectual skill and encompasses an emotion or sensibility. Recognition of the harmonyand simplicity of correct mathematical arguments enables mathematical creativity.
Aesthetic appreciation of mathematical beauty is that which distinguishes the mathematician from the logician, according to Poincaré. Because that intuition is notdeductive, mathematics is not purely deductive and the paradox that Poincaré sets
before himself at the beginning of the chapter is resolved.
Two features of Poincaré’s discussion of mathematical ability are particularly
important.
First, his argumentation is purely psychological, involving examples from his own
personal experience as well as from the general experience of mathematics instruction.
Psychological principles provide the foundation for arguments in favor of metaphysical conclusions. That aesthetic appreciation plays an important role in the basic
functioning of this ability reinforces Poincaré’s psychological basis.
Second, intuitive mathematical ability, is the recognition of unifying general principles and a perception of their truth from observation of individual cases via an aesthetic appreciation for the beauty and harmony of mathematics. Logic serves only
to verify the accuracy mathematician’s creative proof. It checks each individual stepfrom the starting assumptions to the end conclusion. Mathematical genius is thecomprehension of the general structure of mathematics beyond the individual moves
permitted by the rules of logic; it takes specific permitted deductive steps and marshals them into a proof of the general structure of mathematics. This particular facetof intuition is manifested clearly in Poincaré’s view on mathematical induction. It
becomes the ability of the mathematician to move from the particular syllogisms of21

P (1)
P (1) ⊃ P (2)P (2)P (2) ⊃ P (3)
P (3)
..
.

to the general statement ofP (n)for all n. We will return Poincaré’s conception of induction in detail later in §2.3,
once the two additional notions of intuition have been discussed.

2.2.2

Temporal Progression

Poincaré’s conception of intuition as mathematical ability is relatively explicit in his
text; his notion of intuition as an immediate grasp of time as a succession of instants
is more subtle. I argue that Poincaré viewed an understanding of a sufficiently dense
ordering of the passage of moments of time as a necessary presupposition to any
mathematical reasoning. Mathematics for Poincaré is not atemporal manipulationsof eternally unchanging quantities. Rather, it is dynamic and persists through thepassage of time. Any mathematical reasoning is done against the backdrop of thesuccession of moments. Immediate access to intuitions of temporal succession underlies Poincaré’s philosophy of mathematics. Poincaré’s writing on two mathematical22

concepts—geometrical space and infinity—will serve to illuminate his conception ofthis aspect of intuition.

Time and Geometric SpacePoincaré discusses at length how it is that we come to conceptualize geometric space[49] and [46, pp.51-88]. In fact, an entire part of his collection Science and Hypothesis
is dedicated to the examination of the theory of space and geometry. In order tounderstand the basic theory of space as it relates to geometry, Poincaré examines theinitial way in which humans learn about geometry. In order to understand geometry’s
epistemic status (whether it is a priori or contingent) Poincaré examines how humans
learn about geometry in the first place. This psychological learning process necessarily
accords for Poincaré with the epistemic and metaphysical justification of geometry.
Psychological justification is bedrock.
For this reason, Poincaré calls the following a paradox,
Beings whose minds were made as ours, and with sense like ours, but
without any preliminary education, might receive from a suitably-chosen
external world impressions which would lead them to construct a geometry
other than that of Euclid, and to localise the phenomena of this external
world in a non-Euclidean space, or even in space of four dimensions.[46,p.51]
This constitutes a paradox for Poincaré because geometry (being a merely mathematical subject) is not empirical in any way because it is not verified by experiment.
One does not go about measuring triangles in the world to verify the Pythagorean
Theorem. It is thus a product purely of our minds and of our psychology. But if it ismathematical—and thus completely reflective of the internal psychological structuresof our mind—then a being with identical psychological structures could only generate
an identical geometry. The internal psychological structures pre-determine the mathematical structures created by that mind. Yet, Poincaré claims that external factors
23could result in the development of non-Euclidean geometry as the primary way inwhich the world is apprehended. Beings with the same psychology cannot conceive ofa different geometry as primary (in the way that we might think Euclidean geometry
is primary) unless geometry itself is empirical or somehow contingent. Hence, theparadox.
To resolve this problem, Poincaré claims that although the axioms of Euclidean
geometry are “mere” conventions in the sense that they are only one of many potential sets of syntactic rules for geometry, they are impressed upon us by our external
physical surroundings. An analogy can be drawn between the rules of geometry andunits of measurement. Just as there are many different sets of rules for geometry (hyperbolic, Euclidean, parabolic, etc.), there are many different units of measurementthat accurately describe the world. However, different situations impress the use ofdifferent units upon us. For instance, we cannot measure distances between cities
in nanometers or microchip specifications in light-years. The impossibility is not alogical one, for surely we could, in some sense, measure the distance between stars
in millimeters. Rather, the impossibility is a psychological and pragmatic one. We
can’t measure star distances in millimeters because our minds could not hold thatmany numbers in our head and we fail to have any useful perspective.
Similarly, it is not a logical impossibility that we would generate a non-Euclidean
geometry as the standard geometry used in ordinary situations. Poincaré even goesso far as to show a conversion from non-Euclidean to Euclidean geometry.However,
it is a psychological impossibility, given our external experiences. The psychologicalrequirement arises because of the external environment. Pragmatic reasons force
humans to use Euclidean Geometry while different reasons force those strange beings
to construct non-Euclidean geometry even though they share our psychology. The
paradox is resolved by the fact that geometries, Euclidean and non-Euclidean alike,are merely different tools to help understand the external world. Different geometries
24are appropriate in different situations, based on external impressions.
A key step in this argument, however, is to show how the phenomena impressed
upon us actually bring about an impression of geometrical space. Poincaré provides
a short summary of his explanation:
None of our sensations, if isolated, could have brought us to the conceptof space; we are brought to it solely by studying the laws by which thosesensations succeed one another.[46, p.58, original emphasis]
Poincaré argues that there is no way static physical impingements on a retina
could bring about a conception of space.6 His thesis is that what enables the construction of a correct account of geometric space is a succession of sensations. Thesesensations need not be continuous, but they must sufficiently dense so as to constructa conception of space. Moreover, they must be properly ordered, or else the rules ofgeometric space will fail to be comprehended.
To witness the plausibility of this view, let us isolate vision as the only sense andconsider a subject who is blind from birth. Moreover, our subject lacks the intuitive
conception of time; she cannot perceive the passage of time whatsoever. Suppose wewere to suddenly give her sight. If we hold her visual field completely static, she will
be unable to form even the most basic rules of perspective or distance. For instance,
if in her visual field one man is standing near while another is hundreds of feet away,
she has no reason not to conclude that the two men are the same distance away, but
the former can hold the latter in the palm of his hand. Thus, Poincaré concludes,
there must be a succession of sensations rather than just the sensation itself to create
the conception of geometric space.
Now, however, let us move the subject around the room, but only let her open
her eyes for half a second every minute. Recall, also, that she does not have any
conception of the passage of time, so all these sensations are not ordered, nor is6

Poincaré gives an argument similar to the following in [49]25

she aware of how much time passes between opening her eyes. Again, she will beincapable of rendering a coherent conception of space. Objects will appear to blinkin and out of existence or jump from one place to the next without occupying thespace in between. The impressions must be sufficiently dense in order for her to forma correct conception.
Finally, suppose that we give her a sufficiently dense series of sense impressions,but that we do not put them in any order. That is, a lifetime of sense impressions
appear randomly on her eyes. Certainly, no matter how dense this collection of sense
impressions is, the subject will be unable to form a meaningful mental conception ofgeometric space. Instead, the sense impressions would merely represent a nonsense
jumble with no inherent structure.
Thus, concludes Poincaré, the construction of the mental concept of geometricspace require sense perceptions that are sufficiently dense and properly ordered. While
the sense perceptions themselves come from biological organs (eyes, ears, nose, mouth,etc.), understanding their density and order presupposes some notion of the passage oftime. This is precisely what Poincaré has in mind for the intuitive grasp of temporal
succession. One’s understanding of time must be immediate, sufficiently dense, andproperly ordered. For Poincaré, failing to have such immediate access results in an
inability to represent geometric space, and more generally, do mathematics at all.
The presupposition of the understanding of the passage of time is, for Poincaré, an
aspect of human intuition.
Time and Infinity
The instinctual grasp of temporal progression also underlies Poincaré’s conception of
infinity. His notion of infinity as an uncompleted potential stands in stark contrast
to the Cantorians’ (so-called by Poincaré himself) view of infinities as actual, existing, completed quantities. He attacks, “There is no actual infinity The Cantorians
26

forgot this, and so fell into contradiction. . . Like the Cantorians, the logisticians haveforgotten the fact, and they have met with the same difficulties”[48, p.195, originalemphasis]. The debate arises because while speaking of all the natural numbers is
acceptable to Cantorians or to the logicists, such talk was not permitted by Poincaré.
For him, infinity was merely a way of expressing the possibility of going on forever.
Infinity was a quantity that could surpass all other quantities, rather than one that
did surpass all others. He writes,
The notion of infinity had long since been introduced into mathematics,
but this infinity was what philosophers call a becoming. Mathematicalinfinity was only a quantity susceptible of growing beyond all limit; it wasa variable quantity of which it could not be said that it had passed, but
only that it would pass, all limits[45, p.143, original emphasis].
Poincaré’s conception of infinite implicitly assumes the passage of time in mathematics. This assumption is present in his definition of infinity as a ‘becoming’,
because it must then be changing (or at least have to possibility to change). The
original French, ‘devenir ’ has the same connotations of a process occurring throughtime. However, in order to change (or have the possibility of change), a thing must
exist within a framework of time. In order for change to occur, the object in question
must exist in at least two distinct times.
In contrast to constant mathematical concepts like 0 that are eternal and unchanging for Poincaré, infinity is not a static quantity. Infinity is a process, a merely
potential quantity. Any talk of infinity as a whole, completed quantity is inadmissible
for Poincaré. Infinity is not something that remains constant, but rather it is a dynamic, changing thing that is constantly in flux. We might say that it is constantly
growing. The infinity now, as you read the first word of the sentence, is smaller
than the infinity now as you read the last word. Both of these “infinities” will besmaller than what infinity will be if you re-read this paragraph. Poincaré points tothis process of ever-increasing size in the last sentence of the quotation. If it could
27not be said that it “had passed ” in the past, but only that it “would pass” in the
future, then the implication is clear that Poincaré’s notion of infinity is firmly withina temporal framework. Not only does infinity exist in such a framework, but all ofmathematical reasoning exists within a temporal framework. To grasp mathematics,
Poincaré holds that one must possess a pre-existing understanding of the temporalframework. The centrality of this position helps explain why mathematical induction
is so vital for Poincaré in the sciences.

2.2.3

Psychological Necessity

The third and final aspect of Poincaré’s intuition is to indicate knowing a proposition
via an immediate acceptance of it or its self-evidentness. Moreover, this access topropositions has epistemic power for Poincaré; having an intuitive grasp of something
counts as an epistemic basis, though not absolutely so, and ultimately is the only
possible justification. This particular usage is most similar to the non-technical, pretheoretical, or popular usage. Phrases like ‘women’s intuition’ or ‘mother’s intuition’echo this meaning, as do the ethicist’s questions about our ‘moral intuitions’ about
immolating babies. The mother can point to no ostensible reason that she knows that
her child is lying. We cannot say exactly how we know that burning infants alive is so
obviously morally wrong. It simply is, and there is no way to answer how we know,
yet it is beyond doubt that we do in fact know.
Poincaré’s employs this immediacy criterion of intuition in his discussion of distance and the relativity of space. In it, he argues that we have no direct access to
distance itself, but merely to relative distances. So, for example, I know that I am six
feet tall only by dint of the fact that the measuring tape that I use is itself six feettall. If one night, while I sleep, my height was radically changed to that of six miles,
surely I would notice this change. However, were similar changes to be made to every
28item in existence, including, but not limited to, my bed, the measuring tape, door
frames, etc., then there is no way for me to determine the change in my size. The
point here for current purposes is to recognize Poincaré’s denial of immediate accessto distance and his use of ‘intuition’ to describe such access.
We have so little the intuition of distance in itself that, in a single night, as
we have said, a distance could become a thousand times greater without
our being able to perceive it, if all other distances had undergone the same
alteration [45, p.99].
‘Intuition of’ can most easily be replaced here by the phrase ‘direct access to’.
Though this usage is not identical to the immediacy notion that characterized thelast instance, direct access and self-evidentness are intimately related.
Poincaré continues this usage when examining the epistemic status of mathematical axioms. He writes,
Every conclusion presumes premises. These premises are either self-evident
and need no demonstration, or can be established only if based on otherpropositions; and, as we cannot go back in this way to infinity, every
deductive science and geometry in particular, must rest upon a certain
number of indemonstrable axioms [46, p.35].
This quotation presumes that psychological self-evidentness possesses justificatory power.

Moreover, Poincaré claims that all conclusions have a justification

grounded ultimately in psychological inevitability, because the justification claim
leads back to undeniable propositions. So, not only is self-evidentness a justification,in the end it is the ultimate epistemic justification for any proposition to Poincaré.
The indemonstrability of the axioms is replicated in Poincaré’s response to Russell’s theory of types in Science and Method. This time, however, the indemonstrability is attributed to intuition, rather than self-evidentness:
He [Russell] similarly introduces principles which he declares to be undemonstrable. But these undemonstrable principles are appeals to intuition, a priori synthetic judgments [42, p.162].
29

Thus, intuitive grasp of the axioms, for Poincaré, is simply a recognition of theirself-evidentness. A third notion is added in this quotation: synthetic a priori. Recall
that for Poincaré, the synthetic a priori is “the term employed by Kant to designate
the judgments that can neither be demonstrated analytically, nor reduced to identity,
nor established experimentally.”(p. 12
When looking at justification within a mathematical system, Poincaré writes,
No doubt we may refer back to axioms which are at the source of all thesereasonings. If it is felt that they cannot be reduced to the principle ofcontradiction, if we decline to see in them any more than experimental
facts which have no part or lot in mathematical necessity, there is still one
resource left to us: we may class them among a priori synthetic views [43,p.3].
The ultimate justification of these mathematical axioms cannot reside within thesystem (analytically), because they form the basis of the system itself. Empiricaljustification is not possible, as no experiment could demonstrate a mathematicalaxiom sufficiently. The judgment that the axiom is true must be synthetic a priori,or in other words, they must be psychologically necessary and self-evident.
However, the epistemic justification offered by intuition is limited; one cannot
claim an immediate grasp of every mathematical proposition. Poincaré writes,[The mathematicians prior to the rigorization of the calculus in the 19th
century] trusted to intuition, but intuition cannot give us exactness, noreven certainty, and this has been recognized more and more. It teaches
us, for instance, that every curve has a tangent—that is to say, that every
continuous function has a derivative—and that is untrue. As certainty was
required, it has been necessary to give less and less place to intuition[45,p.123].
At first glance, it may seem that he is speaking about mathematical ability,
as in our first interpretation of ‘intuition’ (p. 17). While this notion is central to
Poincaré’s thought, the last sentence points to a slightly different meaning, because
he would most certainly disagree that “mathematical ability has less and less place”
30

in proofs. Instead, what must have a less important place is the initial appearance ofthings that seem obvious. The rigorization of the calculus demonstrated that what we
immediately accept as obviously correct in mathematics need not, in fact, be correct.
This quotation imposes a limitation intuition’s justificatory power. Poincaré denies that all mathematical propositions are justified by intuition alone, in essence
denying Kant’s position on the matter (Cf. §2.4.2). Only the most basic principles
are supported by an intuitive grasp of their truth. Moreover, due to their fundamental nature, the only support these principles could have is intuition. The farther
one gets from these basic principles, the farther one proceeds from the situations inwhich an immediate grasp of truth is reliable. In more complex areas of math, logic
and deductive procedures must guarantee the truth of proofs, rather than what is
immediately thought to be true.
Intuition, under this construal of Poincaré, provides the epistemic basis for theaxioms of scientific inquiry. It serves as the only justification for these base axioms,
but this justification cannot be extended to the higher, more complicated propositions
save through the proper application of logical rigor. Again, this position is fundamentally different from Kant’s position, wherein intuition is a mode of representationthat can only be directed towards individual objects (§2.4.2).

2.3

Induction

A clear statement of the principle of mathematical induction (PMI) must first be
given, before an examination of Poincaré’s conception. For some property of thenumbers P , if
• P holds for the base case of 0, and• If P holds of n, then P holds of n + 1, for any n
31then P holds for all natural numbers. We can rewrite this principle, again inmodern form, in second-order logic as∀P (P(0) ∧ ∀n (P(n) → P(n + 1)) → ∀n (P(n))) .
Poincaré elucidates his conception of induction by defining addition recursively
and then deriving associativity and commutativity from that definition alone. In sodoing, the reader is meant to witness the centrality of induction to mathematicalreasoning. He assumes that the operation x + 1 is already defined and goes onto
define addition of x + a via the equationx + a = [x + (a − 1)] + 1(I)Note that this definition is recursive and thus employs mathematical induction.
Poincaré remarks that through this process,. . . we can define successively and “by recurrence” the operations x+2, x+
3, etc. This definition deserves a moment’s attention; it is of a particularnature which distinguishes it even at this stage from the purely logicaldefinition; the equality (I), in fact, contains an infinite number of distinct
definitions, each having only one meaning when we know the meaning of
its predecessor.[43, pp.6-7]
There are three implicit notions of induction at work in the quotation. First,
induction is shorthand for an infinite number of definitions. It is merely a convenient
way of writing the following:x + 2 =(x + 1) + 1
x + 3 =(x + 2) + 1
x + 4 =(x + 3) + 1..
.
32

Inductive definition prevents the mathematician from actually having (for epistemic
certainty and complete formality) to go on indefinitely.
Second, by being shorthand for an infinite number of definitions, inductive definition, for Poincaré, allows mathematicians to move from particular instances—x + 1,
x + 2, etc.—to a general definition. Definition by recursion is taking an infinite set ofindividual definitions to be one overall general definition.
Third, this procedure is what prevents mathematics from being purely deductive.
Moving from the particular to the general gives mathematics the ability to make newtruths for Poincaré.
These three aspects secure what mathematical induction does; this topic will be
explored thoroughly in §2.3.1. The second focus of the section will be how induction
is justified. The fundamental position induction holds in Poincaré’s system indicatesthat its epistemic justification can only be psychological. Both what it does and how
it is justified, I claim, are the culmination of the three aspects of intuition discussed
earlier—mathematical ability, temporal progression, and immediate grasp—which inturn rely on Poincaré’s psychologism at base. Let us begin with an examination of
how Poincaré views induction as a progression from the particular instances to thegeneral rule.
Induction will provide the solution for Poincaré to the earlier seeming contradiction that begins Science and Hypothesis:“The very possibility of mathematicalscience seems an insoluble contradiction” (p. 17) This contradiction results from two
apparent truths about mathematics.
First, mathematics appears to be purely deductive in form. Mathematicians do
not argue for a mathematical result as a philosopher does a philosophical result, nor domathematicians venture out into the world in search of empirical evidence to support
their mathematical results, nor are experiments conducted in laboratories to verify
geometrical truths. Rather, mathematicians start from some basic principles and33

follow certain prescribed rules in order to arrive at the intended result. Mathematical
proofs can be valid. However, implicit in Poincaré’s notion of ‘deductive’ is that it isnon-creative. Deductive reasoning can verify the accuracy of mathematical reasoning,but that is all it is able to do. It is, perhaps, best thought of as an officious copy
editor incapable of writing on its own. Deduction is, recall, a complete obedience toa set of rules.
Second, mathematics is capable of generating new truths, by which Poincarémeans that mathematics can have results that are new and count as knowledge in a
way that deductive consequences do not. Mathematics can create new true statements
in a non-trivial way. For instance, the proof of the Heine-Borel theorem taught us
something new about closed and bounded sets: every open cover of such a set hasa finite subcover. This fact, one of pure mathematics, was genuinely new knowledge
that was not known prior to the 19th century. In some way, it was not merely arearrangement of old knowledge, but genuinely new. Mathematics thus generates
new truths.

2.3.1

Induction as Moving from the Particular to the General

The paradox for Poincaré is that rote obedience to rules seems to result in a closed,
complete system. For Poincaré, nothing not inherent in the premises. How can sucha system generate new knowledge on its own?
To escape the paradox, one might be tempted to claim that logic generates its
own new truths, thereby undermining Poincaré’s second half of the contradiction,
since logic is the epitome for Poincaré of deductive systems. If logic also generates
new truths, then the paradox fails. One might argue by considering the basic rule ofModus Ponens: p, p ⊃ q, therefore q.
If we then interpret p as ‘Plato is an ancient Greek’ and q as ‘Plato is dead’, then34

we might think that we have “learned” the new fact that ‘Plato is dead’.
However, this result comes not from the formal logic per se. Rather, it comesfrom the application of logic to empirical facts; the truth of ‘Plato is an ancient
Greek’(p) and ‘If Plato is an ancient Greek, then Plato is dead’ (p ⊃ q). Thus,
the new knowledge is generated from this application, not logic qua logic. Similarly,
mathematics can be applied to the architectural plans and physical information abouta building to learn that it would withstand a wind of 100 mph. However, Poincaré
would deny that this truth is purely mathematical, but instead applied mathematicallearning to empirical facts. “Learning” that q is true independent of any semanticinterpretation of q is not really learning anything at all to Poincaré. Mathematicaltheorems, on the other hand, do count genuinely as new knowledge.
Instead, Poincaré disputes the first half of the paradox. He argues that mathematics is not merely deductive at all,What is the nature of mathematical reasoning? Is it really deductive, as
is commonly supposed? Careful analysis shows us that it is nothing of
the kind; that it participates to some extent in the nature of inductive
reasoning, and for that reason it is fruitful[46, pxxiv].

Induction is central to Poincaré’s conception of mathematics. It is the creative
spark in the mathematician that permits new knowledge to be generated even thoughmathematics is highly abstract, independent of empirical study, and obedient to given
rules. Deductive reasoning is subservient to it in that deductions exist in mathematics
only to verify that mistakes were not made in the creative mathematical reasoning.
Poincaré highlights the primacy of inductive reasoning in his notion of mathematics:
I asked at the outset why we cannot conceive of a mind powerful enoughto see at a glance the whole body of mathematical truth. The answer is
now easy. . . he cannot conceive of [arithmetic’s] general truths by direct
intuition alone; to prove even the smallest theorem he must use reasoning
by recurrence [induction], for that is the only instrument which enables
us to pass from the finite to the infinite [43, p.11].
35

Note that for Poincaré, induction is the instrument—the only one according to
his thinking—that allows the mathematician to pass from the particular instances ofa deduction to proving it for all possible cases. To repeat, induction is the only tool,according to Poincaré, that permits the mathematician to make universal claims. He
continues on to show that induction lies not only at the heart of mathematics, but
all of science:In this domain of Arithmetic we may think ourselves very far from theinfinitesimal analysis, but the idea of mathematical infinity is alreadyplaying a preponderating part, and without it there would be no scienceat all, because there would be nothing general.[43, p.11]
Since arithmetic is necessary for any scientific study, the fundamental role induction plays in arithmetic is similarly played by induction for all sciences. Without
the principle of induction, reasoning by recursion, there would be no science at all,because induction is required to make universally general claims.

2.3.2

Induction and Temporal Progression

There are a number of threads to trace here from the earlier discussion of intuition.
First, the notion of continuing on indefinitely is made much clearer when viewed
through the lens of temporal succession. Recall the comparison of Poincaré’s conception of the sequence of moments to the rational numbers embedded in the realline (p. 15). The moments are distinct and isolated from each other, yet there canalways be found another moment between two given moments. It is not coincidence,
I claim, that Poincaré constructs the rational numbers via the same procedure. Thatis, he constructs what he calls the first continuum by arguing that the process of incalation between each set of distinct numbers can be carried out indefinitely. Taking
the process of intercalating as taking a moment in time (as Poincaré understandsit), then a one-to-one correspondence can be imagined between moments in time and36the rational numbers. Just as time plods on moment after moment, so too does theincalation process and so too does any inductive process. Induction is the realizationthat the given operation could be carried out forever. Or, alternatively, that there isno good reason to stop at any particular point.
Recall that Poincaré’s view of infinity was not as a static quantity, but rather acontinually dynamic, ever increasing entity that would eventually surpass any quantity, but that had not done so yet (p. 27). Again, the notion of infinity continuallyplodding on its course to ever increasing numbers, undergirded as it is by the intuitionof temporal progression, lies behind Poincaré’s induction. When a mathematician employs the principle of induction, she merely recognizes that for some property thatthe same inference from some number n to the next number n + 1 could be carriedout over and over again. That the mathematician can in principle do this stems
from the same intuitive basis for her understanding of infinity: the progression oftime. She grasps immediately that each inference can be done in succession in an
appropriate way. As such, the principle of mathematical induction simply becomes arecognition that going on so indefinitely would result in concluding that the property
would be shown to hold for every number. Therefore, the mathematician concludes
that it does hold. The Principle of Mathematical Induction is a transcendence of thisinfinite process to its final result.

2.3.3

Induction and Mathematical Ability

The second thread of intuition that must be pulled free is intuition as mathematicalability. The first similarity is to the aesthetic recognition of mathematical unity that
Poincaré claimed underlay the ability of the mathematician to create new mathematical proofs (see p. . It is vital that Poincaré, in describing the construction of the real
numbers, speaks of “feel[ing] that [a recursive] operation may be continued without37

limit, and that, so to speak, there is no intrinsic reason for stopping.”[46, p.25] This
“feeling” echoes the sensorial language used to describe the unconscious recognitionof the correct mathematical structure (witness “real aesthetic feeling” (p. 20)[45,p.59]. It is not that we know or comprehend the absence of an intrinsic reason tostop; rather, we feel it. We feel the unity that persists throughout the process, thateach subsequent step is identical in kind to the one that preceded it and that there
is a harmony to how each subsequent step relates to the step before it. This sort of
explanation, of course, is warranted only by Poincaré’s psychologism: appealing toour feelings as a legitimate indicator of mathematical truth is permitted only if one
accepts, as Poincaré does, the primacy of psychological principles.
The second aspect of similarity is the very notion of progressing from the particular to the general. Poincaré, remember, compared mathematical ability to theappreciation of architecture. Someone who only understood the individual deductive
steps was someone who appreciated only “the mason’s work in the edifices erected bygreat architects,” while those who could appreciate the whole proof—those who possessed mathematical and not just logical ability were able to “understand the generalplan of the master”[45, p.126]. So, Poincaré takes mathematical ability as that whichallows mathematicians to move from the minor details to the general plan. Similarly,
induction simply is the ability to move from the individual instances of an inference
to the general truth that underlies it. Progressing from the unending collection ofequalities that defined addition recursively to a universal definition of addition is a
progression from the particular to the general. In this sense, mathematical induction
simply is mathematical ability.

38

2.3.4

Induction Justified by Psychological Necessity

The third thread of intuition—immediate psychological necessity—justifies, in themind of Poincaré, the principle of mathematical induction. The principle is supported
only by the fact that we feel it to be unquestionably true.
Why then is this view [reasoning by recurrence] imposed upon us with suchan irresistible weight of evidence? It is because it is only the affirmationof the power of the mind which knows it can conceive of the indefinite
repetition of the same act, when the act is once possible. The mind has
a direct intuition of this power, and experiment can only be for it anopportunity of using it, and thereby of becoming conscious of it.[43, p.13]

This quotation contains two key interpretive rewards. First, Poincaré equates
reasoning by recurrence with a power of the mind. This is perhaps the most explicit
rendering of induction as mathematical ability. Second, Poincaré explicitly calls upon
intuition to explain our access to this power. Induction is an element of human psychology that enables the application of mathematical induction. Similarly, Poincarédraws an explicit connection when he claims that the rule of induction, “inaccessible
to analytical proof and to experiment, is the exact type of the a priori synthetic
intuition”[43, pp.12-3]. Recall that synthetic a priori for Poincaré consists of those
things that “can neither be demonstrated analytically, nor reduced to identity, norestablished experimentally”[47, p.145]. Classifying induction as synthetic a priori
simply is, as I argued previously, appealing to the psychological necessity of it inorder to provide justification.
Induction then, should be seen as the combination of these three notions of intuition. It presupposes an intuition of the progression of time, consists of a formalizationof the mathematical ability to move from particular instances of a deduction to thegeneral truth, and is justified solely by its synthetic a priori status.
39

2.4

Kant’s Philosophy of Arithmetic

The obvious background against which to view Poincaré’s philosophy of arithmetic
is Kant’s philosophy of mathematics. This fact should come as no surprise, since
Poincaré thought himself to be defending Kant against those who, at the Kant jubilee discourse, he accuses of whispering, “I well see this is the centenary of Kant’s
death”[47, p.146]. And while Poincaré develops a philosophy of geometry wholly independent of Kant, he largely agrees with Kant’s conception of arithmetic. However,to dismiss Poincaré’s philosophy of arithmetic as purely Kantian—and thus not worth
study in its own right—is too hasty.
In particular, I want to highlight two central concepts of Poincaré’s philosophy ofarithmetic that differ significantly from Kant’s position: intuition and syntheticity. In
order to witness the difference, we must first explicate Kant’s position. However, doing
so comes at the risk of becoming overly involved in Kant’s philosophy of mathematics.
In order to avoid becoming so entangled, I will restrict the sketch of Kant’s position
to just those two topic and compare them to Poincaré’s stance. The goal of thissection is simple: to demonstrate that Poincaré, despite being sometimes labeled apure Kantian, in fact had a different philosophy of mathematics, one worthy of study
independent of Kant.

2.4.1

Syntheticity

Kant argues for the syntheticity of mathematical propositions in two steps, first forarithmetic propositions and second for geometric propositions. Taking up the first,
he argues that propositions like7 + 5 = 1240

are synthetic because they are the result of the construction of a new predicate concept
not present in the original subject, yet one that is shown to apply to the original
subject nonetheless [52, p.98]. In the case of the above equation, 12 is not a predicate
concept that is present in the original subject, which is the sum of 7 and 5. Rather,
the reasoner comes to exhibit in intuition (more on this in a moment) the concept 12
and that this concept applies to the sum of 7 and 5.
Shabel uses the helpful example of constructing a triangle from its definitive
conceptual parts. That is, if I am told that a triangle is a rectilinear, three-sided
figure, I construct the concept of the triangle out of the concepts I already have for‘figure’, ‘rectilinear’, and ‘three-sided’ [52, p.99]. The concept of a triangle is notpresent in the definition of those terms, and the fact that I have constructed the newconcept means that it is synthetic.
Returning to the arithmetic example, the left hand side of the proposition has
the concepts of 5, 7, and the concept of summing the two together, but it does nothave the concept of the sum itself, namely 12. No amount of analysis of these three
concepts will result, according to Kant, in the concept of 12. Instead, I must constructthat concept via a synthesis of concepts into something new.
The picture for Poincaré, I argue, is different. Poincaré agrees with Kant that allof mathematics is synthetic. However, this syntheticity stems not from the synthetic
construction of arithmetic concepts in each and every arithmetic judgment. Rather,
it is based on the requirement that fundamental mathematical axioms must be considered synthetic (in fact synthetic a priori) propositions because their truth is notdetermined by their meaning. Poincaré writes that synthetic a priori judgments arethose “that can neither be demonstrated analytically, nor reduced to identity, nor established experimentally”[46, p.146]. Later, he accuses Russell of asserting the truthof his axioms because of their synthetic character (p. 29). Poincaré’s conception ofsyntheticity is purely epistemic, and flows through the axioms to all later proposi41

tions. Moreover, Poincaré’s notion of syntheticity is highly psychological in character,
resembling psychological necessity, rather than the constructive character of Kant’s
syntheticity. Whereas “7 + 5 = 12” must be apprehended synthetically for Kant, forPoincaré it is a trivial consequence of the axioms, which have the synthetic status.
It is also important that Poincaré’s conception of syntheticity necessarily relies
on an axiomatic structure. Kant is opposed to axiomatic structures in general.
As regards magnitude (quantitas), that is, as regards the answer to begiven to the question, ’What is the magnitude of a thing? ’ there are noaxioms in the strict meaning of the term, although there are a number of
propositions which are synthetic and immediately certain (indemonstrabilia) [36, A 163-44].
Because of this denial, a Poincaréan style conception of the synthetic is notavailable to Kant. This divergence marks a key way in which Poincaré’s philosophyof mathematics is not purely Kantian.

2.4.2

Intuition

Both Kant’s and Poincaré’s respective conceptions of syntheticity follow from theirrespective conceptions of intuition. Given that their understanding of the synthetic
is different, it should be no surprise that their notions of intuition differ as well.
For Kant, intuition is “a species of representation (Vorstellung) of, in the languageof Descartes and Locke, an idea” [41, p.111]. Intuition is a way for the mind to relate
to objects. There are criteria that differentiate intuition from concept:7[Knowledge] is either intuition or concept (intuitus vel conceptus). The
former relates immediately to the object and is single, the latter refers to
it immediately by means of a feature which several things may have incommon [36, A320]
For Kant, intuition is both singular and immediate. The immediacy criterion
7

Parsons cites the following passage in his [41].

42mirrors Poincaré’s conception of intuition as psychological necessity(§2.2.3)—though
it lacks the explicitly psychological basis.8 The singularity criterion, however, isdistinct to Kant; Kant holds that intuitions are about particular objects; this is incontrast to concepts that can relate to generalities.
Particularity makes every mathematical judgment a judgment about intuitions
for Kant. If intuition is the only method to obtain mathematical knowledge (as Kant
claims), then we must come to learn about each mathematical judgment individually.
Intuition is how we know that 5 + 7 = 12 is true, as is 3 + 4 = 7, but not how weknow the rules of addition.
Poincaré’s notion of intuition fails to have this particularity condition. In fact,
I argue in §2.2.1 that a main theme of Poincaré’s conception of intuition is as anability. Poincaré claims that intuition enables us to recognize the uniformity in theprocedures of mathematics to gain a full understanding of addition.
Poincaré’s philosophy of mathematics was largely Kantian, but two of the central features to his philsophy of arithmetic—intuition and syntheticity—difffer significantly from Kant.

8

Parsons’s interpretation of the immediacy criterion could be read in a psychological manner. He
writes, “[The immediacy condition] means that the object of an intuition is in some way directly
present to the mind, as in perception, and that intuition is thus a source, ultimately the only sourceof immediate knowledge of objects”[41, p.112]. However, this discussion is beyond the scope of thecurrent inquiry.

43

Chapter 3
HilbertPoincaré’s attack on those who sought to ground mathematics in logic was bothwidespread as well as indiscriminate. There was little differentiation between Hilbert,
Dedekind, Cantor, Russell, or Frege in any given attack. All were simply derided as“logisticians” who had made little progress in the foundations of mathematics. Given
that he did not discriminate between wholly distinct philosophers, Poincaré did not
distinguish their continually evolving views. Of course, in the case of Hilbert, suchdifferentiations were not possible, as Poincaré died in 1912, before the revision ofHilbert’s main views in 1918 and later.
There are three primary aims of this chapter. First, to elucidate the discourse
between Hilbert and Poincaré in the first decade of the 20th century, arguing that
Poincaré’s criticism of Hilbert’s nascent program were warranted and effective. Second, to interpret Hilbert’s later project by developing four distinctions made byHilbert: theory vs. metatheory; finite vs. transfinite; idea vs. concrete; and formal
vs. contentual. These four contrasts serve to be a substantial, novel interpretationthat explains Hilbert’s response to Poincaré. Third and finally, to evaluate whether areformulation of Poincaré critique is viable against Hilbert’s fully-fledged system. In
so proceeding, the chapter will refer to the Hilbert of 1900 and 1904 as the “early”
44

Hilbert and the post-1918 Hilbert as the “later” Hilbert.
To achieve the stated goals, I will first outline the early Hilbert’s attempt at anaxiomitization of arithmetic as outlined in his 1904 On the foundations of logic andarithmetic. Second, I will examine how Poincaré’s broad petitio argument is viableagainst this early line of thought by Hilbert. Third, I will reconstruct later Hilbert’s
evolution from a tentative position to a fully formed program and develop a response
to Poincaré’s objection out of this system. Owing to the sparseness of Hilbert’s own
reply to this objection—a mere few polemical paragraphs—the discussion will evaluate
both Michael Detlefsen’s [13] and Marcus Giaquinto’s [22] respective reconstructionsof Hilbert’s argument, ultimately concluding that ultimately both positions have fatal
flaws.

3.1

Early Hilbert

Hilbert’s work on the foundations of mathematics can be separated into two distinct
eras: his early work from 1895-1904 and his later work, which began in 1917 butrounded into shape only in the mid to late twenties. Hilbert’s return to a foundational
focus in his later period brought about a precisification of the sketches in his early
work into a fully developed project to prove the consistency of arithmetic via solely
finitistic methods. Understanding this relationship between the early and later work
can only be obtained via a thorough grasp of both periods. We begin with the early.

3.1.1

Hilbert’s Foundation of Geometry

Hilbert’s first project in mathematical foundationalism was to give a novel axiomaticcharacterization of geometry that was divorced from any interpreted meaning. Hilbert’s
geometry was unlike Euclid’s in that it was not a description of points, lines, and othersuch geometrical objects. Rather, it was a characterization of the relationships be45

tween arbitrary objects. This approach is in contrast to Euclid, whose use of ‘point’and ‘line’ actually refers to those very things. Thus, when Euclid claims that any 2
points define a single line, he is making a claim about the nature of points and lines:
for any two we make two (idealized) dots on a piece of (idealized) paper there is asingle (idealized) line that connects them.
However, Hilbert’s use of ‘point’ and ‘line’ does not refer to things commonly
understood to be points and lines (or rather he does not speak merely of these things).
In fact, Hilbert’s usage is abstracted from the interpreted meaning of the words.
In my opinion, a concept can be fixed logically only by its relations to otherconcepts. These relations, formulated in certain statements, I call axioms,
thus arriving at the view that axioms (perhaps together with propositions
assigning names to concepts) are the definitions of the concepts.[20, Letterfrom Hilbert to Frege, 7/11/1903]
The words ‘point’, ‘line’, and the rest are meaningless prior to axioms being laid
down; they receive meaning only through the system of relations to other concepts.
It is only by choosing a given set of axioms that the concepts of ‘point’, ‘line’, ‘circle’,‘distance’, and any other geometrical notion gain a definition through their relations
to each other.
The procedure of laying down an axiomitization without an explicit interpretation was a radical shift from the mathematics that came prior to Hilbert. That theconcepts contained within the axioms are implicitly defined wholly and solely by theaxioms themselves—that they do not have a meaning outside this axiomitization—
shapes Hilbert’s later quasi-metatheoretical approach to proving the consistency ofarithmetic.
The best way to witness Hilbert’s conception of axioms operating as implicit
definitions of the concepts contained therein is to follow an example from Hilbert’s
text. For exegetical clarity and brevity, it is sometimes advantageous to employ
modern terminology, such as ‘syntactic’, ‘interpretation’, and the like. Such uses46

should not imply an attribution of such views to Hilbert himself. Following thisexegesis, I will return to Hilbert’s lack of these fundamental distinctions in his early
work—and how this absence renders him vulnerable to Poincaré.
Taking the elements point, line, plane, etc. as primitive, the axioms of Group II,
according to Hilbert, “define the concept of “between” and by means of this concept
the ordering of points on a line, in a plane, and in space is made possible.”[29, p. 5]
Axioms introduce conditions under which particular relations hold and define theirproperties. The meaning of ‘between’ is simply a set of rules under which are licensed
certain assertions about points and the relation of between. Axiom II, 1 states,
Axiom II,1: If a point B lies between a point A and a point C then the points A,
B, C are three distinct points of a line, and B also lies between C and A.[29]This axiom determines that ‘between’ is “symmetric” in the sense that if B lies
between A and C, then B also lies between C and A and, moreover, that A, B, andC are all distinct points. However, given only Axiom II, 1, the full, standard meaningof betweenness is absent; betweenness is an impoverished definition, for even theability to determine, given three points, whether any of them lie between the others
is impossible. Only the addition of further rules for the use of the relation ‘between’makes it resemble our ordinary definition of betweeness.
In this example, Hilbert has provided a single rule governing the use of the relation‘between’, but not defined in the traditional sense, what ‘between’ means. In modern
notion, Hilbert’s “definition” defines a three-place relation ‘MABC ’ as ‘B lies between
A and C’. Axiom II, 1 can be recast as a statement about the relation MABC rather
than about ‘between’ in any traditional sense. Axiom II, 1 would be phrased:
If a point B is such that MABC holds, then the points A, B, C are three
distinct points of a line and MCBA also holds.
Abstracting further away from the “traditional” notions of point and line, we can47

create a one-place predicate ‘Px ’ defined as ‘x is a point’, define a three-place predicate‘Lxyz ’ as ‘x, y, and z are colinear’, and represent Axiom II, 1 wholly symbolically:(PA ∧ PB ∧ PC ∧ MABC ) −→ (MCBA ∧ ¬(A = B) ∧ ¬(A = C) ∧ ¬(B = C) ∧ LABC )Understanding Axiom II, 1 thus written divorces the three place relation M from any
prior intuition or knowledge of points, lines, or what it means for a point to lie between two others. Under the Hilbertian conception, this axiom (in combination withthe others) defines what it is for a point to be ‘between’ two others.1 In particular, it
depends on the definition of colinearity. By itself, it is unable to permanently fix thedefinition of ‘between’. Hilbert’s goal in providing axioms is to give a purely syntactic
account that can be interpreted as the ordinary conception of geometry. However,this axiomitization does not rely on any traditional understanding of Euclidian geometry; it does not presuppose, for instance, familiarity with points, lines, and thelike. Rather, Hilbert’s axioms provide a framework within which one is permitted towork. Hilbert himself says,
But it is surely obvious that every theory is only a scaffolding or schemaof concepts together with their necessary relations to one another, andthat the basic elements can be thought of in any way one likes. If in
speaking of my points I think of some system of things, e.g. the system:
love, law, chimney sweep. . . and then assume all my axioms as relations
between these things, then my propositions, e.g. Pythagoras’ theorem,
are also valid for these things.[20, p. 40]

In short, Hilbert uses axioms to establish a framework within which we can use
concepts that are being “defined”. Geometry can be about chimney sweeps, laws,
cats or whatever on likes insofar as they obey the axioms—geometry has been recast
1

I want to here acknowledge that this notion of implicit definition via axiom was and remains
quite controversial. Frege for one, in his correspondence with Hilbert, objects strenuously to thisidea. I want only to highlight this aspect of Hilbert’s perspective, and not argue for or against its
correctness.

48

as not the study of points and lines but rather the study of the relationships between
arbitrary objects that accord with a set of given axioms. Definitions of terms are
not isolated in single lines, as in a dictionary. Rather, Hilbert believes that “only thewhole structure of axioms yields a complete definition” and to “try to give a definitionof a point in three lines is to [his] mind an impossibility.”[20, p. 40] Definitions are
only grasped in the context of a system of axioms and are made at (in modern
terminology) a syntactic level.
“Grounding” geometry is a matter of lay down a set of purely syntactic axioms
that can be interpreted semantically as the truths of traditional Euclidean geometry, and then proving that this set is consistent. Hilbert begins his Grundlagen derGeometrie by describing his goal to set forth
a complete, and as simple as possible set of axioms and to deducefrom them the most important geometric theorems in such a way thatthe meaning of the various groups of axioms, as well as the significance ofthe conclusions that can be drawn from the individual axioms, come tolight.[29, p. 1, emphasis in the original]
The first desideratum is that the axioms must be complete; they must entail
every geometrical result that follows from the Euclidean axioms. Hilbert’s axiomatic
system must be able to recapture all the standard proofs of the mathematical corpusthat were then known as Euclidean geometry. For instance, from Hilbert’s axioms, heshould be able to prove that “In any triangle two sides taken together in any mannerare greater than the remaining one.”2 If Hilbert’s axioms fail to produce a Euclidean
result, then his characterization will have failed to be “geometry”.
Hilbert’s second desideratum is that the axioms of his system be “as simple aspossible.” This has a dual meaning for Hilbert. On a shallow, psychological reading,Hilbert means that the axioms must not appear more complicated than those which
they intend to supercede. To provide a more obscure account of a subject whose2

Proposition 20, Euclid’s Elements, as translated by Heath, 286

49original exposition was a paragon of clarity would not be progress. The second more
technical and important interpretation of Hilbert’s requirement for simplicity is logicalindependence: no axiom (or its negation) should be derivable via inference rules froma subset of the others. This second interpretation makes Hilbert’s project so ambitious
and mathematically fruitful. While Euclid had come close to outlining a system of
axioms that were complete for (Euclidean) geometry, people had long denied the
independence of these axioms and attempted to prove the parallel postulate from theother four axioms. For Hilbert to undertake a proof of the independence of this axiom
and succeed was a major step in geometry.
Fundamental to proving the independence of the axioms was to prove their consistency. For if a set of axioms is inconsistent, then one can (trivially) derive any
conclusion from that set. Consequently, the negation of any the axioms can be derived from the original set of axioms, violating Hilbert’s requirement that the axioms
be independent. The axioms must be shown to be consistent in order for them to beshown to be independent. However, proving the consistency of the axioms was morethan a simple pre-requisite for Hilbert’s independence criterion. Instead, Hilbert saw
a proof of their consistency as a proof that established their truth.
. . . if the arbitrarily given axioms do not contradict one another with all
their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence[20, p. 39-40]
For Hilbert, a proof of the consistency of his axioms for Euclidean geometrysimply is a of their truth.
Hilbert’s ultimate aim of grounding and/or justifying geometry can be achieved
only through two distinct goals: a reconstitution of Euclidean geometry using his
axioms and a proof of their consistency.3 The former goal is an exercise in for3

A similar two-part strategy is employed by the neo-logicist. The neo-logicist must implicitly
introduce terms via abstraction principles and then show a reconstruction of traditional mathematics
from these new terms. See §5.2

50

mal geometry, and thus outside the purview of the current investigation. However,in order to achieve the latter, Hilbert proves the equiconsistency of his axiomitization with arithmetic via a translational procedure that recasts geometrical notions
as arithmetical ones. Hilbert provides an arithmetic interpretation of his axioms
that proves the equiconsistency of his axiomatic geometry with arithmetic. By thismethod, Hilbert shows that any proof of the inconsistency of his geometry would
likewise prove arithmetic inconsistent as well. Thus, Hilbert’s proof of the consistency of geometry depends on the existence of a positive proof of the consistency ofarithmetic.4

3.1.2

Initial Sketches of the Consistency of Arithmetic

Hilbert first attempted to sketch a positive proof of the consistency of arithmetic ina speech to the Third International Congress of Mathematics in 1904 titled Foundations of Logic and Arithmetic. In it, Hilbert laid out his axioms for arithmetic
and demonstrated how he thought consistency might be proven via a syntactic proof.
Even though the formalization of such a consistency proof by purely arithmeticalmeans is impossible, as G´’odel later showed, an explicit exegesis of Hilbert’s attempt
will show the efficacy of Poincaré’s naive petitio.
Viewed through the lens of a modern logician, Hilbert’s attempt at a consistency
proof is wholly syntactic and metatheoretical. Hilbert first claims that all contradictions have a certain form—the formal representation of any contradiction follows aparticular pattern. The proof argues that applying the axioms and rules of inference
always results in a sentence that does not exhibit the pattern characteristic of a contradiction. Therefore, no contradiction can result from Hilbert’s axioms conjoined
4This model theoretic method is extremely fruitful mathematically and heretofore unknown, so
even if the ultimate aim of his project fails, Hilbert made incredible contributions. After Hilbert
however, these proofs common (for example, Boolos uses one in §5.1.3).

51

with the rules of inference, and thus the axioms are consistent.
Though metatheoretic in that it is reasoning about the symbols, it is inappropriate to call it such for Hilbert. Though his axioms are wholly syntactic, there remains
minimal differentiation between the syntactic and semantic. At this point in his
project, there is formalized metatheory or proof theory. This murky understanding
and lack of distinction at this early stage—because everything is ostensibly within thetheory—makes Poincaré’s petitio objection forceful. This objection will be returned
to in §3.1.3 after fully examining Hilbert’s proof. However, it is important to notethat Hilbert qualifies the proof in his address, saying,
In the brief space of an address I can merely indicate how I conceive of
this common construction. I beg to be excused, therefore, if I succeed
only in giving you an approximate idea of the direction my researches aretaking.[27, p. 131]

In responding to Poincaré’s objection in the later period, Hilbert claims that
Poincaré’s objection is only effective against this nascent position:
Under these circumstances Poincaré had to reject my theory, which, incidentally, existed at the time only in its completely inadequate earlystages.[27, p. 473]

To critique early Hilbert is unfair given that he categorizes his position as provisional and incomplete. However, such a critique is important for two reasons. First,
it shows how Poincaré’s petitio argument can be successful. Hilbert’s later return
to foundationalist questions are significantly refined, no doubt in part because of
Poincaré’s objection. In particular, Hilbert is forced to distinguish between the object language of the theory and the metatheory. For now, we turn to the early Hilbert
and the effectiveness of Poincaré’s objection.
52

Hilbert’s Formal System
Hilbert’s overall strategy for proving the consistency of his axioms is based on theseparation of all propositions into what he calls ‘entities’ and ‘nonentities’, the former
understood to be true propositions, while the latter are false propositions.5 Hilbert
must prove that it is not possible to derive from his axioms the inclusion of both a
and ā in the class of entities [27, p. 132].
Hilbert’s notation in On The Foundations of Logic and Arithmetic is quite cumbersome owing to the fact that it is merely a sketch of a more fully fledged notionof proof theory. However, it is vital that full exposition be given in the original notation, because Hilbert’s proof relies on the precise syntactic form of his formulae.
He argues that application of the rules of inference and the axioms to the primitive
symbols results only in formulas that do not have the syntactic form necessary to bea contradiction—within the space of possible combinations of the strings of symbols
that count as formulas, there is a limited number of possible combinations that are theresult of correct applications of the axioms and all these combinations have a given“shape”. All contradictions, Hilbert argues, have a different “shape” and fall outsideof possible results of reasoning from the axioms. Specifically, all contradictions havea negated clause. Because the axioms and rules of inference do not result in a formulawith a negation, no series of legitimate maneuvers within the system will result in acontradiction.
We now want to see the argument in full detail. Hilbert’s first primitive notionis ‘1’, which is merely an object of our thought. It is without meaning and signifies
nothing. It is divorced of ordinary notions we ascribe to the symbol ‘1’. We can combine it with itself to form longer collections of marks, what Hilbert calls combinations,

5Cf. “We call a a true proposition if a belongs to the class of entities; on the other hand, let ā
be called a true proposition if a belongs to the class of nonentities.”[27, p. 132]53

or strings. For instance,
11
11111
1111111111111
are all combinations for Hilbert. Differentiation between two strings is achieved byexamining them to see if “the combinations deviate in any way from each other withregard to the mode and order of succession in the combinations of the choice andplace of the objects 1 and [the second primitive object] = themselves”[27, p. 132].
For two combinations to be the same is simply for them to have the same marks inthe same order. Two combinations that are not the same length—for instance thefirst and second combinations above—cannot possibly be the same. When examining
combinations containing only ‘1’, identity is nothing more than having the samelength. This sort of reasoning about the “shape” of combinations permeates both
Hilbert’s axiomitization and his consistency proof.
Any combination of these two primitive thought-objects is likewise considered athought-object and forms the domain over which Hilbert’s variables in his axioms will
range. The first two objects are demarcated as simple objects to differentiate themfrom the combinations. This gives us sufficient background to understand the firsttwo of Hilbert’s axioms:
x=x

(1)

{x = y a.w(x)}|w(y)6(2)

The first of these is the statement of identity. The second axiom is Leibniz’s lawof the indiscernibility of identicals that states that any two things that are identical

6‘a.’ here denotes the conjunction.

54

share all of the same properties. In Hilbert’s own words, “from x = y and w(x), w(y)follows,” meaning that if a predicate ‘w’ is true of x and x is equal to y, then w istrue of y as well. With these two axioms, Hilbert claims that he has formed “thedefinition of the notion = (equals).”7 Wishing to avoid any controversy on whether
such axioms can define implicitly, I will not make such a bold claim and instead saythat Hilbert has (at least) governed the use of identity within his formal system.
Before moving on to the next three axioms, the remaining primitive notions must
be defined. They are u, f , and f 0 (as well as parentheses, though not mentioned
explicitly). ‘u’ is called by Hilbert ‘infinite set’.[27, p.133, emphasis in original] The
string ‘ux’ is read as the claim that x is a member of the infinite set.
The two functions f and f 0 are more obscure. Hilbert calls them the ‘following’and ‘accompanying’ functions respectively. Jointly, they play the role of the modern
successor function. However, for Hilbert, they are two different functions with different domains. f 0 has a domain of all possible thought-objects, whereas f has a domain
restricted only to elements of the set U. As it turns out, Hilbert’s axiom 3 is thestatement that the two functions are identical over the domain of the infinite set.
He advances three additional axioms:

f (ux) = u(f 0 x),(3)f (ux) = f (uy)|ux = uy

(4)

(f (ux) = u1)(5)

Axiom 3 should be read literally as, “The object which follows the element (of7FLA, 132, emphasis in the original55

the infinite set) that is x is identical with the element (of the infinite set) that is theresult of the accompanying function applied to x.” Axiom 4 should be read as “If theelement that follows x equals the element that follows y, then the element x equals
the element y.” Finally, Axiom 5 should be read, “There is no element x such thatthe element that follows it equals the element 1.”
Note that these axioms have been wholly syntactic in nature. They govern therules by which one may reason within Hilbert’s formal system. As such, they havebeen given no fixed semantic meaning and consequently only produce a structure,
which is what Hilbert wants to prove consistent. If the syntactic structure is shownconsistent, then any interpretation of the rules will also be consistent. Of course one
interpretation—the intended interpretation—is standard arithmetic.
Proof of Consistency of Hilbert’s System
The five axioms listed above are not the only axioms Hilbert introduces to his system.
Notably, I have omitted the induction axiom, from which the Poincaréan objectionof circularity springs. I made this omission because Hilbert’s proof of the consistencyof the first five axioms is a necessary presupposition for the proof of the consistencyof every further axiom. By sketching how Hilbert intends to prove this basis—andlater arguing that it presupposes induction—I will show why a Poincaréan objection
is a good one.
To prove the consistency of the first five axioms is simply to show syntacticallythat a contradiction cannot be derived from repeated application of the laws of inference and the axioms. A contradiction in the system can only have the form ‘a ∧ ā’,where a is some combination of the five primitive thought objects. Because Axiom 5 is the only axiom that results in a negation (something of the form ‘ā’), theonly contradiction that can be derived is the conjunction of Axiom 5 with the statement ‘f (ux(0) ) = u1’, where ‘x(0) ’ represents some thought-object in the system (e.g.
56

f 1 = 1, ff 1 = 111, fff 1 = 1111, etc.). In order to show that this sort of contradictoryformula cannot arise, all Hilbert must show is that a statement of the formf (ux(0) ) = u1

(6)cannot be derived from Axioms 1-4.
In order to prove that no statements of the form of equation 6 can result from
Axioms 1-4, Hilbert introduces the notion of “homogeneity of equations”, which hedefines as follows:
we call the equation (that is, the thought object) a = b a homogeneous
equation if a and b are combinations of two simple objects each, likewise
if a and b are combinations of three simple objects each or of four and soforth[27, p. 134, my emphasis];
An equation is homogeneous when it has the same number of primitive objects
on both sides of the ‘=’; an equation is heterogeneous whenever it has a different
number of primitive objects on either side of the ‘=’. Though I have chosen to use‘number’ to explain hetero- and homogeneity, Hilbert deliberately avoids doing so via
the use of examples and the phrase “and so forth”. An example of a homogeneous
equation is f 1 = f 1. An example of a (false/non-entity) heterogeneous equation is
f 1 = ff 1, since there are two simple objects on the left of the equation and three on
the right.
Hilbert claims that only homogeneous equations follow from applications or reiterations of Axioms 1 through 4. This claim is trivially true for Axiom 1 and for Axiom
2, since the former is the smallest of all homogeneous equations and the latter does nothave as its main statement an equation. For Axiom 3, it is also evidently true sincef (ux) = uf 0 x is homogeneous for any object x. Finally, both the premise of Axiom 4
and the conclusion are homogeneous (f (ux) = f (uy) and ux = uy respectively)given
Hilbert’s definition of equality.
57

Thus, any statement of Axioms 1 or 3 or an application of Axiom 2 or 4 must
result in a homogenous equation. Thus, a heterogeneous equation like 6 cannot be
a logical consequence of Axioms 1 through 4. To repeat, the left side of an equationof the form of 6 must have at least three simple objects (f , u, and whatever x(0)represents) while the right side has at most two (u and 1). Therefore, since equations
exhibiting the form of equation 6 are the only way for a contradiction to arise, there
can be no contradiction derived from Axioms 1-5.
Consistency Results
Hilbert thinks his result successful and from its success, he draws a number of powerfulconclusions. First, due to his position that the consistency of a mathematical entity
guarantees its existence, the thought objects u, f , and f 0 have been proven to exist.
This result is particularly important in the case of u, since by demonstrating its
existence, Hilbert thinks he has proven the existence of the infinite:
So far as the notion of the infinite u, in particular is concerned, the assertion of the existence of the infinite u appears to be justified by theargument outlined above; for it now receives a definite meaning and acontent that henceforth is always to be employed [27, p. 134, emphasis inoriginal].
The major result, however, is a proof of the consistency (and therefore existence)of mathematics. As noted above (p. 56), the intended interpretation of Hilbert’s
axioms results in standard arithmetic. Axioms 1 and 2 govern the use of ‘equals’
within Hilbert’s system and mirror how ‘equals’ is used in arithmetic. Interpreting u
as the set of integers and f and f 0 as the successor function, Axiom 3 states that every
integer has a successor that is also an integer. Axiom 4 states that if the successorof x and the successor of y are equal, then x = y. Axiom 5 states that 1 is the
successor of no integer, or, alternatively, 1 is the first integer. Interpreting the axioms
in this way does not wholly prove the consistency of arithmetic, but does establish
58

the foundation from which, proceeding in a similar manner, one can demonstrate theconsistency of all of arithmetic. To get to the next step—a proof of the smallest
infinite or ω—Hilbert himself says,
If we translate the well-known axioms for mathematical induction into
the language I have chosen, we arrive in a similar way at the consistencyof this larger number of axioms, that is, at the proof of the consistent
existence of what we call the smallest infinite (that is, of the ordinal type1, 2, 3, . . . ) [27, p. 135, emphasis in original].8
By continuing to prove that ever growing sets of axioms are consistent, Hilbert
has outlined how he believes the consistency of arithmetic will be proven. The mathematical heavy lifting is present in his initial proof of the consistency of Axioms 1-5.

3.1.3

Poincaré’s Petitio against Early Hilbert

Poincaré dedicates a brief section in his Mathematics and Logic: II to addressing
Hilbert’s attempt at proving the consistency of arithmetic [42, pp 166-170]. There islittle explication of how Hilbert is supposed to have presupposed mathematics in his
derivation of mathematics. Poincaré only offers that Hilbert has used ‘two’, ‘three’,or ‘several times’ repeatedly in defining a combination of primitive symbols [42, p.
167]. This objection is somewhat uncharitable to Hilbert, as these uses of number
are only for the purposes of elucidating what is being defined, rather than in theformal definition itself. However, Poincaré’s objection can be refocused on anotherpart of Hilbert’s program—one in which he implicitly assumes number in a formal
definition—and effectively challenge Hilbert’s Proof.
In the petitio’s simple form, Poincaré accuses Hilbert of presupposing a conceptof number in his purely syntactic proof. He argues that Hilbert’s consistency proof
implicitly assumes the principle of mathematical induction. Since the axiom of math8

Note that this definition of the smallest infinite is a more restrictive notion that that of thepreviously defined infinite u.

59ematical induction is one of the things of which Hilbert proves the consistency(andthus also the existence), there is circularity in his reasoning. Due to this circularity,
Hilbert fails to prove the consistency of his axioms.
A revised Poincaréan critique attaches itself in two different places in Hilbert’s
above proof. First, he necessarily uses induction to prove that whenever Axioms
1-4 are applied to a homogeneous equation, it results in a homogeneous equation.
When Hilbert says, “Axiom 3 yields only homogeneous equations if in it we takeany thought-object for x,” he implicitly relies on induction to guarantee the truth ofthe claim [27, p. 134]. Recall that Axiom 3 states f (ux) = u(f 0 x). Hilbert claims
that substituting any thought object for x into Axiom 3 results in a homogeneous
equation. The proof of this claim must first show that it holds for thought-objects
x composed of one simple object, then for x composed of two simple objects, andso forth. This “so forth” is an implicit application of the principle of mathematicalinduction. Hilbert relies here on an intuitive understanding that if the homogeneityof Axiom 3 holds for a thought-object of length n, then substituting a thought objectof length n + 1 also results in a homogeneous equation. In order to progress fromthe truth of this claim to the statement that it holds for all possible thought objects,
Hilbert must employ PMI.
The second place that a refocused Poincaréan style objection is pertinent against
Hilbert is in the well-definedness of the concept of homogeneity in the first place.
Hilbert’s definition of homogeneous is recursive. Recall that every recursive definition necessarily relies on an inductive inference to demonstrate that the definition
is well formed for definiendum of arbitrary length. Hilbert’s definition relies on thereader’s intuitive grasp of induction and inductive mathematical reasoning to define
homogeneity.
Both of these instances, it is important to note, do not occur in explanatory
remarks or otherwise non-essential places. The first instance involves a claim central60

to Hilbert’s proof that only homogeneous equations result from applications or statements of the axioms. The second instance occurs in a formal definition essential forthe proof. It is not possible, therefore, for Hilbert to do what Poincaré would call a“patching-up” and remove these instances of mathematical induction.
The intuitive leap that Hilbert relies on in these two instances is precisely what
Poincaré argues is inseparable from the foundations of mathematics; it is the crux of
his objection (see §2.3.1). By relying on this intuition, Hilbert has failed to give an
acceptable, non-circular proof of the consistency of Axioms 1-5. Further consistency
proofs for additional axioms rely on similar strategies that could be attacked in asimilar fashion, but there is no need to do so, because they necessarily rely on a
successful proof of the consistency of Axioms 1-5. Thus, Hilbert has not successfully
produced a proof of the consistency of arithmetic.

3.2

The Development of Hilbert’s 1920s Program

Each of four major papers throughout the 1920s mark a significant development in
Hilbert’s program to justify and clarify the foundations of mathematics. The first
three papers all solidify a substantial differentiation in Hilbert’s proof theory while
also providing inklings of the distinction to be drawn in the next one. The current
section will impose a partially artificial structure on these papers in order to present a
clear account of four vital, yet often implicit contrasts in Hilbert’s system. This clear
account will then enable the construction of a proper Hilbertian reply to Poincaré’s
objection.
The new grounding of mathematics: first report [31] introduces a boundary between the theory and the metatheory that was only implicit in Hilbert’s earlier papers.
The logical foundations of mathematics[30] distinguishes between finite and transfinite reasoning, equating the former with the object language and the latter with the61metatheory introduced in the prior paper. On the Infinite [28] establishes ideal propositions in contrast to concrete propositions. Ideal propositions are axioms adjoined
to the concrete propositions of the object language for the purpose of expanding themathematical power of the object language. Finally, all three distinctions are employed most clearly in The foundations of mathematics [26], wherein Hilbert thought
his critics answered. Throughout these four papers, one final contrast is implicit andnever formally mentioned: purely formal reasoning verses contentual (inhaltliche)
reasoning.

3.2.1

Theory vs. Metatheory: 1922

Early Hilbert’s attempt at a proof of the consistency of mathematics employed aproto-proof theory.

The attempt at a positive proof examined the structure of

the proofs, defined a purely syntactic property (heterogeneity), and demonstrated
first that any contradiction had this property and second that no proposition withthis property could result within the proof system (§3.1.2). It was vulnerable toa Poincaréan style objection because the system lacked a formalized differentiation
between reasoning done with the symbols and reasoning done about the symbols themselves. A semantic/syntactic distinction was absent, as was differentiation between
the object language and the proof theoretic language.
In 1922, Hilbert made this distinction. He introduced the object level, in which
there were only the ‘number-signs’ that “are themselves the object our consideration,but otherwise they have no meaning [Bedeutung] of any sort” [31, p. 29]. The formal
system is restricted at first to definitions of formulas, signs, and the length of signs.
As an example of the sort of proof possible in such a limited system, Hilbert gives
a sketch of a proof of the statement ‘a + b = b + a’ for any strings a and b. By so
restricting the system, Hilbert confidently claims, “. . . no contradictions of any sort62are possible. We simply have concrete signs, objects, we operate with them, andwe make contentual [inhaltliche] statements about them”[31, p. 31].9 Though thisrestricted system cannot prove much, it provides the basis to which the addition of
axioms allows Hilbert a derivation of arithmetic.
However, the introduction of any new axioms immediately requires a proof of
those axioms’ consistency. The way that Hilbert endeavors to perform such a proof
is by
mov[ing] to a higher level of contemplation, from which the axioms, formulae, and proofs of the mathematical theory are themselves the objectsof a contentual investigation. But for this purpose the usual contentual
ideas of the mathematical theory must be replaced by formulae and rules,
and imitated by formalisms. . . and at the same time it becomes possible
to draw a sharp and systematic distinction in mathematics between theformulae and formal proofs on the one hand, and the contentual ideas on
the other [31, p.33]

Postponing a full discussion of the contentual/formal contrast until §3.2.4, these
two terms for the time being should be read as shorthand for ‘having some meaning’and ‘pure manipulations of symbols’.10 Hilbert not only differentiates the subject
matter of the theory and the metatheory, he also differentiates between the type of
reasoning that takes place in each. The subject matter of the theory is solely composedof the basic symbols. It is the instantiation or application of axioms to form proofs.
9Note the use of inhaltliche even in this early paper. There will be numerous uses of the contrast—
particularly in the following quotation—throughout the next three sections, but discussion of this
distinction will be postponed until §3.2.4
10
I do not want to impute to Hilbert the sort of formalism wit which he is often labeled , something
akin to thinking that mathematics is simply a game that one plays with symbols devoid of meaning.
In fact, the inclarity with which he uses the terms ‘contentual’ and ‘formal’ suggests his reluctance toaccept that particular view of mathematics. In particular, I think that Hilbert wants to say that theoperations in the object language are purely formal insofar as they have no given content. However,in reading them and doing the manipulations, I can impute some content or meaning to the proof,
even if it lacks that “official” meaning. For example, when I prove that ‘a + b = b + a’ in the object
language, it has no meaning. What I have shown is merely something about strings. However, I
might whisper to a student who is not understanding, “Pssst, what I’m doing here is proving thecommutativity of addition of integers.” In this way, I can think or reason contentually about purely
formal notions.

63

Reasoning within this object language is wholly and completely determined by theaxioms and is solely the manipulation of eligible objects according to given rules. In
contrast, Hilbert has as the subject matter of the metatheory those very things thatare taking place in the object language: proofs, formulae, and axioms. Reasoning
in the metatheory has content since it is about something, whereas object language
manipulations are uninterpreted and purely syntactic. This distinction is brought out
explicitly by Hilbert:
To reach our goal[of proving the consistency of the axioms in the object
language], we must make the proofs as such the object of our investigation;
we are thus compelled to a sort of proof theory which studies operations
with the proofs themselves. For concrete-intuitive number theory, which
we treated first, the numbers were the objectual and the displayable, andthe proofs of theorems about the numbers fell into the domain of thethinkable. In our present investigation, proof itself is something concreteand displayable; the contentual reflections follow the proofs themselves[31, 59, emphasis in original].11

The purpose of the metatheory, or as Hilbert calls it the ‘metamathematics’, isto prove the consistency of the axioms in the object language. Thus, he envision theprocess of deriving mathematics as a two step cycle. Step 1 is to derive provable formulas within the object language using the existing axioms. This first step provides
the desired mathematical results. However, once the fruitful results of the current setof axioms are exhausted, proceed to Step 2, which adds new axioms to the alreadyexisting axioms and proves their consistency via metamathematical means. Having
proven the new axioms’ consistency, Step 1 begins again to obtain more mathematically fruitful results. Once all the necessary axioms are introduced, the whole ofarithmetic is a consequence.
By explicitly bifurcating the theory and the metatheory, Hilbert has a potential
response to the Poincaréan style objection of circularity. To the charge that he em11

Note also the use of ‘concrete’. This subject will be taken up in §3.2.3

64ploys induction in the proof of the consistency of the axioms (on of which is induction
itself), Hilbert could reply that because the purported consistency proof is done in
the metatheory, the principle of induction employed is metatheoretical rather thantheoretical. Metatheoretical induction is utterly different from the formal principle
of induction. Hence, Poincaré’s objection is defanged. On the face of it, this rebuttal
seems effective and complete. However, this efficacy will become our topic later insection §3.3.

3.2.2

Finite vs. Transfinite: 1923

Hilbert’s next paper clarifies the proof theory of 1922. Of particular importance ishis differentiation between finite and transfinite reasoning. Given that the theory isintended to reproduce all existing mathematical results, a limitation to purely finite
logic is too restrictive. Elements of mathematics are excluded from this system. However, the epistemic basis for Hilbert’s work arises out of its simplicity and surveyability(see p.62) Hilbert writes,
But in our proof theory we wish to go beyond this domain of finite logic,
and we wish to obtain provable formulae that are the images of the transfinite theorems of ordinary mathematics [30, 13].
In an attempt to resolve this dilemma, Hilbert introduces the single transfinite
axiom that is to be the sole source of any non-finite conclusion: The axiom reads:A(τ A) → A(a)(TF Axiom)

In non-technical language, this axiom says that if some predicate A holds of the object
τ A, then it holds for all objects a. The τ operator picks out a purported counterexample, designating an object of which the predicate A does not hold. Therefore,
the axiom states that if the predicate holds even for a purported counter-example—it65

holds even for τ A—then it holds for every object [14, p.1135]. If we imagine thepredicate ‘A’ to mean ‘bribable’, then τ A is a man so morally upright that if he isbribable, then so are the rest of the human rabble [30, 19].
The τ operator is the precursor (and dual) to the more well-known  operator that
Hilbert introduces in 1925. Whereas the τ function picks out a purported counterexample, the  operator is intended as a choice function.
This lone axiom “is to be regarded as the original source of all transfinite concepts,
principles, and axioms”[30, 20]. By this limitation, Hilbert hopes to preserve theepistemic security garnered from the earlier limitation to purely finite reasoning.
The question arises as to what type of reasoning is permitted in the metatheory.
Presumably, it might seem that if the theory permits transfinite reasoning, then soshould the metatheory. However, the metatheory must guarantee the epistemic basisof the theory, and the only reliable epistemic basis for Hilbert is finite reasoning. The
objects of the metatheory are concrete displayable proofs. These objects are all finite,
and thus there is no need to involve transfinite reasoning.
This differentiation is borne out in the text, in Hilbert’s sketch of a consistency
proof of the axioms. This proof proceeds along familiar ground, closely resembling
the proof offered in 1904: show via considerations of the form of the proof that acontradiction cannot be derived from the given axioms. In Hilbert’s own words,
The basic idea of such a proof [of consistency] is always as follows: weassume that we are presented with a concrete proof having the end-formula
0 6= 0 . . . Then, by considering the matter in a finite and contentual way,
we show that this cannot be a proof satisfying our requirements[30, 23].

Hilbert’s strategy is to provide a metatheoretical proof schema wherein any purported (finite) derivation of a contradiction in the theory is shown to fail to be a
proof. Because all proofs are finite, this proof schema will suffice to demonstrate thatthe theory is free from contradiction. Moreover, because the metatheory is finite, it
66

attains the secure epistemic footing required to eliminate doubt in the foundations ofmathematics.

3.2.3

Ideal vs. Concrete: 1925

There are two apparent problems with Hilbert’s project in 1923. First, it is prima
facie implausible that it is possible to prove the consistency of a set of axioms of atheory using a weaker theory.12 Particular to Hilbert’s program, it seems unlikelythat a finitist metatheory has a theorem stating the consistency of a transfinite theory. Second, the transfinite theory stands at odds with Hilbert’s attempts to restrict
foundational work to the most epistemically simple and clear cases. Hilbert holds theview throughout the 1920s that the only area within which we are able to safely infer
and reason are finite domains. How is the introduction of a transfinite axiom not in
violation of this prohibition?13
These two worries are answered in his 1925 paper, On the Infinite, wherein heclarifies the epistemic status of transfinite propositions and axioms within the theory.
These transfinite propositions—what he dubs ‘ideal’—are distinguished from the finite
propositions of the theory—what I will call ‘concrete’.14 Through this differentiation,
the hope is that transfinite reasoning within the theory is accessible even though the
12

Clearly the finite metatheory is weaker than the transfinite theory, as the theory contains all
finite results plus all those results obtained by the addition of the lone transfinite axiom.
13
Gentzen, a student of Hilbert’s school, was attempting to do something of this sort even into
the 1930s after G´’odel’s result. Gentzen recognizes that an absolute consistency proof is impossible,
writing that “A consistency proof can merely reduce the correctness of certain forms of inference tothe correctness of other forms of inference” [21, p.138]. He then goes on to formulate a proof of theconsistency of elementary number theory along Hilbert’s lines, wherein he employs induction to showthat he can reduce the instances of the only connective that could lead to contradiction. In this way,
it is very similar to Hilbert’s 1927 approach. Gentzen holds that his proof is a reduction because hedoes not employ complete induction, but rather only induction up to 0 . This notion of induction,
he asserts, can be construed finitarily and therefore is suitably a reduction. However, it is importantthat this is still induction, and it is still the case that Poincaré would argue that understanding thatthe proof is universally true presupposes an understanding of complete induction.
14
Though this usage is not extensive throughout Hilbert’s papers, it does agree with his usage
and is the most illuminating in terms of contrast with ideal. See, “We simply have concrete signs orobjects” [31, 31].

67

theory itself retains the privileged epistemic status of the purely finite.
The distinction is born out of a method, common in mathematics, of introducing
new elements to a system to complete the rules by making them more symmetric,
harmonious, or universally applicable. The introduction of these ideal elements createa more powerful system able to address a broader range of topics. It is not surprising,
given his axiomitization of geometry, that Hilbert harkens back to the Euclidean plane
to introduce this method of ideal elements as a solution to the problems facing his
finitistic project.
Consider what Hilbert dubs the “Axiom of Connection”:(AC)Any two points on the plane are connected by one, and only one, line.
The geometric dual of this axiom—the statement obtained by replacing ‘point’
with ‘line’, ‘line’ with ‘point’, and ‘are connected by’ with ‘intersect at’—however,fails to be true, since it is:(AC ∗ ) Any two lines on the plane intersect at one, and only one, point.
This dual must be couched with the caveat that (AC ∗ ) fails in the case of parallellines. The axiomitization of ordinary plane geometry has a fundamental asymmetry:
any two points determine a single line without exception, while the same is not true
for any two lines determining a single point. Hilbert outlines how to correct thisasymmetry.
But, as is well known, the introduction of ideal elements, namely, points at
infinity and a line at infinity, renders the proposition according to which
two straight lines always intersect each other in one and only one point
universally valid[28, p.372].
The introduction of ideal elements in plane geometry is aimed at a very particularpurpose: banishing asymmetries in the system and thereby making the geometrical68rules universally applicable without exception. Their advantage—and therefore theirraison d’être—is that they make “the system of the laws of connection as simple andperspicuous as is at all possible”[28, pp.372-3]. Introducing both a point and a line
at infinity makes the Axiom of Connection and its dual true. Let the point at infinity
be ω and the line at infinity be λ. We now prove that both (AC) and (AC ∗ ) are truein this new, expanded geometry that includes ideal elements.
(AC) For any two points {a, b}, if a, b 6= ω, the line l that connects {a, b} is connected by the line guaranteed to connect a with b by the axiom of connection in
traditional plane geometry. If, without loss of generality, a = ω, then λ connects
a = ω with b.
(AC ∗ ) There are three cases. For any two lines l and m,
1. If l, m 6= λ and l is not parallel to m, then l and m intersect at the point
guaranteed by the logical consequence of the original axiom of connection in
traditional plane geometry.
2. If l, m 6= λ and l is parallel to m, then l and m intersect at point ω.
3. If, without loss of generality, l = λ, then l intersects m at point ω.
The addition of these two ideal elements has made possible the duality principleof geometry—that concepts have a dual with which they may be interchanged producing a new result. A similar strategy appears in the field of analysis. Prior to theadjunction of complex numbers to the reals, polynomial equations such as y = x2 + 4
lack any roots. Any claim made about roots of a polynomial must, in this weakersystem, be made with caveats regarding the existence of such roots. However, oncethe complex numbers are introduced, these caveats may be done away with. All
polynomial equations have as many roots as their degree (counting the repetition of69

roots). The theorems and results of analysis are universally applicable because of theaddition of these ideal elements.
However, a discussion ‘adjunction’, ‘addition’, and ‘introduction’ of these ideal
elements necessitates an examination of their ontological status. Are they on the samefooting as the elements to which they are being introduced? Does the new system
replace completely the old system? Is a distinction between these new elements andthe old elements worth maintaining?
On this matter, Hilbert draws a sharp line separating the old from the new,
between the concrete and the ideal. In discussing the geometric example, Hilbertsays, “. . . the points and straight lines of the plane are initially the only real, actuallyexisting objects”[28, p.372] This quotation suggests that Hilbert holds that these newelements at infinity are different from the original ones. However, he uses the word
‘initially’, so perhaps the actual existence of the ideal elements is marked by theirintroduction to the system.
The proper context, however, clarifies that Hilbert is vehemently opposed to assigning the same ontological status of the original elements to the new ideal elements.
The introduction of this example comes in the midst of a lengthy discussion of theinfinite. More precisely, it comes at the conclusion of a passage wherein Hilbert argues against the actual existence of the infinitely small or infinitely large. For theformer, Hilbert argues that considerations of relativity made it necessary to relinquish
Euclidean geometry and replace it with elliptical geometry, doing away with ideas of
the infinitely large actually existing in nature. As for the latter, Hilbert says,. . . we do not find anywhere in reality a homogeneous continuum that
permits of continued division and hence would realize the infinite in thesmall. The infinite divisibility of a continuum is an operation that ispresent only in our thoughts; it is merely an idea, which is refuted byour observations of nature and by the experience gained in physics andchemistry [28, p.371].

70

Hilbert denies the actual existence of infinite quantities, either infinitely smallor large, and concludes that there is nothing to recommend their actual existence
in mathematics. Rather, instead of actually existing, Hilbert assigns an essential
heuristic role to the infinite in mathematics.
Yet it could very well be the case that the infinite has a well-justified placein our thinking and plays the role of an indispensable notion [28, p.372,
emphasis in original].
The ontological justification for the existence of the infinite extends only as far as
it helps our reasoning and thinking. Returning to the geometry example, the point at
infinity does not exist in the same way as the rest of the points in the plane. Rather,
it is a kind of useful fiction—we can talk about it, reason with it mathematically,and even use it to prove interesting results. Ultimately, its existence is justified solely
by the extent of its usefulness, whereas ordinary points are ontologically secure andbasic. Note that the introduction of the point and the line at infinity does not makethe principle of duality true. Instead, it clarifies its use by creating a more powerfulsystem within which the principle of duality has no exceptions. It is by making these
sorts of calculations simple, the ideal elements are said to exist for Hilbert.15
This sort of existence allows Hilbert to introduce ideal elements while keeping theonly epistemically relevant elements concrete. Concrete propositions are the ontological simples of the theory; their existence is actual, in contrast to ideal elements thatexist only as helpful tools for proving things about the concrete objects. The difference between the ontological status (and thus epistemic status) of ideal and concrete
permits Hilbert to address the two potential issues raised at the beginning of thissection. First, there is not a finite metatheory proving the consistency of a transfinite
theory; instead, the finite metatheory proves that the adjunction of ideal elements
to a finite metatheory does not cause contradictions. The theory remains fundamen15

There is resonance here with the role of the infinite and Poincaré’s own view of the infinite asa non-existing, mental construct. Compare to §2.3.2

71

tally finite in nature. Hence, Hilbert can classify the transfinite axiom (page 65) as
an ideal element of the system and employ it without sacrificing epistemic certainty.
Second, Hilbert’s introduction of ideal elements brings the theory in accordance withhis prohibition of infinite reasoning, because what appears to be infinite reasoning
using ideal propositions is actually indirect reasoning about finite propositions. Ideal
propositions are employed only because of convenience of communication —they holdno epistemic status beyond indicating how authentic, formal, finite reasoning should
proceed.

3.2.4

Formal vs. Contentual

Unlike the first three, there is no seminal paper that draws this contrast into sharp
relief. Instead, the distinction becomes clear over the course of Hilbert’s papers.
The formal/contentual distinction appears straightforward, but only in light of theideal/concrete distinction. Once the contrast between ideal and concrete propositions
is established, it is clear how to characterize Hilbert’s conception of formal versus
contentual.
Formal reasoning is reasoning done solely through the manipulation of symbols
according to a set of rigid rules. It is defined purely by its form, lacking any interpreted
meaning behind the symbols: it is purely syntactic.
Contentual reasoning on the other hand, has content. There is an intended
interpreted meaning. For Hilbert, a contentual proposition is one that attempts tocommunicate an idea to the reader and does so with an intended meaning. The mark‘1’ is trivially formal, because it is a basic element of the theory, a concrete element
devoid of meaning. Just as trivially, ‘2’ is contentual, since it has a meaning: it isintended to communicate to the reader the mark ‘11’. Similarly, ‘24’ is contentual
because it means ‘111111111111111111111111’. Formal reasoning takes place when72

and only when concrete objects are involved; contentual reasoning takes place in allother cases, but can be marked out positively by the presence of some identifiable
meaning.
An example of the differentiation between contentual and formal reasoning will
help illuminate this dichotomy—necessarily, the distinction mirrors the distinction
between ideal and concrete. Hilbert [28] distinguishes between propositions like1 + (1 + 1) = (1 + 1) + 1

and
1+2=2+1anda + b = b + a.
The first counts as a concrete, formal proposition. Its epistemic value derives fromthe fact that it is finite, surveyable, and epistemically secure. The second is also
concrete, but strictly speaking, fails to be formal because the ‘2’ means something.
It is a method of conveying a formal equation in a shorter, easier to understandmanner. Its epistemic value trades on the fact that it communicates a finite, formal
proposition, but is not a finite, formal proposition itself. Finally, the third proposition
is composed of ‘a’ and ‘b’, which are uninterpreted formal objects without finite
content. It is an ideal, transfinite proposition. Its epistemic value derives not fromimmediate grasp of its content but from its derivability from the transfinite axiom(page 65). Formal reasoning must involve only propositions that resemble the firstequation. Contentual reasoning takes place involving equations that resemble thesecond and third equations.
A proof using ideal propositions will necessarily be contentual. It can provide a73

proof-sketch of properly formal, finitary proofs, but does not formally prove anything.
For instance, a proof of the statement 1 + a = a + 1 can be accomplished through theapplication of the mathematical induction axiom. We show that 1 + 1 = 1 + 1 is true.
Then, we assume that 1 + n = n + 1 holds and show that 1 + (n + 1) = (n + 1) + 1
holds. Via the ideal principle of induction, we will have then shown that for all a, thestatement ‘1 + a = a + 1’ is true. But for Hilbert, the proof just given is not a real
proof in the sense that it is not wholly formal and finite. It is merely a useful way ofcommunicating both that there is a finite, purely formal proof of that equation andgiving an outline of how to proceed.16 It is short hand for an infinite collection ofreal proofs, for example, the proof that shows that ‘1 + 57 = 57 + 1’ is true. Hilbertdraws the distinction thus:
If we generalize this conception [of treating ideal propositions as communicating finite propositions], mathematics becomes an inventory of
formulas—first, formulas to which contentual communications of finitary
propositions [hence in the main, numerical equations and inequalities] correspond and, second, further formulas that mean nothing in themselves
and are the ideal objects of our theory[28, p.380, emphasis in original]
The first set of formulas Hilbert mentions here is the set of contentual propositions
composed of numerals and normal mathematical symbols that serve to communicate
formal propositions made up of concrete objects. Their referents are the subject of thetheory make the goal of Hilbert’s project—the elimination of epistemic doubt in thefoundations of mathematics—possible by their secure epistemic status. The second
set of objects are the ideal propositions that provide the mathematical machinery toderive arithmetic from Hilbert’s foundation.
How do these four distinctions help to clarify Hilbert’s project in the 1920s? They
do so by providing us a precise vocabulary to describe the details of Hilbert’s project.
16

I should mention here that this interpretation of what Hilbert is doing in terms of the status of
proofs using ideal propositions is somewhat controversial. Detlefsen disagrees, but I will argue laterthat Detlefsen’s reconstruction fails to avoid a serious Poincaréan objection, so the question of its
faithfulness to Hilbert in this regard becomes moot.

74

Hilbert must provide an axiomatic foundation and then prove two things about it:1) that it actually is a foundation for mathematics, and 2) that the foundation isitself without doubt.17 The foundation that Hilbert provides is a finite, formal axiom
system about concrete, purely syntactic marks. Because it deals only with thesefinite marks, Hilbert holds that it is beyond epistemic doubt.18 He proves that it isa foundation by reconstructing arithmetic via the introduction of ideal axioms to thetheory, thereby allowing a contentual derivation of all mathematical results. However,
the introduction of these transfinite ideal elements means that the theory has driftedfrom the epistemic safety of finitism into the danger of the transfinite. In order toprove that these axioms have not led the project astray, Hilbert creates a proof schemain the purely finite metatheory using solely finitary that the axioms can lead to nocontradiction. The proof proceeds by demonstrating that any purported proof of acontradiction is not in obedience with the ideal axioms. Because any such purported
proof will be finite, this proof sketch is a contentual demonstration of a set of finite
proofs. Because of this finitism, Hilbert thinks the epistemic status of the theory issecure.

3.3

Hilbert’s Reply to Poincaré

Hilbert has laid out a complicated structure in refining his initial sketches, yet theseeds of the 1920s project lay in his early work—the metatheory/theory and to a lesser
extent finite/transfinite distinctions are present in undeveloped forms; the general
structure of the foundation, the commitment to proceeding via the axiomatic method,
17

This two step process is nearly identical (in general terms) to the project of the neo-logicist, tobe examined in chapter 5.
18
A Poincaréan would seize upon this claim and argue that Hilbert is holding a psychologicalprinciple as bedrock (see §2.1.3). To some extent, I think there is a resonance here, but I think theHilbertian would respond that if finite reasoning is not beyond doubt, then total skepticism aboutmathematics is the only viable position. Once the complete skeptical position is rejected, there must
be something taken as epistemically basic, though not because of psychology.

75

and the attempt to prove the impossibility of the derivation of a contradiction by
purely syntactic means are all present. The pertinent question for current purposes
is whether or not this refinement can address a Poincaréan style petitio objection.
Hilbert certainly seems to think so when he writes,
Poincaré already made various statements that conflict with my views;
above all he denied from the outset the possibility of a consistency proof
for the arithmetic axioms, maintaining that the consistency of the methodof mathematical induction could never be proved except through the inductive method itself. But, as my theory shows, two distinct methods
that proceed recursively come into play when the foundations of arithmetic are established, namely, on the one hand, the intuitive constructionof the integer as numeral. . . that is contentual induction, and, on the other
hand formal induction proper, which is based on the induction axiom andthrough which alone the mathematical variable can begin to play its rolein the formal system.
Poincaré arrives at his mistaken conviction by not distinguishing between
these two methods of induction, which are of entirely different kinds. . . [26,p.473, emphasis in original]
However, this quotation is the entirety of Hilbert’s response to Poincaré. Though
he is right to say that there are two forms of induction—the finite, intuitive inductionof the metatheory and the transfinite, formal axiom of induction in the theory—
Hilbert fails to show how this differentiation defuses Poincaré’s objection.
In order to attempt to answer this question, I will look at two modern defenses ofHilbert against the Poincaréan objection, by Marcus Giaquinto and Michael Detlefsen.
Giaquinto appears to misconstrue the heart of the objection, rendering his reply
insufficient. However, Detlefsen’s response appears effective in rebutting Poincaré—
the justification of the Axiom of Induction is, under this interpretation, non-circular.
However, despite being the best chance Hilbert has to respond to Poincaré, I willargue that the response comes at too high a cost. The finite metamathematical proofof the axiomatic consistency of the theory implictly invokes non-finitary induction
as support for one of his premises. Thus, Hilbert invokes a meta-metatheoretical76statement that is non-finitary. This implicit appeal, coupled with Hilbert’s statedgoal of removing any doubt from all of the foundations of mathematics opens Hilbert
to a refocused Poincaréan objection. Thus, while Hilbert can reply to Poincaré’s firstobjection, the goal of his project him to be vulnerable to a second-order Poincaréanobjection. First we turn to Giaquinto and then to Detlefsen.

3.3.1

Giaquinto’s Hilbert

Giaquinto’s reconstruction of Hilbert’s program in many ways agrees with the one presented here (summarized on page 74) in most ways, with the differences attributed todifferent emphases. In particular, his account focuses on the distinction between finite and transfinite propositions and reasoning. A fundamental question for Giaquinto
becomes how apparently infinite laws can be recast in solely finitary methods. For
example, Giaquinto identifies what he calls “finitary general statements” such as [22,p.146]:
c+1=1+c(FG)

The problem with such statements, according to Giaquinto, is that they appear “toinvolve a tacit reference to an infinite totality,” and thus there is reason to believe thatthey should be excluded from a purely finitistic reasoning [22, p.147]. He addresses
this concern by arguing that the meaning of statements like FG can be understoodas a general disposition to assent to any instance of FG with a numeral in place of‘c’. Giaquinto highlights that someone can have this disposition without having any
reference to an infinite totality or even having been exposed to an infinite object. The
point then, is that these sorts of statements are legitimated through this restricted
reading.
Proofs of these finitary general statements can be salvaged in a similar way for
Giaquinto—a proof of a finitary general statement like FG is considered legitimate
77

whenever the ideal proof of FG gives an outline of how to finitarily prove any specificinstance of the generality. Giaquinto phrases this as “when we have a finitary procedure for obtaining from the purported proof a finitary proof of any given instance ofthe general proposition”[22, p.155].
The principle of induction in Hilbert’s theory is salvaged in the same way. Rather
than being a quantified principle of induction, Giaquinto considers it as a schema,
free-variable induction, or as a rule:ϕ(1), ϕ(y) −→ ϕ(sy)ϕ(y)

Note that this rule, when used in a proof of a generality, can be employed to
produce a finite proof of any instance of the rule. For instance, once shown that thisrule holds for some predicate ϕ, then for any given number c, one can construct a
proof that ϕ(c) by simply constructing a proof involving c − 1 instances of modus
ponens. We (the mathematically educated) recognize that the hypothetical person
cognizant of only finite quantities—the idealized person from above never introduced
to infinite totalities—is able to construct with this rule a proof for any instance ϕ(c).
From this, we, along with Giaquinto, conclude that the finite induction schema in themetatheory is justified and well-grounded.
Giaquinto thinks that this distinction between the unquantified rule of inductionin the finite metatheory and the formalized quantified principle of induction resolves
the Poincaréan objection. However, he misconstrues the objection, taking the forceof the objection to be that the principle of induction is not accessible from the finite
point of view:
The argument is that induction on the natural numbers would be neededin any proof of the consistency of a theory and yet induction is something78

that is not evident from the finitist point of view[22, p.162].
Giaquinto thinks his argument that the unquantified rule of induction is acceptable from the finitist point of view rules out this objection. However, the Poincaréanobjection is not that induction fails to be finite; rather, the objection’s force lies in
the claim that Hilbert has presupposed induction—finite or infinite—in proving thereliability of induction, either finite or infinite. A use of finitely acceptable induction
to demonstrate the consistency of a finitely acceptable form of induction is just as
illicit as using infinite induction to prove the acceptability of infinite induction. Simply showing that Hilbert has a finitistically acceptable form of induction at hand does
not answer the question.
Giaquinto’s position has painted himself into something of a corner, regarding
the actual petitio challenge. The emphasis on the finite/transfinite distinction meansthat Giaquinto is committed to saying that ideal propositions are finitely accessible.
In particular, this commitment means that the induction axiom (which is included in
the consistency proof that Hilbert needs) is finitely accessible. But then, according
to his response to what he took to be Poincaré’s objection, so is the principle inthe meta-theory—indeed this property hews closely to Hilbert’s own language. But
then, Giaquinto construes Hilbert as using a finite principle of induction to prove theconsistency of a finitely accessible induction axiom. Regardless of the fact that one isin the metatheory and one in the theory, it still remains prima facie circular, because
they share the same epistemic justification: finitism.

3.3.2

Detlefsen’s Hilbert

Michael Detlefsen’s reconstruction of Hilbert’s response to Poincaré’s example clearly
differentiates between the formal, transfinite principle in the theory and the contentual, meta-theoretical principle of induction that is employed in its justification.
79

In fact, Detlefsen’s argument that the use of contentual induction to non-circularly
justify the formal principle of induction succeeds in rebutting Poincaré. However,though Hilbert is vindicated initially, I will argue that even Detlefsen’s reconstruction must admit a meta-metatheoretical statement that employs a use of transfinite
induction. This admission, coupled with the Hilbertian goal, again opens the door toanother petitio type objection.
Detlefsen’s argument hinges on a particular notion of circularity. Specifically, heclaims that an argument is circular whenever any reason for doubting the conclusion“is an equally strong reason for doubting the premises or inferences which lead up to
the conclusion”[13, p.60]. Operating from this fundamental definition of circularity,
Detlefsen reconstructs Poincaré’s argument as having two premises:
1. Any reason for doubting the real-soundness of I’s use of induction is an equallystrong reason for doubting the truth (truth-preservingness) of any contentual
meta-mathematical induction that might be used to prove I’s real-soundness.
2. Any reason for doubting the real-soundness of I is an equally strong reason for
doubting the real-soundness of its use of induction.[13, p.60]. 19
Poincaré’s claim, according to Detlefsen, is a combination of these two premises;
the strategy to reject a Poincaréan objection is to critique both of these premises.
Detlefsen rejects the second premise because any ideal proof I that invokes the induction axiom will also invoke other, non-inductive premises that one might also
doubt. One’s reason to doubt the non-inductive premises would also count as reason
to doubt the soundness of I as a whole, so there are some reasons for doubting thereal-soundness that are not doubting the use of induction. Thus premise two is not
true.
19

‘I’ denotes any ideal proof that involves induction and “real-soundness” means that the transmission of the epistemic finite status from the premises to the conclusion.

80However, this objection misses the mark on a number of fronts. First, it isreliant on a particular notion of circularity, that could just as easily be reconstituted
in terms of reasons to believe rather than reasons to doubt. That is, if, instead of
Detlefsen’s account of circularity, we posit that an argument is circular wheneverreasons to believe the premises presuppose reasons to believe the conclusion, thenDetlefsen’s account of premise two and his objection can be turned away. His accountof circularity is contentious and lies at the heart of this objection.
Even granting his account of circularity, he still misconstrues Poincaré’s argument
in requiring premise two at all. For premise two to be needed, the conclusion ofHilbert’s metatheoretical argument must be that ideal proof I is sound, since this is
the conclusion to be doubted in Detlefsen’s premise. Because the overall soundness ofa proof guarantees the soundness of each of its constituents, the use of induction in I
is real-sound. That being the case, there is then a parallel meta-theoretical argumentwith the conclusion that the use of induction is sound. The Poincaréan can direct
his attack there. The objection need not accept Detlefsen’s second premise since it istrivially true that doubting the conclusion of this parallel argument (that I’s use of
induction is real-sound) provides good reason to doubt the real-soundness of I’s useof induction.
However, Detlefesen’s main response to the Poincaréan is an objection to the firstpremise. His argument can be paraphrased as follows:

1. Finitary logic is a sub-theory of classical reasoning;
2. The closure of any set of propositions in finitary logic is a subset of the closureof the same set of propositions within classical reasoning (from 1);
3. It is conceivable that a use of induction in classical reasoning would result in
an inconsistency, while the same use of induction in finitary logic would not;81

4. A reason to doubt the soundness of a use of induction in classical reasoning is
not necessarily a reason to doubt the soundness of the same use of induction in
finitary logic (from 2 and 3)[13, p.62].
Note that this argument is not at all vulnerable to a recasting of the account ofcircularity, for if we use my suggested alternative above, we could similarly conclude
it is not the case that any reason to believe the premise (that finite induction issound) presupposes a reason to believe the conclusion (that induction within classicalreasoning is sound).
One problem with this argument is that it equivocates on a tension that arises
from Hilbert’s introduction of ideal elements to the theory. On the one hand, the ideal
axioms are introduced so that they can extend the results in the concrete theory.
The very reason that Hilbert introduces these ideal elements is because the set ofconclusions that can be drawn by the concrete theory does not reconstitute all ofarithmetic. Hilbert writes,
And analysis cannot be constructed by a concrete procedure of the sort
we have just given for elementary number theory. For we cannot comeclose to exhausting the essence of analysis merely by using that sort ofcontentual communication; rather, we need real, actual formulae for its
construction [31, 32].
On the other hand, these ideal principles must be included only in the theory—
they must in some way be translatable or verifiable by the concrete objects in thetheory. Hilbert wants to use the epistemic certainty of the finite, concrete language
to guarantee the certainty of the ideal part of the theory. Thus, the results derived
by use of the ideal axioms must be finite truths, or at the very least what Giaquinto
calls finitely general statements. The tension is that the axioms appear to need to go
beyond finite domains in order to achieve their purpose (reconstructing mathematics),
while at the same time must be restricted to finite conclusions in order to retain their82

epistemic certainty.
The way that Detlefsen’s response equivocates on this tension is that it hews close
to the one side of the tension—that the ideal axioms must introduce or go beyond
the purely finitary logic—in order to justify premise 3. What makes this possibility
conceivable is that the ideal propositions like the induction axiom extend the finite
system; if they did not do so, then the failure envisioned in premise 3 would not be
conceivable.
However, Detlefsen’s own ascription to an instrumentalist view of the ideal propositions—roughly that they are a tool by which one can derive new finite truths even though thetools themselves are not solely finite—requires that his ideal axioms and non-finitary
reasoning be finitarily sound in the finite domain. Detlefsen’s own position requiresthat the ideal axioms must not result in any contradictory finite statements, whichmeans that it should not be conceivable that applications of induction in classicallogic lead to a contradiction while failing to lead to a contradiction in finitary logic.
This particular commitment is deeply seated in Detlefsen’s program.
However, the equivocation on this tension is a problem solely because of Detlefsen’s other commitments. One might not subscribe to Hilbertian instrumentalism
and have something like Giaquinto’s position, wherein ideal proofs are mere proof
schemata rather than at the same epistemic level as proofs themselves.20 . There
would need be no equivocation and something like Detlefsen’s argument would apply.
However, this patch up job ignores a much deeper problem that is present in any
response that models itself on Hilbert’s while maintaining Hilbert’s stated goals.

20

Detlefsen is committed to this latter position. Cf. Detlefsen [13, pp.54-57]

83

3.4

Poincaré Redeemed

There is a general problem with any attempt at reconstructing Hilbert’s project in
an attempt to avoid Poincaré’s petitio objection. Any such attempt will necessarily
run afoul of Hilbert’s stated goal for the project: the dismissal of all questions oflegitimacy from the foundations of mathematics. First, I will provide some brief
evidence expounding on Hilbert’s goal and then show how any attempt to prove theconsistency of the ideal axioms (which include induction) non-circularly will resultin an implicit appeal to a full principle of induction. Such an appeal in the metametatheory would not be damning to the project, save that Hilbert’s goal precludes
such an appeal.
Hilbert’s goal remained constant throughout the 1920s—the dismissal of the question of doubting the foundations of mathematics. Early in his 1922 paper, he writes,
But that is what I require: in mathematical matters there should be inprinciple no doubt; it should not be possible for half-truths or truthsof fundamentally different sorts to exist . . . I am of the opinion that thefoundations of mathematics are capable of full clarity and knowledge, andthat the problem of grounding our science is difficult but neverthelessconclusively solvable [31, 1].

He begins the 1923 paper,
My investigations in the new grounding of mathematics have as their goal
nothing less than this: to eliminate once and for all, the general doubt
about the reliability of mathematical inference [30, 1].

He writes in 1925,
That, then, is the purpose of my theory. Its aim is to endow mathematical method with the definitive reliability that the critical era of theinfinitesimal calculus did not achieve [28, p.370].

And finally, he writes in 1927,
84

I pursue a significant goal, for I should like to eliminate once and for all
the questions regarding the foundations of mathematics, in the form inwhich they are now posed, by turning every mathematical proposition
into a formula that can be concretely exhibited and strictly derived, thus
recasting mathematical definitions and inferences in such a way that they
are unshakable and yet provide an adequate picture of the whole science[26, p.464].
The uniting theme in each of these quotations is that Hilbert aims to solidify notjust the consistency of his axioms, nor just the status of finitary arithmetic. His goal
is to establish the whole of mathematical reasoning. He aims to banish any and allquestions of justification and to demonstrate once and for all that it is impossible tohave reasonable doubt about the foundations of mathematics.
However, this goal will blunt any attempt to give a non-circular justification of
induction in Hilbert’s theory. Hilbert attempts a sketch of this proof in 1923; it
resembles in many ways his earlier attempts from 1904. Hilbert endeavors to showthat the formula 0 6= 0 is not derivable within the system. He proceeds by reductio,
where he first assumes a proof with a conclusion of 0 6= 0. Then he performs a numberof transformations on the proof to bring it into a logical normal form: he makes every
line have only one conclusion, eliminates all variables, and translates it into the object
language. From this “refined” proof, he then claims that each formula can be checked“to determine whether it is ‘correct’ or ‘false’ in a certain sense that can be precisely
stated”[30, 29]. The pertinent quotation immediately follows:
Now, if the supposed proof were to satisfy all our requirements, thenclearly each formula of the proof would have to pass this test in turn.
Thus the end-formula 0 6= 0 would also have to be ‘correct’; but it is notcorrect [30, 29].
Even setting aside the issue of the definition of ‘correct’, there remains a controversial claim. Namely, Hilbert claims every proof can be transformed into this linear
structure and it can be determined if the formulae are correct. But the fact that this85

claim is true for every proof is based on induction. Its truth is guaranteed by therealization that for any given proof P , with n-many formulas, we are able to checkthat the first formula is correct, that the second formula is correct, and so on up to
the nth formula. That such a procedure is possible for every proof—for all finite
proofs— is then merely an inductive claim. This time, the application of induction
is at the meta-metatheoretical level.
Giaquinto has a similar problem in his account. He argues that Hilbert has
finitary general statements, and that these statements amount to a general disposition to assent to specific instances. Similarly induction, both theoretical and metatheoretical, becomes an unquantified rule. Giaquinto’s argument that this rule can begeneralized to provide a finite proof of any particular instance, however, also invokes
an implicit inductive inference. Giaquinto gives an example of how to employ therule to create a finite proof, and on providing such an example claims, “Obviously,
the prescription can be generalized to obtain a finitary proof of any instance of ϕ(n)”[22, p.155] What he claims is true—we can generalize his procedure and thereby havea finite procedure for producing finite proofs from the finite induction schema. However, this claim itself is metatheoretical (and in this case meta-metatheoretical since
Hilbert will employ this induction schema in the metatheory). That we know such a
procedure exists is only possible via appeal to a transfinite principle that allows us
to conclude that proofs for all numbers can be generated from this rule.
Finally, suppose that we grant a form of Detlefsen’s reconstruction. That is,
suppose that there is a proof in the finite metatheory of the ideal transfinite principle
of induction in the theory. We could even concede that this derivation in the metatheory of the theory’s consistency is non-circular. However, this proof would have toshow that for every proof in the theory, ‘0 6= 0’ is not its conclusion. This amounts
to a procedure in the meta-theory of how to show, given a purported proof of ‘0 6= 0’,
that it is not actually a proof, that it fails to obey the rules laid down in the theory.
86Regardless of how this proof proceeds, it must be that it is applicable to every proofin the theory that purports to show a contradiction. However, the claim that thisprocedure will work in every case is, again, another meta-metatheoretical claim. Sucha procedure would have to be defined recursively in the finite metatheory, along thelines of some rule formulation as in Giaquinto. However, because the procedure isrecursively defined, the only way to justify the universal meta-metatheoretic claim
would be via induction. In order to know that the proof procedure in the metatheory
does, in fact, prove the consistency of the theory, one would have to use, either
implicitly or explicitly, induction in the meta-metatheory.
There is good reason to think this problem is endemic. Any consistency proof
that purports to increase the epistemic strength of the axioms must use a principle ofinduction that is at least as weak as the principle within the theory so that the epistemic status is improved. If the metatheoretical principle of induction is not strictly
weaker than the theoretical principle, then it is difficult to see how the epistemic
security of the theory has been improved.
However, if the metatheory employs a weaker principle of induction than thetransfinite principle in the theory, the question arises as to how it is known that thegiven derivation of consistency is, in fact, a good one. Any explanation of this sort
will necessarily appeal to a higher metatheory that employs a stronger principle ofinduction.
It is Hilbert’s stated goal of dismissing all of the questions of doubt that makes
this appeal unavailable to the Hilbertian. The explication of all the distinctions
Hilbert has made and all the progress seemingly gained by the formalization wash
away in the face of such an appeal; appealing to a higher level of induction once
again makes the project circular. Poincaré’s objection, refocused and made metametatheoretical, arises again.
In this way, the problem with epistemic justification resembles a Tarski-style87regress of metalanguages. The only way to ascertain that the epistemic justificationof a meta-theoretical proof is correct is by advancing to a yet higher metatheory.
A Hilbertian reply that such a justification is not needed because finitary reasoning is unquestionable appears dogmatic. Any attempt by the Hilbertian to justify
finitary reasoning will lead to a regress, so he must simply accept the epistemic security of finitism. Unless this position is intuitively appealing, the Hilbertian has little
to offer as argument for his position.

88

Chapter 4
Poincare against the Logicists
Poincare’s objections to foundationalism in the early 20th century failed to differentiate between Hilbert and bona fide logicists like Frege, Russell, or Dedekind. Moreover,
his critiques openly mock the work of those to whom he is objecting, even going so
far as to say regarding the petitio objection for the logicists, “I have not seen it in thepages I have read, but I do not know whether I should find it in the three hundred
pages they have written that I have no wish to read.”[48, 183].
Among these objections, the petitio objection stands out. Recall that this objection amounts to an accusation of begging the question by employing induction in thederivation of mathematical axioms, one of which is induction itself. In chapter 3 wesaw how the original petitio was effective against early Hilbert and how it retained its
force in a revised form for the later Hilbert. In this chapter, I will show 1) why theoriginal petitio is irrelevant to the logicist; 2) give a revised, sophisticated version of
the petitio; 3) argue, according to Goldfarb [23], why even this sophisticated version
fails to touch the logicist.
It should be noted that Goldfarb’s argument rests upon a particular interpretationof the logicists’ project. More precisely, it relies on a certain interpretation of thelogicists’ view of logic—that there was no conception of a metatheory in their writings.
89Though I find this view compelling, whether or not one objects to it will not affect the
purpose of the current chapter. The goal is twofold: first, to show that the logicist isable to wriggle free of Poincaré’s grasp only by lights of his peculiar view of logic asinescapably universal—to deny the existence of a metatheory—thereby preventing arevision of the original petitio to the sophisticated petitio; and second, to show thatthe neo-logicist cannot respond to Poincaré in the same manner as the logicist. If one
disagrees with Goldfarb’s interpretation, the sophisticated petitio remains a viableobjection, both for the logicist and the neo-logicist.
One final terminological note. For this chapter, I will operate under the assumption that Hilbert, regardless of which time period one is talking about, is not alogicist. So, when I speak of the “logicist”, I do not mean to include within this group
Hilbert. Logicists are those who generally followed a Fregean framework to provide
an undoubtable basis in logic for mathematics. Hilbert abandoned this project early
to pursue a finitist method of removing doubt from mathematics.

4.1

Original Petitio and the Logicist

We have seen the original petitio objection previously in reply to early Hilbert (§3.1.3).
In it, Poincaré accuses Hilbert of presupposing a principle of induction in his proof ofthe consistency of induction with the other axioms and given rules of inference. Any
justification or proof of such consistency could only be made via an invocation of theprinciple of induction. As Goldfarb puts it,
The clearest [petitio] is directed against the notion that mathematicalinduction is not a principle with content, but is just an implicit definitionof the natural numbers. Poincaré notes that such a definition must be
justified by showing that it does not lead to contradiction; yet any suchdemonstration would have to rely on mathematical induction [23, 64].
90

Having to provide a demonstration that given axioms—or implicit definitions—do
not lead to a contradiction is a uniquely Hilbertian requirement. The logicist does notgive axioms and then guarantee the safety of reasoning with them via a consistency
proof. The origin of this difference stems from their underlying assumptions about
existence and consistency. We saw earlier that for Hilbert, mathematical consistency
guaranteed the existence of the items in question. Recall:. . . if the arbitrarily given axioms do not contradict one another with all
their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence [20, pp.3940].
However, this is exactly contrary to the position that Frege and the logicists
adopt, wherein the existence of a mathematical object is what guarantees its consistency. Frege writes,
From the truth of the axioms it follows that they do not contradict one another. There is therefore no need for a further proof. The definitions too,must not contradict one another. If they do, they are faulty. The principles of definition must be such that if we follow them no contradiction
can appear[20, p.37].
The fundamental disagreement between logicists and Hilbert is the direction of
implication between consistency and existence. Hilbert believes that consistency is
the more basic of the two notions, and therefore a proof of the consistency of a set of
axioms guarantees their existence; Frege and the logicist hold that the truth of theaxioms (their truth means that they exist as axioms) is the more fundamental notion
and that it implies that they are consistent. As Goldfarb writes,
Frege and Russell (and later on, Couturat) insist that existence is notproved by consistency; rather, consistency is vouchsafed by showing existence.[23,65]This fundamental difference makes Hilbert vulnerable to the original petitio andthe logicists not. Hilbert must somehow show that the axioms he gives for arithmetic91

are consistent because he has no notion of truth of the axioms. There is no way inwhich Hilbert considers his axioms to be “true”. For instance, the parallel postulate
could be laid down or not, with each option resulting in a different geometry. But one
geometry is not the “true” geometry. Indeed, we saw in the previous chapter that heassiduously avoids even giving his axioms even any meaning so that there is only apurely syntactic axiomatic method as the foundation of mathematics.
However, Frege and the logicists have a fundamentally different conception of
axioms. The axioms that they give—the basic laws—have to be true in order toguarantee their consistency. Thus, because they are true as given, they are always
interpreted and never the syntactic axioms that Hilbert gives. There is no syntactic
demonstration of the consistency of the axioms, and thus no application of mathematical induction. The original petitio fails to gain any bite.1

4.2

Sophisticated Petitio

Poincaré makes a more refined and subtle attack on the logicist program which, following Goldfarb, I will call the sophisticated petitio. Similar to the original version,
the sophisticated petitio accuses the logicist of circularity. However, rather than arguing that the induction is implicitly used in a proof of consistency, it argues thatthe logicist presupposes induction in establishing the system within which the reduction of mathematics to logic will take place. Poincaré actually makes this argument
against Russell’s theory of types initially, saying,. . . the theory of types is incomprehensible, if we do not suppose the theoryof ordinal numbers already established. How will it then be possible tobase the theory of ordinal numbers on that of types [44, p.52]?
1

Of course, it should be noted that the axioms chosen by Frege were, in fact, not consistent asshown by Russell’s Paradox. This result should be seen, then, as a proof that Basic Law V is nottrue. There may be very good reasons to doubt the logicist here, but I simply want to show thatthe original petitio is not one of them.

92Poincaré alleges that Russell presupposes a theory of ordinal numbers to differentiate between orders of propositions with the goal of structurally restricting propositions of order n to refer only to propositions of order n − 1. Russell’s theory of types
presupposes a theory of number even to classify what the order of a proposition is; thepresupposition occurs in the formation rules. For Poincaré’s temporal understandingof mathematics and intuition, the presupposition happens before the formal system of
the theory of types is established, and there voids subsequent epistemic simplification.
This objection is generalizable beyond Russell’s theory of types to any logicisttype project. Any proof of the reducibility of mathematics to logic will have to givea derivation of the ordinal number theory. This derivation will necessarily employ
a formal system of logic. In defining this formal system of logic, the logicist must
use induction to define the notions of formula and derivation recursively. Thus, thelogicist relies on mathematical induction for the formation rules of the logical system,within which he intends to derive the principle of induction. He has reasoned in acircle.
Important to note about this sophisticated version is that it makes a fundamentally meta-theoretical claim. While the original petitio argues from a theoreticalstandpoint(from within the formal system), the sophisticated petitio argues from a
metatheoretical standpoint(from outwith the formal system). The former claims thatany consistency proof within the system will employ induction; the latter claims thatthe formation rules for the formal system of logic presuppose induction. The original version is effective against the early Hilbert because his consistency proofs occur
within his system (see §3.1.2). However, it fails to touch the logicist, because thetruth of his fundamental principles guarantees consistency, not a positive proof ofthat consistency. The absence of a consistency proof within the system means that
there is no possibility of an implicit application of induction, nor a presupposition of
number. The original petitio fails, but the status of the sophisticated version remains93

in question.
Goldfarb provides a defense for the logicists against this sophisticated petitio
by invoking the logicists’ universalist conception of logic to blunt Poincaré’s attack
before it starts. The defense says roughly that because there is no meta-theoreticalperspective from which to mount the objection, Poincaré’s sophisticated objection isblocked.
For Poincaré, the objection is made “prior to” the formal system being in place.
This sort of formulation should be no surprise, given Poincaré’s emphasis on temporal
intuition(see 2.2.2). For Poincaré, there is sense in speaking about a time prior toa given formal system. Therefore, he argues that in laying out the rules for any
formal system requires a recursive definition and therefore presupposes the principle
of induction. Establishing rules for a formal system brings into existence that formal
system, for Poincaré. Any justification of this principle from within the system iscircular.
However, this objection is not one that the logicist needs to address. It is mounted
from a metatheoretical standpoint; it comes prior to the formal system of logic. For
the logicist, there is no possibility of such a standpoint; there is, for the logicist, no
notion of a metatheory. The conception of logic is all-encompassing insofar as logic
is not merely a formal system with funny looking symbols and complicated rules, butrather the space within which any and all reasoning takes place [23, 69]. Logic cannot
be decided upon and set forth in formation rules. There is no notion of “prior to” asthere is for Poincaré; for the logicists, logic exists outside the framework of time. It is
eternal and unchanging, instead of being summoned into being by the formation rules.
Logic simply exists and argumentation happens underneath its umbrella. There is no
way to get outside of logic and no Archimedean standpoint from which Poincaré canlaunch his attack.

94

More than being a potent objection, Poincaré’s disagreement with the logicist
highlights a fundamental difference between the two conceptions of mathematics.
Poincaré’s psychologism held that psychological principles are prior to understanding
the formal system of mathematics and therefore also epistemically prior (§2.1.3).
Thus, the formal system itself trades on this intuitive basis.
However, for Frege, no intuition lies at the bottom of mathematics. Logic exists
regardless of our ability to understand it and psychological elements have no placein the epistemic or ontological justification of mathematics. Similarly, formation
rules of the system—definitions of derivations, syntax rules, etc.—do not have any
bearing on justifying the system. They themselves do not need justification nor
examination. They exist merely for our benefit in coming to understand the structureof the system. The rules are similar in kind to laws of physics, which are descriptive
rather than prescriptive. These rules may be necessary for any individual to come tounderstand logic, but this necessity is only a psychological one. Unlike for Poincaré,
this psychological necessity fails to have any justificatory role for the logicists. The
necessity of formation rules has the same epistemic status as the constancy of inkblotson the paper on which we do mathematics. It is necessary that inkblots stay constantthroughout our proofs—that they do not jump around the page to change what we
had written as an ‘x’ to a ‘y’ or worse to nonsense. However, this necessity plays norole whatsoever in the epistemic status of mathematics. Frege says explicitly,
A delightful example of the way in which even mathematicians can confuse
the grounds of proof with the mental or physical conditions to be satisfied
if the proof is to be given is to be found in E. Schr´’oder. Under the heading“Special Axiom” he produces the following: “The principle I have in mind
might well be called the Axiom of Symbolic Stability. It guarantees usthat throughout all our arguments and deductions the symbols remain
constant in our memory—or preferably on paper.”[18, p.viii]

95

While it is true that symbols must be stable—that their meanings do not change
constantly—it is merely a concomitant of proof and not the grounds on which theproof is given. Similarly, the formation rules of logic are required for us to understandlogic, but they do not play a role in forming the system.
With this understanding of the formation rules, it becomes easier to understandFrege’s belief in the truth of the axioms prior to their consistency. Of course therules of logic are consistent because logic is the universal language of thought andargument. The rules by which logic operate simply exist and therefore are consistent.
There will be a question as to whether a given set of axioms is consistent only insofar
as there is a question as to whether that given set of axioms accurately reflects thereal axioms for logic.
These two different conceptions of logic—Poincaré’s and the logicists’—also manifest themselves in the notion of the justification of theorems. For Poincaré, derivation
can be defined recursively, thereby making sense of the claim that a proposition has
been derived. Roughly, pick which rules of inference are allowed and then state thata sequence of (n + 1) many proposition is a derivation if and only if the first n propositions are a derivation and the (n + 1)th proposition in the sequence results fromthe axioms, some subset of the first n lines, and the proper application of the rulesof inference. For Poincaré, a given statement is provable if and only if there exists a
derivation with that statement as its last proposition. The statement that sentences
are in general provable only if there is a derivation of them is a metatheoretic one.
For Frege and the logicists, appealing to the notion of a derivation in order todetermine provability is not possible. There is no assertion to be made that a given
sentence is a theorem. The justification of a theorem is the explicit laying out ofthe derivation. As Goldfarb says, “To give the ultimate basis for a proposition is
to give the actual proof inside the system, starting from first principles; that is, it
is to assert the proposition with its ground, not to assert the metaproposition ’this96

sentence is a theorem.”’[23, p.69] The syntactic checking of a proof—making sure
we have applied the rules correctly—is not a systematic notion for Frege, but rather
another concomitant of logical knowledge. When doing logic, we must check to makesure proofs follow the rules correctly, at which point we may say that a given proof
is indeed valid. But this checking is only something we must do because of ourpropensity to make mistakes; it is not epistemically necessary. It is something wehave to do only out of psychological necessity, just like our need to write down proofs
because we cannot hold their entireties in our heads at once. Our inability to work a
proof without aid of paper and pencil does not make paper and pencils epistemicallynecessary to mathematics. Our need to check proofs does not make such syntactic
statements such as “This is a proof,” necessary to logic, nor does it legitimize it as an
actual statement in the language of logic. Instead, it’s something informal meant toindicate that no human mistakes were made. There is, for the logicist, no formalized
notion of a metatheory nor an uninterpreted system—there is no syntax absent ofsemantics.
Poincaré’s critique, then, is defused before it gets off the ground. Metatheoretic
statements are impossible for the logicist, and Poincaré’s attack is launched from a
metatheoretical standpoint. As Goldfarb notes, because there is no metatheoretic
standpoint from which Poincaré’s argument can take hold, the sophisticated version
fails to be formulable.
Poincaré’s petitio objection to the logicist seems to not be a legitimate objectionthat engages with the logicist, but rather a natural consequence of his fundamental
ideas. Similarly, the logicists’ reply is not one that shows why Poincaré’s objection
should be rejected, but rather a demonstration of how their conception of logic does
not cede any ground to Poincaré’s objection. The problem becomes that there isnothing more for either side to say. The Poincaréan finds this objection to completely
convincing, while the logicist thinks no more needs to be said.
97

Poincaré has made a number of objections to the logicist, of which we havefocused on two: the original and sophisticated petitio. Both made accusations of presupposing induction: the former within the system and the latter outwith. However,the universality of the logicist’s conception of logic allowed both to be turned backand a stalemate reached. The always interpreted, non-syntactic nature of his conception of logic allows the logicist to turn away the original petitio and the absence of ametatheory blunts the sophisticated.
For our purposes, this fact is significant because the neo-logicist cannot invoke
this universal conception of logic. Besides ignoring more than a century’s advance inproof theory and the like, the neo-logicist must make one significant metatheoreticalclaim. Namely, he must claim that his formulation of the rules in the proof of Frege’s
theorem—including Hume’s principle—does not meet the same fate as Frege’s set of
axioms; the neo-logicist must show that they are consistent and not susceptible toanother “Russellian” paradox. Once they move away from this universal conception,
it becomes natural to again raise a Poincaréan objection. It is to this topic that weturn in the next chapter.

98

Chapter 5
Neo-Logicist
The current chapter looks to revive a Poincaréan petitio objection by refocusing it
against the neo-logicist. Such a refocusing becomes relevant because neo-logicism hasa goal of reviving Fregean approaches to epistemically grounding mathematics while
avoiding the antinomies generated by Frege’s original attempt.
Though there are many competing conceptions of Neo-logicism, I will focus on
the version put forth by Bob Hale and Crispin Wright.1 This choice is made because
the popularity of the view marks it as the de facto neo-logicist position. The plethoraof responses and objections allows the Poincaréan objection to be located in thegeography of other objections, demonstrating that two versions of the petitio principii
objection can be raised—more specifically, the objection can be made in two different
places—both resembling pre-existing objections.
The chapter shall proceed as follows. Section 5.1 will lay out the project of the
Neo-logicist, focusing in detail on those areas vulnerable to a Poincaréan objection.
Section 5.2 will examine the stated goals of the neo-logicist in comparison to theoriginal goals established by Frege. The final section will examine two distinct places

1

For alternative constructions see Tennant ([58], [59], [60]), Fine ([16]), Demopoulos ([11], [12]),
Hodes ([35], [33], [34]), Zalta ([68], [67]) or Linsky and Zalta ([37])99where a Poincaréan objection threatens—the formation of the formal system and theconsistency proof of Hume’s Principle with second order logic. These two forms will becompared and contrasted with existing objections put forth by Shapiro ([54], [55]) andby a host of people under the umbrella of the so-called “Bad Company” objections.
Bad company objections are so-called because they accuse Hume’s Principle of too
closely resembling principles that lead to contradiction, for example, Basic Law V.
Objections of this sort have been voiced by Field [15] and Boolos [5], among others.

5.1

Neo-Logicism: Reviving Frege

The project of neo-logicism is in one sense easy to summarize. Let Hume’s Principle(HP) to be the following:(∀F )(∀G)[(N x : F x = N x : Gx) ↔ (F 1 − 1G)]2(HP)

Neo-logicism is a proof of Frege’s theorem,
FT From second order logic and Hume’s Principle, Peano’s Postulates follow,
with the assumption of a special epistemic status for second order logic and HP.
Because of FT, the neo-logicist concludes that mathematics holds the same epistemic
status as logic and HP.
However, like all sweeping generalizations, this one obliterates intricate specifics.
In particular, it ignores several extensive philosophical theses that the neo-logicist
must adopt. The following will give a brief historical background of the failure of theoriginal logicist program, thereby motivating the neo-logicists’ replacement of Basic
2

In giving the formalizations of these principles, I will follow the presentation of MacBride [38].
The formalized version of the principle warrants a brief description of the symbols. ‘N x’ is anoperator on the extensions of concepts, so ‘N x : F x’ can be read as ‘the Number of xs that are
F ’. Similarly, ‘1 − 1’ is a relation between concepts, so ‘F 1 − 1G’ should be read as ‘there exists a
one-to-one correspondence between F and G’100

Law V by HP; spell out the commitments of Hale and Wright’s position; and finally
examine the question of HP’s consistency with second order logic. For this third
section, the chapter will sketch George Boolos’s proof of the equiconsistency of HP
and second order logic with analysis [3].

5.1.1

The Failure of Frege and Basic Law V

Frege initially introduces what came to be called Hume’s Principle as a definition for
number in the Grundlagen. The first attempt in §§62-4 amounts to Hume’s Principle:
However, as MacBride notes, this proposal fails to address the so-called ‘Julius
Caesar’ problem, wherein the definition of number via HP—though adequate to introduce numbers via Frege’s Theorem—does not give an acceptable criterion for determining what is and what is not a number. In his analogous case of the directionof parallel lines, Frege highlights the difficulty thus:
In the proposition “the direction of a is identical with the direction of b”
the direction of a plays the part of an object, and our definition affords
us a means of recognizing this object as the same again, in case it should
happen to crop up in some other guise, say as the direction of b. But thismeans does not provide for all cases. It will not, for instance, decide for
us whether England is the same as the direction of the Earth’s axis. [19,§66]. 3Returning to the numerical case, the problem is that one is able to determine
whether or not a given object b is the same number as a only if it is presented ina certain way: HP provides a way to determine if the number of F s is the same as
the number of Gs, but not a way to determine if the number of F s is the same as
Julius Caesar. One is only able to evaluate the equality of numbers if they are bothpresented as the number of a concept.
Seemingly in light of this problem, Frege gives a revised definition of number in3

Heck [25] has a transposition of this quotation into the context of HP.

101§§68-9. He writes,
My definition is therefore as follows: the Number which belongs to theconcept F is the extension of the concept “equal to the concept F.” [19,§68]Frege now reformulates Hume’s Principle into what will be generalized into Basic
Law V in the Grundgesetze:
But now the proposition: the extension of the concept “equal to the concept F” is identical with the extension of the concept “equal to the concept
G” is true if and only if the proposition “the same number belongs to theconcept F as to the concept G” is also true.[19, §69]4
This reformulation of the definition hopes to avoid the Julius Caesar problem
by defining numbers as the extensions of concepts, which excludes Julius Caesar as
a potential number, since he is not allegedly an extension of a concept but rather aconcrete object.5 However, avoiding the Julius Caesar problem comes at a high cost,for when codified into Basic Law V, this revised definition enables Russell’s paradox
to rear its head.
Basic Law V states that the objects falling under one concept are the same as theobjects falling under another concept if and only if the two concepts are co-extensive.
Symbolically—following MacBride [38]—this is written:(∀F )(∀G)[(Ext : F x = Ext : Gx) ↔ (F x ↔ Gx)]

(Basic Law V)

Taking this formulation, Russell defines the predicate Rx as ‘x is a predicate
which cannot be predicated of itself’, and claimed that, “From each answer its opposite follows,” for if R is predicated of itself, then it should not be so, and if it is
not predicated of itself, then it should be so[51]. This formulation, however, is not
4Here, the translation has used ‘equal’, but a more accurate translation may be ‘equinumerous’.
Note that there is a question as to whether or not this step actually solves the problem or merely
pushes it back one more step.
5102

formalizable in Frege’s system. Frege took this ill-formed objection and produced aformal argument within the system.
First, Frege proves that every concept had an extension in his system:

BLV = (∀F )(∀G)[(Ext : F = Ext : G) ↔ (F x ↔ Gx)]
BLV ` {x : F x} = {x : F x} ↔ ∀x (F x ↔ F x)
` ∀x (F x ↔ F x)
BLV ` {x : F x} = {x : F x}
BLV ` ∃y (y = {x : F x})

Now, from the comprehension principle(CP)—∃F ∀Z (F x ↔ Xx), where Xx isany concept not involving F —Frege defined the Russell set:Rx ↔ ∃F (x = {y : F y} ∧ ¬F x)(Russell Set)

This is a formalization of the Russell paradox that says that x is in the Russell set
if and only if x is the extension of some concept F and x does not fall under thatconcept. The paradox arises when one considers r = {x : Rx}, that is the extension103

of the Russell set itself.

Assume Rr
iff ∃F (r = {x : F x} ∧ ¬F r)
iff r = {x : F0 x} ∧ ¬F0 r
but r = {x : Rx}
so {x : F0 x} = {x : Rx}
applying BLV from left to right∀x (F0 x ↔ Rx)
F0 r ↔ Rr
¬F0 r
¬Rr

Therefore, BLV, CP, Rrvdash¬Rr. Thus, BLV, CP ` Rr → ¬Rr. Finally, BLF,
CP` ¬Rr.

Now assume ¬Rriff ¬∃F (r = {x : F x} ∧ F r)
iff ∀¬F (r = {x : F x} ∧ F r)
iff ∀F (r 6= {x : F x} ∨ F r)
iff ∀F (r = {x : F x} → Rr)
iff (r = {x : Rx} → Rr)
By the definition of r, Rr

104

Therefore, BLV, CP vdashRr. BLV and CP together jointly prove Rr ∧ ¬Rr,
and we have the contradiction.
The historical importance of this story is three-fold. First, Russell’s paradox
caused Frege to ultimately forsake his project of providing a derivation of mathematics from logic, thereby solidifying the epistemic ground mathematics stood upon.
Frege despaired the possibility of demonstrating that mathematics could be derived
from logic alone, thereby inheriting the epistemic strength of logic. The neo-logicist
position can be seen as an attempt to avoid this despair and resurrect at least part ofFrege’s program by returning to Frege’s initial attempt at defining number via HP.
Second, the only necessary application of Basic Law V in Frege’s derivation is thederivation of HP. That is, Basic Law V is not used essentially in Frege’s derivationof mathematics (see Heck [25]), except for proving HP. Once Frege has HP in hand,
he is able to continue with the derivation of Peano’s Postulates.6 Important to notehere is that whereas Basic Law V conjoined with second order logic is inconsistent,
HP conjoined with second order logic is consistent: thus, the neo-logicist strategy ofresurrecting Frege’s project by replacing Basic Law V with HP.7
Third, the introduction of Basic Law V was an attempt to avoid the Julius
Caesar problem arising from a defining numbers via HP alone. Thus, if the neologicist replaces Basic Law V with HP, they need to supply an answer to the Julius
Caesar problem that is both adequate to dismiss the objection as well as not run
afoul of contradiction.

6

More precisely, he is able to derive Fregean Arithmetic, within which Peano’s postulates can beinterpreted.
7
I will return to this issue in the section 5.1.3.

105

5.1.2The Neo-Logicist Project

Fraser MacBride identifies three distinct bodies of doctrine that the neo-logicist iscommitted to in order to make the proof of Frege’s Theorem achieve the epistemic
goals of the project:8
1. a general conception of language and reality;
2. a particular method for introducing novel expressions into language;
3. a specific understanding of the scope of logic[38, p. 107].9
Language and Reality
The neo-logicist’s conception of the connection between language and reality is wholly
different from the standardly held notion. An ordinary interpretation of the relationship holds that reality is distinct from language and is independently determined.
However, for the neo-logicist, language and reality are inextricably and necessarily
entwined. As MacBride says, the two “are so related that, if we speak truly, the structure of reality inevitably mirrors the contours of our speech” [38, p. 108]. Therefore,
any true statements cannot fail to reflect the way the world actually is.
MacBride further subdivides this doctrine into three components—Syntactic Decisiveness, Referential Minimalism, and Linguistic Priority [38, p.108].10 The result
8

These goals will be examined in detail in section 5.2.
The following discussion is based on MacBride[38, pp.108-115]. My presentation is made withan eye to highlighting features of the last two doctrines—the use of abstraction principles and thestatus of second-order logic—so that objections in §5.3 can be properly placed. As such, my analysis
will for the sake of space pass quickly over some distinctions drawn out by MacBride.
10
These three are described respectively:
Syntactic Decisiveness: if an expression exhibits the characteristic syntactic features of a singularterm, then that fact decisively determines that the expression in question has the semantic functionof a singular term (reference).
Referential Minimalism: the mere fact that a referring expression figures in a true (extensional)
atomic sentence determines that there is an item in the world to respond to the referential probingof that expression.
Linguistic Priority: linguistic categories are prior to ontological ones; an item belongs to the categoryof objects if it is possible that a singular term refer to it. [38, p.108]
9106

of this doctrine is that “the syntactic form of our (true) sentences cannot deceive us;
reality cannot fail to include the objects and concepts which the sentences apparently describe” [38, p. 108]. By adopting this position, the neo-logicist argues in thefollowing manner:

1. Ordinary mathematical practice can be replicated using numerical terms.
2. These numerical terms are parts of true sentences.
3. The Numerical terms have syntactic features characteristic of singular terms,
and therefore can be treated as referential in the same way as singular terms.
(Syntactic Decisiveness)
4. Because they can be treated (semantically) like singular terms and are parts oftrue sentences, these numerical terms refer to something in the world. (Referential Minimalism)
5. That “something in the world” to which they refer is an object. (Linguistic
Priority)
6. Terms that exhibit the same pattern of use refer to the same objects. (Meaning
Supervenes on Use)
7. Therefore, ordinary mathematical objects exist as objects in the world.11
The rough synopsis of this argument is that the reconstructed mathematical practicehas certain properties that guarantee that the terms refer to existent objects in theworld. The argument proceeds from a purely syntactic basis to a claim about theexistence of objects in the world. It moves from a statement about the syntactic
features of numerical terms to the substantive claim that ordinary numbers exist as
objects in the world. Obviously, each of these theses must be defended in detail tocomplete the neo-logicist project. We will not do so here, because it would take us
from our purpose. What is relevant is how the neo-logicist introduces the syntactic
terms used in the reconstruction of mathematical practice: the method of abstraction.
11

Drawn from Macbride [38, pp.108-9]

107

Abstraction Principles
For the neo-logicist, abstraction principles are the mechanism by which one is able tointroduce new terms to a language solely via reference to pre-existing and understoodterms. It is the method by which one is able to fix the use and meaning of new
operators in a language, bootstrapping from existing resources. For example, in alanguage that already contains the terms (‘α1 ’, . . . , ‘αk ’) and some sort of equivalence
relation (‘≈’) that holds between those terms, we can introduce a new operator (‘Σ’)
that goes from concepts to objects via the following general principle AP:(∀αϕ )(∀ακ )[(Σ(αϕ ) = Σ(αϕ )) ↔ (αϕ ≈ ακ )]12

(AP)

The neo-logicist is committed to two claims about principles of the form of AP:
semantic and syntactic novelty. The former states that the method of abstraction
introduces genuinely new operators to the language that retain the syntactic featuresof singular terms, while the latter states that the new terms, provided they actuallyrefer, refer to items to which the αi s do not refer.13
Notice that HP is an instance of AP obtained by replacing ‘Σ’ with ‘Nx’, ‘αϕ ’ and‘ακ ’ with ‘Fx’ and ‘Gx’ respectively, and the ‘≈’ with the one-to-one relation ‘1-1’:(∀F )(∀G)[(N x : F x) = N x : Gx) ↔ (F x1 − 1Gx)]14

12

Taken from MacBride [38, p. 110]
MacBride also identifies Referential Realism as a principle to which the neo-logicist is dedicated[38, p. 112]. I omit it here because the potential issues surrounding this position would take me far
a field from the objections I wish to examine.
14
Note that the language within which ‘Fx’ and ‘Gx’ are terms is second order logic. The notionof one-to-one correspondence can be spelled out in a purely second-order logic manner. Rossberg
and Ebert [50] give the full statement:
13∀F ∀G [N x : F x = N x : Gx ↔ ∃R (∀x [F x ⊃ ∃y (Gy ∧ Rxy ∧108

∀z (Gz ∧ Rxz ⊃ z = y) ∧ ∀y [Gy ⊃ ∃x (F x ∧ Rxy ∧ ∀z (F z∧ ⊃ z = x))]. (HP)Thus, HP inherits all of the properties ascribed to generic abstraction principles.
In particular, the stipulation of HP gives numerical terms the syntactic propertiesof singular terms, which is required for the argument in §5.1.2. Thus, via HP, theneo-logicist claims that one can advance from our knowledge of second-order logic toknowledge of new operators ‘Nx’. The patterns of use of the standard mathematicalpractices mirror the uses of these new operators, and due to their inheritance of thesyntactic characteristics of singular terms, the new operators refer to bona fide objects
in the world. Thus, via HP and knowledge of second-order logic, one may advance toknowledge of numbers as objects in the world.

Scope of Logic
The final commitment of the neo-logicist is to hold that second-order logic is basically understood and epistemically fundamental. MacBride identifies this as theSecond-Order Logic is Logic commitment, wherein the neo-logicist is committed tothe existence of a class of inferences involving quantification over first-order notions
like property and relation whose validity is transparent to us [38, p. 113]. In otherwords, the neo-logicist is committed to the claim that we have special epistemic abilities to recognize the validity of second-order patterns of inferences, not just first-orderpatterns.
The reason why such a commitment is necessary is clear: the neo-logicist requires
some ability to determine when the right-hand side of HP is satisfied (recall that oneto-oneness is representable only in second-order logic). Only through this special
ability are we able to gain knowledge of the right-hand side of HP and thereby gain109

knowledge of number on the left-hand side. This ability to recognize the truth ofsecond-order statements allows the neo-logicist to claim immediate knowledge of thetruth of the following:15(x : x 6= x)1 − 1(x : x 6= x)(A)This statement—the things that fall under the predicate ‘is not identical with itself’ can be put in one to one correspondence with that selfsame collection of objects—
is an instance of the right-hand side of HP, meaning that the following is true by HP:(N x : x 6= x) = (N x : x 6= x)(B)This equation essentially “says” that the number of things that are non-identical isequal to the number of things that are non-identical. Then, since it is both true andhas the syntactically relevant features (they are the same as singular objects), thismeans that there is something in the world to which ‘N x : x 6= x’ refers, since ourtrue linguistic uses cannot fail but to refer. Because there is such a thing, we are
permitted to existentially quantify and say:∃y (y = N x(x 6= x))(C)This statement “says” that there exists something that is identical with the numberof non-identical things; it “says” that zero exists.
Thus, the neo-logicist is committed to the three positions here outlined. Hiscommitment to the epistemic priority of second-order logic allows the admission of(A) into epistemically warranted judgments. His commitment to the abstraction
principle schema AP—in particular the abstraction principle HP—permits the move
15

The following uses the presentation of MacBride [38, p114], but it replicates Frege’s original
procedure for defining (if I may be permitted a rough characterization) the number zero as thatwhich is equinumerous with the empty set.

110to (B), and his view of the connection between language and reality permits the moveto (C). Only by each of these inferences is the neo-logicist able to prove the existenceof numbers and obtain the desired result.

5.1.3

Proof of the Consistency of HP with Second-Order
Logic

An immediate question arises for the neo-logicist program: how can we know that HP
and second-order logic will not lead to another contradiction like Basic Law V andsecond-order logic did? How is one assured that the resultant system is consistent?
As Boolos vividly writes, “How do we know that some Super-Russell of the 22nd
Century won’t find some ingenious derivation of a contradiction from the number
principle, the way our Russell derived a contradiction from the set principle” [7, p.
151]?
Boolos provides a proof that consistency of his Frege Arithmetic(FA)—the system
within which Boolos is able to reconstruct Frege’s program in the Grundlagen—is
equiconsistent the strictly weaker theory of analysis as constructed out of the natural
numbers via the rationals.16 That is, Boolos proves that FA is consistent if and only ifanalysis is consistent. I will focus on the proof in which Boolos provides a constructive
account of how a proof of ⊥ in FA can be directly translated into a proof in analysisof ⊥, since analysis is not the theory in question.
The proof proceeds by first demonstrating a model M for HP that also satisfies
the principles of second-order logic. We must give a characterization of FA and theninterpret those notions within ZF, and then finally translate this interpretation into
an interpretation in analysis. This final step must be done by changing the domainfrom U = {0, 1, 2, . . . , ℵ0 } in ZF to U = {0, 1, 2, . . .} in analysis, while rejiggering
16

It should be noted that Boolos’s proof builds on ideas in Hodes [32] and Burgess [9], but I will
focus on Boolos’s proof in [3]111some of the related notions.
FA, recall, is the system generated by the conjunction of HP with second-orderlogic, so the fundamental objects of FA will be nearly identical to the fundamental
objects of second-order logic. There are three types of variables: object variables(a, b, c, . . . m, n, o, . . . x, y, z), unary-predicate variables (F, G, H, . . . , R, S, T, . . .), andbinary-predicate variables (ϕ, ψ, . . .). These variables are simply the standard components of second-order logic, with an added restriction of predicate variables to only
two places. The only non-logical component of FA is dubbed by Boolos as ‘η’, “atwo-place predicate letter attaching to a concept variable and object variable. (ηis intended to be reminiscent of ∈ and may be read ‘is in the extension.’ ”[3, p.
185] Atomic formulas are the standard ones in second-order logic, plus formulas of
the form ‘F ηx’, and these formulas can be combined in the usual ways by the usual
connectives. The final part of FA is the only non-logical axiom:17∀F ∃!x∀G(Gηx ↔ F eq G)(Numbers)

Notice that Numbers is a formal analogue of Hume’s Principle. Numbers is the claimthat there is the unique object x that is the extension of some higher level conceptthat G falls under. In the case of Hume’s Principle, that higher level concept ismarked out by a unary function sign ‘N’, which has as its domain concept variables
and its range object variables. Thus, ‘N F ’ should be read as ‘the number of F s’.
So, to demonstrate a model of Hume’s Principle and second-order logic in ZF set
theory, let the domain of the model be U = {0, 1, 2, . . . , ℵ0 }. We can interpret theconcept variables as subsets of U and similarly interpret the binary predicate variables
as sets of ordered pairs.18 η is easily rendered as ∈. The last to be interpreted is the17
18‘eq’ here means equinumerous.
This second interpretation can be generalized for arbitrary n-ary predicate variables.

112

function letter ‘N’, which we are able to interpret as the cardinality of a set. Thus,
the number of things falling under a concept in FA becomes the cardinality of the set
in ZF. The proof of the truth of Hume’s Principle in M follows almost immediately
thereafter.19
The key to the success of this model is the fact that the cardinality of any of
the subsets of U is in U itself. Because this fact fails to be true for the restricteddomain of the natural numbers, we were forced to include ℵ0 . However, this presents
a problem for the attempt to give a model in analysis, as the domain of analysis is
only the subsets of the natural numbers. We cannot use ℵ0 as a point in the model.
The trick, however, is to reconsider ℵ0 as zero—what Boolos calls “coding”—andthe remaining numbers as their successor (that is, n as n + 1). Boolos outlines thetranslative procedure by which to convert any proof of ⊥ in FA into a proof of ⊥in analysis[3, p.190], but he presents a more dialectically clear—if less rigorous—
version in “Gottlob Frege and the Foundations of Arithmetic”[7] In §5.3.2, I will
present in detail that argument to highlight where a Poincaréan petitio seems viable.
This objection will be equally applicable to both the more rigorous and less rigorous
versions. As such, I want to give a sketch of the rigorous proof of Boolos here.
The basic picture of the coding into analysis is that by interpreting ℵ0 as 0 andevery finite n as n + 1, one can give an expression of the relation “exactly z natural
numbers belong to the set F” through the analysis relation “there exists a one-one
correspondence between the natural numbers less than z and the members of F”[3, p.
190]. This expression becomes a way of expressing the FA relation ‘η’ as a notion ofanalysis. Then, we are able to form the analysis analogue of Numbers by replacement:∀F ∃!x∀G(Eta(G, x) ↔ F eq G)
19(Analogue)

I pass over this proof to look more in detail at the analogous proof in analysis. Boolos performs
this proof on p. 188 of [3].

113

By providing this translation of Numbers into analysis, combined with the fact thatthe basic comprehension principles of second-order logic are provable in analysis, wehave a direct translation procedure for any proof of ⊥ in FA into analysis.
The contradiction arising from Basic Law V and second-order logic meant that
Frege forsaked his ambition of deriving mathematics from logic. The neo-logicist looks
to replace Basic Law V with HP and derives in FA an analog to Peano’s Postulates.
Finally, because of the threat of contradiction, the neo-logicist provides a proof of theequiconsistency of FA with analysis. Having achieved this result and presuming thatthe original argument of §5.1.2 is supported, the neo-logicist has achieved his goals.
We turn not to examine these goals in more detail.

5.2

Respective Goals of Logicism and Neo-Logicism

In the introduction to the collection Reason’s Proper Study, Hale and Wright briefly
describe the neo-logicist program:. . . our efforts in the service of the neo-Fregean programme have, so far,been concentrated . . . on the development and defence of the thesis thatarithmetic, as codified in the Dedekind-Peano axioms, does have a basis in
logic and definitions of a kind which, although not exactly as called for byFrege’s conception of analyticity, coheres with and underwrites a soberly
platonistic conception of its ontology combined with an intelligible andrelatively inexpensive epistemology.[2, p.23]

This quotation highlights the three main theses for the neo-logicists: what I
will call the mathematical, the ontological, and the epistemic. They may each besummarized thus:
Mathematical Thesis(MT) Arithmetic, as codified by the Peano Axioms, is derivable in the system generated by adjoining Hume’s Principle to Second-Order
Logic.
114

Ontological Thesis(OT) Numbers are mind independent, abstract objects.

Epistemic Thesis(ET) We possess a privileged knowledge of arithmetic throughour privileged access to second-order logic and HP.20

The mathematical thesis is so-called because it is a question of a systematic
proof within a system. Given that this result is purely technical and proven as alogical consequence within a given system, it can be considered settled (positively)by the work of the neo-logicists and Frege himself.21 As Wright says, “Frege did atleast establish a new mathematical foundation of number theory: a subsumption of
the laws for finite cardinal numbers under a single principle . . . The Dedekind-Peano
Famous Five can be reduced to One”[64, 273]. This goal is also the one which Fregehimself set out to outline in the Grundlagen and later hoped to prove formally in theGrundgesetze, yet despaired over after Russell’s paradox.
The other two goals are also in accordance with Frege’s original intentions. Certainly, he held the ontological thesis that numbers were mind independent objects in
a Platonist sense. One need only look at his anti-psychologistic derision in the introduction to the Grundlagen to understand his commitment to this ontological status of
numbers. Additionally, Frege’s project looked to provide an epistemic foundation formathematics—that the content of mathematics was derivable as a consequence frombasic principles of logic. Basic Law V was considered logical in nature, had Frege’s
derivation not been inconsistent, he would have shown a logical basis for mathematics and the privileged access we have to logic would guarantee privileged access tomathematics.22
20

I purposefully remain vague as to what I mean by “privileged knowledge” here. I will take up
the issue in §5.2.2
21This result was first noticed by Parsons [40], rediscovered independently by Wright [63], andBoolos gives an in depth discussion in [3] and a rigorous proof in the appendix to [4]22For a discussion of Frege’s epistemic and ontological goals, see Ebert and Rossberg [50].

115The neo-logicist, unlike Frege, faces an additional epistemic challenge. Even if we
grant that there is a special epistemic access to second-order logic, there is no prima
facie reason to grant the same access to HP; HP is not logical for the neo-logicist as
it was for Frege. For the logicist, it was a logical consequence of Basic Law V, which
was itself logical. . Therefore, the neo-logicist actually has two epistemic theses, one
weak and one strong:
Weak Epistemic Thesis (WET): Epistemic justification for HP is transferable
to all of arithmetic. The non-logical axioms of mathematics can be reduced to
HP.
Strong Epistemic Thesis (SET): HP is analytic and second-order logic is transparent to us in a way different from arithmetic. Thus, knowledge of arithmetic
is a result of logical transparency and analyticity.
The weak epistemic thesis is of course, a reclothing of the mathematical thesis in
epistemic garb, though nonetheless important. It simplifies the non-logical commitments of mathematics and allows consideration of the strong epistemic thesis in thefirst place. Supposing that the weak epistemic thesis is settled, we need to address
the specifics of the ontological and strong epistemic theses and compare them withthe original goals of Frege.

5.2.1

Ontological Thesis

On first glance, it appears that the ontological thesis is proven by the neo-logicist’s
employment of abstraction principles to introduce number via their conception of
language and reality (recall §§5.1.2 – 5.1.2). Using abstraction principles, the neologicist introduces new terms to the language that they claim cannot fail but to refer
to objects in the world; in some sense, it appears that they have “created” the very
things that witness the truth of OT.
116

Such a claim seems to beg the question, for it looks as if what guarantees the existence of objects is the neo-logicist’s own conception of language and reality. Moreover,the endorsement of the use of abstraction principles as a viable method for introducing new terms appears to have the fatal flaw of being ontologically inflationary.23
This objection has particular bite on the neo-logicist for the dilemma appears to be
to either accept that these principles introduce new things into the ontology (andthus should not be regarded as definitional) or that they should be accepted as definitional and legitimately introducing new concepts (and thus not able to guarantee
the existence of independent mathematical objects).
The neo-logicist has a response that he takes to be in accordance with Frege’s
original position. While it is true that concepts are created via abstraction principles,
it is not the case that objects are created in this way. What happens is not an
ontological inflation but an ontological redivision, called alternatively a “carving”(Hale, [24, 103]) or a “reconceptualization of the type of state of affairs depicted on
the right [of the abstraction principle]” (Wright, [65, 312]). The stipulation of, forinstance, HP does not create new ontologically unique objects but rather re-divides
what is already in existence into new arrangements. One is able to talk about the“new” thing called a number because it is “new” in the conceptual sense—having justbeen organized as such by a new concept—but fails to be new in an ontological sense.24
One analogy that may be useful is the assembly of furniture pieces. Combining fourlegs and a flat surface makes a table that did not exist prior to assembly, but no new
physical (ontological) matter has been created.

23

For a more detailed explanation of this objection, see MacBride [38], §5.
MacBride argues that this response from the neo-logicist is not sufficient to address the concernof what he calls the ‘rejectionist’, who may argue that the neo-logicist is making an un-substantiated,
yet significant, claim that two concepts—that on the right-hand side and that on the left-hand side
may be “necessarily correlated in the way in which the stipulation demands” [38, 126]. He attributes
this disagreement to fundamental differences in the conception of the relation of language to reality,and even more fundamentally on whether metaphysical accounts of reality are tractable.
24117

5.2.2Strong Epistemic Thesis

Strong Epistemic Thesis (SET): HP is analytic and second-order logic is transparent to us in a way different from arithmetic. Thus, knowledge of arithmetic
is a result of logical transparency and analyticity.

Whereas the WET was simplification of epistemic commitments (from five Peano
Postulates to one), the SET acts as a bedrock epistemic principle upon which the
neo-logicist looks to reliably ground mathematical knowledge. Additionally, SET
serves as a means of support for the ontological thesis, as it describes the manner inwhich we may come to know and have access to mind-independent abstract objects.
The argument given in §5.1.2 allows for privileged access to second-order logic to betransmitted through abstraction principles and onto the newly carved out concepts;
in this way, the neo-logicist is able to answer how it is that one is able to come toknow Platonic numbers.
The neo-logicist owes an account of the form of the transparency of second-orderlogic and of the analyticity of abstraction principles. For the latter, it is sufficient for
the neo-logicist to discuss the epistemic status of HP, rather than the overall status ofgeneralized abstraction principles. As MacBride notes, “And if the neo-logicist is not
simply to replace one mystery with another, a grasp of second-order logical truth andconsequence must be more epistemologically tractable than a grasp of mathematical
truth”[38, p. 136].

Second-order logic Though both second-order logic and HP require an epistemic
justification, it appears at first that the epistemic transparency of second-order logic
is more plausible.
In fact, the potential problem arises in the claim that second -order logic, ascompared to first-order logic, is epistemically basic and uncontestable. The neo118

logicist needs to argue that “logic”, construed as second-order logic, retains the same
status as first-order logic. He needs to claim that second-order logic simply is logic.25
Objections have been raised that there fails to exist a firm distinction that separates
second-order logic from mathematics; second-order logic, according to this objection,is inherently mathematical in nature, and alleging epistemic privilege for second-orderlogic is tantamount to claim the same privilege for arithmetic.26

Hume’s Principle The second-epistemic claim the neo-logicist must make to satisfy the SET is that HP is somehow epistemically basic. Wright claims that HP is,in fact, analytic. However, the definition of analytic must be revised slightly fromthe traditional one. First, it needs to account for attacks on the notion of analyticity
writ large, such as Quine’s ‘Two Dogmas of Empiricism’. But, leaving this issue tothe side, there still needs to be tightening up of HP and its status.
We have already touched on one of Wright’s responses to criticisms from Boolos(the notion of recarving, rather than ontologically stipulating p. 117), and Wright
responds in kind to Boolos in “Is Hume’s Principle Analytic?”([65]).27 Wright distills
the issue to which Boolos objects:. . . that Hume’s Principle may be laid down without significant epistemological obligation: that it may simply be stipulated as an explanation of themeaning of statements of numerical identity, and that—beyond the issueof the satisfaction of the truth-conditions it thereby lays down for suchstatements—no competent demand arises for an independent assurance
that there are objects whose conditions of identity are as it stipulates.[65,p. 321, emphasis in the original]

The problem that Boolos sees with this claim is that too much is being carriedby the notion of explanation.
25

See §5.1.2
Shapiro argues along these lines in [53], [57], and [56]. I will return to compare this objection
to a Poincaréan version in §5.3.1.
27
We will return to Boolos’s ‘Bad Company’ objection in §5.3.2.
26

119

However, it is hard to avoid the impression that more is meant, thatWright holds that to call a statement an explanation of a concept is toassign it an epistemological status importantly similar to the one it wasthough analytic judgments, including definitions, enjoy. It is to this further suggestion that I wish to demur[6, p. 310, emphasis in original].
The difficulty with such an objection is that it does not seem to meet the neo-logicist
head on. Boolos objects that there is some extra epistemological weight being granted
HP by calling it an explanation, and that it is upon this basis that HP gains thestatus that the neo-logicist wants. However, this fails to be an objection insofar as it
is a reservation—if one buys into the neo-logicist project, such a concern will not be
troubling. Unless one share’s Boolos’s intuitive worry, one need not bear the same
concern. Because the neo-logicist is committed to a conception of how language andreality are connected (§5.1.2), such a worry would have no effect on them. Indeed,
Wright even argues that if it is possible to gain an understanding of abstract entities
at all, then
. . . it has to be because we have so fixed the use of statements involving reference to and quantification over such entities as to bring the obtaining of their truth conditions somehow within our powers of recognition. . . something we did by way of determination of meaning. [65, p.
323]
Wright argues that unless one wants to dispense with the view of numbers as
abstracts, one has to accept something resembling the neo-logicist position.
The disagreement between Boolos and Wright seems to stem from a fundamental
disagreement about the status of language and reality. Given that, neither presents aparticularly strong reason for the rejection of their opponent or the adoption of theirown view. That there are disagreements about the intuitions is hardly surprising andmeans that this particular point seems at an impasse.
The important point to recognize here is that much of the epistemic status of HP
stems largely from whether or not one believes the neo-logicist account of language’s
120inability to fail to mirror reality.
Whether or not one thinks the neo-logicist has done enough to demonstrate SET(and therefore the entirety of his goal) depends on whether or not one shares the
neo-logicists’ starting intuitions. However, this problem is precisely why the WET
exists independent of SET. Regardless of the epistemic status of HP, the proof of
WET is a significant result in the foundations of mathematics.

5.3

Poincaréan objections posed to Neo-Logicism

This section shall focus on two potential Poincaréan style objections to be madeagainst the neo-logicist project. The first resembles the sophisticated petitio as described by Goldfarb, wherein any formal system necessarily presupposes knowledgeof the principle of induction(see §4.2). The objection against the neo-logicist is that
second-order logic has a necessary sub-fragment comprised of first-order logic, thefoundational rules of which require determination by recursion. Moreover, these rules
are not merely descriptive as they were for the logicist but prescriptive in that they
establish the logic. Therefore, since induction is used in the establishment of thesystem, the epistemic goals of the neo-logicist go unrealized. This particular objection is of a kind with Stuart Shapiro’s objection that second-order logic is, in fact,
mathematical in nature.
The second objection arises in the consistency proof of HP. This proof is required
because of the presence of bad-company objections that argue that because HP closely
resembles other abstraction principles that lead to contradiction, a verification of HP’s
legitimacy is required. In the proof of the equiconsistency of FA with analysis, there
is an application of induction in order to demonstrate that HP is true in a model ofanalysis. Given that there is such an application, a mathematically ignorant subject
cannot come to be assured of the legitimacy of the stipulation of HP and thereby
121cannot gain mathematical knowledge out of nothing.

5.3.1

The Sophisticated Petitio and the Mathematical Quality of Second-Order Logic

The neo-logicist must claim both that second-order logic simply is logic (§5.1.2) andthat logic has a special epistemic status (§5.2.2); from these two claims, the neologicist is able to derive mathematics from an assumption of Hume’s Principle, thereby
endowing mathematics with the epistemic standing of logic and HP.
However, Shapiro counters that there is no boundary between what should beclassed as “mathematical” and what should be classed as “logical” within secondorder logic. Second-order logic has, under a standard semantics, predicate variables
that range over the powerset of the domain of discourse (recall that Boolos’s translation of second-order logic into set theory had monadic predicates as analogous tosubsets of the domain). Thus, there appear to be set-like objects presupposed in thenotion of predication in second-order logic. Thus, Shapiro denies “that there is a
sharp boundary between mathematics and logic.”[56, p. 59]
If such a division cannot be made, the neo-logicist project is perilously in trouble,
as he would be ascribing to logic a special epistemic status and hoping to transmit thatwarrant to mathematics via Frege’s Theorem. However, he would have presupposedthat mathematics had such a status when ascribing it to logic. Thus, a circle.
Here is the similarity to a Poincaréan petitio objection. The petitio at base isan accusation of begging the question. It was an accusation against Hilbert and thelogicists that they were presupposing an understanding of number and elementaryarithmetic in their derivation of the Peano Postulates. Seen in this general light,
Shapiro’s objection that second-order logic is mathematical in nature is Poincaréanin spirit.
122

One can make a slightly more focused objection that better resembles the original
petitio argument. Recall that the sophisticated petitio suggested by Goldfarb in §4.2
accused the logicists of presupposing number in the formation rules of the formal
system. This particular objection did not effect the logicists because the logicist
explicitly denies a meta-theoretical perspective from which to make the argument.
The same defense cannot be argued for the neo-logicist. Their rules for characterizing the first-order fragment of second-order logic are recursive, and therefore
presuppose some notion of induction for a thinker. The employment of induction
for the comprehension of the formation rules within the system comes prior to thequestion of whether second-order logic is fundamentally mathematical, prior even to
the question of the epistemic status of logic. In this sense, it is an objection madebefore the logical system gets off the ground. The Poincaréan objection alleges thatthe neo-logicist, in his recursive formation rules, has already presupposed knowledgeof number.
One particular aspect of the Shapiro objection is phrased by MacBride thus:
Semantics: in order to provide a semantic theory for second-order logic, a considerable body of mathematics must be called upon[38, p. 136]
MacBride suggests a response for the neo-logicist to this semantic objection that
seems extendable to defend against the petitio.
More importantly, there is a distinction to be drawn between the tools
one employs to investigate a given subject matter and the nature of thesubject matter itself. One cannot immediately conclude from the fact that
one has to employ tools of such and such a sort that the subject matter
itself concerns items of that sort [38, p. 137].
To extend this defense against the petitio, one could argue that induction is used
as a tool to understand and lay down the formation rules of logic does not mean that
logic presupposes mathematics or induction. The tools of comprehension should be123strictly separated from the subject matter itself.
This remark, however, does not seem successful on examination. There is a disanalogy between the objection lodged by Shapiro and this new petitio; namely, theformer is a concern leveled within the system, whereas the latter is made outwith
the logical system. Set theory may be used after the system is in place in order toclarify and investigate more fully the theory in question, for instance, in investigating
questions of relative consistency. Shapiro emphasizes this practice as what mathematicians commonly do in order to illuminate the nature of structures in question[56]. In this way, the richer theory is used as a tool to examine the sub-theory.
But the employment of induction in the formation rules, however, is not suchan investigatory use. The use of induction does not serve as a method by whichmathematicians or logicians learn more about second-order logic; rather, it serves
as a way to introduce a fragment of second-order logic and bring about its proper
formation. Thus, it is not a tool used to investigate, but rather a tool used to create.
To that extent, when the questions of epistemic foundations are raised, it does beg
the question to claim that the neo-logicist can dismiss this concern as being merely aconfusion of the tools used to investigate the theory with the theory itself.
A second response given by MacBride to the semantic objection seems more
fruitful against both the semantic objection and the petitio objection. This response
claims that the reconstruction of mathematics does not rely on the totality of secondorder logic, which the neo-logicist might concede requires a full blown set-theoretic
semantics. Rather it only relies upon the “recursively enumerable fragment relevantto the derivation of Frege’s theorem”[38, p. 139]. In this way, the neo-logicist denies
that the fragment within which he derives mathematics is necessarily mathematicallyproblematic.
A similar reply appears available to the petitio objection. The neo-logicist canclaim that the derivation of Frege’s Theorem need not rely on a full-blown under124

standing of first-order or second-order logic. Rather, all that needs to be understood
are that the given formulae in the theorem are in fact well-formed, that the actual
derivation of the theorem is an acceptable chain of deductions, and so on. In this way,
the neo-logicist might be able to claim that induction is not used in the foundational
way as envisioned before.
The neo-logicist is even able to frame the petitio objection, if stretched and pulled
slightly, as a psychologistic objection. The neo-logicist could argue that the force of
this objection lies not in the logical presupposition of mathematics, but in the psychological presupposition. An intuitive grasp of elementary arithmetical operations
is required to understand even what constitutes distinctness between formulae, presupposing recursive definition. The notion of what constitutes a syntactically wellformed formula supervenes on the ability to recognize that the recursive constructionof formulae—as laid out by the recursively enumerated fragment of second-orderlogic—could continue on forever or rather that it need not stop at any given point.
This supervenience, the neo-logicist could offer, is psychological rather than logical.
Because he does not subscribe to psychological principles as fundamental as does
Poincaré, the objection holds no weight.
One need not accept psychological principles as basic in order for the neo-logicist
response to fail. Should one hold that the understanding of rules of derivation andsyntax in particular cases is parasitic on a generalized grasp of these rules, then the
neo-logicist response fails to be successful. In fact, MacBride suggest just such acommitment by the neo-logicist:
Moreover, the neo-logicist may allow that a systematic understanding of
second-order logic requires the exercise of mathematical concepts whilst
nevertheless maintaining that a mathematical novice might follow the
proof of Frege’s theorem even when unable to explicitly formulate or theorise about the specific rules employed [38, p. 139].
Any mathematical novice who is so led through the proof must already have an125understanding of recursive rules for the generation of formulae. This “novice” mayindeed fail to have a knowledge of the totality of second-order logic, but seems torequire having a knowledge of induction in order to understand that the derivation
she is being shown is, in fact, a derivation. Recognition that the proof of Frege’s
theorem is good depends upon a pre-understood notion of recursiveness and therebyof induction. Indeed, the original response from the neo-logicist called upon the“recursively enumerable fragment relevant to Frege’s Theorem.”
For the novice to recognize that the applications rules in the derivation of Frege’s
Theorem are instances of a general rule, the novice must first know the general rule.
For suppose that there are n many instances of modus ponens in the derivation. For
the novice to recognize these as n-many applications of the same rule—and not nmany different individual rules—she must already know the general rule. If one thinksthat this idealized mathematical novice need not have an understanding of induction
prior to being shown the proof of Frege’s theorem, then there seems to be no problem
with the neo-logicist response. The neo-logicist owes an explanation of how the proof
of Frege’s Theorem is not circular.

5.3.2

Consistency Proofs and Bad Company Objections

Neo-logicism claims that via a use of HP, an abstraction principle, one is able to deriveknowledge of mind-independent abstract objects. Via stipulation of the principle andknowledge of second-order logic, the strong epistemic thesis follows. More generally,
the neo-logicist claims that abstraction principles—one of which is HP—are legitimate
epistemic means by which to obtain knowledge of new terms on the left-hand side of
the principle via a grasp of the right-hand side, which is composed solely of terms
antecedently understood (see §5.1.2).
However, abstraction principles are not generally acceptable. For example, Hartry
126

Field has proposed the following principle:(G) The God of x = the God of y iff x and y are spatio-temporally related.
Thus, proceeding in the same manner as the neo-logicist, Field claims that knowledge that x and y are temporally related results in knowledge that the God of x is thesame as the God of y. Then, given the neo-logicist conception of language and reality,
Field can conclude the existence of God via existential generalization. Certainly, sucha result is anathema to the neo-logicist.
Though there is a response available for this sort of objection—the neo-logicist
can argue that the concept of “God” introduced in (G) is not of the omnipotent deity
in the sky but simply a “God” of spatio-temporal relation—there are more subtle
objections put forth by Boolos in the form of the Parity Principle (what Wright calls
the Nuisance Principle), an abstraction principle that is consistent with second-orderlogic, yet is true only in finite domains:(P) ∀F ∀G (Parity(F ) = Parity(G) iff F and G differ evenly )where differ evenly means that the objects falling under F and not G or G and
not under F is even and finite ([5, pp. 214-5]). Though neo-logicist has anotherpossible response to this objection by requiring conservativeness of abstraction principles (see MacBride [38, p. 145]), the upshot is that the neo-logicist requires somemeans of differentiating HP from those principles in the neighborhood that lead tocontradiction. One of the first requirements, of course, is the differentiation between
HP and Basic Law V.
To this end, the neo-logicist proves the consistency of HP with second-order logic.
More precisely, he proves the equiconsistency of Frege Arithmetic with analysis. We
have already outlined the proof in §5.1.3. The revised Poincaréan objection runs thus:127

1. The neo-logicist requires a proof of the consistency of HP with second-orderlogic in order to differentiate it from the bad-company HP keeps.
2. One of the postulates derived within Frege Arithmetic to reconstruct arithmeticis the principle of induction.
3. The proof given by Boolos on behalf of the neo-logicist employs an applicationof the principle of induction.
4. Therefore, the neo-logicist fails to reduce the epistemic commitments of arithmetic and fails to show the SET.
Claims 1 and 2 are straightforward descriptions of the status of HP and theoutcome of Frege’s Theorem. The onus for the Poincaréan objection is to demonstrate
1) that there is an application of induction within the proof of consistency and 2) thatsuch a use of induction is illicit insofar as it undercuts the epistemic simplificationthat the neo-logicist wants to achieve. In order to accomplish this first goal, we turnto look in detail at Boolos’s proof of the equiconsistency.

Induction in the Proof of Equiconsistency
Recall that Boolos provided a constructive account of how a proof of ⊥ in FA could
be translated into a proof of ⊥ in analysis. This process was done in part by demonstrating how a model of FA could be constructed in ZF set theory, which in turn
could be coded into a model in analysis. Recall also that the difficulty in achieving
this coding was that in the model in ZF, we employed the notion of cardinality tomodel the operator ‘Nx’ in FA, thereby requiring the expansion of the domain of themodel to include ℵ0 . Such a move is not possible in analysis, where ℵ0 is not in thedomain of analysis.
In order to circumvent this difficulty, Boolos defines number by:
Number by If the number of F s is n, then the number by F s is n + 1, and if thenumber of F s is infinite, then the number by F s is 0.
128

So, for instance, the number by strikes in an at-bat in baseball is 4, the number
by US senators is 101, and the number by the even numbers is 0. We can now
reformulate HP into HPa, for analysis:
HPa The number by Fs = the number by Gs iff F 1-1 G
Having interpreted FA in analysis thus, Boolos must prove that HPa is true inanalysis. Assume first that the F s and the Gs are in a one-to-one correspondence.
There are either finitely many F s or infinitely many F s. If there are finitely many
F s, then ex hypothesi there is a mapping from F to G that is finite. Thus, if there
are n F s then there are n Gs. So,

the number by F s = n + 1 = the number by Gs

If there are infinitely many F s, then that means there are infinitely many Gs.
But this means thatthe number byF s = 0 = the number byGs.

Thus, the right-to-left direction of the biconditional.
Assume now that the number by F s = the number by Gs. Either the number
by F s= 0 or the number by F s= n + 1, for some finite n. If the latter, then thenumber of F s (in the standard interpretation) equals the number of Gs, and we cancreate a mapping from a finite set to a finite set. If the former, then the number ofF s and Gs are both infinite. However, because the vocabulary of analysis consists
only of the natural numbers, we can establish a function mapping the least elementin the F s to the least element in the Gs, the second least in the F s to the second
least element in the Gs. Since they are both infinite, the process can continue to map
every F to only one G and vice versa. we have a one-to-one mapping and a one-to-one
129correspondence. Therefore, the left to right direction of the biconditional holds.28
However, this proof of the model has a number of applications of the principle ofinduction. In the left to right direction, Boolos writes, “If [the number by F s = thenumber by Gs] is positive, n+1, say, then there are n F s and n Gs and the F s and Gs
are in one-one correspondence.” Though true, this argument relies on mathematicalinduction. The claim is that for any finite number n, for two concepts F and G, with
the number of F s = n = the number of Gs, then there is a mapping (one-to-one)from F to G. To prove this claim, we show that there is a mapping in the base case(between two concepts with only one item falling under each). Then, we show that
if we assume a mapping for concepts of size n, we can construct a mapping for setsof size n + 1 simply by taking the pre-existing mapping for the first n objects andextending it by mapping the remaining thing that is F to the remaining thing that is
G. Thus, we are able to conclude that a mapping can be made between two concepts
that have the same number by. Of course, this claim is merely an application of
induction.
Moreover, in addressing the possibility that the number by F s = the number by
Gs is infinite, Boolos writes,
But if the number [by] is 0, then there are infinitely many F s and infinitely many Gs. But in this case too, the F s and Gs are in one-one
correspondence: the least F corresponds to the least G, the second least
F to the second least G, the third least, etc. [7, p. 153].
But this “etc.” is precisely the conception of induction that Poincaré has in mind.
It is a recognition that the process described can be carried out indefinitely. One
recognizes that the procedure of mapping the least F to the least G, the secondleast F to the second-least G, and so on can continue indefinitely only because of an
intuitive grasp of induction. It is a grasp of the extensibility of the given procedure.
28

This proof follows Boolos in [7, pp.152-3]. He writes there that he has “just shown how to interpret analysis in analysis the result of adjoining the number principle to the system of Begriffsschrift”

130Should a neo-logicist feel compelled to be worried about this sort of inductive
procedure? Does he need to buy into Poincaré’s psychological conception of induction? It might be that the neo-logicist can dismiss Poincaré’s induction in this matteras purely psychological, whereas Boolos’ and the neo-logicists’ concern is epistemic.
The neo-logicist might reply that while it is a psychological fact of our nature that wemust understand induction in order to recognize the truth of what Boolos writes, thisfact does not bear any epistemic weight. It is a concomitant fact, similar to Frege’sview on the constancy of inkblots (p. 95)
That response, however, misses the point. The reason that Boolos’s proof has
any value at all to the neo-logicist is because it helps to mark out what is epistemically responsible; it demonstrates that HP does not run afoul of any inconsistency.
However, part of the epistemic thesis that the neo-logicist wants to advance is that,as MacBride writes,“(HP) may be employed to introduce a mathematically ignorantsubject (‘Hero’) to number theory”[38, p. 146]. Hero’s employment of this abstraction principle in some manner in order to verify that he is not actually employing an
illicit principle like Basic Law V. Note that this purpose is precisely why Boolos gives
his proof.
However, as a mathematically ignorant subject, Hero is not in a position to verify
the consistency of HP via Boolos’s proof, and therefore unable to even differentiate
HP (in terms of its consistency) from Basic Law V. Hero can blindly follow HP, but
he cannot assure himself of the safety of its use. The situation echoes the dilemmafor Hilbert’s program; if the neo-logicist wants to retain his stated goal of the SET—
simplifying the epistemic commitments of mathematics—then the prescribed method
for achieving that aim appears to fail.
MacBride notes a similar point and phrases it as a dilemma between an internaland external justification; on the one hand, Hero’s use of HP cannot be justified
internally, as the mathematics required to provide epistemic safety are beyond Hero’s
131ability, while on the other had the attribution of external epistemic warrant to modesof reasoning is implausible. Wright sketches a path between the internal and externalhorns of the dilemma by noting that even uncontroversial notions of concept formationor sense fixing run into similar problems (see Wright [64, pp.286-8]), so the issues
raised by this objection are more universal that first seems. However, he admits
that such reflections do “not excuse the Fregean the work of offering some principledrestriction where instances of [abstraction] may be excluded”[64, p. 288]. No suchinternal, principled restrictions are on offer. Therefore, it appears that the dilemma
and the Poincaréan objection remains active.

132

Chapter 6
Conclusion
This project declared a number of goals at the outset. Some were interpretive (e.g.
providing an interpretation of Poincaré’s ‘intuition’); some historical (showing why
the petitio was not effective against the logicists); and some looked to contribute toongoing debates (refocusing the petitio to the neo-logicist).
Chapter 2 argued for a novel interpretation of Poincaré, identifying four elementsof his psychologism that informed his conception of intuition, which in turn, informed
his notion of induction. The secondary goal was to demonstrate that Kant’s philosophy of mathematics differed from Poincaré significantly, perhaps suggesting that a
Poincaréan philosophy of arithmetic is worthy of further investigation.
Chapter 3 addressed both Early and Later Hilbert with the petitio objection,
arguing that even under multiple interpretations of Hilbert’s project—a novel one ofwhich was advanced—a revised version of Poincaré’s objection still holds.
Chapter 4 argyed that the only way for the logicist to wriggle free of Poincaré’s
original objection was to hold a particular view of logic as all-encompassing anduniversal—to deny any metatheory.
Finally, Chapter 5 argued that two forms of a Poincaréan objection are possible.
Moreover, because of particularities in the neo-logicist platform, the responses of133the logicists are not available. These two reformulations were compared to extant
objections in the literature, and it was argued that the neo-logicist has not supplieda potential response. As such, these objections remain viable.

134

Acknowledgments
First and foremost, I would like to extend my appreciation and deep gratitude toProfessor Peter Clark for his supervision, comments, and mentoring throughout thisproject from early questions through final drafts. Thank you also to Professor Stephen
Read for his comments on my draft and further supervision. Thank you to Michael
Hoffman and George Grech for extensive comments on earlier drafts. Thank you also
to Dan Labriola and Fergus Bruce for helpful remarks during the writing process.
Thank you to the Rotary International Foundation, George McIntosh, and Anne Tait
without whose help I would have been unable to complete this project. Thank you to
Trudy Fraser for her patience with my late nights and support throughout the year.
Finally, thank you to my mother for her constant support over the past three years.

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