A non-model theoretic realism in mathematics∗
Minao Kukita†
May 16, 2009
at Kyoto University

1

Introduction

Why are many mathematicians realists, while many philosophers are not? This question
becomes even more striking when one considers the fact that modern mathematics iscarried out totally in the axiomatic manner. The main aim of this article is to give ananswer to this question.
One reason lies in diﬀerent conceptions of reality between philosophers and mathematicians. For example, Abelson and Sussman wrote[a computational process] cannot be seen or touched. It is not composed ofmatter at all. However, it is very real. It can perform intellectual work. It cananswer questions. It can aﬀect the world by disbursing money at a bank or bycontrolling a robot arm in a factory. (Abelson and Sussman, [1], p.1)
They think that a computer process is real because it has certain eﬀects and functions.
On the other hand, many philosophers seem to think that in order for something to be real,∗†This is a draft version.
minao kukita@hotmail.com

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it must exist in space-time; it must be independent of our senses, perceptions, thoughts,
etc.; it must have causal eﬀects, etc. Let us call the reality in the former sense functional,
and that in the latter substantial. When philosophers are discussing reality, it is often asubstantial one that is intended. However we certainly use the word “reality” in many
senses, and there is no reason to conﬁne the word in any one of them, or to regard any oneof them as central or normal. It is clear that mathematical objects are not substantiallyreal. One may be led to think that the question about the reality of mathematics is only
concerned with how we deﬁne the word “reality,” and that mathematics is not substantiallyreal, but functionally real. However this is not the case. The question of reality has acertain signiﬁcance within mathematics.
Abelson and Sussman demarcate computation from mathematics, and do not extend
their view to mathematics in general. However, the development of intuitionistic mathematics over the past century has shown that a fairly large part of mathematics can beviewed as computational. How far we can extend this view, i.e., to what extent mathematics can be thought of as (functionally) real is an interesting question. Therefore reality
has it own signiﬁcance apart from philosophical or metaphysical discussion.
A diﬀerence between computation and mathematics is that the former makes more ofprocesses, while the latter makes more of results. For example, in computation, a function
is viewed as a program that prescribes how to process one piece of data into another. In
mathematics, however, it is viewed as a set of ordered pairs of inputs and outputs, nomatter how the outputs are obtained. In other words, a function is reiﬁed and seen as a
substantial object in mathematics. This, I think, is a major source of the controversy overrealism in mathematics.
Most philosophical arguments against realism seem to be based, for one thing, on the
non-existence of mathematical objects such as numbers, sets, etc. and, for another, on
the irrelevance of the state of actual aﬀairs to the truth and falsehood of a mathematicalsentence. Thus, many realist replies consist in claiming some other form of existence thanthat of ordinary perceptible objects, and then associating with it the truth of a statement.

2

Such realists look into the realm of the thoughts, forms etc. for mathematical objects towhich mathematical statements refer. In doing so, they are tacitly employing a form ofmodel-theoretic interpretation for mathematical statements.
This is understandable and even natural, because the model-theoretic interpretation
is so dominant in current logic that we are quite accustomed to interpreting formulas
or statements by means of models. However, this is not the only way of interpreting
mathematical language. One alternative is the operational interpretation, where the notionof truth is redundant and hardly seems to demand that mathematical statement refer tosomething outside mathematics. The meaning of an expression is fully determined by
the formal system, and the notion of truth is replaced by the notion of provability orderivability. It might at ﬁrst seem that realism is out of place when one is employing
this interpretation. I shall argue that some form of realism can still be maintained in thealternative.
My conclusion is that objects of mathematics are procedures prescribed by mathematical terms, formulas and proofs, and they can be viewed as real by weaker criteria thanthose of philosophers. I formulate the followings as criteria employed by mathematicians:(1) identity, (2) manipulability and (3) force.
Therefore the notion of reality is diﬀerent between mathematicians and philosophers.
A mathematician’s notion of reality has its own signiﬁcance in mathematics, particularly
when one considers to what extent mathematics is real rather than whether mathematics
as a whole is real or not.

2

Mathematical truth

A typical argument against mathematical realism goes as follows:
1. Mathematical statements are about numbers, sets, etc.
2. Numbers, sets, etc. are abstract but not concrete; they do not exist in space-time;they have no causal eﬀect, etc.

3

3. Therefore they are not real.
There are two assumptions behind this type of argument. One is that mathematicalstatements are saying something about numbers, sets, etc. Another is that if somethingis real, it must be concrete; it must exist in space-time; it must have causal eﬀects, etc.
The ﬁrst assumption has much to do with a model-theoretic conception of the meaning, or model-theoretic interpretation (hereafter referred to as MTI). Roughly speaking,
MTI is a doctrine that each well-formed expression in a theory should be associated withsomething that belongs to the intended model. Names and singular terms are associatedwith individual objects, predicates with relations, and sentences with conditions. Moreformally, a model-theoretic interpretation consists of a set of individual objects D, functions I1 : Term → D, I2n : Predn → Dn , where Term is the set of terms, and Predn isthe set of n-ary predicates.
LanguageModelterms: a1 , a2 , . . . , anobjects: I1 (a1 ), I1 (a2 ), . . . , I1 (an )n-ary predicate: P

n-ary relation: I2n (P )sentence: P (a1 , a2 , . . . , an )condition: hI1 (a1 ), I1 (a2 ), . . . , I1 (an )i ∈ I2n (P )A sentence “P (a1 , a2 , . . . , an )” is said to be true if and only if it is the case thathI1 (a1 ), I1 (a2 ), . . . , I1 (an )i ∈ I2n (P )For example, consider the equation “A ∪ B = B ∪ A.” In MTI it is interpreted asindicating the condition that the union of
and A

( A

and B

A

and

B

is equal to the union of

are the sets denoted by “A” and “B” respectively).

4BLanguage

Model

A∪B

the union of A

and B

B∪A

the union of B

and A

=

equalityA∪B =B∪A

the condition that the union of

A

B

and A

is equal to the union of B

andAnother way to describe the situation is by saying that the terms “A ∪ B” and “B ∪ A”
denote the same object.
“A ∪ B”

ZZ“B ∪ A”½½½ denotes
Z
½
Z ½
~
=

denotes Z•

In MTI, it is necessary that there should be a set denoted by both “A ∪ B” and“B ∪ A,” but independent of these expressions. Here one views sentences as declarative
or descriptive: they mention some objects, describe their conditions, and are true or false
depending on whether they are describing the state of these objects correctly or not.
Most arguments around realism in mathematics, whether for or against, implicitly
presuppose MTI. Antirealists claim that mathematics is not real because there are nomodels for mathematical theories anywhere in the real world, while realists claim that themodel exists in the realm of forms, or in our thoughts, which are real in some sense. Theyboth take it for granted that in order to claim the reality of mathematics, we should seekout real entities outside mathematics to which mathematical statements refer.
Is this really what happens in mathematics? Modern mathematics does not seem verymuch to care about what lies outside of the mathematical language, particularly not about
what mathematical terms may mean. In fact, the axiomatic method is one of the mostremarkable features of modern mathematics. If one asserts that such-and-such is the case,it implies only that there is a formal proof of it, and nothing more. Therefore, whatever5

a mathematical expression may refer to is entirely irrelevant for mathematics.1 If so, why
do many mathematicians claim the reality of mathematics?
Let me draw an analogy. Suppose someone says “Cars must keep to the left.” This
statement is true in such countries as the U. K., Australia and Japan, and false in manyother countries. Whether it is true or false depends on the traﬃc law of each country. So
the statement is not about a particular situation of things, but about the system of law.
The law itself is neither true nor false.
The law is a result of our language activity and therefore not independent of us. It
is not a physical entity that exists in space-time. Nor is it a psychological entity thatexists someone’s mind, since if it is so, it will be subjective, but the very nature of the law
requires it to be objective.
The same thing can be said about mathematics. Syntactical rules, deﬁnitions, axioms
and inference rules are neither true nor false. They are not descriptions of things, but
prescriptions for constructing terms, formulas and deductions. However, we can maketrue statements about the formal system they determine.
It is important to distinguish formulas in an axiomatic system from statements about
it. For example, (xy)−1 = y −1 x−1 is not a mathematical statement, but simply a formula.
“For every group G, (xy)−1 = y −1 x−1 is deducible for any x, y ∈ G” is a mathematicalstatement that is true. However, this distinction is only relative, and not a kind of distinction between object language and metalanguage. The latter statement is also within themathematical language, which is in turn an object of other mathematical statements suchas “It is provable that for every group G, (xy)−1 = y −1 x−1 is deducible for any x, y ∈ G.”
In fact, there is no clear distinction between object language and metalanguage inmathematics. A statement about mathematics is often a statement within mathematics.
In other words, the mathematical language provides mathematics with objects, and where
there is no language, there are no mathematical objects either.

1

It may help mathematicians to get some insight, though.

6

Mathematical objects are products of linguistic activities. However, it does not follow
from this that mathematical objects are simply meaningless symbols. In the next chapter
we will consider the nature of mathematical objects.
Before proceeding to the next chapter, I should brieﬂy mention semantics for formal
systems. Model-theoretic semantics is an almost mundane part of logic. However, it is notreally a theory of meanings, but a theory of translationfrom logic into mathematics,

in particular into set theory, algebra, topology or category theory. They are also formal
systems determined by their own axioms and inference rules. Moreover the means oftranslation are also given in axiomatic manner. Therefore, model-theoretic semantics does
not establish semantic relation between language and things. This fact does not sully thesigniﬁcance of ﬁnding a translation from a logic into a mathematical system or structure.
Such translation is not only an interesting — sometimes even amazing — mathematical
discovery in itself, but also may have epistemic values. For example, Kripke frames area useful tool in examining whether a formula is provable or not, and so is denotationalsemantics in examining whether a program is correct or not.

3

Mathematical objects

It is obvious that mathematics does not deal with concrete objects. Nor does it withempty symbols. Mathematical symbols do have meanings, but not in the way words ofordinary language do. For example, a name such as “Caesar” or “Romans” may have
Caesar or the set of Romans as their meaning. Symbols in mathematics do not have suchconcrete objects or even sets of objects as their meaning. They represent functions in thesymbol manipulating system, and their functions are determined by syntax, axioms andinference rules.
Consider a simple example from group theory. A group is any quadraple hG, ·, ², −1 i,where G is a non-empty set, · is a binary operation on G, ² is an element of G (called unit),
and −1 is a unary operation on G satisfying the following conditions: for all x, y, z ∈ G,

7

(A1) x · (y · z) = (x · y) · z,(A2) x · ² = x = ² · x,(A3) x · x−1 = ² = x−1 · x.
We write xy for x · y. Owing to (1), parentheses can be omitted.
It follows immediately from (1)-(3) that:(F1) for all x, y ∈ G, (xy)−1 = y −1 x−1 .
(F2) for all x, y ∈ G, if xy = ² = yx, then x−1 = y and y −1 = x.
Now let G = hG, ·, ², −1 i be a group satisfying the following conditions.2(GA1) a, b, c ∈ G where a, b and c are mutually distinct.
(GA2) for all x ∈ G, there are some z1 , z2 , . . . , zn ∈ {a, b, c}(n ≥ 0) such thatx = ²z1 z2 . . . zn .
(GA3) aa = bb = cc = ².
(GA4) ab = bc = ca.
(GA5) ac = cb = ba.
Since we do not know what a, b and c stand for, we can see elements of G such as ab,cabaa, etc. as mere strings of symbols (we call them terms), and the axioms as rules forterm rewriting.3 Let us see what can be said about this formal system.
It follows that:2

G is an axiomatization of the groups of permutations on lists of three elements.

3

Cf. [5].

8

(GF1) a = a−1 ,b = b−1 ,c = c−1 .

(GF2) for all z1 , z2 , . . . , zn ∈ {a, b, c}(n ≥ 1), (z1 z2 . . . zn )−1 = zn . . . z2 z1 .
(GF3) G = {a, b, c, ab, ac, ²}.
(GF3) may not seem so obvious but it is easy to prove by induction on the length ofterms.
For example, Let t be ab(bac)−1 . Then,t = ab(bac)−1 = abc−1 (ba)−1 = abc−1 a−1 b−1 = abca−1 b−1 = abcab−1 = abcab =bccab = bab = cbb = c
So term t can be viewed as having the term c as its value. The terms which appear in thecourse of the rewriting of t (including t itself) may also be values of t. Therefore we could
not speak of the value of t. Fortunately, however, in G, every term is equal to just one of², a, b, c, ab and ac as stated in (GF3), and these are among the simplest of such sets ofterms. We will call these terms normal forms. If t = v and v is a normal form, v is called
the value of t. Also we will say that a term t reduces to t0 if t = t0 and t0 is not longer
than t.
The following conditions are equivalent for every t, t0 ∈ G:(1) t = t0 ,(2) t and t0 has the same value,(3) t and t0 reduce to the same term.
For example abc = bcc = b, and cba = baa = b. So we can infer abc = cba.
abc

ZZ
reduces to Z

cba

Z
½
~ ½
Z
=

½
½
reducesto½bThus a term of G has two aspects. One is the value, which determines equality between
terms. Another is the form (the conﬁguration of symbols), which indicates how the term9

is constructed as well as how to calculate its value. The process in which we conﬁrm theequality of abc and cba is indicated in their composition, and if one knows the syntax,
axioms, and rules of G, one can construct a proof of the equality. These two aspects
resemble Frege’s Bedeutung and Sinn, but we have to be aware that t’s value is also aterm of G and not independent of G. The relation between a term and its value is not
semantic but proof-theoretic or computational. When we talk of the value of a term t, wereally talk of a procedure to deduce t = v (a proof or a construction of t = v) in G forsome v ∈ N .
Moreover, a term g ∈ G can also be viewed as a function, since for all x ∈ G, gx ∈ G.
Here we think of g as a procedure that is applied to arbitrary x ∈ G and produces thevalue of gx. Among such functions, some are of interest. For example, deﬁne α := ab andβ := ac. Then α is a “one step backward” operator and β a “one step forward” operator,i.e.,α(a) = cα(b) = aα(c) = bβ(a) = bβ(b) = cβ(c) = a

It is obvious that αα = β, ββ = α, ααα = αβ = βββ = βα = ², etc. These facts
are useful when we carry out calculation. For example, suppose we have to calculate on
acbcacbcababc. Recalling that ab = bc = ca and ac = cb = ba,
acbcacbcabcac = βαβαααc= ²²βc= aα and β are abstractions which result from neglecting the forms of terms and focusing
only on their eﬀects on the atomic terms a, b and c next to them. This example showsthat each term has particular functions in particular contexts. Like combinators in lambda
calculus, they are names for procedures or programs described in G’s terms. And just as
programs are meaningful to its interpreter, so are terms of G to those who are working
at G. However, to those who are not familiar with G’s axioms or rules, it is no wonder

10

these terms appear meaningless.
The example of G shows that even if each atom of language is devoid of meaning,
every term of a formal system is not empty. It has its own value as well as functions inparticular contexts. In order to know the value and the functions of a term we should
calculate it. The way of calculation is shown in its form, together with the syntax, axioms
and inference rules of the system. In short, a term expresses the procedures we have tocarry on. The same thing can be said about other mathematical objects such as formulasor proofs.
It is such procedures that mathematics is really about. Mathematicians do not deal
with mere empty symbols, any more than chess players play with chess pieces. Chessplayers are playing chess, and the essence of playing chess is to combine possible moves
into eﬀective strategies. Similarly, mathematicians combine symbols according to rules
into signiﬁcant terms, formulas and proofs. Once they have been constructed, they canbe used to carry on the construction further.

4

Criteria for reality

In this section, we will examine the second assumption of anti-realist arguments that inorder for something to be real, it must be concrete; it must exists in space-time; it musthave causal eﬀects, etc. These are criteria for what I called substantial reality. It is obviousthat the objects of mathematics do not meet these criteria. However, mathematicians seemto be adopting a weaker conception of reality which I called functional. For example, Georg
Cantor wrote
First, we may regard the integers as actual in so far as, on the basis of deﬁnitions, they occupy an entirely determinate place in our understanding, arewell distinguished from all other parts of our thought, and stand to them indeterminate relationships, and thus modify the substance of our mind in a
determinate way [...]. ([2], p. 895)11

[concept of mathematics] must both be consistent with each other and also
stand in exact relationships, ordered by deﬁnitions, to those concepts whichhave previously been introduced and are already at hand and established.
[7] In particular, in the introduction of new numbers it is only obligated togive deﬁnitions of them which will bestow such a determinacy and, in certain
circumstances, such a relationship to the older numbers that they can in any
given instance be precisely distinguished. As soon as a number satisﬁes allthese conditions it can and must be regarded in mathematics as existent andreal. ([2], p. 896)
Cantor’s criteria for reality include identity, relationships with other things, and eﬀects
on other terms and on our activities.4 . He then claims that they are existent and realin our mind. This is probably because he also felt the need to ﬁnd out the references ofmathematical expressions independent of mathematical language, i.e., he also presupposeda sort of model-theoretic interpretation for mathematical language.
Once we have done away with the model-theoretic view, we no longer have to look forthe references of mathematical expressions. In addition, as we have already seen, identity
in a formal system is determined only by means of its syntax, axioms and rules. Whether
x = y holds or not depends only on whether there is a proof of it or not, which has nothing
to do with our state of mind.
I want to reformulate the criteria without referring to any psychological entity. I
include the followings in the list of what seems to be mathematicians’ criteria for reality:(1) identity, (2) manipulability, and (3) force. I do not claim these to be a necessary
and suﬃcient set of conditions for reality. Nor do I claim a novelty for them. I only hopethat they are natural requirements that we expect every real object to fulﬁll. Let me
explain these in order.

4

Recall also the quotation from Abelson and Sussman [1] in the begining of the article.

12

(1) Identity of something means its having clear enough characteristics to be qualiﬁed
as a member of some established category or classiﬁcation, and to be identiﬁed with itself
and distinguished from any other object in the category.
For example, we cannot identify the smallest natural number that cannot be designatedby any English expression less than twenty words long. Therefore the number is not real.
On the other hand, we can identify the smallest natural number that can be designatedby some English expression less than twenty words long. It is, of course, zero.
In mathematics, the way to identify something is contained in its deﬁnition, and identity is to be proved formally. For example, to identify two functions f : A → B andg : C → D, it is necessary to prove A = C, B = D, and for all x ∈ A, f (x) = g(x).
(2) Something is manipulable when we can manipulate it according to deﬁnite procedures that yield certain (not necessarily deterministic) results when applied to any object
in the same category. Here I use the word “manipulate” in a broader sense. For example, construction, application, calculation, detection, measurement etc. are kinds ofmanipulation.5
One important form of manipulation here is making an assertion or judgement aboutsomething. Judgement requires some justiﬁcation for making it. The proposition that acertain apple is sour needs no justiﬁcation, while if you are to judge so, you need to eat
it, to test its acidity, or perhaps to observe someone eat it and frown. There are a great
variety of procedures that justify judgements, but in mathematics, there is only one kind:
proof.
Manipulability in my sense does not imply that we can do anything we like; on thecontrary, manipulations should be understood as quite restricted. The restriction may be
imposed either by physical conditions or by conventions. In either case, the stricter the
restriction is, the more real the result feels.

5

Examples of manipulation which add to reality are abundant in physics, chemistry or medical science.

13

Axioms and rules of mathematics are among the strictest forms of restriction, because if
we break them we can no longer carry on mathematical studies. Therefore mathematics is
not a product of our imagination. It is this strictness of mathematics that made Bertrand
Russell write:
Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created theIndians. The number 2 is not purely mental, but is an entity which may bethough of. Whatever can be thought of has being, and its being is a precondition, not a result, of its being thought of. ([3], §472)(3) Force is the disposition to invoke in something an action or a change of state, orto prevent it from making an action or changing its state. Again this process need not be
deterministic or causal. A typical example of a force that may not be causal is imperative
force, i.e., that of (or due to) laws, rules, orders, contracts, appointment, etc. Like thegravity of the earth or a dose of medicine, they have certain eﬀects on us and can throw
us into or out of an action.
The intensity of the force that a law or a rule has may diﬀer from person to person,
probably depending on one’s nature or on how one has been conditioned so far. This diﬀerence may account for the fact that many mathematicians are realistic about mathematicalobjects, while many philosophers are not.
Another example is the force of plans, methods or strategies. For example, think ofchess strategies. If a player knows an eﬀective strategy that can be applied to the present
situation, he will employ it. So chess strategies have a great eﬀect on chess players.
Mathematical objects have this kind of force. Once we have constructed an object, it
facilitates further mathematical acivities. For example, if it had not been for a deﬁnitionof the primes, it would not have been proved that there are an inﬁnite number of primes.

14

When one employs these criteria, the reality in question is subjective and empirical.
It comes in diﬀering degrees from person to person, and therefore diﬀerent mathematicalobjects have diﬀerent degrees of reality. For example, the power set of {0, 1} is real forevery mathematician, while the power set of the set of all natural numbers is not. The
question about reality in mathematics is about to what extent mathematics is real, notabout whether mathematics as a whole is real or not.
The word “real” in ordinary language has a wide range of meanings. Philosophers andmathematicians are talking about diﬀerent meanings of the word. So philosophers do nothave to feel oﬀended when mathematicians insist on the reality of mathematical objects,
and vice versa.
Some may be afraid that the criteria above are too weak and that almost everything
will be qualiﬁed as real. I do not insist on reality for every procedure. For example,
consider a chess strategy that begins with “Take the opponent’s queen at the openingof the game, and then...” The most likely response will be “But how can I take theopponent’s queen at the opening?”
It is meaningful to ask whether a procedure is feasible, realizable or eﬀective. Thesenotions have something to do with the notion of reality as used by mathematicians. It isuseless to dispute whether mathematics as a whole is real or not, for, as we have seen, it
is a mere matter of deﬁnition of the word. A more signiﬁcant question is to what extent
mathematics is real. For it is on this question that disagreement between mathematiciansarises. Few mathematicians are sheer anti-realists who do not admit the reality of natural
numbers, or full-blooded realists who admit the existence of the set of everything. Disputes
over reality in mathematics are about where to stop. Finitism, intuitionism or Platonism;predicative or impredicative analysis; ﬁrst-order, second-order or higher-order predicate
logic; ZF, ZFC or more; at which level of the arithmetical hierarchy; typed or untyped
lambda calculus; context-sensitive or phrase structure grammar; etc.

15

5Conclusion

Mathematical objects are essentially diﬀerent from physical or everyday objects. Therefore, when mathematicians say they are real, a diﬀerent meaning of reality is intended.
Rather than blaming mathematicians for being careless in the use of the word, it is useful
to consider in what sense mathematical objects are real as well as what the nature ofmathematical objects is.
My conclusion is that objects of mathematics are procedures expressed by a formal system, not something independent of or referred to by mathematical language. Neverthelessthey can be regarded as real in some weaker sense.
Whether you call procedures real or not may be only a matter of diﬀerent viewpoints.
However, discoveries of diﬀerent viewpoints have been triggering important advances inmathematics. In particular, to view a procedure as an object gave rise to lambda calculus,
related programming language, and model theory such as denotational semantics6 . In thissense also, realism in mathematics has certain signiﬁcance.

References[1] H. Abelson and G. J. Sussman. Structure and Interpretation of Computer Programs.
The MIT Press, Massachusetts, second edition, 1996.
[2] G. Cantor. Foundations of a general theory of manifolds: a mathematico-philosophicalinvestigation into the theory of the inﬁnite. In W. Ewald, editor, From Kant to Hilbert:
A Source Book in the Foundations of Mathematics, Vol. 2, pp. 878–920. Oxford University Press, New York, 1883.
[3] B. Russell. The Principles of Mathematics. Routledge, London, second edition, 1992.

6

Cf. [4]. In Japanese, see [6].

16

[4] J. E. Stoy. Denotational Semantics: The Scott-Strachey Approach to Programming
Language Theory. The MIT Press, Massachusetts, 1977.
[5] Terese. Term Rewriting Systems, Vol. 55 of Cambridge Tracts in Theoretical Computer
Science. Cambridge University Press, Cambridge, 2003.
[6] 横内寛文. 『プログラム意味論』, 情報数学講座, 第 7 巻. 共立出版, 1994.

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