Structuralism and Category Theory
Minao Kukita
12th March 20091

Introduction

Category theory has proven to be an eﬀective tool for describing a variety of mathematical structures from topological to algebraic to logical ones. Moreover, topos
theory can be used to construct enough of set theory within category theory. Thus,
some category theorists have claimed that category theory can provide an alternative
conceptual framework for understanding mathematics. In particular, Steve Awodey[1] suggests that philosophers use category theory in order to clarify the notion ofmathematical structure.
On the other hand, Geoﬀrey Hellman [7], a philosophical structuralist, put somequestions to the idea of using category theory as a structuralist framework for mathematics. In replying Hellman’s objections, Awodey [2] argues that Hellman, adhering
conventional foundationalism, overlooks an essential feature of mathematical practice,namely, shcematic and top-down character.
This debate between Awodey and Hellman illustrates well what is at issue in thephilosophy of mathematics, and what obstacles lie in the way of establishing a theoryof meaning acceptable both philosophically and mathematically.
In the following we shall review the debate between Awodey and Hellman, see what
is at issue, and identify what prevents philosophers like Hellman from appreciating
what mathematicians like Awodey suggest. Then we shall draw some morals fromthe debate, and propose a way to interpret mathematical language. It stands in sharp
contrast with those considered by philosophers like Hellman in that it is based on aquite anti-foundationalist view of mathematics. Yet it has contituity with at least
some part of our ordinary language.

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The structuralist controversy

One of the major tasks in the philosophy of mathematics is to propose a way of interpreting mathematical language. Thus, the question often arises about the referencesof such expressions as 0, −1, i, ∅, ω, etc. Although they seem to refer to something,
we cannot directly identify their references in the way that allows everyone to agree
on what they are.
An early structuralist approach to this question was made by Dedekind [5], where
he defined natural numbers as any structure (a triple of a set, a function and anelement) that satisfies certain conditions. However, quite diﬀerent approaches tonatural numbers were also taken by Cantor, Frege, Russell, and later by Zermelo, von
Neuman, etc. They constructed, in one way or another, objects which behave likenatural numbers out of objects or notions which they thought were at their disposal;that is, sets or properties (concepts), things which they believed were suﬃciently
evident. Set theory was particularly popular, and became regarded as the foundationof mathematics on which every branch of mathematics is to be built.
Structuralism has gained popularity in the philosophy of mathematics since Paul
Benacerraf [4] raised the question about set-theoretic definitions of natural numbers:
if natural numbers can be defined in several diﬀerent ways in set theory, which natural
numbers are intended, when we talk about them? The structuralist answer is: “Anything will equally do as long as it shares the structure essential to natural numbers.”
The basic slogans of structuralism are as follows:• Mathematical objects are only identified by relations in some structure.
• A mathematical theory is not about internal properties of individual objects,
but about relations in the structure expressed by the theory.
• A theorem of a theory states some fact about the common structure of arbitrarymodels that satisfy the theory.
Structuralism saves us the trouble of thinking about the essence of mathematicalobjects

what they really are, etc. Moreover it is obvious that present-day

mathematicians treat mathematical objects structurally. On the other hand, to bea structuralist means that one inevitably undertakes the task of explaining what amathematical structure is. Some philosophers like Shapiro and Hellman explain thenotion of structure basically in terms of model theory. However this seems to bringabout no less complicated problems than their earlier ones: problems of the existenceof, the reference to, the underdeterminacy of a structure, etc.

2Meanwhile, in mathematics, with the development of category theory purely structural approaches to mathematics flourished. Some see this movement as “revolution”
that will turn over the dominance of set theory as foundational framework for mathematics.1 However, despite the great success of category theory, philosophers of mathematics do not seem to pay much attention to it.
It is thus that Awodey [1] suggests that philosophers of mathematics use category
theory if they want to approach mathematics structurally. The reasons for his suggestion are: that the notion of mathematical structure is better clarified by category
theory than by set theory, or any other known method; that using category theory means doing everything structurally; and that category theory makes it possible
to treat a variety of diﬀerent mathematical structures and notions in a uniform way,
making it easier to see connections among them. In [1], Awodey explains how familier
notions in set theory and logic are derived and connected within topos theory.
However, Hellman [7] objects to the suggestion in favor of his modal structuralism.
Here we will consider two of the questions he raises. One is about what the assertory
axioms are. Hellman distinguishes two kinds of axioms: one is “formal” or “schematic”
and plays only the role of definition, while the other is “assertory” i. e. asserts some“truth” about something. He thinks that a suﬃcient account for structuralism must
provide a set of assertoric axioms. For example, set theory has one, while group theory
does not. A set-theoretic axiom, say, “if x ∈ A and x ∈ B then x ∈ A ∩ B,” expresses
some truth about concrete objects, namely sets, while a group-theoretic axiom suchas “(xy)z = x(yz) for all x, y, z ∈ G” only constitutes a part of the definition of whata group is. In particular, the axioms of category theory are of this character, and tell
us nothing true of any concrete structure.
Related to the above question is the question about “home address” or about themathematical existence. Hellman asks “where do categories come from and where
do they live?” (p. 136) He says that category theory simply does not address thisquestion. Thus he concludes that category theory is insuﬃcient as a framework forstructuralism. Behind this diagnosis, there is a belief that a suﬃcient account forstructuralism must answer the following questions2 :1. what primitive notions and background logic are employed?
2. what are the assertory axioms?
1

Cf. Goldblatt [6], §1.3.

2

Cf. Hellman [8], pp. 537f.

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3. are stuructures-as-objects eliminated?
4. how are the mathematical existence of structures and their indefinite extendability explained?
5. how is reference to structures, or epistemic access to structures possible?
which category theory hardly does.
Awodey [2] does not deny Hellman in that category theory does not address thesequestions, but retorts that it is the very nature of mathematics: the “schematic” and“top-down” character. Mathematics has nothing to say about any concrete object inany absolute fashion, but only in a hypothetical and indeterminate fashion. In other
word, mathematical statements are always of the form “if A(X) then B(X)” leaving
X unspecified and even unquantified. In addition, axioms of mathematics are given
from top down rather than from bottom up. Take group theory for example. We
do not have the whole universe of groups to begin group theory with, and of which
its axioms and theorems are true. Instead, the axioms give a condition under whichanything is qualified as group. And this is the way mathematical practice is usually
carried out.
Of course Hellman would not deny the schematic and top-down character in mathematical practice. What he would deny is perhaps the idea of mathematical language
being totally ungrounded, so to speak. This is understandable, since we are accustomed to the time-honored principle called compositionalityor the “bottom-up”

principle: the meaning of a complex expression is determined by the meanings of its
components and the configuration in which they are arranged. Therefore, in order
for a statement to be meaningful, each of its components should be meaningful to
begin with. This principle seems too obvious to doubt. Even when a mathematical statement seems to contain indeterminate components, there must be something
they “really” mean. So the task of the philosophy of mathematics is to find out thatsomething, and get mathematics grounded on it.
This perhaps is what is at stake in the controversy.

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The obstacle to reconciliation

We are long accustomed to assuming that a meaningful expression should not leave
any of its component indeterminate. In other words, if an expression contains an indeterminate component, then so is the meaning of the whole expression. In particular,4

if the expression is a proposition, its truth-value is indeterminate. But a proposition
without definite truth-value is not a proposition at all!
This line of thought is aﬀected by the concern for “having a homogeneous semantical theory in which semantics for the propositions of mathematics parallel thesemantics for the rest of the language” (Benacerraf [3], p. 403). Benacerraf considers
this concern to be one of the two main motivations for the account of mathematical
truth.3 But he also confesses that he is ‘indulging here the fiction that we have semantics for “the rest of language, ” or, more precisely, that the proponents of the views
that take their impetus from this concern often think of themselves as having suchsemantics, at least for philosophically important segments of the language’ (p. 403n).
In addition, Benacerraf says he assumes that “truth conditions for the language (e.
g., English) to which mathematese appears to belong are to be elaborated much along
the lines that Tarski articulated” (p. 410). No doubt “philosophically important segments of the language” in his fiction is such that we can apply a usual analysis ofterms, function symbols, predicates, logical connectives and quantifiers to them. He
calls such language “referential.” This assumption is not unique to Benacerraf, butquite common to philosophers of mathematics including Hellman.
I will call “the referentiality assumption” the assumption that mathematical language must be interpreted referential as our ordinary language is. It is this assumption (or “fiction”) that I think prevents philosophers like Hellman from appreciating
Awodey’s suggestion. This assumption seems to be so wrong as to be harmful at leastin two respects. For one thing, it disguises the fact that we have no homogeneous
semantical theory even for our ordinary language. In fact, a substantial proportion of
our ordinary language is not referential. I am doubtful whether it is possible to dismiss
it as philosophically unimportant. For another, it is not obvious that mathematicallanguage must be treated as referential. I will explain these in the next section.

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How to get rid of the obstacle

In the previous section, we saw that the referentiality assumption is an obstacle in
the way of Hellman accepting categorical structuralism. Thus, we shall try to refute
it in this section.

3

The other is “the concern that the account of mathematical truth mesh with a reasonable

epistemology” (ibid.).

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4.1

Are non-referential segments of language negligible?

What is in the non-referential part of language? We sometimes use language toestablish some institutional facts. Such usage is called “performative,” and indeed
paid much attention to by philosophers of language. We name persons, things andnotions; we make promises, rules and laws; and in particular, we define new terms.
In these cases we use sentences of which it is useless to ask whether they are true ornot, or what is the reference of each of their components. For example, consider a signthat reads, “If you park here, you are fined $50.” The function of this sentence is not
to report some information about something, but to set out a rule that may be applied
anything that satisfies a certain condition. What truth-value or truth-condition does
this sentence have? What is the reference of “you” here? These questions do notseem to be sensible to ask.
Focusing on definition, the role of definition is multifold. Sometimes definition
serves as a convenient abbreviation to refer to something. For example, we use “abrother of mine” instead of saying “a child of a parent of mine who is male andis diﬀerent from me.” Sometimes a definition makes clear a subtle idea that has
been unnamed but only vaguely perceived: in the way Horace Walpole invented theword “serendipity.” Sometimes we use definition to introduce a totally new idea by
combining ideas that is already known. For example, I define the word “chilrent” asa person who is both a child and a parent of another person. More precisely:
x is called a chilrent of y if and only if x is a child of y and x is a parentof y.
Then we can use this word and deduce a lot of statements about chilrenthood.
What is important here is that before my definition, the string “chilrent” has no
meaning at all, but it has now become a fully meaningful expression that we can use
in our ordinary discourse.4 I can ask one, “Do you have a chilrent?”, and one will
reply “No.” The pragmatic concern set aside, this coversation is as meaningful as
when I ask one if one has a cousin. In short, definition is a performative speech act
by which a term acquires a new definite meaning for our later use. It makes it an
institutional fact that the term has that particular meaning.
We use language in order to extend or modify language and the environment inwhich sentences are evaluated. This is a striking feature of human language. And I
4

In many cases in mathematics, definition gives a new specific meaning to an already existent

term.

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do not regard it to be philosophically unimportant. But if the word “philosophicallyimportant segments” here is to mean, in the first place, something worth the attentionof philosophers of mathematics, we should go on to the next question.

4.2

Must mathematical language be viewed as referential?

In the previous subsection, we saw examples of non-referential segments of language,
and among them is definition. Definition deserves a special attention from us because it is part of mathematical activity. In fact, definition plays a central role inmathematics. We simply cannot start mathematics without definition.
As a performative speech act, definition makes an institutional fact, and its truth
is not to be questioned. Hellman argues that mathematics must have some axioms
that are assertory in its foundation. We do not try to refute his argument directly,
but see how if we interpret mathematics otherwise, that is, think of all axioms inmathematics as merely definitional.
Once we have done away with the idea that some axioms are true of something,
no question will arise if axioms in mathematics contain indeterminate components.
As we saw, it is natural that rules contain indeterminate component. When we see
the sign that reads “if you park here, you are fined $50,” we do not wonder who this“you” is. Nor do we think that this sentence is quantified, because quantificationneeds a fixed range and perhaps that is not intended by this rule. The same holdsof definition. When I defined the word “chilrent” neither a particular person nor afixed range was intended. What is more, it does not matter that there exists no one
who satisfies the condition for being chilrent. By this fact, Hellman and some others
would worry that everything can be said of chilrenthood. This happens only when weinterpret the “if” part of the definition truth-functionally. We will discuss this later.
Seeing axioms as definitional makes the situation simple. At least three of fivequestions Hellman listed

2, 4 and 5

become irrelevant. This is substantial

gain. What then is the loss? Some may become at a loss what mathematics isall about. Others may dislike the idea that mathematical truth is all by definition.
However they have not escaped from the referentiality assumption.
Awodey, trying to show diﬀerence between his position and old-fashioned “if-thenism,” says:
The truth of the consequent statement doesn’t depend on some unknown
or unknowable antecedent conditions; rather it applies only to those cases
specified by the antecedent description. ([2], p. 9)

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Thus, he does not interpret if-then truth-functionally. The antecedent works to pickup those cases to which the consequent can be applied. If-then in mathematics expresses applicability condition, so to speak. One may be reminded of the BrouwerHeyting-Kolmogorov interpretation of if-then. In the BHK interpretation, “if A thenB” means “there is a procedure p that turns any proof x of A into a proof p(x) ofB.” Thus, p is applied only to proofs of A to produce proofs of B, and not anythingelse. Note that one need not be an intuitionist or a constructivist to employ the BHK
interpretation of if-then, because a proof x of A or a procedure p can be classical.
Again, this usage of if-then is not marginal in “the rest of language.” If-then thatappears in rules, promises, laws and so on is of this character. One may also recall
that in a programming language, if-then is usually not a Boolean expression but acommand in a special form. For example, the expression if x ̸= 0 then y := y/x
end tells us to let y be y/x if x is not 0, and do nothing otherwise. This is also
a good example of an exception to the principle of compositionality. If x is 0, thenthe expression y/x is meaningless. However, the whole expression is still evaluatednormally, and the interpreter does nothing, as expected.
To sum up this section, non-referential segments of language is worth our consideration, to explain mathematics in non-referential terms has several advantages, andeven then continuity with some part of ordinary language is still maintained.

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conclusion

Given what impact category theory has had on mathematics, logic and computerscience, it is rather surprising that structuralists such as Shapiro, Resnik and Hellman
pay scant attention to category theory. Reviewing the debate between Awodey andHellman, we saw that the reason for the scantness can be trace back to the traditional
concern for having homogenious semantical theory for mathematics and the rest of thelanguage. Behind this concern lie the referentiality assumption that both our ordinary
language and mathematical language are referential. Awodey’s reply to Hellman,emphasizing the schematic and top-down character of mathematical practice, casts
doubt on this assumption. Thus we tried to show that non-referential aspects oflanguage is as important, and that viewing mathematics as non-referential has several
advantages.

8References[1] S. Awodey. Structure in mathematics and logic. Philosophia Mathematica, Vol. 4,
No. 3, pp. 209–237, 1996.
[2] S. Awodey. An answer to Hellman’s question: “does category theory provide aframework for mathematical structuralism?”. Philosophia Mathematica, Vol. 12,
No. 1, pp. 54–64, 2004.
[3] P. Benacerraf. Mathematical truth. In P. Benacerraf and H. Putnam, editors,
Philosophy of Mathematics: Selected Readings, pp. 403–420. Cambridge University
Press, Cambridge, second edition, 1983.
[4] P. Benacerraf. What numbers could not be. In P. Benacerraf and H. Putnam,
editors, Philosophy of Mathematics: Selected Readings, pp. 272–294. Cambridge
University Press, Cambridge, second edition, 1983.
[5] R. Dedekind. Was sind und was sollen die Zahlen? In R. Fricke, E. Noether, andÖ. Ore, editors, Gesammelte Mathematische Werke, Zweiter und Dritter Band,
pp. 335–391. Chelsea Publishing Company, New York, 1962.
[6] R. Goldblatt. Topoi. Dover Publications, Inc., New York, 2006.
[7] G. Hellman. Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica, Vol. 11, No. 3, pp. 129–157, 2003.
[8] G. Hellman. Structuralism. In S. Shapiro, editor, The Oxford Handbook of Philosophy of Mathematics and Logic, pp. 536–562. Oxford University Press, Oxford,
2005.

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