Is the philosophy of mathematics most fruitfully pursued as a philosophical investigation 
into the nature of numbers as abstract entities existing in a platonic realm inaccessible 
by means of our standard perceptual capacities, or the study of the practices and activities 
of mathematicians with special emphasis on the nature of the fundamental "objects" that are 
the concern of actual mathematical research? Platonists promote the first approach, while 
structuralists and category theorists advocate the second. This paper will trace the develop-
ment of the structuralist perspective by critically evaluating the positions advocated by 
Benacerraf, Harman, White, Resnik, Shapiro and Jubien. Since one of the conditions of an 
adequate philosophy of mathematics is an explanation of the relationship between pure and 
applied mathematics, the mathematical structuralists positions will then be compared with 
the ideas promoted by the philosophers of science who advocate the semantic conception of 
theories, such as Sneed, Suppe, Giere and Van Fraasen, whose primary concern is the appli-
cation of mathematical structures to empirical phenomena. While category theory has been 
promoted as an alternative to set theory as a foundations for mathematics, it can also be 
considered compatible with, and an extension of, the structuralist's point of view. An 
evaluation of category theory as a possible foundations for mathematics will be presented 
through an analysis of the papers of Bell. 

Paul Benacerraf's paper, "What Numbers Could Not Be", is often credited with initiating the 
philosophical thesis of structuralism, although the structuralist attitude had been prevalent 
among working mathematicians since the 1930s. The lead quotation to his article makes this 
dominant attitude explicit, and is worth quoting in full.

	The attention of the mathematician focuses primarily upon mathematical structure, 
      and his intellectual delight arises (in part) from seeing that a given theory 
      exhibits such and such a structure, from seeing how one's structure is "modelled" 
      in another, or in exhibiting some new structure, showing how it relates to previously 
      studied ones...But...the mathematician is satisfied so long as he has some "entities" 
      or "objects" (or "sets" or "numbers" or "functions" or "spaces" of "points") to work 
      with, and he does not inquire into their inner character or ontological status.
	The philosophical logician, on the other hand, is more sensitive to matters of 
      ontology and will be especially interested in the kind or kinds of entities that 
      are actual...He will not be satisfied with being told merely that such and such 
      entities exhibit such and such a mathematical structure. He will wish to inquire
 	more deeply into what these entities are, how they relate to other entities...
      Also he will wish to ask whether the entity dealt with is sui generis or whether 
 	it is in some sense reducible to (or constructible in terms of) other, perhaps more 
      fundamental entities.-- R.M.MARTIN, INTENSION AND DECISION [272]

If this article does reflect the prevailing point of view of mathematicians, then a sharp 
distinction should be drawn between the philosophy of mathematics on one side, and ontology 
or philosophical logic on the other (philosophical logic being the activity of philosophers 
who presumably analyze the presuppositions and activities of logicians). Very often philosophical 
logicians are really logicists who are promoting the program of reducing mathematical objects 
to logic and set theory. This is clearly distinct from the activities of mathematicians who 
are not interested in a reductionist program, but who instead explore the consequences of their 
formal constructions, along with the relationship of these constructions to other mathematical 
structures and "objects."

Benacerraf argues for the structuralist position by first presenting an example in which the 
sons of two militant logicists, Ernie and Johnny, first learn logic and set theory instead of 
elementary number theory. When it comes to learning about numbers, they merely learn new names 
for familiar sets. They count members of a set by determining the cardinality of the set, and 
they establish this by demonstrating that a specific relation holds between the set and one of 
the numbers. To count the elements of some k-membered set b is to establish a one-to-one 
correspondence between the elements of b and the elements of A less than or equal to k. The 
relation "pointing-to-each-member-of-b-in-turn-while-saying-the-numbers-up-to-and-including-k"
establishes such a correspondence. [Benacerraf,275]

The problem, as posed by Benacerraf, is that Ernie and Johnny have learned different versions 
of set theory, the referents assigned to the terms are different, and we are faced with the 
following problem: [Benacerraf,280]

		(A) Both are right in their contentions: each account contained conditions each 
                of which was necessary and which were jointly sufficient. Therefore 3=[[[0]]],
                and 3=[0,[0],[0]]].
		(B) It is not the case that both accounts were correct; that is at least one contained 
                conditions which are not necessary and possibly failed to contain further 
                conditions which,taken together with those remaining, would make a set of 
                sufficient conditions.

Since Benacerraf holds that (A) is absurd, he considers (B) to be the only alternative worth 
considering. The two accounts disagree in terms of the referents of the terms, but they agree 
in over-all structure and the relations and functions that hold between the referents: both 
have a recursive progression and successor function which follows the order of that progression.
But then, how is one to distinguish the correct referents from the incorrect one. Is there any 
set which has a better claim to be the numbers than any other set? Benacerraf considers Frege's 
solution.[282]

Frege chose as the number 3 the extension of the concept "equivalent with some 3-members set"; 
that is, for Frege a number was an equivalence class - the class of all classes equivalent with 
a given class. Although an appealing notion, there seems little to recommend it over, say Ernie's. 
It has been argued that this is a more fitting account because number words are really class 
predicates, and that this account reveals that fact. The view is that in saying that there are 
n F's you are predicating n-hood of F, just as in saying that red is a color you are predicating 
colorhood of red. I do not think that this is true. And neither did Frege.

Benacerraf concludes that there is no one account which conclusively establishes which sets are 
the "real" numbers, and he doesn't believe that there could be such an argument. Any "object" or 
referent will do as long as the structural relations are maintained. Frege believed it necessary 
to find some "objects" for number words to name and with which numbers could be identical. From 
his point of view there is a world of objects which are the referents of names and descriptions 
of which the identity relation holds. It made sense for Frege to ask of any two names whether 
they named the same object. Benacerraf argues that Frege's belief stemmed from his inconsistent 
logic. Since all objects of the universe were on par, the question whether two names had the 
same referent always had a truth value. But identity conditions make sense only in contexts where 
there exist individuating conditions.

		If an expression of the form "x=y" is to have sense, it
 		can be only in contexts where it is clear that both x
 		and y are of some kind or category C, and that it is
 		the conditions which individuate things as the same C
 		which are operative and determine its truth value.
 		[Benacerraf,287]

One could construe that identity is systematically ambiguous, or, like Frege, hold that identity 
is not ambiguous, but always means sameness of object. But Benacerraf argues that what constitutes 
an object varies from theory to theory, category to category, and that Frege failed to realize 
this fact. It is a thesis that is supported by the activity of mathematicians, and is essential 
to the philosophical perspective underlying category theory, as we shall discuss later. The search 
for urelements, fundamental objects of the mathematical universe, is a mistaken enterprise that 
underlies an absolute theory of identity and the platonic philosophy of mathematics. This does 
not relativize logic for Benacerraf, because he still asserts that 

		Logic can then still be seen as the most general of
 		disciplines, applicable in the same way to and within
 		any given theory. It remains the tool applicable to
		all disciplines and theories, the difference being
		only that it is left to the discipline or theory to
 		determine what shall count as an "object" or
		"individual." [Benacerraf, 288]

	Benacerraf concludes that numbers could not be sets at all on the grounds that there are 
no good reasons to say that any particular number is some particular set, for any system of 
objects that forms a recursive progression would be adequate. He also points out the results of 
Takeuti who has shown that the Godel-von Neumann-Bernays set theory is reducible to the theory 
of ordinal numbers less than the least accessible number. This supports the thesis that sets are 
really ordinal numbers, but leaves us with the question of which is really the more fundamental 
object: sets or ordinal numbers. Benacerraf refers back to Martin's quotation that "the mathematician's 
interest stops at the level of structure. If one theory can be modeled in another (that is, 
reduced to another) then further questions about whether the individuals of one theory are really 
those of the second just do not arise."

	While the philosophical logician continues to ask questions about the objects, Benacerraf 
believes that this is a mistake, for "what is important is not the individuality of each element, 
but the structure which they jointly exhibit." Therefore numbers could not be objects at all 
because there is no good reason to identify any individual number with any one particular object 
than with any other. 

		Therefore, numbers are not objects at all, because in giving the properties 
		(that is, necessary and sufficient) of numbers you merely characterize an 
		abstract structure - and the distinction lies in the fact that the "elements" 
		of the structure have no properties other than those relating them to other 
		"elements" of the same structure. If we identify an abstract structure with a 
		system of relations (in intension, of course, or else with the set of all relations 
		in extension isomorphic to a given system of relations), we get arithmetic elaborating 
		the properties of the "less-than" relation, or of all systems of objects(that is, 
		concrete structures)exhibiting that abstract structure. That a system of objects 
		exhibits the structure of the integers implies that the elements of that system 
		have some properties not dependent on structure. It must be possible to
 		individuate those objects independently of the role they play in the structure. 
		But this is precisely what cannot be done with numbers. [Benacerraf,291]

The properties that numbers possess outside of the properties of the structure (in virtue of 
being arranged in a progression) are of no consequence to the mathematician, nor should they 
therefore by of concern for the philosopher of mathematics. There is an activity which does 
theorize about the unique properties of individual numbers separate from the progressive structure; 
its called the discipline of numerology, and it does have its followers and publications, which 
usually can be found on the magazine racks of drug stores. But no mathematician that I am aware 
of has ever received a Ph.D. degree or had any article published in a mathematical journal 
discussing the non-structural properties of a unique individual number such 4, 7, or 11. Such 
theories of the properties of numbers are of no concern to mathematicians.

	Arithmetic is the science exploring the abstract structure that all progressions have in 
common merely in virtue of being progressions. It is not concerned with particular numbers, and 
there are no unique set of objects which are the numbers. Number words don't have a singular 
reference, because the theory is elaborating an abstract structure and not the properties of 
individual objects.

		In counting, we do not correlate sets with initial
 		segments of the sequence of numbers as extralinguistic
 		entities, but correlate sets with initial segments of
 		the sequence of number words.The central idea is that
 		this recursive sequence is a sort of yardstick which we
 		use to measure sets. Questions of the identification of
 		the referents of number words should be dismissed as
 		misguided in just the way that a question about the
 		referents of the parts of the ruler would be seen as
 		misguided.[Benacerraf, 292]


	Gilbert Harman, in his article "Identifying Numbers", supports Benacerraf's position that 
number words don't have singular reference. Harman's position is that numerals are best analyzed 
as function symbols rather than names. "Any T-sequence of different sets (or other things) can 
be used as the numbers." Knowing that 3+4=7, means that for any such sequence s, the sum operation 
for that sequence applied to the third and fourth members of the sequence yields the seventh member, i.e,

	3s +s 4s = 7s	

Here '3', '4' and '7' are not names but function symbols. The letter 's' names a sequence and '3s' 
names the third member of that sequence. '3' by itself does not name anything. Harman elaborates 
by saying that numerals are not names of functions; they are not names at all, but instead function 
symbols, and therefore not names of objects.

	Category theory, as we shall see later, is compatible with this position since it takes as 
fundamental arrows or morphisms, which are generalizations of functions. A category is a collection 
of objects and the arrows between these objects which satisfy certain axioms to be discussed in 
detail later. But what is an "object" and what is an "arrow" is relative to the category under 
investigation (or the context of discourse), with the added insight that the object of any category 
can be considered the identity-arrow on that object. This identity arrow is also isomorphic to 
the object, which enables category theorists to reduce their "ontology" to consisting of only of 
morphisms, and morphisms between morphisms. The point to be emphasized here is that Harman's 
abstraction of the referent of a numeral from a number, i.e.,"object", to a function, lays the 
ground work for the development of the intuition that functions (morphisms), structures, and 
transformations(including isomorphisms) between structures is the essential concern of mathematicians, 
and therefore should be the primary concern of the philosophy of mathematics.

	Nicholas White, in "What Numbers Are", disputes Benacerraf's conclusion that numbers are 
not objects. Instead he argues that numbers are objects which occupy positions in progressions, 
but there are an infinite number of them for each position. He agrees that arithmetic can be 
modeled in set theory, and further agrees that it can be modeled in an infinite number of ways, 
with any specific number turning out to be radically different sets. But he rejects the argument 
offered by Benacerraf that this profusion of set-theoretic models of arithmetic demonstrates 
that there is no good reason for saying that numbers are sets. Because if numbers are sets, then 
they must be some specific set, but there is no good reason to choose one particular set over 
another.

	White takes the heroic position that "there are multiple full-blown series of natural 
numbers. Thus, for example, instead of there being only one three, there are after all many 
threes, and many thirty-sevens, and so on." [White, 112]

		...there is nothing in the discourse of arithmetic
 		itself which requires us to fix on a single object,
 		whether set or otherwise, as the number three, or on
 		any single progression as the progression of natural
 		numbers. As far as that discourse is concerned, we may
 		have more threes, and more sequences of numbers than we
 		had thought we had. [White, 116]

	White's first argument in support of his thesis relies on the assumption that since 
Benacerraf continues to use number-words and numerals, they must be interpreted as singular 
terms, and therefore have referents in the "world". But Harman's interpretation of numerals 
as function symbols can be used to meet this objection.

	White next considers the objection that while there may be several different progressions 
which satisfy the laws of arithmetic, there is one unique progression considered to be the 
sequence of natural numbers. But White believes this to be false, since he considers that all 
progressions are on par, "...the point can be put by saying that unintended models of a formal 
system, however unintended they may be, satisfy the system just as much as the intended models 
do." [White, 117]

	White argues that even if in one's counting and cardinality-assignments one uses numerical
singular terms, there is no reason to suppose that there is only a single sequence of numbers.
Counting is a mapping process in which elements of sets are mapped onto the initial sequence of
numbers. But even if this is so, it does not follow that there is only one sequence of numbers 
because any T-sequence will do, since all sequences of natural numbers are isomorphic to each 
other. White quotes Benacerraf as stating that

		in counting we do not correlate elements of a set with
 		initial segments of numbers as extra-linguistic
		entities but correlate sets with initial segments of
 		the sequence of number words", and that for purposes of
 		counting we may perfectly well do without the
		supposition that 'two','2','zwei', and 'deux', all
 		stand for the same object. [White, 118]

White objects to this because he believes that Benacerraf wants to adopt some single sequence 
of expressions as basic to all others. White rejects the need for one string of expressions 
to be more fundamental to all the others in order for us to understand both of them.

		All that seems to be required is that we be able
		uniformally and with facility to map each sequence onto
 		the other in an order-preserving way. Thus, not only
 		could we use different linguistic progressions from
 		any one which we in fact use; we actually do use
		different such progressions on different occasions.[White, 121]

In mathematical terms, White is saying that what is necessary is the ability to construct an 
isomorphism between the two linguistic progressions. An isomorphism is a one-to-one onto mapping 
which has an inverse and is order preserving in the sense that any relation the holds in the 
first structure holds for the images of the objects mapped into the second structure. In set 
theory these are also known as order-preserving bijections, and in category theory functors are 
generalizations of isomorphisms that allow us to "translate" from one category to another in 
a way that preserves the categorial structure of its source. 

	White's philosophical mistake is to assert that because we have multiple numeric 
expressions in various languages, and that they all have the syntactic form of singular terms, 
they must refer to different objects and therefore we have many natural number sequences. From 
a structuralist point of view, what is important is mapping the physical objects that are to 
be counted, or the members of the set to be counting, with a subset of the sequence of the 
numeric-expressions. The mapping is what is important, not putative referents to the linguistic 
terms. Since all these linguistic sequences are isomorphic to each other, instead of having a 
multiplicity of objects that are the referents, we have none at all. All that we really have 
are the physical objects or members of the set to be counted, the sequence of expressions, and 
the mapping process.

	The structuralist philosophy of mathematics was further developed in the papers of Michael 
Resnik. In his paper, "Mathematics as a Science of Patterns: Ontology and Reference", he states 
his purpose as developing a philosophy of mathematics in which the logical forms of mathematical 
statements are taken on face value, i.e., the numerical expressions are singular terms that 
refer. He intends to work within the framework of the referential semantics developed by Tarski. 
He therefore is a platonist because he believes that this referential semantics entails that 
mathematics is a science of abstract entities: "immaterial and nonmental things which do not 
exist in space and time." But like so many other platonists, he fails to give an adequate 
account of how singular terms, which are linguistic expressions existing in space and time, 
could "refer to" and be "satisfied" by abstract objects which do not exist in space and time. 
He does acknowledge that platonism does have to overcome some critical objections before it can 
be taken as a serious philosophy of mathematics.

		Many philosophers of mathematics who would like to be
 		platonists are bothered by two rather deep problems:
 		the first is that since platonic mathematical objects
 		do not exist in space or time the very possibility of
 		our acquiring knowledge and beliefs about them comes
 		into question. The second arises from the fact that no
 		mathematical theory can do more than determine its
 		objects up to isomorphism. Thus the platonist seems to
 		be in the paradoxical position of claiming that a given
 		mathematical theory is about certain things and yet be
 		unable to make any definitive statement of what these
 		things are. [Resnik, 529]

	Resnik attempts to overcome these objections by asserting that the problems arise due to a 
misconception of what mathematics is really about. It is a mistake to think that numbers can be 
given to us in isolation from the other numbers. With this conception it is natural to think of 
knowledge of numbers involving some sort of interaction between the perceiver and the number. 
Resnik hopes to appeal to structuralism in order to support his platonist leanings, and therefore 
agrees that we are not given mathematical objects in isolation but rather in structures.

		In mathematics, I claim, we do not have objects with an
 		"internal" composition arranged in structures, we have
 		only structures. The objects of mathematics, that is,
 		the entities which our mathematical constants and
 		quantifiers denote, are structureless points or
		positions in structures. As positions in structures,
 		they have no identity outside of a structure.
		Furthermore, the various results of mathematics which
 		seem to show that mathematical objects such as the
 		numbers do have internal structures, e.g., their
		identification with sets, are in fact interstructural
 		relationships. [Resnik,530]

Resnik takes mathematical objects, whether they be numbers, sets, functions or points, to be 
structureless entities that occur in mathematical structures. These objects serve as only 
positions within these structures, with their identity determined only by their relationships 
with other positions within that structure. His basic underlying metaphor is that of geometric 
points, and he claims that we do not have knowledge of mathematical objects given in isolation 
but rather as "pieces of structures."

	Resnik prefers to speak of patterns, rather than structures. Our knowledge of patterns 
starts with experiencing something as patterned, then recognizing structural equivalence 
relations among the data now experienced as patterned. We then introduce predicates for the 
equivalence and other relationships that we recognize as holding. In a second paper, "Mathematics 
as a Science of Patterns: Epistemology", Resnik discusses the process of acquiring beliefs about 
patterns.

		Another phenomena which often accompanies our thinking
 		about abstract entities is the retention of the visual,
 		auditory or other sensory images of the concrete things
 		which led to our abstract conceptions...Apparently the
 		early Greeks used arrays of dots to represent the
 		numbers and to arrive at truths about them. Today, it
 		is not unusual to use a linear array of dots to
		represent the natural number sequence or a tree diagram
 		to represent the iterative hierarchy of sets, and
 		category theorists are completely dependent upon their
 		arrow diagrams. The fact that images may figure in our
 		thoughts about abstract structures does not imply, of
 		course, that we perceive these structures with some
 		sort of mental eye, but I suspect that it lies behind
 		the fairly widespread belief among mathematicians that
 		we have intuitions of mathematical structures...So I
 		have no objection to talk of intuition as long as
 		it is not construed literally. [Resnik, 99]

For Resnik, we go through stages during which we conceptualize our experiences in more abstract 
terms. At the last stage we leave our perceptual experiences so far behind that we think of our 
theories as being about abstract entities. He does not believe that we have a special mental 
faculty that we use to acquire knowledge of patterns, nor does he claim that the abstractive 
process yields necessary truths or a priori knowledge. There is no way to place a priori 
limitations on the patterns that mathematicians can study which is not arbitrary. But he does 
take it for granted that there are mathematical truths, that they are about abstract entities, 
and that some of them are known to be true. He does distinguish between an applied theory of a 
pattern and a pure one. 

		An applied theory of a pattern will consist of a pure
 		one together with claims stating how the pattern is
 		instantiated. An applied theory can be falsified by
 		showing that the data it covers does not fit its
		pattern. A pure theory can be falsified by showing that
 		it fails to characterize any pattern at all, that is,
 		that it is inconsistent, and it can be shown inadequate
 		by showing that it does not characterize a unique
 		pattern, that is, that it is not categorical.[Resnik,101]

	There are two kinds of objects that Resnik talks about: structures or patterns, and 
objects(or positions) in structures. Resnik believes that it makes no sense to say of 
structure A and structure B that A=B or that A is not identical to B, implying that structures 
are excluded from the identity relation. Also, when it comes to objects in different structures, 
identity is not applicable. There is no such thing as trans-structural identity for objects 
(places) within different structures. Since Resnik motivation is to offer a plausible platonist 
philosophy of mathematics, he hopes that this solution will solve Benacerraf's multiple reduction 
problem, because across structures, one cannot ask if an object in one structure is or is not 
identical to an object in another structure. 

	Chihara, in his book, Constructibility and Mathematical Existence, critically analyzes 
Resnik's positions, which he finds to be implausible. He argues that Resnik's doctrine that 
two different structures cannot be said to be identical or not identical is incoherent since 
it presupposes the notion of two structures which are non-identical.

		Resnik makes heavy use of the notion of congruence of
 		structures. But how is congruence to be defined?
		Evidently, it would go something like the following: 
 		A is congruent with B iff there is a one-one correspondence 
		between objects in A and those in B such that...But notice 
		that we are here using a language in which we speak of objects 
		in both A and B. As Resnik notes: "This seems to require a 
		universe which contains positions from different patterns." 
		This seems to undermine Resnik's position, because in such a 
		language, it would seem, that we would have to have the identity
 		relation well-defined over the universe of this language. [Chihara, 144]

	One way out of this problem for Resnik is to have all patterns studied by mathematicians 
to be sub-patterns of one big pattern: the universe as the mega-pattern of which others are 
just a portion. But such a mega-pattern within which contradictory and incompatible patterns 
can be integrated doesn't seem conceptually possible. Another solution advocated by Resnik is 
to have a many-sorted logical language in which each universe of the language corresponds to 
the totality of objects of some structure talked about. In this language, one can have infinitely 
many identity relations, one for each universe of discourse. In this way one can avoid having 
any identity relation between the objects of different structures. But this seems to be a radical 
and extreme change in our conceptual framework, motivated by the desire to offer a plausible 
form of platonism. 

	In "Mathematics and Reality", Stewart Shapiro offers a non-platonist version of structuralism. 
He takes as the fundamental problem in the philosophy of mathematics the accounting for the 
relationship between mathematics and non-mathematical reality. His version of the structuralist 
philosophy proposes that "mathematics applies to reality through the discovery of mathematical 
structures underlying the non-mathematical universe." 

Shapiro believes that a scientific explanation of a physical event often is nothing more than a 
mathematical description, or the construction of a mathematical model of the phenomena under 
investigation.

		Strictly speaking, a mathematical description, model,
 		structure, theory, or whatever, cannot serve as an
 		explanation of a non-mathematical event without an
 		account of the relationship between mathematics per se
 		and scientific reality per se. Without such an account
 		it is not clear how  "scientific explanations" succeed
 		in explaining anything. That is, one cannot begin to
 		account for how science contributes to knowledge
		without an account of what mathematical-scientific
 		activity has to do with the reality of which science
 		contributes knowledge. This problem becomes particularly 
		important in the context of philosophy of science and 
		philosophy of mathematics. [Shapiro, 525]

Shapiro believes that the philosophy of any discipline is partly a branch of epistemology, 
because its purpose is to provide an account of the goals, methodology, and subject matter 
of the field. A philosophy of a discipline tries to describe the activity of that discipline 
and demonstrate how that activity accomplishes its goals. Therefore, a philosophy of a 
discipline is not to be separated from the practice of the discipline.

		Thus conceived, the philosophy of X is not understood
 		as a field isolated from the practice of X. Rather, the
 		philosophy of X is engaged in by people who care about
 		X in order to describe and account for its activity,
 		its success and failures, and its importance. A
		practice of X, or an Xist, who adopts such a
		philosophy should gain something thereby. The adopted
 		philosophy should orient the Xist to his work by
		providing a clear account of what he is trying to
 		accomplish and how his practice contributes to this.[Shapiro,525]

	Shapiro states that within mathematics the distinction between "pure" branches and 
"applied" branches is artificial. He believes that there are more similarities than differences 
concerning the aims, techniques, logic and even subject matter of the two branches. His position
is that there is "no significant or philosophically illuminating distinction to be made between 
branches of pure mathematics and applied mathematics."

	Shapiro states that none of the traditional philosophies of mathematics adequately explains 
the relationship between mathematics and scientific reality, and some even imply that there is 
no such relationship. He intends that his structuralist philosophy of mathematics will provide 
a more fruitful account of the relationship between mathematics and scientific reality. 
Structuralism holds that the "subject matter of mathematics consists of patterns or structures 
and not collections of mathematical objects." 

		A mathematical structure can, perhaps, be similarly
 		construed as the form of a possible system of related
 		objects, ignoring the features of the objects that are
 		not relevant to the interrelations. The structure is
 		completely described in terms of the interrelations... 
		A typical beginning of a mathematical text consists of
 		the announcement that certain mathematical objects,
 		such as real numbers, are to be studied. In some cases
 		at least,the only thing we are told about these objects
 		is that there are certain relations among them and/or
 		operations on them...One easily gets the impression
 		that the objects themselves don't matter; the relations
 		and operations are what we study. The idea that the
 		subject matter of a given branch of mathematics is a
 		certain structure (or class of structures) is a 
		straightforward interpretation of this observation.[Shapiro, 535]

	A given structure is abstract because it can have more than one exemplification. Shapiro 
holds that this is a special case of the problem of universals "a structure is a universal and 
a system of objects exemplifying it is an instance." Mathematical logic and model theory 
quantify over structures, so by the Quinian criteria of ontological existence, these theories 
are committed to the existence of structures.

		Mathematical logic and, in particular, model theory can
 		perhaps be construed as a theory that quantifies over
 		structures. To say that a class of sentences is
		satisfiable may be to say that there is a structure
 		that satisfies it. It is more common to construe the
 		variables of model theory as ranging over sets. Sets
 		are currently taken to be places within the set-theoretical-
		hierarchy-structure. Thus, like any other branch of mathematics, 
		model theory is seen as the study of a particular structure...
		the thesis that mathematics can be reduced to set theory amounts 
		to a claim that any given mathematical structure (except that of 
		set theory itself) can be modeled in the set-theoretic structure and, 
		moreover, that the latter captures the relevant relationships between 
		structures. [Shapiro, 537]

	This problem does not arise for category theorists, who are also concerned with structures 
categories) and relationships between structures (functors between categories), because they do 
not use the language of first-order predicate calculus with quantifiers. Quine's criteria for 
ontological existence cannot be expressed within the language of category theory, although 
structures can be constructed and morphisms between structures can be exhibited. It therefore 
seems that implicit within the category theory is a constructibility requirement for the 
existence of mathematical objects. But these are issues that Shapiro doesn't deal with since 
he relies on the concepts of mathematical logic and model theory to explicate his version of
structuralism.

	Shapiro's account of the relationship between mathematics and science rest on his belief 
that "the contents of the non-mathematical universe exhibit underlying mathematical structures 
in their interrelations and interactions." This becomes a special case of the instantiation of 
universals. "Mathematics is to reality as universal is to instantiated particular...mathematics
is to reality as pattern is to patterned." Some applications of mathematics to the phenomena can
be explained by the "claim that science proceeds by discovering exemplifications of mathematical
structures among observable objects." But Shapiro then qualifies this position by stating that 
"the discovery is often indirect and involves the postulation of theoretical entities. The 
situation might best be described as scientific theories incorporating mathematical structures."
In a footnote to his paper he discusses an idea offered by Kvart that instead of talking about 
mathematical structures "underlying" physical reality, one can speak of "isomorphisms between 
systems of mathematical objects and systems of physical objects." This would be a more fruitful 
approach for a non-platonist structuralist, and is actually fully developed by structuralist 
philosophers of science, like Sneed. Sneed's formal work concerning the relationship between 
mathematics and empirical science will be presented in a later portion of this paper. A detailed
presentation of this structuralist approach to the philosophy of science can be found in Balzer,
Moulines and Sneed, An Architectonic for Science.

	Shapiro views structuralism as providing a more holistic view of mathematics and science 
that can account for the interaction of these disciplines, as opposed to a platonistic philosophy 
that can't even begin to account for how mathematical objects in a separate "mathematical world" 
not accessible through human perceptual capacities could interact with or represent the phenomena 
of the empirical world. For Shapiro there is no sharp distinction between the structures studied 
by "pure" mathematics, and the structures studied in "applied" mathematics. 

		Virtually any structure can be a mathematical structure
 		if mathematicians (qua mathematicians) study it (qua
 		structure)...the difference lies more in the way that
 		structures are presented and studied. Mathematical
 		structures are described abstractly--independent of
 		what the structures may be structures of--and studied
 		deductively. 

	One can conceive of "applied mathematics" as having additional constraints which deal with 
the adequacy of the structure as a representation of the physical phenomena under consideration. 
These constraints are studied by philosophers of science and usually have to do with compatibility 
with existing evidence, data and successful prediction of future phenomena inferred from the 
structures used in the model. Shapiro concludes by stating that an additional advantage of 
structuralism is that it accounts for the interconnections among the various branches of 
mathematics. "The interplay between them is a result of the modeling of one structure within 
another or, in other words, the use of one structure to study another structure."

	Chihara, in his chapter on "Mathematical Structuralism" in Constructibility and Mathematical 
Existence, criticizes Shapiro's position for not taking into account the work done in the 
philosophy of science by philosophers working within the semantic approach to scientific theories, 
an approach which emphasizes mathematical structures instead of the formal sentences of predicate 
logic.

		Under the semantic view, theories are characterized by
 		specifying a class of mathematical structures to be
 		used for the representation of empirical phenomena; so
 		one would have thought that Shapiro would have been
 		sympathetic with such an approach. The semantic
 		approach, however, suggests that applications of
		mathematical structures to the physical world are
 		frequently not made in the straightforward way
 		described by Shapiro. For example, in evolutionary
 		theory, structures are frequently used to describe
 		ideal populations of organisms. In such cases, there
 		may be no actual population that exemplifies the
 		structure. But the idealization may be useful just the
 		same in making certain kinds of estimates, in designing
 		experiments, in making predictions, in explaining
 		certain features of actual population growth, etc.[Chihara, 139]

	For these philosophers of science who are actually working on the specific problems of 
"applying" mathematical structures to empirical phenomena, the approach is not to seek these 
structures as somehow "underlying" or embedded in the physical phenomena itself, but as useful 
constructions that can be mapped onto aspects of the phenomena, and which are useful for explanations.

	In "Reduction, Interpretation and Invariance", Joseph Sneed presents a formal method of 
describing reduction relations among theories as a special case of the structure of related 
empirical theories. Category theory and mathematical structures play a key role in explicating 
these relations.

		The global structure of empirical theories is represented
		as a net of linked theories. Central to the understanding 
		of empirical science are interpreting links. These links 
		provide a kind of "empirical semantics" for the mathematical 
		apparatus associated with individual theories. Interpreting 
		links are characterized and distinguished from reducing links.
 		The concept of an interpreting link provides us with a
 		formal characterization of the distinction between
 		theoretical and non-theoretical concepts, relative to a
 		given theory and the net in which it lives, as well as
 		a formal characterization of the intended applications
 		of a theory in a net. The role of invariance principles
 		in relation to interpreting links is described.[Sneed,95] 

	Sneed represents empirical theories as a net of linked theory elements. A theory element 
consists of some "concepts" - call them K - that are used to say something about some array of 
things, the intended applications of the concepts - call them I.
Thus a theory element is a certain kind of ordered pair

		T = 

On Sneed's view the "conceptual apparatus" K of the theory element consists of categories of 
set-theoretic structures, where a category ||X|| is an ordered triple of classes

		||X|| = <|X|,X,Xc> 

where |X| is the class of "objects", X the class of morphisms and Xc the class of morphism 
compositions.

|X| is a class of set-theoretic structures or models associated with the theory. 


X is a function:
		
		X: |X| x |X| -> |SET|

For x,y 0|X|, X(x,y) is the set of morphisms "from x to y". 
Morphisms are things like structure preserving transformations from the domain of x into the 
domain of y. 

In categories associated with empirical theories Xc will be the composition of functions. 
Among these morphisms which may be in X(x,y) are "isomorphisms". A morphism :0X(x,y) is an 
isomorphism iff there is a :-1 0 X(x,y) so that : and :-1 composed both ways yield the 
identity morphisms. We denote the set of isomorphisms by 'Ix' so that

		Ix(x,y) fX(x,y)

In the categories associated with empirical theories the members of  Ix(x,y) are things like 
"scale transformations" and "coordinate transformations" that relate different models of the 
same theory. They are essential to describing the "invariance properties" of the theory's claim. 
Sneed wants to view the conceptual apparatus of mature empirical theories as categories of 
set-theoretic structures - classes of models for the theory, together with morphisms which 
relate "empirical equivalent" structural features of these models. The models for a theory 
element are to be identified with the theory's empirical laws. [Sneed, 97]

	Invariance properties of theories have become important to philosophers of science working 
within the model-theoretic semantic approach to scientific theories. Category theory is the 
language best suited for this type of representation because it avoids the incommensurability 
problems which result from the Tarskian semantics essential to mathematical logic and model-
theory for which satisfaction relations and truth definitions can only be defined for a specific
language and the structure used to explicate the semantics. Within Tarskian semantics you only 
have truth relative to a model and a specified language. Truth cannot be defined across different 
languages and different structures. Therefore, the notion of invariance is in-expressible within 
this formal semantics for mathematical and empirical theories. Category theory does not appeal 
to Tarskian truth semantics, but instead uses functors between categories to express invariance 
relations. This insight underlies Sneed use of category theory to explicate empirical theories, 
because it allows him to express invariant properties for these theories across transformations 
and representations in alternative languages.

	Invariance properties have replaced the notion of laws of nature for Bas Van Frassen, as 
discussed in his recent book, Laws and Symmetry. Van Frassen argues that metaphysicians speak 
of laws of nature in terms of necessity and universality, while empirical scientists speak in 
terms of symmetry and invariance. He develops an empiricist view of science as a construction 
of models to represent phenomena, in which concepts of symmetry, transformation, and invariance
illuminate the structure of such models.

	In "Ontology and Mathematical Truth", Michael Jubien, in attempting to find a non-platonist 
reasonable theory of mathematical truth, focuses on the notion that "the subject matter of 
mathematics is abstract structure per se, and that the apparent range of informal mathematical 
quantifiers does not in any way constitute that subject matter." A specific mathematical theory 
would be a theory of a specific mathematical structure, but not any realization of that structure. 
This is reinforced by the mathematical practice of ignoring differences between isomorphic systems. 
He intends to make a distinction between abstract structures and concrete instances of the 
structure, perhaps to distinguish "pure" mathematics from "applied" mathematics.

	Jubien considers the problem of specifying structures without appeal to exemplications of 
the structure, a problem that Sneed's explication of structures in terms of category theory 
would resolve.

		...it is reasonable to view model-specifications as
 		consisting of two distinct parts: an attempt to specify
 		a certain structure and an attempt to pick out certain
 		entities to serve as an exemplification of that
 		structure. We should not infer that the first attempt
 		has failed from the failure of the second, for the two
 		are truly independent of each other. The reason for
 		this independence is that in a model the primitive
 		relations are understood to be interpreted extensionally 
		(normally as sets of ordered n-tuples). Thus isomorphic 
		models have the same structure, in our sense, regardless 
		of the identity of the elements of the domains from which 
		they are constructed and regardless of the details of their 
		construction. It therefore seems reasonable to expect that a 
		formal notion of interpretation could be developed from the
 		notion of a model-specification by ignoring or somehow
 		"factoring out" the fated attempt to provide a particular 
		exemplification of the structure intended.[Jubien, 145]

	Jubien states that a formal interpretation be any maximal set of pairwise isomorphic 
fundamental models of a theory. This could explicate the view of an interpreted theory as a 
theory of structure divorced from a specific realization of that structure. Interpreted within 
Sneed's conceptualiztion, where a theory element is an ordered pair, 

		T = 

The I(K) represents the intended interpretations and specific realizations of the structure, 
while K consists of categories of set-theoretic structures. K a category ||X|| is an ordered 
triple of classes ||X|| = <|X|,X,Xc> as explicated above.

	By now it should be clear that category theory has much to contribute to the stated program 
and goals of the structuralist philosophers of mathematics. It is a significant theory that has 
much to offer for both the philosophy and foundations of mathematics, for category theory's 
primary concern is with the explication of mathematical structure. Structuralism, as advocated 
by mathematicians, and category theory have not developed in isolation from each other. Leo 
Curry, in his paper "Nicolas Bourbaki and the Concept of Mathematical Structure", analyzes the 
interaction between Bourbaki's work and the first stages of category theory. But instead of 
discussing the historical roots of category theory, it would be more fruitful to understand the 
philosophical motivations for the development of the theory, and its implications for the 
philosophy of mathematics, as presented by one of it's founders, Saunders Mac Lane.

	Mac Lane correctly states that many interesting questions cannot be settled on the basis 
of the Zermelo-Fraenkel axioms of set theory. 

		Various additional axioms have been proposed, including
 		axioms which insure the existence of some very large
 		cardinal numbers and an axiom of determinacy (for
 		certain games) which in its full form contradicts the
 		axiom of choice. This variety and the undecideability
 		results indicate that set theory is indeterminate in
 		principle: There is no unique and definitive list of
 		axioms for sets; the intuitive idea of a set as a
 		collection can lead to wildly different and mutually
 		inconsistent formulations. On the elementary level,
 		there are options such as ZFC, ZC, ZBQC or intuitionistic 
		set theory; on the higher level, the method of forcing 
		provides many alternative models with divergent properties. 
		The platonic notion that there is somewhere the ideal realm 
		of sets, not yet fully described, is a glorious illusion. [Mac Lane, 385]

MacLane believes that this situation is similar to that of geometry after the proof of consistency 
for non-Euclidean geometry demonstrated that there are many geometries, and not just one. In a 
similar manner, the intuitive idea of a collection leads to different versions of set theory. 
For Mac Lane, this is sufficient reason to consider alternatives to set theory as a foundations
for mathematics. The alternative that he proposes is category theory.

	In "Category Theory and the Foundations of Mathematics", J.L. Bell critically examines the 
thesis that category theory should replace set theory as the "official" foundations for mathematics. 
Bell first presents the formal definitions of category and functor. [Bell, 350]

		A category E consists of two classes, the members of
 		the first of which -- denoted by letters X, Y, ...--
 		are called objects (structures) and the members of the
 		second of which -- denoted by the letters f,g,... --
 		are called arrows (morphisms). Each arrow f is
		assigned an object X as domain and an object Y as
 		codomain, indicated by writing f: X -> Y. If g is any
 		arrow g: Y -> Z with domain Y, the codomain of f, there
		is an arrow fg: X -> Z called the composition of
		f and g. For each object Y there is an arrow idY:Y -> Y
		called the identity arrow of Y. These notions are
 		assumed to satisfy the following identity and
		associativity axioms:

			f C idY = f,  idY C g = g, f(gh) = (fg)h
		for any arrows f: X -> Y, g: Y -> Z, h: Z -> W .

		Given two categories D and E, a functor F from D to E
 		consists of a pair of functions(both denoted by F), one
 		from the class of objects of D to that of E, and the
 		other from the class of arrows of D to that of E, such
 		that
			if f: X -> Y in D, then F(f): F(X) -> F(Y) in E;
			F(idX) = idF(X) 
		and 
			F(fg) = F(f)F(g) for composable arrows f,g of D.


A functor can be thought of as a morphism of categories. Categories and functors are found in 
many seemingly diverse branches of mathematics. Some categories are:

 Set: objects - the sets; arrows - the (set) functions
 Grp: objects - the groups; arrows - group homomorphisms
 Top: objects - topological spaces; arrows - continuous functions
	
An example of a functor is the "forgetful" functor from Grp to Top of Set which assigns to each 
group or topological space its underlying set. This functor has the effect of "forgetting" the 
structure and just maintaining the elements. [Bell,351]. (NOTE:A discussion of the application 
of categories and functors to general systems theory and economics can be found in Donald Katzner's 
book, Analysis without Measurement, "Categories of Systems", pp.131-136.)

	Bell discusses the historical background of category theory before evaluating its philosophical 
significance. It grew out of the rise of abstract algebra in the 1930s when it was recognized 
that the notions of isomorphism, homomorphism and substructure had a universality that was 
independent of their set-theoretical origin. Many of the fundamental notions of abstract algebra 
could be derived from the notion of a structure preserving function, also known as a morphism. 
The relationships between mathematical structures as embodied in the network of morphisms came 
to be seen as more significant than the objects which constitute the elements of the structures. 
It also came to be seen that the notion of identity appropriate for structures is not set-theoretic 
equality but isomorphism.

		From the philosophical point of view, a category came
 		to be thought of as an embodiment of the 'abstract
 		structure' that all its constituent objects exemplify,
 		or even literally to be that 'abstract structure'...
		So category itself came to be viewed as a theory of
 		(mathematical) structure. [Bell,352]

	For Bell, there are two possible sense in which category theory could serve as a foundations 
for mathematics:

	(1) the strong sense: all mathematical concepts, including 	those of the current 
	logico-metatheoretical framework for mathematics, are explicable in category-theoretic 
	terms.
	
	(2) the weaker sense: one only requires category theory to serve as a (possibly superior) 
	substitute for axiomatic set theory in its present foundational role. [Bell,354]

Bell argues that it is implausible that category theory could function as a foundation in the 
strong sense, because even set theory doesn't serve this function. This is due to the fact that 
set theory is extensional, and the combinatorial aspects of mathematics, which is concerned with 
the finitely presented properties of the inscriptions of the formal language is intensional. 
This branch deals with objects such as proofs and constructions whose actual presentation is 
crucial.[Bell, 354]

		So if category theory is to furnish a foundation for
 		mathematics in the stronger sense, it must provide
 		convincing accounts of both of these aspects. But a
 		category is defined to be a class of a certain kind,
 		and classes are extensional, while combinatorial
		objects are generally not. Since there is no reason to
 		suppose that a satisfactory account of intensional
  		objects can be given solely in terms of extensional
 		ones, it seems to me that category theory as currently
 		formulated in terms of classes must fail to provide a
		faithful account of the combinatorial aspect at least.

But as stated above, for similar reasons this 'weakness' is also shared by set theory. Bell goes 
on to say that no foundational scheme at present is capable of providing an explication of both 
combinatorial and semantical objects. So the most that Bell's argument shows is that set theory 
and category theory are equivalently weak, and neither has the advantage over the other in this 
respect.

	Concerning the weaker sense, the question is whether category theory can serve as a 
substitute for axiomatic set theory in its current foundational role,

		One possible means of achieving this would be to
		construct a formal interpretation of some
		'foundationally adequate' first-order version of set
 		theory (e.g. Zermelo-Fraenkel set theory with choice,	ZFC) 
		in a suitable consistent extension T of first order category 
		theory in such a way that the interpretation of any theorem 
		of ZFC is provable in T. Now this in fact has already been achieved. 
		T may be taken to be the theory of elementary toposes (a finite
 		extension of the first order theory of categories),augmented by 
		certain other axioms, notably category-theoretic versions of the 
		axiom of choice and the axiom scheme of replacement. [Bell,354] 

Bell concludes that 

		it would be technically possible to give a purely
 		category-theoretic account of all mathematical notions
 		expressible within axiomatic set theory,and so formally
 		possible for category theory to serve as a foundation
 		for mathematics insofar as axiomatic set theory does.[Bell,355]

For a more detailed formal presentation of a category-theoretic axiom system that can serve as 
an alternative to set theory, the reader is referred to G. Osius,"Categorical Set Theory: A 
Characterization of the Category of Sets", Saunders Mac Lane,Mathematics: Form and Function, 
chapter XI, "Sets, Logic and Categories", or Robert Goldblatt, Topoi, the Categorial Analysis 
of Logic.

	In his recent book, Toposes and Local Set Theory, J.L. Bell continues to elaborate on the 
philosophical significance of category theory by contrasting its approach to set theory, with 
category providing an "element-free" version of mathematics.

		Both set theory and category theory transcends the
 		particularity of mathematical structures. Set theory
 		strips away structure from the ontology of mathematics
 		leaving pluralities of structureless individuals open
 		to the imposition of new structure. Category theory, by
 		contrast, transcends particular structure not by doing
 		away with it, but by taking it as given and generalizing it. 
		It may be said that the success of category theory as a 
		unifying language for mathematics is due to the fact that 
		it, and it alone, gives direct expression to the centrality 
		of form and structure in mathematics. [Bell, 237]

	Category theory is essentially anti-platonistic, for it undermines the received idea that 
the meaning of any mathematical concept is fixed by referring it to the context of a unique 
absolute universe of sets. 

		...it becomes natural, indeed mandatory, to seek for
 		the set concept a formulation that takes account of its
 		underdetermined character, that is, one that does not
 		bind it so tightly to the absolute universe of sets
 		with its rigid hierarchical structure. Category theory
 		furnishes such a formulation through the concept of
 		topos, and its formal counterpart, local set theory.[Bell,238]

Any topos may be regarded as a mathematical domain of discourse or 'world' in which mathematical 
concepts can be interpreted and mathematical constructions performed. Bell develops an analogy
between mathematical frameworks and local coordinate systems of relativity theory. Each serves 
as the appropriate reference frame for fixing the meaning of mathematical or physical concepts 
respectively.

		The topos-theoretical viewpoint suggests that the
 		absolute universe of sets be replaced by a plurality of
 		'toposes of discourse', each of which may be regarded
 		as a possible 'world' in which mathematical activity
 		may (figuratively) take place. The mathematical
		activity that takes place within such 'worlds' is
 		codified within local set theories; it seems appropriate, 
		therefore, to call this codification local mathematics, 
		to contrast it with the absolute (i.e., classical) mathematics 
		associated with the absolute universe of sets. Constructive 
		provability of a mathematical assertion now means that it is invariant,
 		i.e., valid in every local mathematics. Thus, from the standpoint of 
		local mathematics, the use of constructive proof procedures, far from 
		hobbling mathematical activity, has instead the opposite effect
 		of extending the validity of mathematical reasoning to the widest 
		spectrum of contexts. [Bell,245]
	
If category theory accomplishes nothing else, it enables us to conceive familiar mathematical 
concepts from a radically new perspective, and undermines the "essential" ontological commitments 
that we believe follows from "the foundations of mathematical reality". New mathematical objects 
are generated by constructing variations on established mathematical objects or existing formal 
rules and procedures, and reasoning about the consequences of our formal constructions, not by 
discovering objects that exist in a platonic realm inaccessible to empirical perceptions.