Fillion, Mathematical Truth, Ontology and Structures Category Theory

7/29/2019 Fillion, Mathematical Truth, Ontology and Structures Category Theory

1/15

Mathematical Truth, Ontology, and Structures:

The Point of View of Category Theory

Nicolas Fillion

The University of Western Ontario

May 5, 2009

Abstract

This paper will examine a structuralist philosophy of mathematics

based on possibilities op ened by category theory. Specifically, by sketch-

ing general philosophical projects meant to provide foundations for math-

ematics in a global philosophical context, the paper will clarify the justi-

ficatory status of the argument pros and contra this view.

Contents

1 Nature of a Philosophy of Mathematics 2

2 Truth, Knowledge, and Ontology 5

2.1 The Traditional Conception . . . . . . . . . . . . . . . . . . . . . 5

2.2 Structuralism: Moving Away from Substantivalism . . . . . . . . 8

3 The Open Mathematical Universe and Its Ennemies 11

This paper will discuss some philosophies of mathematics bearing the namestructuralism from the point of view of the problem of mathematical truth.I assume that the problem of mathematical truth can rightly be regarded asthe central problem for philosophies of mathematics. Indeed, it seems to bea categorical imperative, for any philosophy of mathematics, to account forthe fact that, in some sense, mathematical statements are true. Otherwise,mathematics would lack the objective character that it is almost universallyacknowledged to have, and would then have to be ranked on a par with strictlycreative artistic disciplines.

Many of the ideas explored in this paper are borrowed from the works ofMarquis, Landry, and Awodey. In fact, this paper is meant to provide further

justification for their views or at least for a common core of philosophical viewsthey share. Their arguments rely on many deep and fascinating conceptual

1


2/15

Term Paper N. Fillion

distinctions, some of which will be presented here. Nonetheless, some deepobjections have been put forward by philosophers such as Hellman and Shapiro.

The contribution of this paper is in attempting to counter those objections atthe outset by determining which questions the proponents of this view maylegitimately be required to answer, and which one may be dismissed.

Claims about legitimacy of questions are very delicate, since there is a con-stant threat of merely be begging the question; for this reason, a huge conceptualpreliminary work is required. A sketch of it will be presented here.1 To do solegitimately, it will be important to take a step back and look at what a philoso-phy of mathematics is supposed to do. To this effect, a general characterizationof what a global philosophy of mathematics is will be presented. Building onthis characterization, I will discuss various structuralist philosophies of math-ematics from the point of view of the problem of mathematical truth which, Isuggest, is a natural anchor.

Using this strategy, I will suggest that not only have I made it clear thatphilosophies of mathematics along the line of Marquis, Landry, and Awodey areto be preferred, but also I will suggest that I have explained why some otherapproaches are essentially misguided. The motivation for undertaking this ex-amination is that, from there on, we will perhaps be justified in discarding somephilosophers annoying insistence on telling them what ontological foundationsmathematics has.

1 Nature of a Philosophy of Mathematics

Philosophy of mathematics involves two intertwined and complementary tasks,which are nonetheless distinct. On the one hand, philosophy of mathematicsconsists in producing global views about the nature of mathematics, hereafter

referred to as (global) philosophies of mathematics. This sort of project is veryaccurately described by Shapiro (2005b, 5-6):

For any field of study X, the main purposes of the philosophy ofXare to interpret X and to illuminate the place of X in the overallintellectual enterprise. The philosopher of mathematics immediatelyencounters sweeping issues, typically concerning all of mathematics.Most of these questions come from general philosophy: matters ofontology, epistemology, and logic. [. . . ] Some problems and issues onthe agenda of contemporary philosophy have remarkably clean for-mulations when applied to mathematics. Examples include mattersof ontology, logic, objectivity, knowledge, and mind.

At this level, we have general, all-encompassing philosophies of mathematics,often driven by interests lying outside of the field of mathematics. Typically,global philosophies of mathematics have a programmatic role, and may be seenas research programmes.

1I am aware that many of the arguments presented here are very close to hand-waving,but it is why I call it a sketch.

PHI-9302: Category Theory 2


3/15


Global philosophies of mathematics are very involved philosophically due totheir highly speculative nature.2 From the global perspective, the task does

not consist in attempting to interpret specific mathematical or logical results,but in identifying features common to the whole of mathematics, i.e., invariantfeatures. Famous global philosophies of mathematics are Platonism, Kantism,Logicism, Naturalism (in many senses), Nominalism, Intuitionism, Fictionalism,Structuralism, Formalism, etc. To be sure, those philosophies have the weaknessof their strength, which is the inattention to specific mathematical and logicaldetails.

On the other hand, philosophy of mathematics deals with local matters. Thepoint here is to examine technical results in mathematics and logic, to interpretspecific theorems, to put forward doubts concerning global philosophies by find-ing the devil in the details, etc. It is easy to see that this task is complementaryto the global one, but that they are still very different. This paper will dealwith global philosophies of mathematics. So, our reference to category theoryas a support of certain structuralist views will not be to specific results. Theargument developed here will be about what a global philosophy of mathematicsshould be like.

Now, let us take another step back and consider how global philosophiesof mathematics are articulated. Typically, philosophies of mathematics try tospecify what the objects of mathematics are, what the basic concepts and ax-ioms of mathematics are, what the correct inference to be made on this basisare, what the sense and reference of fundamental terms is, what the ontology ofmathematics is, how mathematics is practiced, how mathematical knowledge isacquired, what epistemological status mathematical theories have, etc. Follow-ing Marquis (1995), we can see those concerns as corresponding to six differentsenses of the phrase being a foundation of relevant to mathematics.

To begin with, P

is a foundation forQ

has a logical sense. In this case, wetake P to be an (axiomatized) theory for Q expressed in a completely specifiedformal language. This theory consists in a class of axioms (or axiom schemata)together with a specified underlying logic captured by rules of inferences. Pmakes explicit the deductive structure ofQ and, as such, it can be seen as a sys-tematic reconstruction ofQ. The key concepts involved in a logical foundationare definability and provability in P, as well as satisfaction ofP by Q. Notethat, ifQ is meant to be a significantly large part of mathematics, it will notin general be a set and, as such, the model theory will have to be very general. 3

Closely related to the logical sense of foundation is the semantical sense of Pis a foundation for Q. The relation between those two senses of foundations isone of duality:

Thus, at first sight, it seems that the relation of semantical founda-tions is the converse of the relation of logical foundations:

SemFound(P,Q) = LogFound(P,Q)op.

2They live in the kingdom of isms.3As Marquis (1995) and Hatcher (1982) mentions, they are many serious difficulty with

this characterization. Given our purpose, we judge it to be unnecessary to go in the details.



4/15


This should not be surprising, for, since Tarski, a semantics, that isa theory of reference and truth, is based on a satisfaction relation.

However, as in the case of the relation of logical foundations, as soonas we try to be more exact, the situation becomes more delicate.(Marquis, 1995, 429-30)

The complications relative to semantical foundations are just the dual compli-cations that we face with respect to logical foundations. However, if logicalfoundations are mostly insensitive to strictly linguistic issues, semantical foun-dations will naturally be very sensitive to them. This is why semantics is thenatural bridge between logical foundations and other types of foundation.

Thirdly, P is a foundation for Q has a ontological sense. In this case, therelevant questions are typically taken to be concerned with what the entitiesof Q are made of, and what kind of existence of they possess. It is a commonpractice in philosophy to connect ontological foundations to semantical andepistemological issues in a way that will be explained in the second section ofthis paper.

The epistemological sense of P is a foundation for Q consists in linkinga certain body of knowledge P to Q. P is a body of knowledge specificallyselected for possessing certain epistemological properties, such as analyticity,certainty, objectivity, self-evidence, etc. The nature of the link is meant toexplain that, given P possesses a certain epistemological property, Q also does.Three prototypical examples of such a link are reducingQ to P, demonstratingthatQ is a conservative extensionofP, and embeddingQ inP. Often confusedwith the epistemological sense of foundation is the cognitive sense of P is afoundation for Q. Cognitive foundations are usually developed along threelines: (1) a strong cognitive interpretation, stating that one cannot know or

understandQ

without possessingP

(here,P

is a mental faculties andQ

is amathematical system); (2) a weak cognitive interpretation, which may includespedagogical and heuristic foundation; (3) a transcendental interpretation.

Finally, there is a methodological sense of P is a foundation for Q, whichinvestigates the principles and methods applied in a mathematical field in a waythat guarantees that certain methods are legitimately dealt with. As emphasizedby Marquis (1995, 431), the logical foundations are in a sense constructedfrom the methodological foundations. The methodological foundations appearduring the construction of a field, whereas the logical foundations constitute areconstruction of the given field from a specific standpoint. This importanceof this point will be seen in the last section.

Now, let us come back to our characterization of global philosophies of math-ematics. On the basis of those six foundational relations, it becomes possible to

characterize a philosophy of mathematics as follows:

A philosophy of mathematics is an ordering of the above relations.Thus, within a philosophy of mathematics, some of these relationslose their foundational status, since they are presumably shown tofollow from one or a few others, and some are ignored altogether.



5/15


(Marquis, 1995, 431)4

From the point of view of the problem of mathematical truth as traditionallyconceived, philosophies of mathematics are articulated around only four sensesof being a foundation of. So, in the next section, I will present the traditionaltreatment of the problem of mathematical truth and sketch how it orders foun-dational relations. On this basis, I will approach the debate between differentkinds of structuralism, and finally return to a category-theoretically inspiredconception of structuralism.

2 Truth, Knowledge, and Ontology

It is now time to give a formulation of the problem of mathematical truth. Curryhas done it in those words:

The central problem in the philosophy of mathematics is the defini-tion of mathematical truth. If mathematics is to be a science, thenit must consist of propositions concerning a subject matter, whichpropositions are true in so far as they correspond with the facts. Weare concerned with the nature of this subject matter and these facts.(Curry, 1951, 3)

If no account of mathematical truth is provided, then it will appear that math-ematics is purely aesthetic.

The reason for which this problem is central to logical and semantical foun-dations is rather obvious, namely considerations of satisfaction. One of thepoint of interest is to understand how it also relates (or not) to epistemologicaland ontological foundations.5

2.1 The Traditional Conception

In this section, I will discuss a traditional approach to the problem of mathemat-ical truth by characterizing the relation between logical, semantical, ontological,and epistemological foundations underlying it. This view was first articulatedsystematically by Aristotle in the Analytics. It has found many very influentialappropriations by various mathematicians, most prominently by Euclid in theElements. We find variations of this theme through the whole history of phi-losophy of mathematics. More recently, the father of modern logic, Frege, alsoadopted this conception of mathematical knowledge and truth. It must thus beno surprise that it is still very influential.

4Marquis (1995, 421-2) also rightly claims that [.. . ] some of the arguments given either in

favor or against category theory are based on different conceptions of what should be includedin, or what should be meant by, the foundations of mathematics. It is hoped that this willallow us to see precisely where the different parties disagree and, from there, orient the debateappropriately. From the point of view adopted in this paper, it is hoped that it will allow usto identify the source of illegitimate questions.

5Again, I ignore the methodological and cognitive aspects for the sake of characterizingthe traditional approach.



6/15


To begin with, let us say a few things about Aristotles conception of sciences.For Aristotle, logic is not a science, since it has no genus. That logic has no

genus follows from three fundamental features of Aristotelian deductive logic(Corcoran, 2008). Firstly, deduction is cognitively neutral, i.e., we use the sameprocess of deduction whether we know that the premises are true, false, orwhether we ignore which one is the case. Secondly, deduction is topic neutral,i.e., the same process of deduction is used for any genus. Thirdly, deductionis content independent, i.e., it is not necessary to know anything to be trueabout the genus to deduce. Aristotles conception of logic involves no ontologicalcommitment. The function of logic is strictly epistemological, in that it providesknowledge of the validity of arguments, but not of the truth of premises orconclusions. On the other hand, demonstrations have to provide knowledgeof the truth of conclusions. A demonstration is a deduction with premisesknown to be true.6 As a result, the conclusions will also be known to be true.Consequently, the function of demonstrative logic is both epistemological andsemantical, in that it provides knowledge of the truth of conclusions. If weconsider a certain specific science, this task is neither cognitively neutral, topicneutral, nor content independent. This is why, for Aristotle, demonstrative logicis the theory of science. Those considerations constitute the background of theconnection to ontological foundations.

The important point, in the case of mathematics, is that a demonstrationrequires to firstly accept first principles (primitives), which are truth aboutthe genus of mathematics. Without first principles known to be true, thereis no demonstrative mathematical knowledge. Since those first principles areassertions in the sense required by the Aristotelian conception of truth by cor-respondence, there must be a corresponding fact of the mathematical subjectmatter. Those mathematical facts serve as truth-makers. The bridge from

this semantical consideration to an ontological one is made via an argument ofregress ad infinitum. The claim argued for is that, if no truth-makers have on-tological import, then there is no objective knowledge of the genus of a science(a fortiori of mathematics). Take an arbitrary mathematical statement thathappens to be semantically true (except first principles). Is it assertoric, i.e.,is it ontologically true? We can answer yes if it can be demonstrated fromfirst principles. There must be first principles ontologically true, for other-wise the objective truth of no mathematical proposition whatsoever would beobtained. So, there are immediately knowable, apodictic first principles forcingan ontological commitment.

There is a distinction between the Aristotelian and the Fregean approachesin that the formers logic is a formal epistemology, whereas the latter is a formalontology (Corcoran & Scanlan, 1982). However, their conception of mathemat-

ical truth and knowledge are very similar. Freges view of science is in manyrespects closer to the traditional conception of science than to a more modern(say, twentieth-century) conception. Science has an epistemic role to fulfill to

6Note that this requirement is significantly stronger than the claim that a demonstrationis a deduction with true premises, since it involves a bridge between semantical and episte-mological considerations.



7/15


LogFound SemFound EpistFound

OntFound

Figure 1: Traditional Foundations Based on Ontology

produce objective knowledge and it may be presumed to fulfill it in actuality.Given this epistemic function, it is necessary for scientific discourses to have auniquely determined content being in correspondence7 to the realm it refers to:

A science is a system of truths. A thought, once grasped, keeps uspressing us for an answer to the question whether it is true. Wedeclare our recognition of the truth of a thought, or as we may alsosay, our recognition of a truth, by uttering a sentence with assertoricforce. (Frege, 1899-1906, 168)

For Frege, the task of science is to grasp objective contents (thoughts), to dis-cover truths (i.e., to judge of the truth of these thoughts), to systematize them,and finally to express these true thoughts in assertions. Note that the epis-temological position Frege takes forces him to maintain that the semantics ofmathematics commits one to an ontology:

Just as the geographer does not create a sea when he draws boundarylines and says: the part of the oceans surface bounded by these linesI am going to call the Yellow Sea, so too the mathematician cannotreally create anything by his defining. (Frege, 1893, 11)

For Frege, this amounts to give a certain degree of existence to the truth-makersof mathematical statements.

There is a common theme in the way the traditional conception orders thefoundational relations, with respect to the problem of mathematical truth.8

Logical foundations are discussed for the sake of semantical elucidation. Thesemantical foundations are coupled with epistemological foundations requiringobjectivity, and this coupling is taken to be possible only by making both ofthem rely on a solid, unshakable ontological ground (see figure 1). Notice the

absence of cognitive and methodological foundations. Notice, also, that the

7Strictly speaking, Frege does not endorse the correspondence view of truth. But the pointis mostly immaterial here.

8A similar theme is discussed by Wilson (2006), under the name classical gluing. Itwould be worth exploring the analogy in more details, but it lies outside of the scope of thispaper.



8/15


ontological ground is unshakable; as a result, mathematics can only grow byexpanding on its basis, not by radical conceptual transformation. At the end

of this paper, I will suggest that it is important to avoid such consequencesand that, consequently, the objectivity of semantical foundations required byepistemological foundations should come, not from ontological foundations, butfrom weak cognitive foundations and from methodological foundations.

2.2 Structuralism: Moving Away from Substantivalism

The scheme of argument presented above has often been used to support strongontological claims about the existence of mathematical objects. This viewcan accurately be called entity realism. However, the commitment to a realmof mathematical entities has appeared inappropriate to many philosophers ofmathematics. It has seemed inappropriate for two different kinds of reasons:firstly, because it seems unreasonable to say that mathematics is about objects,

and secondly, because it may seem that one must be more parsimonious con-cerning the ontology of a discipline such as mathematics.

Let us consider the first reason, that it seems unreasonable to say thatmathematics is about objects. This worry is found, to take a classical exam-ple, in Benacerrafs claim that numbers cannot be objects, since arithmetic isconcerned with systems that share a common structure, and not with any par-ticular ontology of objects. So, one must think of numbers in terms of sharedstructures. Category theory also suggests that the really important aspect ofmathematics is not the objects that could be taken to constitute structures, butthe structures themselves. There are many excellent detailed expositions of thispoint,9 and so I will not elaborate on this point.

Whether one endorses a structuralist view as a foundation for mathematics,it must be conceded that talks of structure are everywhere in mathematics.Mac Lane (1996, 176) describes the situation as follow:

This notion of structure is clearly an outgrowth of the widespreaduse of the axiomatic method in mathematics. This method was ini-tially deployed primarily to give a rigorous description (called anaxiomatization) of some unique mathematical object. [. . . ] Theseaxioms were categorical, in the sense that they had, up to isomor-phism, only one model [. . . ]. Then one may say that these axiomsdescribe the structure of the system [. . . ].

In mathematical logic, the notion has a broader acceptation. For instance,Johnstone (1987) describes any set equipped with relations and functions cor-responding to the operations and predicates of a language L an L-structure.

Under such a reading, there is not very much in mathematics that cannot beconsidered to be a structure. To adopt a terminology that corresponds closelyto the one in the literature discussed here, we will take the definition given byMac Lane (1996, 174):

9Awodey (E.g., 1996, 2004); Bell (E.g., 1986, 2001); Hatcher (E.g., 1982); Landry & Mar-quis (E.g., 2005); Mac Lane (E.g., 1986); Marquis (E.g., 2006, 2007, 2009).



9/15


[. . . ] a structure is essentially a list of mathematical operations andrelations and their required properties, commonly given as axioms,

and often so formulated as to be properties shared by a number ofpossibly quite different mathematical objects.

So, from this point of view, a structure is whatever system of operations andrelations that satisfies a specified list of axioms (or axiom schemata). Thestudy of structure is not the study of relata, but the study of relations per se.Consequently, an ontological analysis of things simply called mathematicalobjects is likely to miss the real point of mathematical existence. (Mac Lane,1996, 182)

The second reason (or perhaps should we say motivation) underlying struc-turalist philosophies of mathematics is to relax the ontological commitmentto something that would be more readily acceptable. From the point of viewadopted here, structuralism is a philosophical strategy consisting in trying to

provide a satisfying answer to the problem of mathematical truth without com-mitting to entity realism. Various kinds of structuralism will thus be variousclaims as to the degree to which one should make ontological commitment, andto what ontological kinds one should legitimately commit.

There are two basic stances that one can take with respect to ontology inmathematics: substantivalism and formalism (Curry, 1977). To put it simply,substantivalism is the claim that, in one way or another, one will still have tocommit to some sort of ontology of structure; for proponents of subtantivalistviews, this is unavoidable. Formalists, on the other hand, prefer to stay awayfrom such ontological commitment and, accordingly, they try to solve the prob-lem of mathematical truth without any appeal to ontological foundations. I willcall the structuralist views relying on a formalist stance schematism, in referenceto Hilberts famous view that it

is surely obvious that every theory is only a scaffolding or schemaof concepts together with their necessary relations to one another,and that the basic elements can be thought of in any way one likes[. . . ]. Any theory can always be applied to infinitely many systemsof basic elements. One only need to apply a reversible one-one trans-formation and lay it down that the axioms shall be correspondinglythe same for the transformed things. [. . . ] But the circumstanceI mentioned can never be a defect in a theory (it is a tremendousadvantage), and it is in any case unavoidable. (emphasis of mine,Hilbert, 1899, 40)

If we look at the philosophical details, the story is more complex. If pressedto say what exactly the structures are, the structuralist will have two basic op-tions: (1) the structure belongs to all particular systems that satisfy the axioms;(2) the structure is just the properties and relations required by the theory, andnot something that belongs to particular systems. In the first case, we willsay that the subject matter of mathematics is systems that have structures,10,

10I will not elaborate on this view. The reason is that I mainly want to examine views di-



10/15


whereas in the second case, we will say that the subject matter of mathematicsjust is structures.

The claim that mathematics is the study of structured systems (as opposedto the study of structures per se) does not involve considering structures asgenuine objects. However, given the motivation to stay away from consideringterms in axioms as singular terms referring to particular objects one refuses tobe assertory with respect to objects , the structuralist is brought to considerstructures abstractly. This is the algebraic approach, which does not considerplaces in the structure as made up of objects, but as the genuine subject matter.Here, the axioms are not true because they adequately capture some particularsystem. Rather, they are true by definition a key point of this approach. So,schematism accepts the fact that mathematics produces its objects, in somesense. Such an approach is obviously at odds with one that presupposes apreexisting immutable mathematical realm.

Now, for many mathematical purposed, particularly metamathematical ones,there is a need for a theory of abstract structures. This theory may be, totake the most common examples, set theory, modal logic, second-order logic, orcategory theory. Must this theory be considered algebraically (structures beingwhatever satisfies this theory of structures), without assertory force, or mustit be considered assertory? In other words, can the algebraic perspective bemaintained all the way down, or does it stop when we reach metamathematics.If the later option is chosen, then we are adopting an ante rem realism. AsShapiro (2005a, 67) says,

The idea is that places in a structure are bona fide objects, and wecan have quantifiers ranging over them. The structure itself is achunk of reality, and the theory is about it.

This background provides us with assertions about the existence of structures,and thus necessitate an ontology of structure. In this case, the improvementover Platonism is of a very limited nature. It will be unacceptable for a schema-tist, since we still have a background ontology, a reification of structures, ora hypostatization of a theory of relations. This ontological burden to asser-torically characterize structures reintroduce traditional universals and Platonicforms. What motivates philosophers such as Shapiro to make such an ontolog-ical commitment is that they organize their relational foundations in the sameway the traditional conception does. It is taken to be necessary to provide onto-logical foundations in order to say what a structure is, and when two structuresare identical or distinct. Metamathematical claims must not be understood inthe algebraic way, but in the assertory one. So, there is a requirement of amathematical and/or a philosophical theory of structures with an ontological

commitment. The point is only that the ontological commitment should beabout structures, instead of objects.Nonetheless, there has been a worry about the possibility of genuinely pur-

suing a structuralist programme. As the structuralist program goes, there is

minishing or eliminating ontological commitments. However, as explained by Shapiro (2005a)this view requires a heavy ontology of structured systems, or otherwise it would be vacuous.



11/15



OntFound

entities


OntFound

structures

Figure 2: Resemblance ofante rem structuralism and the traditional conception

really no such things as mathematical objects to study. Now, following theworks of Bourbaki, set theory has become the canonical backdrop in which mostmathematical questions were formulated. So, set theory became the canonicaltheory of structure, i.e., the theory specifying what structures are made of, what

structures can be built, what structures are identical, etc. The point is that,in set theory, there is a study of objects, since a set is uniquely determinedby its elements. It thus seems to be impossible to develop a genuinely struc-turalist programme based on a theory of relations such as set theory. This iswhere category theory enters the philosophical scene: understood as a top-downapproach to mathematics (Awodey, 1996), category theory seems to be a frame-work fulfilling the two motivations of structuralist philosophies of mathematics:to talk exclusively about structures, and to avoid ontological commitment to ob-

jects. Moreover, as will be explained below, it provides strong support for theschematist approach according to which no ontological commitment whatsoeveris necessary (nor desirable).

3 The Open Mathematical Universe and Its En-nemies

As we have seen, the classical approach maintains that it is necessary to have anultimate background to provide foundations for mathematics. To put it bluntly,there is no foundations at all if there is not first an ontological foundation. Thetop-down approach of category-theoretic inspiration denies this claim:

As opposed to this one-universe, global foundational view, thecategorical-structural one we advocate is based instead on the ideaof specifying, for a given theorem or theory only the required orrelevant degree of information or structure, the essential features ofa given situation, for the purpose at hand, without assuming some

ultimate knowledge, specification, or determination of the objectsinvolved. (Awodey, 2004, 56)

This philosophy of mathematics acknowledge the central character of the prob-lem of mathematical truth, and considers the logical, semantical, and epistemo-logical dimensions of the problem to be legitimate. This approach refuses to



12/15


transfer the philosophical challenges that we find at these levels to the ontolog-ical level. Passing the buck is not admissible.

Providing foundations for mathematics in terms of a one-universe immutabletheory of relations is only possible and beneficial if the traditional conceptionof mathematical truth and knowledge is accepted. If there is no rigid gluingbetween semantico-epistemological considerations and ontological ones, then theuniverse of relations cannot be uniquely fixed. As a result, it cannot bepresumed that such a universe exists, and it can even less be presumed thatsuch a universe is knowable. As a result, the foundational strategies promotedby the schematist of category-theoretical inspiration will refuse to pursue thequest for an unshakable ground. The big picture is beautifully formulated byLawvere (1969, 1):

Foundations will mean here the study of what is universal in mathe-matics. Thus Foundations in this sense cannot be identified with any

starting-point or justification for mathematics, though partialresults in these directions may be among its fruits. But among theother fruits of Foundations so defined would presumably be guide-lines for passing from one branch of mathematics to another and forgauging to some extent which directions of research are likely to berelevant.

As explained by Lawvere, Bell (1986, 2005), and Mac Lane (1986), there is animminently dialectical dimension to foundational tasks. Attempts to provide aninvariable starting points in terms of an ontology of objects are vain, becausethe supply of mathematical objects studied in mathematics constantly expands.The point is similar with respect to ontologies of structures. Whereas it had beenaccepted that the classical world constituted all the structures there were, it is

now recognized that they only constitute a minuscule part of the universe ofstructures. Moreover, the fact that classical principles once considered essentialto mathematical structures are merely contingent properties is very significant.

The schematist perspective tries to give an answer to the problem of math-ematical truth that will not systematically undermine attempts to solve otherphilosophical problems concerning mathematics. One of the most fascinatingand difficult problems, as explained by Marquis (1998), is the problem of theapplicability of mathematics to itself. The conceptual loops involved in thisapplication are literally impossible to understand in a hierarchical ontologicalframework. Understanding those loops requires one to stay away from quests forunshakable grounds. The universe of mathematics is open, and the enemies ofthe category-theoretic perspective base their critique on an illegitimate appealto a total, closed universe.

However, the initial worries motivating the appeal to ontological foundationshave not been completely addressed yet. One the one hand, there may still bean infinite regress that would prevent us from accounting for the objectivity ofmathematics. This problem is addressed in two ways. Firstly, the application ofmathematics to itself allows one to characterize mathematical invariants (Bell,



13/15


1986). As a result, the only ground there is is provided by invariants. More-over, some of them serve as fix-point, in the sense that some theories are their

own semantics (this is the case for category theory, but set theory has beentreated in this way too). Secondly, it is addressed by making a significantlydifferent logico-semantical analysis of the meaning of mathematical statements.That brings us to the second problem: how can there be a satisfying account ofthe meaning of mathematical statements in a formalist framework? Isnt it thecase that, for a formalist, mathematical statements are strictly meaningless?The answer to this last question is no, and the explanation for the previousone is complex. It is more complex than the story traditionally given, preciselybecause it does not oversimplify the problem.11 The formalist attitude of aschematist is to say that axioms determining structures are true by definition.This move solves the problem of mathematical truth in the simplest possibleway. It also thereby blocks the path to ontological foundations. Still, thereare important foundational resources that can be use to explain the meaningof mathematical statements. The appeal to cognitive considerations, includingpedagogical and heuristic ones, and to methodological considerations seems tobe obviously appropriate. Explaining the connection, however, will be the topicof another paper.

References

Awodey, S. (1996). Structure in Mathematics and Logic: A Categorical Per-spective. Philosophia Mathematica, 4: 20937.

(2004). An Answer to Hellmans Question: Does Category TheoryProvide a Framework for Mathematical Structuralism?. Philosophia Mathe-

matica, 12: 5464.

Bell, J.L. (1986). From Absolute to Local Mathematics. Synthese, 69: 40926.

(2001). Observations on Category Theory. Axiomathes, 12.

(2005). The Development of Categorical Logic. In: Handbook of Philo-sophical Logic, vol. 12. Springer.

Corcoran, J. (2008). Aristotles Demonstrative Logic. History and Philosophyof Logic, 00: 120.

Corcoran, J. & Scanlan, M. (1982). The Contemporary Relevance of An-cient Logical Theory. The Philosophical Quaterly, 32(126): 7686.

Curry, H.B. (1951). Outlines of a Formalist Philosophy of Mathematics. Am-sterdam: North-Holland.

11I think that a formalist philosophy is not the the gospel of despair, as suggested by Bell(1986). True, it does not give us any ontological meat to put on the bones, but this may verywell be the condition to appreciate the cognitive and methodological meat!



14/15


(1977). Foundations of Mathematical Logic. New York: Dover, 2 edn.

Frege, G. (1893). The Basic Laws of Arithmetic: Exposition of the System.Berkeley and Los Angeles: University of California Press. 144 p.

(1899-1906). On Euclidean Geometry. In: Posthumous Writings. Oxford:Basil Blackwell, pp. 1679.

Hatcher, W.S. (1982). The Logical Foundations of Mathematics. New York:Pergamon Press.

Hilbert, D. (1899). Letter to Frege dated 29 December 1899. In: G. Gabriel,H. Hermes, F. Kambartel, C. Thiel, & A. Veraart (Eds.), GottlobFrege: Philosophical and Mathematical Correspondence. Chicago: The Uni-versity of Chicago Press, pp. 3843.

Johnstone, P.T. (1987). Notes on Set Theory and Logic. Cambridge: Cam-bridge University Press.

Landry, E. & Marquis, J.P. (2005). Categories in Context: Historical, Foun-dational, and Philosophical. Philosophia Mathematica, 13: 143.

Lawvere, F.W. (1969). Adjointness in foundations. Dialectica, 23: 28196.

Mac Lane, S. (1986). Mathematics, Form and Function. Dordrecht: Springer-Verlag.

(1996). Structure in Mathematics. Philosophia Mathematica, 4: 17483.

Marquis, J.P. (1995). Category Theory and the Foundations of Mathematics:Philosophical Excavations. Synthese, 103: 421447.

(1998). Epistemological Aspects of the Application of Mathematics toItself. In: D. Anapolitanos, A. Baltas, & S. Tsinorema (Eds.), Phi-losophy and the Many Faces of Science. Lanham: Rowman & Littlefield, pp.18395.

(2006). Categories, Sets and the Nature of Mathematical Entities. In:J. van Bentham, G. Heinzmann, M. Rebuschi, & H. Visser (Eds.), TheAge of Alternative Logics: Assessing Philosophy of Logic and MathematicsToday. Springer, pp. 18192.

(2007). Category Theory. In: E.N. Zalta (Ed.), The Stanford Ency-clopedia of Philosophy. Revised version.

(2009). From a Geometrical Point of View: A Study of the History andPhilosophy of Category Theory. Dordrecht: Springer.

Shapiro, S. (2005a). Categories, Structures, and the Frege-Hilbert Contro-versy: the Status of Meta-Mathematics. Philosophia Mathematica, 13(1):6177.



15/15


(2005b). Philosophy of Mathematics and its Logic: Introduction. In:S. Shapiro (Ed.), The Oxford Handbook of Philosophy of Mathematics and

Logic. Oxford: Oxford University Press, pp. 327.

Wilson, M. (2006). Wandering Significance: An Essay on Conceptual Be-haviour. Oxford University Press.


Documents

Fillion, Mathematical Truth, Ontology and Structures Category Theory