The Genetic Versus the Axiomatic Method: Responding to Feferman 1977

THE REVIEW OF SYMBOLIC LOGIC

Volume 6, Number 1, March 2013

THE GENETIC VERSUS THE AXIOMATIC METHOD: RESPONDINGTO FEFERMAN 1977

ELAINE LANDRY

Department of Philosophy, University of California, Davis

Abstract. Feferman (1977) argues that category theory cannot stand on its own as a structuralistfoundation for mathematics: he claims that, because the notions of operation and collection are bothepistemically and logically prior, we require a background theory of operations and collections.Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of theaxiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is,in reply to Shapiro (2005), we can be structuralists all the way down; we do not have to appeal tosome background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert;I borrow his (Hilbert, 1900a) distinction between the genetic method and the axiomatic method toargue that even if the genetic method requires the notions of operation and collection, the axiomaticmethod does not. Even if the genetic method is in some sense epistemically or logically prior, theaxiomatic method stands alone. Thus, if the claim that category theory can act as a structuralistfoundation for mathematics arises from the organizational use of the axiomatic method, then itdoes not depend on the prior notions of operation or collection, and so we can be structuralistsall the way up.

§1. Introduction. Feferman’s 1977 paper, “Categorical Foundations and Foundationsof Category Theory,” has been appealed to often to argue that category theory cannot standon its own as a structuralist foundation for mathematics (Bell, 1981; Hellman, 2003).Others have argued that a category-theoretic structuralist foundation is still possible byclaiming that Feferman misses his mark (Landry, 2006; Marquis, 2006, 2009; McLarty,2004, 2005). In any case, Feferman (1977), had become, and remains still, the litmus testfor arguments for and against category-theoretic structuralist foundations.

In this paper I will explore the ways in which Feferman’s (1977) arguments have beenused (and misused) in the philosophical literature to argue both for and against category-theoretic structuralism. Having navigated the philosophical landscape, my aim will be todirectly reply to Feferman (1977) by tying together three threads: a) Hilbert’s distinctionbetween the genetic and axiomatic method; b) Awodey’s distinction between top-down,algebraic, and bottom-up, assertory, ways of working; and c) my distinction between theorganizational versus constitutive1 use of the term ‘foundation’. Thus, whereas previouslyI was content use the terms ‘language’2 or ‘framework’,3 my intent here is to retake the

Received: December 18, 2011.1 See Landry (2006, 2011).2 See Landry (1999).3 See Landry (2006).

c© Association for Symbolic Logic, 2012

24 doi:10.1017/S1755020312000135

THE GENETIC VERSUS THE AXIOMATIC METHOD: RESPONDING TO FEFERMAN 1977 25

term ‘foundation’ and argue that one can be a category-theoretic structuralist all theway up. 4

§2. Philosophers’ uses of Feferman 1977. As noted, philosophers of mathematicshave used, and as we will see misused, the arguments and claims found in Feferman (1977)to argue both for and against category-theoretic structuralist foundations. I begin withBell (1981) who, citing Feferman (1977), argues that category theory cannot stand as thefoundation5 for mathematics since the notions of class, operation, and set are epistemicallyprior. He first considers the strong, “logico-metatheoretical,” sense in which we mightconsider category theory as a foundation. He then considers two aspects of this strongsense:

the combinatorial [or proof-theoretic] which is concerned with theformal, finitely presented properties of the inscriptions of the ambientformal language, and the semantical, which is concerned with theinterpretation and truth of the expressions of that language. [p. 353]

While noting that, given the Godel results, “[n]either of these aspects is – at present –reducible to the other” [p. 353], he then continues to consider each aspect in its turn. Inrelation to the combinatorial, he notes that “a category is defined to be a class of a certainkind,6 and classes are extensional, while combinatorial objects are generally not.” [p. 353]

Turning next to consider the semantical aspect, he notes that

. . . the concept of class is epistemically prior to the concept of (classical)logical truth, [since interpretation involves reference to classes orpluralities in an essential way (as the range7 of the variables in theexpression)].” [p. 353]

Thus, category theory cannot serve as a foundation in this semantical sense eitherbecause

. . . it is surely the case that the unstructured notion of class isepistemically prior to any more highly structured notion such ascategory; in order to understand what a category is, you first have toknow what a class is.1 [Footnote 1 reads: For a similar conclusion, seeFeferman (1977). My argument here owes much to this article.] Thisalso applies, mutatis mutandis, to the notion of functor whose explication

4 It is in light, then, of Feferman’s claim that the structuralist view “has been favored particularly byworkers in category theory because of its successes in organizing substantial portions of algebra,topology, and analysis” [p. 149; italics added.] and is “best expressed” [p. 149] by Lawvere(1966), that my aim should be understood as showing that in virtue of this organizational role,we should take category theory as a foundation for mathematical structuralism.

5 Bell, unlike Feferman (1977) and Hellman (2003), seems not to be just considering the question ofstructuralist foundations but is rather concerned with what one might term ‘logical’ foundations.

6 See Bell’s definition of a category: “A category E consists of two classes, the members of the firstof which – denoted by the letters X, Y,. . . are called objects (structures) and the members of thesecond of which – denoted by the letters f,g,. . . - are called arrows. . . ” (Bell, 1981, p. 350). Soreally its not all that surprising that for Bell, the notion of class is presumed!! I will say moreabout this in the last section.

7 The notion of a range does not have to be given in terms of a class; it can be understood moregenerally as a codomain.

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involves grasping the epistemically prior notion of operation. . . It seemsto be that these considerations show that category theory as currentlyconceived is not capable of serving as a foundation for mathematics inthe strong [logic-metatheoretical] sense. [p. 354; italics added.]

He next considers whether category theory can serve as a foundation in the weaker sense,that is, the sense in which we consider whether category theory, as framed by both theETCS and the CCAF axioms, can “serve as a (possibly superior) substitute for axiomaticset theory in its present role” [p. 353]. As regards the theory of elementary toposes, asframed by the ETCS axioms, he concludes,

it would be technically possible to give a purely category-theoreticaccount of all mathematical notions expressible within axiomatic settheory. . . [but] the actual translation is awkward and has (unlike the basiccategory-theoretic notions themselves) a factitious character. . . What thistranslation of set theory really amounts to is the replacement of thenotion of ‘mathematical object as set’ by the notion ‘mathematical objectas pair of arrows (of a certain kind) in a category’ [p. 355].

Next, he notes that

it would seem, however, that a more convincing and naturalformalization of mathematics within category theory would be obtainedif mathematical objects could be construed as categories tout court.(This would also be more in keeping with the structuralist view thatmathematical objects are given as structures and that categories providean embodiment of the idea of structure). [p. 355]

However, when considering Lawvere’s (1966) theory �, framed by the CCAF axioms,for just this task, he claims that

. . . in developing the notions of workaday mathematics within � itseems to be necessary to bring in the notion of set ‘through the backdoor’. . . This, it seems to me, will inevitably make a system like �appear artificial as a ‘foundation’ for mathematics, despite the beautyand naturalness of the category-theoretic notions themselves. [p. 355;italics added.]

Finally, he considers whether a framework for “full” category theory could reasonablybe sought within the theory of arbitrary properties, suggesting, again with a nod toFeferman (1977) and his intensional theory T , that “the notion of a category would inall probability continue to be a derived notion and not a primitive one” [p. 356]. But,contra Feferman (1977), Bell accepts “framework” (if not “foundational”) pluralism andthereby chooses topos theory, as framed by the ETCS axioms, as his preferred “generalframework”

. . . for dealing with mathematical structures. . . while still dependent onset theory as the ultimate source of mathematical entities, [categorytheory as expressed by a ‘model’ of set theory called an (elementary)topos] nonetheless frees mathematics from the particular form imposedon it by having to regard these entities as pluralities of elements. [p. 356;italics added.]


Claiming as a result that

[t]his [relativistic] attitude involves abandoning (or, at least, reservingjudgment about) the idea that mathematical constructions should beviewed as taking place within an ‘absolute’ universe of sets with fixedand predetermined properties. . . . . . the paramount achievement of topostheory is to have identified the basic core of set theory in such a way thatthe set concept becomes manifest in contexts (such as algebraic geome-try of constructive mathematics) where before its presence was at mosttacit. Thus category theory far from being in opposition to set theory,ultimately enables the set concept to achieve a new universality. [p. 358]

Hellman (2003),8 citing both Feferman (1977) and Bell (1986, 1988),9 argues that“[c]ategory theory and topos theory are found wanting both a prior, external, theory ofrelations as well as substantive axioms of mathematical existence” [p. 130]. Neither canact as a foundation, in the sense of being “strongly autonomous,” because, as Bell (1981)notes, they either require a “detour through set theory” [p. 133] or, as Feferman (1977)notes, they “require a prior theory of operation and collection (given in terms of a theory ofrelations)” [p. 135]. Furthermore, if we attempt a reply to Feferman by acknowledging10

that “the notion of function is presupposed, at least informally” [p. 133], in much the sameway that it can be taken as presupposed for various set theories, the result is foundationalpluralism, that is, “. . . we get two theories based on the notion of function making theirrespective foundational claims” [p. 134].11

In any case, according to Hellman, the problem that remains is that any such reply bythe category theorist “relies on an intended interpretation of ‘composition’, as a binaryoperation on functions” [p. 134] which itself is “diametrically at odds” [p. 134] with bothAwodey’s (1996) algebraic account of category theory and the structuralism that “actuallyunderlies” [p. 134] Feferman (1977).12 Hellman’s conclusion, then, to the question oftaking category theory as an autonomous algebraic, or structuralist, foundation is that yetstill some notions must be taken as prior:

. . . a structural understanding of category theory actually underliesFeferman’s critique; somehow we need to make sense of talk ofstructures satisfying the axioms of category theory, i.e., being categories

8 Hellman is considering the question of what constitutes a structuralist foundation but claims thatthis “. . . is naturally viewed in the context of Mac Lane’s repeated claim that category theoryprovides an autonomous foundation of mathematics as an alternative to set theory. . . if categorytheory is not autonomous but rather must be seen as developed within set theory, then Awodey’ssuggestion could not be realized. . . ” [pp. 129–130]

9 Hellman’s appeal to Bell (1986, 1988) is misplaced, for unlike Bell (1981), in these papers Belldoes take topos theory as a foundational framework, though he does maintain the pluralism of hisearlier work, taking each topos as a “‘possible world’. . . codified within local set theories” (Bell,1986, p. 245).

10 It is Hellman, not Feferman, who claims that the notion of function is presupposed. Hellmanclaims that “There is frank acknowledgement that the notion of function is presupposed, atleast informally, in formulating category theory; indeed, category theory has been described asinvestigating the behavior of functions under the operation of composition” [p. 133]. It is unclear,however, who Hellman has in mind here.

11 This, however, would be no reply to Feferman whose aim is to reject pluralism.12 See Feferman (1977) footnote #4.

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or topoi, in a general sense, and it is at this level that an appeal to‘collection’ and ‘operation’ in some form seems unavoidable. Indeed,one can subsume both these notions under a logic or a theory ofrelations (with collections as unary relations); that is what is missingfrom category and topos theory both as first-order theories and, crucially,as informal mathematics. . . [p. 135]

Issues of foundational autonomy aside, Hellman next notes that, if we try to construethe axioms purely structurally, or schematically as Hilbertian “defining conditions,” as inAwodey (1996),13 we are left to philosophically face the “home address problem” [p. 136]or worse still, at a metalogical level, are left in the position of the ‘‘if-then-ist” [p. 138].In the first instance, then, even if we could somehow side-step Feferman’s “priorityobjections,” we are still left to face “questions of existence”:

[m]oving beyond the question of autonomy proper, we turn now toan equally important, intimately related, problem that of mathematicalexistence. . . . (We might dub this the problem of the ‘home address’:where do categories come from and where do they live?) . . . these[the category axioms] are to be read ‘structurally’, that is, asdefining conditions, not [as is the case with set theory] as absoluteassertions or (putative) truths based on established meanings of primitiveterms. . . ([this distinction] lays at the root of the famous debate incorrespondence between Frege and Hilbert). . . Or, more formally, whataxioms govern the existence of categories or topoi? [p. 137]

Now Hellman, in a footnote, does consider whether the CCAF axioms can be taken asproviding an answer, but notes, and, in so doing, references Bell (1981),14 that

[a] detailed assessment of Lawvere’s axioms would examine this issueof credibility, which in turn rests on some prior, not merely structural,understanding of the primitives and intended interpretation, which doesbuild in the notion of ‘category’ itself, and thereby presupposes thenotion of ‘collection’ as well as ‘functor’. (Cf. Bell, 1981)15 [footnote#8, p. 137]

Turning next to consider the Hilbertian “metalogical” strategy of letting “[relative]consistency imply existence,” Hellman notes that

[a]nything proved from such ‘axioms’ only establishes the conclusionas holding ‘in any (or any possible) structure satisfying the axioms’,which is a generalized conditional assertion. . . We are then in the old ‘if-then-ist’ predicament that plagues deductivism; what we thoughtwe were establishing as determinate truths turn out to be merely

13 Awodey’s characterization is given in term of the EM axioms.14 And further, he notes that Bell’s (1981, 1986, 1988) “many topos” view requires “integrating

category theory in a modal structural framework, which would supply both the wanted priortheory of relations and substantive existence postulates.” [p. 130]

15 What Bell (1981) claimed was that the notions of ‘class’, ‘operation’, and ‘set’ seemed to bepresupposed; it was Feferman who claimed that the notions of ‘collection’ and ‘operation’ arepresupposed. Neither, however, made mention of the notion of ‘functor’ as presupposed.


hypothetical, dependent on the mathematical existence of the very struc-ture we thougth we were investigating, and threatening to strip mathe-matics of any disctinctive content (cf. Quine, (1976), 82 ff.) [p. 138]

Finally, Hellman considers the status of Bell’s (1986, 1988) foundational, yet pluralistic,“many topoi view,” in which topoi are taken as frameworks in the sense of “possibleworlds” for mathematical activity (See Bell, 1986, p. 245). His conclusion, which againis claimed to build upon the arguments of Feferman (1977), is that

[c]ategory theory does offer an interesting structuralist perspective onmost mathematics as we know it. But it needs to be supplementedand set within a yet wider framework. As explained above, categorytheory – with its non-assertory, algebraico-structural axioms – dependson prior notions of structure (collection and relations) and satisfactionby structures to make sense of the very notions of ‘category’ and‘topos’. . . Although this usually manifests itself through an introductoryappeal to an unspecified ambient domain of sets, that is not necessary.Instead, one can develop a modalized theory of large domainsrelying on the more neutral and general or schematic notions of‘part/whole’ and plural quantification, as described above. . . [which]while accommodating the structures defined by both set theory and topostheory. . . removes any dependence on actual existence [thus avoidingthe home address problem], as it is only possibilities that matter forpure mathematics; and it even reconstructs a rich theory of relations[which can be used to frame our talk of functions, or operations andcollections], via the language and logic of plurals. . . [pp. 154–155]

McLarty (2004, 2005) and Awodey (2004) offer sustained and detailed replies toHellman (2003) and, in so doing, McLarty further offers a direct reply to Feferman(1977).16 McLarty (2004) notes that while Hellman was right to claim that Mac Laneused a set-theoretic foundation, he missed that this was, in fact, a set theory founded on theETCS axioms. Next McLarty offers detailed replies to Hellman’s claims that a set theory sofounded is: i) too weak, ii) too complicated, iii) presupposes the notion of function,17 iv)cannot address the question of mathematical existence, and v) requires an external theory ofrelations. I will only here consider those replies that bear weight on the claims of Feferman(1977). Of iii) he admits that the ETCS axioms were motivated by the informal ideas of setand function, and the EM axioms were motivated by a yet different informal idea of func-tion, but, he argues, motivation is not presumption [pp. 38–41]. Of iv) McLarty notes that

each proposed categorical foundation is a proposed answer. Categoricalset theory says: the sets and functions posited by the axiomsexist. . . SDG, taken as a foundation, says: the smooth spaces and maps

16 McLarty (2005) is an exception; he does directly respond to Feferman (1977). But, as we willsee, his “Feferman” is looked at though the lenses of Hellman (2003).

17 Here it is important to remind ourselves of Feferman’s claim: “. . . let me repeat that I am not argu-ing for accepting current set-theoretical foundations of mathematics” (Feferman, 1977, p. 154).So McLarty is mistaken when he states that “Hellman follows Feferman (1977) saying categoricalfoundations presuppose a theory of sets and function” [p. 41]. Feferman’s claim is that categoricalfoundations require the notions of ‘collection’ and ‘operation’ and it is only the topos axioms,specifically, the replacement scheme, that is claimed to presuppose set-theoretical notions.

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posited by the axioms exist. . . So Hellman’s claim the ‘categorytheory. . . lacks substantial axioms for existence’ is a misunderstanding.[p. 42]

The EM axioms, according to McLarty, are thus the only axioms that are purelyalgebraic in Awodey’s (1996) sense; “these axioms make no existence claims and theyare constantly used in many different interpretations. . . but no one proposes those asa foundation” [p. 42]. Of v) McLarty notes that “Feferman has never yet given thisgeneral theory [of operation and collection] and this probably explains why “Mac Lanenever responded directly to Feferman’s critique (Hellman, (2003), 133)” [p. 45]. Theseclaims, however, are odd for two reasons; Mac Lane clearly did offer “in correspondence”rejoinders to Feferman, in fact, he is noted as having done so in Feferman (1977), andFeferman’s theory T is clearly offered there as the theory of operations and collections.18

In any case, in so far as Hellman’s claim that category theory requires an external theoryof relations is based on an algebraic reading of the category axioms, and given that, asexplained above, the ETCS and CCAF axioms are intended as assertory, they, accordingthe McLarty, require no such external theory.

McLarty (2005) continues this argument thread with the aim of showing what wecan learn “from Hellman’s questions, and from the ones he invokes from SolomonFeferman’s. . . (1977)” [p. 45]. Here he claims that

[t]he deepest point of Feferman’s paper as it seems to me is to showthat we want much more from a foundation than formal adequacyand practical efficacy. In his [Feferman’s] metaphor, to accept a givenfoundation merely because it is formally adequate and practicallyproductive is like ‘not needing to hear, once one has learned to composemusic’ (Feferman (1977), 153). We want to hear the music. [p. 45]

Offering, then, a direct reply to Feferman’s critique he says:

. . . there had been no explicit reply to Feferman until now, and itis worth giving it because his position is more subtle than manypeople realize. . . [Feferman’s position] can be summed up in threepoints: Category theory cannot be a logical foundation; it is alsopsychologically derivative; and it is unmusical. What are logically andpsychologically prior, he says, are notions of operation and collection.He says categorical notions of arrows cannot be logically prior to set-theoretic account of objects. [p. 45]19

Where things get confusing, however, is that while in this 2005 paper McLarty doesconcede that Feferman (1977) offers “a non-extensional theory of his own as progress

18 As we will see, McLarty (2005), however, does recognize that Feferman (1977) offers such atheory.

19 McLarty next gives the following Feferman quote “My use of ‘logical priority’ refers not torelative strength of formal theories but to order of definition of concepts, in cases where certainof these must be defined before others. For example, the concept of vector space is logically priorto that of linear transformation. (Feferman (1977), 152).” McLarty then proceeds to show howfor the concept of a vector space this was not the case both historically and in modern algebraic,and for current category-theoretic presentations in terms of Abelian categories. Thus, categorytheorists “hear the music just fine.”


towards a correct non-platonist foundation for category” [p. 49] he still makes odd claimslike “For him [Feferman] the music lies, almost, in the structure of the iterative hierarchyof ZF sets, more in proof theory, and most of all in philosophic question of realism versusconstructivism, which he wants to build into foundations” [p. 48]. In any case, his reply toFeferman is as follows:

[o]bviously I agree with Feferman that foundations of mathematicsshould lie in a general theory of operations and collections, only I saythe current best general theory of those calls them arrows and objects. Itis category theory. . . The theoretical unity and practical power of modernstructural methods make them, to my ear, actually finer music than prooftheory or realism versus constructivism. [p. 49]

But here one must pause; the general theory of objects and arrows is typically taken tobe that framed by the EM axioms, but in his (McLarty, 2004) McLarty claimed that theseare algebraic and as such that “no one proposes those as a foundation” [p. 42], thus he nowseems to open himself to the “existence” and “if-then-ist” objections of Hellman. However,he does provide “statistical information” showing that, in actual mathematical practice,the EM axioms are taken sometimes as assertory (41/50 publications) and sometimes asalgebraic (7/50 publications).20 Furthering his reply to Feferman, he then turns to arguethat the CCAF axioms do not presuppose any set-theoretic notions,21 noting that

[t]he key point to grasp here is precisely that categorical foundationsfor category theory are not set-theoretic foundation for category theory.When we axiomatized a metacategory of categories by the axioms ofCCAF, the categories are not ‘anything satisfying the algebraic axiomsof category theory’ [the EM axioms]. They are anything whose existencefollows from the CCAF axioms. They are precisely not sets satisfyingthe [EM] axioms. They are categories as described by Lawvere’s CCAFaxioms. [p. 52]

Thus, McLarty’s structuralism (and so his foundationalism) is held as differing fromAwodey’s in the following pluralistic sense:

[o]n my view categorical foundations are not structuralist in thissense [in Awodey’s sense]. Each [foundation] posits a specificcategory. . . They are structuralist in this precise sense: They attributeonly structural properties to their objects, that is only isomorphism-invariant properties. [p. 53]

While Awodey does not make direct use of, or reference to, Feferman (1977), sincemuch has been made, (and indeed, will continue to be made in this paper) of his [1996]and [2004] algebraic view, I will provide a brief outline of his position. Interestingly, formy purposes, Awodey’s (1996) begins with a Dieudonne quote, which situates category

20 Again, things are a bit unclear here. McLarty gives his “statistical information” for researchinvolving the terms ‘category’ and ‘functor’ (see, p. 50), which are typically framed by the CCAFaxioms, but then continues his discussion by making reference to the EM axioms (see, p. 51).

21 Note, however, that Feferman never claimed that theories framed by the CCAF axioms requireset-theoretic notions. His claim was that theories framed by the topos axioms, specifically thoseincluding the replacement scheme, require them (see endnote #17).

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theory in the Hilbertian, axiomatic, tradition. Here Dieudonne claims that “this notion[of structure] has been superseded by that of category and functor, which includes it ina more general and convenient form” [p. 209]. In this light, Awodey notes that one oughtto distinguish “philosophical structuralism” (here he notes the works of Resnik, Shapiro,Hellman, Parsons, and Quine) from “mathematical structuralism” (here he notes the worksof Mac Lane, Stein, and Corry). Of the later he claims that,

[f]rom Dedekind, through Noether, and to the work of Eilenberg andMac Lane, the fact has clearly emerged that mathematical structureis determined by a system of objects and their mappings, rather thanby any specific features of mathematical object viewed in isolation.[pp. 209–210]

His aim, then, is to inform mathematical structuralism by proceeding, “not from modeltheory or from scratch,” [but by] “drawing on . . . the mathematical theory of categories”[p. 210]. Note, however, that his purpose is

not to discuss categorical foundations of mathematics, or to present acomprehensive structuralist philosophy of mathematics. . . but to elabo-rate a notion of mathematical structure from a categorical perspective,so that discussions of other issues may proceed directly. [p. 210]

Next detailing problems with the model-theoretic, Bourbaki, notion of structure,Awodey notes that

[a] category provides a way of characterizing and describingmathematical structure of a given kind, namely in terms of preservationthereof by mappings between mathematical objects bearing the structurein question. For example, topological spaces and continuous mappingsbetween them form a category, which we call Top. . . and Set consists ofsets and functions between them. [p. 212]

After proceeding to define a category,22 by use of the EM axioms, in terms of objects andmorphisms, he claims that “[a] category is anything satisfying these axioms. The objectsneed not have ‘elements’, nor need the morphisms be ‘functions’, although this is the casein some motivating examples” [213]. Again, witnessing McLarty’s distinction betweenmotivation and presumption, Awodey notes that

[g]enerally, suppose we have somehow specified a particular kind ofstructure in terms of objects and morphisms. . . Then that categorycharacterizes that kind of mathematical structure, independently of theinitial means of its specification. For example, the topology of a givenspace is determined by its continuous mapping to and from other spaces,regardless of whether it was initially specified in terms of open sets, limitpoint, a closure operator, or whatever. The category Top thus serves thepurpose of characterizing the notion of ‘topological structure’. [p. 213;italics added.]

22 Awodey’s definition is as follows: A category by definition thus consists of objects A, B, C. . . andmorphisms f, g, h, such that. . . ” [p. 212]. Note, especially, that there is no mention of a class ofobjects, a collection of objects or, indeed, a set of objects.


Resisting the pull of a “zealous structuralist” [p. 214], Awodey next notes that thecategory of abstract sets, as framed by the ETCS axioms, “provides a structural settingin which to conduct virtually any piece of mathematical reasoning that can be modeled inconventional, axiomatic set theory. . . ” [p. 215]. In any case, for my purposes, the crucialpoint is this:

. . . let no misunderstanding arise from the fact that the model-theoretic,Bourbaki notion of structure serves as motivation and provides a sourceof examples. The claim being made here is not that that notion hasbeen pointlessly reformulated in other terms, but that a categorical onedifferent from it has proven more fruitful in the structural approachto mathematics, and that it also serves philosophical purposes better.[p. 215; italics added.]

And to the issue of foundations, he further concludes

the axioms for a bivalent topos with choice. . . provides a structuralaxiomatization of the category of sets and so a structural ‘foundationfor mathematics’, to the extent that one takes set theory as such. I do notthink too much should be made of this. For one thing, the structuralistperspective is at odds with the idea that all mathematical objects existin a single, comprehensive universe of sets. . . Moreover, the very idea of‘foundations of mathematics’ becomes less significant from a categoricalperspective than, say, organizing and unifying the language and methodsof mathematics. . . [p. 235; italics added.]

Continuing in this vein, Awodey (2004) sets out to directly respond to Hellman (2003).That is, having taken his [1996] to have shown that category theory provides the “currentlydominant framework” for mathematical structuralism, he now turns to consider whetherit, likewise, in response to Hellman’s (2003), provides a framework for philosophicalstructuralism. Of Hellman’s question of whether category theory provides an autonomousfoundation, Awodey notes that

this misses the point of my proposal, which is not to prefer that or thatfoundation, but to use category theory to avoid the whole business of‘foundations’. . . Indeed, the view of ‘doing mathematics categorically’involves a different point of view from the customary foundational one,as I shall try to explain in this note. [p. 55]

Thus, Hellman’s “home address” problem, in so far as it’s a ‘foundational’ problem,is dismissed as a misunderstanding of what categorical structuralism amounts to. WhatAwodey then notes about his view is that it

emphasizes form over content, descriptions over constructions. . .characterization of essential properties over constitution of objectshaving those properties. . . the ‘foundational perspective’, to which weare proposing an alternative, is based on the idea of building up specific‘mathematical objects’ within a particular ‘foundational system’. . . .[p. 55; italics added.]

The category-theoretic structural view, in contrast,

is based instead on the ideal of specifying. . . the essential features of agiven situation, for the purpose at hand, without assuming some ultimate

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knowledge, specification, or determination of the ‘objects’ involved. Thelaws, rules, and axioms involved in a particular piece of reasoning, or afield of mathematics, may vary from one to the next, or even from onemathematician or epoch to another. . . The methods of reasoning . . . arethemselves ‘local’ or relative. [p. 56]

To strengthen this point, he then tries a “different tack” by making use of the distinctionbetween “bottom-up” and “top-down” ways of working; he notes that the foundationalist,who works, bottom-up “. . . must ‘construct’ the terms involved. . . and then prove that thespecific entities so constructed do indeed have the stated property” [p. 56; italics added.].The structuralist, on the other hand, works top-down from the structure, which specifiesthe relevant features. His statements are not then universally quantified statements “over aspecific range of specific ‘objects’, presumed or constructed, but somehow fixed and given”[p. 57], rather they are schematic statements “about a structure . . . which can have variousinstances.” [p. 57]

Turning next to consider Hellman’s charge of the algebraic structuralist positionreducing to “if-then-ism,” Awodey notes that

the essential difference between the position being sketched here andold-fashioned, relational structuralism [of, say Russell, in which arelation has to be a relation on some things] is the idea of a top-downdescription, which presupposes no bottom-up hierarchy of things. [p. 61;italics added.]

Now Awodey does not claim that category theory is the only or the best way to talk aboutstructures, though he does say “I know of no better one” [p. 61], explaining that “the reasonfor this broad applicability has a lot to do precisely with their effectiveness at specifyingand manipulating structures” [p. 61]. In detailing why this is so he notes that

such notions as relation, connection, property and operation areall subsumed under the primitive notion of a morphism. . . Needrelations, use products and monomorphisms; operations? morphismson products;. . . connections between structures? use functors betweencategories; connections among connections? categories of functors; andso on it goes.” [p. 61]

Again, in response to Hellman’s “home address” problem, he notes:

. . . the idea [behind Hellman’s ‘home address’ problem] that one is goingup in a hierarchy, and that this requires stronger and stronger collectionprinciples and existence assumptions rests on the ‘foundationalist’conception that the ‘objects’ involved are fixed and determinate. Form acategorical perspective, one is rather ‘going down’, by specifying moreof the ambient structure to be taken into account. . . [p. 62]

And, finally, noting the schematic character of the axioms, that range over variabledomains, he notes the difference between taking category theory as a language forstructuralism, as opposed to as a foundation:

[h]ow, then, do we make precise the notions of schematic statementsabout structures that have different instances? Simply by using theusual language and methods of category theory; they automatically treatmathematical objects as ‘structures’, and categorical statements about


them are inherently ‘schematic’, in the required sense. This is whatmakes category theory a good language for structuralism. It is also whatgives it an essentially different perspective from the foundational one.[p. 62]

Citing Hellman (2003), and building on the work of McLarty (2004) and Awodey (2004),Shapiro (2005) claims, by appeal to analogies drawn from the Frege–Hilbert debate, thateven if the various category-theoretic axioms, either the EM, ETCS, or CCAF axioms,can be taken foundationally as algebraic, or as schematic in Hilbert’s sense, we stillneed a Fregean assertory background metamathematical theory (set, modal or structure,or categorytheoretic) in which we provide our meta-mathematical analyses of logicalstructure that is used to assert or express the acceptability (consistency, satisfiability,coherence, etc.) of the category-theoretic axiom systems themselves.

As a result, Shapiro (2005) claims that category theory cannot act as a foundation fora purely algebraic structuralist account of mathematics; because it requires an assertorymeta-mathematical theory, one cannot be a pure algebraic structuralist, or a structuralistall the way down. That is,

. . . standard set theory, the category-based set theory suggested byMcLarty, my own structure theory, or Hellman’s model set theory arethemselves assertory theories of sets, structures, the possible existenceof systems, etc. As such, each of them is not just another mathematicaltheory, providing implicit definition of some structures, or isomorphismtypes. The reason for this is that set theory, structure theory, etc., has afoundational role to pay concerning the coherence of definitions. Andthis last is an assertory matter. [p. 74]

According to Shapiro, the only other option, as presented by Awodey (2004) is to“kick away the foundational ladder altogether, and take the meta-mathematical set theory,structure theory, or whatever, itself to be an algebraic theory” (Shapiro, 2005, p. 74). Thisoption, however, is represented by Shapiro as a way not to be looked into because itsupposedly has the unwanted consequence that mathematical logic “is similarly liberatedfrom theories. . . our theorist can hold. . . that satisfiability, consistency, or coherence impliesexistence, but she cannot maintain that any of these notions are mathematical matters”[p. 75]. The alleged conclusion being that on such a view meta-mathematical analyses oflogical concepts are, as they were for Hilbert, turned into nonmathematical, or, even worse,philosophical (read Kantian) ones (see Shapiro, 2005, pp. 74–75).

In response, in Landry (2011), I argued that Shapiro (2005), by conflating the rolesof the EM, ETCS, and CCAF axioms, and by confusing the meanings of the categorytheorists’ use of the term ‘foundation’, was mistaken in his claim that one cannot usecategory theory as a foundation for a purely algebraic structuralist account of mathematics.Specifically, I showed that in rationally reconstructing Hilbert’s organizational use ofthe axiomatic method, we can construct a category-theoretic, purely algebraic, version ofmathematical structuralism. Thus, I concluded that we can be mathematical structuralistsall the way down.

While my [2011] paper did not consider responses to Feferman23 (1977), I wouldlike, however, to bring to the fore some distinctions that I made there that may help the

23 See Landry (2006).

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reader to understand my present position. In attempting to pull apart the confusions andconflations between the various category-theoretic levels,24 that is, between how we areto understand the foundational roles of the EM, ETCS, and CCAF axioms, I appealedto three aspects of Hilbert’s foundational programme, and to the Hilbertian distinctionbetween conceptual analyses of mathematical structure and contentual analyses ofmeta-mathematical structure, to note the following:

(1) When conceptually analysing the abstract mathematical structureof the concepts of any given branch of mathematics, we have the EMaxioms as implicitly defining the abstract concept of a category; here ourtask is to present an axiom system qua an abstract conceptual schemafor the facts of any given interpretation (which provides a domain ofobjects, i.e., ‘objects’ and ‘morphisms’, for these concepts) in such a wayas to organize what can be mathematically asserted about such objectsas abstract, cat-structured, concepts, i.e., what can be asserted in termsof anything that satisfies the EM axioms

i. When conceptually analysing the concepts of the branches ofmathematics that are themselves organized set-theoretically, the categorytheorist can take the ETCS axioms as a mathematical conceptual schemefor organizing, in category-theoretic terms, what we say about themathematical or logical structure of these set-structured objects as, cat-structured, concepts, i.e., what can be asserted in terms of anything thatsatisfies the ETCS axioms.

ii. When conceptually analysing the concept of a category itself, thecategory theorist can take the CCAF axioms as both a mathematical anda meta-mathematical conceptual scheme; respectively, in the sense thatthey organize what we say about the concept category itself, and in thesense that are about any object that is a category, including the categorySet, as organized by the ETCS axioms, i.e., what can be asserted in termsof anything that satisfies the CCAF axioms.

(2) When logically analysing axioms or axiom systems themselves,either at the abstract (EM), set-structured (ETCS) or cat-structured(CCAF) level, the category theorist can make use of the resources ofthe many categorical logics to organize what we say about those logicalconcepts, like completeness, independence, consistency, coherence,satisfiability, etc., that are used as “acceptability criteria” for axiomsor axioms systems themselves. Here our task is, again, to give anaccount of those axioms that are necessary and prove, for example, theconsistency of these axioms relative to some stronger theory, and therebyestablish the existence of those objects that satisfy the implicitly definedconcepts.

24 See, for example, Feferman (1977) where he says “It is evidently begging the question to treatcollections (and operations between them) as a category, which is supposed to be one of theobjects of the universe of the theory to be formulated” [p. 150]. There appears to be someconfusion and conflation of levels of the category axioms here: the EM axioms may be takenas talking about collection and operation in terms of ‘objects’ and ‘morphisms’, but it is only theCCAF axioms that take categories, themselves, as ‘objects’, with functors as ‘morphisms’.


Having undertaken both (1) and (2) for the branches of mathematics,category theory included, we thereby establish, via the axiomaticmethod, a conceptual foundation for mathematics, where ‘foundation’is taken in the organizational sense of the term. (Landry, 2011,pp. 448–449)

In response to any Shapiro-like claim that we are still left to face the charge that, atthe meta-mathematical level, a contentual analysis of logical structure is required and so,like Hilbert’s appeal to Kantian intuition to provide the syntactic and finitistic content ofproof-theoretic structure, must “turn to philosophy,” I offered the following reply:

[o]n the category-theoretic pure algebraic structuralist view, a meta-mathematical analysis of the content (semantic, syntactic, or finitistic) ofthe logical structure of the concepts of mathematical logic/mathematicalreasoning, including statements of consistency, coherence, etc., does notrequire a non-mathematical, “philosophical”, analysis. . . [For example]we can use the ETCS axioms to provide a meta-mathematicalanalysis of the semantic content of the various model-theoreticconcepts of satisfiability, interpretation, truth, relative consistency, inso far as we consider these concepts themselves as organized set-theoretically. . . And too, in line with Hilbert’s meta-mathematical, proof-theoretic, analysis of syntactic content, category theory allows us todescribe deductive systems in terms of categories, so we can employcategorical methods for proof-theoretic purposes. For example, onecan analyse the proof-theoretic structure of any system by using Ded,the category of deductive systems, which takes ‘objects’ as formulas,‘morphisms’ as proofs or deductions, and operations on morphismsas rules of inference (See Lambek and Scott 1986). . . . Finally, inline with Hilbert’s preference for finitistic reasoning, we can use theinternal logic of a topos to meta-mathematically analyse the finitisticcontent of the various aspects of constructive mathematics, includingconstructive set theory, the concepts of recursiveness, independence,and models of higher-order type theories generally. (Landry, 2011,pp. 450–451)

Thus, in reply to Shapiro (2005), I argued that category theory has as much to sayabout a algebraic consideration of meta-mathematical analyses of logical structure as itdoes about an algebraic consideration of mathematical analyses of mathematical structure,without either requiring an assertory mathematical or meta-mathematical backgroundtheory as a “foundation,” or turning meta-mathematical analyses of logical concepts into“philosophical” ones. This is the sense in which I showed that we can be category-theoreticpure algebraic structuralist all the way down.

My present aim is to again use a rational reconstruction of Hilbert to argue, now againstFeferman (1977), that one can be a category-theoretical pure algebraic structuralist all theway up. It is to this task that I now turn.

§3. Hilbert’s distinction between the genetic and axiomatic method. Havingreviewed philosophers’ reactions, and their reactions to reactions, I now come to considerthe philosophical purpose of this paper, viz., to revisit Feferman (1977) (and Bell, 1981)

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with the aim of investigating whether, having responded to Hellman25 and Shapiro,26

we are still left to face the claim that some unstructured notions (collection, operation;class, function, set) are prior (logically, psychologically; epistemically) and so we requirea background theory (intensional or extensional; constructive, modal, set, or structuretheoretic) that allows us to talk about (implicitly define; assert) what we say about thestructure of, or between, categories themselves. My aim here will be to directly reply toFeferman (1977) by tying together three threads: Hilbert’s distinction between the geneticand axiomatic method; Awodey’s distinction between top-down27 and bottom-up waysof working; and my distinction between the organizational versus constitutive use of theterm foundation.28 Thus, whereas previously I was content use the terms language29 orframework,30 my aim here, in addition to responding to Feferman (1977), is to retake theterm ‘foundation’ on behalf of the category-theoretic mathematical structuralist.

To this end, I again turn to Hilbert; I borrow his (Hilbert, 1900a) distinction betweenthe genetic method and the axiomatic method to argue that even if the genetic methodrequires the unstructured notions of operation and collection (or, as Bell, 1981, claims, theset-structured notions of function, class, or set) the axiomatic method does not. Borrowingtoo Hilbert’s (1900b) axiomatic account of foundations, I will then argue that if the claimthat category theory can act as a purely algebraic structuralist foundation for mathematicsarises from the organizational use of the axiomatic method, then it does not depend on theconstitutively prior notions of operation or collection (or function, class, or set). So just asI used Hilbert to argue against Shapiro (2005) by showing that one can be a pure algebraicstructuralist all the way down, I now intend to use Hilbert to argue against Feferman (1997)(and Bell, 1981) by showing that one can be a pure algebraic structuralist all the way up.

I will argue that even if the notions of operation and collection (or function and classor set) are in some, perhaps psychological, sense prior, they are prior in their beingappealed to by the genetic method, not by the axiomatic method. And insofar as thecategory-theoretic axioms themselves are taken as top-down implicit definitions, there isno sense of these, again perhaps psychologically, prior notions being either logically orepistemologically prior. Finally, since the axiomatic method aims to structure our concepts,and insofar as the mathematical structuralist works top-down from the axioms, the appeal,by those adopting the genetic method, and so working bottom-up, from unstructured or set-structured concepts, like operation and collection or function, class or set, or to structuredconcepts, like function and set, is not a problem; it is a side issue.

It is often forgotten that Hilbert, as did Frege (1884), distinguished the context ofdiscovery from the context of justification. Hilbert did this by appealing to the differencebetween the genetic and the axiomatic method. That is, in attempting to extend theaxiomatic method to the foundations of real analysis, Hilbert’s (1900a) “On the Conceptof Number,” distinguishes between his axiomatic method (as laid out in his Grundlagender Geometrie) and the genetic method (as employed, e.g., in the works of Dedekind andCauchy). He does this by first noting that typically the methods of investigation of the

25 See Landry & Marquis (2005).26 See Landry (2011).27 See Landry & Marquis (2005) for an account of the historical development and philosophical use

of this top-down approach.28 See Landry (2006).29 See Landry (1999).30 See Landry (2006).


principles of arithmetic and the axioms of geometry are thought to be essentially different;typically it is thought that that latter uses the genetic method and the former the axiomatic.He then details the genetic method or the genetic “manner of introducing the concept ofnumber” as follows:

[s]tarting from the concept of the number 1, one usually imaginesthe further rational positive integers, 2, 3, 4,. . . as arising through theprocess of counting, and one develops their laws of calculation; then, byrequiring subtraction be universally applicable, one attains the negativenumbers; next one defines fraction, say as a pair of numbers – so thatevery linear function possesses a zero; and finally one defines the realnumbers as a [Dedekind]31 cut or a [Cauchy] fundamental sequence,thereby achieving the result that every entire indefinite. . . functionpossesses a zero. We call this method of introducing the concept ofnumber the genetic method, because the most general concept of realnumber is engendered [erzeugt] by the successive extension of thesimple concept of number. (Hilbert, 1900a, in Ewald, 1999, p. 1092;italics added.)

Of the axiomatic method of investigation used in geometry he says:

[o]ne proceeds essentially differently in the construction of geometry.Here one customarily begins by assuming the existence of all theelements, i.e., one postulates at the outset three systems of things(namely, the points, lines, and planes) and then – essentially on thepattern of Euclid – brings these elements into relationship with oneanother by means of certain axioms – namely, the axioms of linking[Verknupfung], of ordering, of congruence, and of continuity. Thenecessary task then arises of showing the consistency and completenessof these axioms, i.e., it must be proved that the application of the givenaxioms can never lead to contradictions, and further, that the systemof axioms is adequate to prove all the geometrical propositions. Weshall call this procedure of investigation the axiomatic method. (Hilbert,1900a, in Ewald, 1999, p. 1092; italics added)

Knowing his preference for the axiomatic method, we find it no surprise to then read

[m]y opinion is this: Despite the high pedagogic and heuristic value ofthe genetic method, for the final presentation and the complete logicalgrounding [Sicherung] of our knowledge, the axiomatic method deservesfirst rank. (Hilbert, 1900a, in Ewald, 1999, p. 1093; italics added.)

Hilbert then proceeds to axiomatically present the theory of arithmetic. That is, inadopting the axiomatic method

[w]e think a system of things; we call these things numbers anddesignate them by a, b, c,. . . We think these numbers in certain reciprocalrelationships whose exact and complete description occurs through the

31 See Ewald (1999) who says: “He [Hilbert] contrasts the axiomatic method with the geneticmethod that had previously been the standard approach in arithmetical investigation, and it wellexemplified by Dedekind’s Habilitation address. . . ” [p. 1090]

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following axioms. . . (Hilbert, 1900a, in Ewald, 1999, p. 1093; italicsadded.)

Now, of course, there’s the “necessary task” of proving the consistency of these axioms,which Hilbert then thought “needs only a suitable modification of familiar methods ofinference” (Hilbert, 1900a, in Ewald, 1999, p. 1095). But, putting these meta-mathematicalissues aside,32 interestingly, Hilbert next points out that

[u]nder the conception described above, the doubts which have beenraised against the existence of the totality of all real numbers (andagainst the existence of infinite sets generally) lose all justification; forby the set of real numbers we do not have to imagine, say, the totalityof all possible laws according to which the elements of a fundamentalsequence can proceed [as we do with say Cauchy sequences], butrather – as just described – a system of things whose mutual relationsare given by the finite and closed system of axioms. . . , and about whichnew statements are valid only if one can derive them from the axioms bymeans of a finite number of logical inferences. (Hilbert, 1900a, in Ewald,1999, p. 1095; italics added.)33

So, as noted in Hilbert (1900b), as one who adopts the axiomatic method, ourfoundational task, is as follows

. . . when we are engaged in investigating the foundations of a science,we must set up a system of axioms which contains an exact and completedescription of the relations subsisting between the elementary ideas ofthat science. The axioms so set up are at the same time the [implicit]definitions of those elementary ideas; and no statement within the realmof the science whose foundation we are testing is held to be correct unlessit can be derived from those axioms by means of a finite number of steps[modulo the independence and consistency of the axioms]. (Hilbert,1900b, in Ewald, 1999, p. 1104; italics added.)

Next we are told how consistency itself implies existence

. . . if it can be proved that the attributes assigned to the concept can neverlead to a contradiction by the application of a finite number of logicalinferences, I say that the mathematical existence of the concept. . . isthereby proved. In the case before us, where we are concerned with theaxioms of real numbers in arithmetic, the proof of the consistency of theaxioms is at the same time the proof of the mathematical existence ofthe complete system of real numbers or of the continuum. . . . The totalityof real numbers. . . . is not the totality of all possible series in decimalfractions, or of all possible laws according to which the elements of a

32 See Landry (2011), where, in reply to Shapiro (2005), I do take-up a Hilbertian analysis ofmeta-mathematical issues.

33 Extending the above analysis to Cantor’s set theory, Hilbert notes: “If we should wish to prove ina similar manner the existence of a totality of all powers (or of all Cantorian alephs), this attemptwould fail; for in fact the totality of all powers does not exist, or – in Cantor’s terminology – thesystem of all powers is an inconsistent (unfinished) set.” (Hilbert, 1900a, in Ewald, 1999, p. 1095)


fundamental sequence may proceed. It is rather a system of things whosemutual relations are governed by the axioms set up . . . In my opinion,the concept of the continuum is strictly logically tenable in this senseonly. . . 34 (Hilbert, 1900b, in Ewald, 1999, p. 1105; italics added.)

Finally we note three things about the axiomatic method: that it is foundational becauseit is organizational; that the notion of structure that it considers is relational, as opposedto constitutive; and that the axioms are to be taken as schematic, as opposed to assertory.Witnessing this first, organizational, aspect of what is meant by foundation, is Hilbert’sclaims that

[e]very science takes its starting point from a sufficiently coherent bodyof facts as given. It takes form, however, only by organizing this body offacts. This organization takes place though the axiomatic method, i.e.,one constructs a logical structure of concepts so that the relationshipbetween the concepts corresponds to the relationship between the factsto be organized. (Hilbert, 1902, in Hallett & Majer, 2004; italics added.)

Explaining the second, relational, aspect of structure is Bernays’ oft quoted remark that

[a] main feature of Hilbert’s axiomatization of geometry is thatthe axiomatic method . . . consists in. . . understanding the assertions(theorems) of the axiomatized theory in a hypothetical sense, that is, asholding true for any interpretation. . . for which the axioms are satisfied.Thus, an axiom system is regarded not as a system of statements abouta subject matter but as a system of conditions for what might be called arelational structure. (Bernays, 1967, p. 497; italics added.)

Finally, noting the schematic aspect of the axioms, is Hilbert’s remark that

. . . it is certainly obvious that every theory is only a scaffolding or schemaof concepts together with their necessary relations to one another, andthat the basic elements can be thought of in any way one likes . . . .(Hilbert, 1899, pp. 40–41; italics added).

Thus we see in Hilbert’s distinction between the genetic and axiomatic method the fabricfor the remaining two threads that I plan to pull together. That is, Hilbert’s distinction goestogether with the idea that the axioms, as schematic implicit definitions (as opposed toassertory truths), structure our mathematical concepts in terms of the relations that bearbetween them (as opposed to in terms of the “subject matter” of which they are constructedor constituted) so that the mathematical structuralist, as Awodey’s distinction between top-down and bottom-up ways of working suggests, begins with the axioms. Second, insofar asa foundation “takes place through” the axiomatic method, which itself aims to “structure”concepts in terms of their relations, and so organizes or founds “the facts” that fall undersuch concepts by beginning with the axioms (as opposed to constructing the structureof concepts by beginning with some fixed domain of facts as its constitutive subject

34 Again, extending the above analysis to Cantor’s set theory, he notes: “I am convinced that theexistence of the latter [Cantor’s higher classes of numbers and cardinal numbers] can be provedin the sense I have described; unlike the system of all cardinal numbers or of all Cantor’salephs, . . . Either of these systems is, therefore, according to my terminology, mathematicallynon-existent.” (Hilbert, 1900b, in Ewald 1999, p. 1105; italics added.)

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matter). This distinction, then, underwrites my distinction between the organizationalversus constitutive use of the term ‘foundation’.

§4. Using Hilbert’s distinction to respond to Feferman ‘77. My aim is to nowshow, against Feferman (1977), how one can use Hilbert’s distinction between the geneticand the axiomatic method, to argue that one can be a pure algebraic structuralist all theway up. More specifically, my aim will be to show that at the mathematical level, thatis, at the level where one uses the category axioms, for example, the EM, ETCS, andCCAF axioms, to implicitly define the concepts of ‘object’ and ‘morphism’, ‘set’ and‘function’, and ‘category’ and ‘functor’, respectively, one does not have to presume eitherthe psychological or logical (or epistemological as claimed in Bell, 1981) priority of thenotions of ‘operation’ and ‘collection’ (or ‘class’, ‘operation’, ‘set’, ‘function’ as claimedin Bell, 1981; Hellman, 2003).

Recall that Feferman claims that, with respect to the category-theoretic structuralist’suse of the CCAF and the ETCS axioms, appeal needs to be made, respectively, to theunstructured notions of operation and collection or to set-structured notions. To provide afully structuralist account, yet one in contrast to any extensional (Platonist) set-theoreticoption, his solution is to “structure” these notions by his constructive and intensional theoryT of operations and collections. My solution is to take these “unstructured” concepts aspart of the genetic method, that is, as part of the heuristic and pedagogical path that mayhave led us to the axiomatically presented, or category theoretically structured, concepts of‘object’ and ‘morphism’, ‘set’ and ‘function’, ‘category’ and ‘functor’, etc., as implicitlydefined by the category axioms, that is, by the EM, ETCS, and CCAF axioms, respectively.

As Landry & Marquis (2005) have pointed out, initially, when formulating the EMaxioms, Eilenburg & Mac Lane (1945) saw the concept of a category itself as a purelyheuristic device, depending perhaps35 on the set-theoretic notions of ‘function’ and ‘set’.They were aware too of the size problems of “large” categories and so considered NGB asbackground theory. As we pointed out, however:

[t]he [1945] introduction of the notions of category, functor, and naturaltransformation led Mac Lane and Eilenberg to conclude that categorytheory “provided a handy language to be used by topologists and others,and it offered a conceptual view of parts of mathematics”; however, they“did not then regard it as a field for further research effort, but just as alanguage of orientation”. (Mac Lane, 1988, pp. 334–335; italics added,in Landry & Marquis, 2005, p. 4)

So certainly, the history tells us that the concepts of ‘operation’ and ‘collection’, perhapsas structured set theoretically in terms of ‘set’ and ‘function’, were needed psychologicallyor heuristically, but it is a far different claim to hold that they are yet still needed logically.

35 I say perhaps, because, in the [1945] paper, the definition of a category was given in terms of‘aggregates’. That is, as noted on Marquis’ (2010) SEP entry on Category Theory, it is definedas follows: A category C is an aggregate Ob of abstract elements, called the objects of C, andabstract elements Map, called mappings of the category. And this with the note, by Marquis,that “The term “aggregate” is used by Eilenberg and Mac Lane themselves, presumably so as toremain neutral with respect to the background set theory one wants to adopt.” As we will see,just how a category is defined is an essential aspect of the uses and abuses of Feferman’s claims.Indeed, my definition will be such as to allow for the Hilbertian reading that I will use to respondto Feferman.


Again, as the history tells us, and as Landry & Marquis (2005) detail, with the workof Grothendieck (1957), Kan (1958), and Lawvere (1964, 1966), and certainly, by 1967category theory was taken as a mathematically autonomous theory. The EM, the ETCS,and the CCAF axioms had, by this time, come to stand alone. As such, they, as implicitdefinitions, needed no prior notions to give meaning or reference, either psychologicallyor logically (or epistemologically), to the concepts of ‘object’ and ‘morphism’, ‘set’and ‘function’, or ‘category’ and ‘functor’. Moreover, it is this mathematical autonomy,viewed now in light of the use of the axiomatic method, that allows for a purely algebraicstructuralist perspective (again, see Landry & Marquis, 2005; Awodey, 2004). That is,it allows us to begin with, and work top-down from, the category axioms, without anypresumption of what pretheoretic concepts might have been, or might still be, taken asprior to, or motivation for these concepts (again, see McLarty, 2004, 2005; Awodey, 2004).

Moreover, these pretheoretic concepts, be they unstructured (as in the case of ‘operation’and ‘collection’ or structured (as in the case of ‘function’, ‘set’, or ‘class’), are not tobe thought of as yielding a constitutive notion of a structure. What we have in its steadis a relational notion of their structure. For example, a group is not to be thought of as“made up” of sets, and sets are not “made up” of elements. Rather a group, as organizedby the EM axioms, is the name we give to an ‘object’ that satisfies the Grp axioms.Likewise for a set, it is the name we give to an ‘object’ which is organized by the Setaxioms, which we may then further structure by the ETCS axioms. Consequently, for thecategory-theoretic mathematical structuralist, no pretheoretic concepts, like ‘collection’or ‘operation’ are needed to stand as foundationally constitutive “atoms” from which weconstruct either meaning or reference. What we have in its stead is an organizationalnotion of foundation; one which relies solely on the use of the axiomatic method whichin itself, and by itself, structures our category-theoretic concepts. For example, the EMaxioms express what we say about anything that is structured in terms of ‘object’ and‘morphism’ so that the categories Set, Grp, etc., organize what we say about, sets andfunctions, groups and homomorphisms, etc., as ‘objects’ and ‘morphisms’. Thus, just asHilbert claims with respect to the number axioms, “[w]e think a system of things; wecall these things numbers and designate them by a, b, c,. . . ” So Mac Lane explains withrespect to the category axioms “[i]n this [axiomatic] description of a category, one canregard ‘object’, ‘morphism’. . . as undefined terms or predicates ranging over things” (MacLane, 1968, p. 287; italics added).

Here I pause, but briefly, to note that it has long been my suspicion that much of theconfusion and conflated claims about the foundational status of category theory has grownmistakenly out of an imprecise, or willfully (set or class) theory laden, explicit definition ofa category, as structured by the EM axioms. This is why I, in all my papers, have taken greatcare to implicitly define a category, in line with both Mac Lane and Hilbert, as follows: Acategory36 C is any abstract system of two sorts of things; objects X , Y , . . . and morphismsf , g,. . . , that satisfy the EM axioms. Notice that there is no use of the terms ‘collection’,‘operation’, ‘class’, ‘set’, ‘function’ to explicitly define what objects and morphisms are.This is the sense, then, in which, following Hilbert, we take the category axioms (herethe EM axioms) as a scaffolding or schema of concepts so that the basic elements (here

36 The wording I typically used was “cat-structured system”; this to further indicate that we shouldgive an in re interpretation of a category as a system that has a structure, as opposed to taking itas an ante rem structure.

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what we take to be objects and morphisms) can be thought of in any way one likes.37 Onewould think from all I have said above, and too from Awodey’s (1996) algebraic definition,that this type of implicit definition would be obvious, if not standard. Alas, Bell’s (1981)definition, which as we’ve seen informed much of the philosophical debates, is given interms of classes, and, moreover, a quick one hundred page Google search for “categorytheory category definition” reveals that all definitions of a category via the EM axioms areexplicitly given in terms of either collections or classes or sets.

Seen now in the light of Hilbert’s use of the axiomatic method, it should be clearthat, by adopting the axiomatic method, the category-theoretic structuralist simply beginsby assuming the existence of a system of two sorts of things (namely, ‘objects’ and‘morphisms’) and then brings these things into relationship with one another by means ofcertain axioms, for example, by mean of the EM axioms, whereby these axioms implicitlydefine, at an abstract level, what is meant by anything that satisfies the axioms, and thusis an object or morphism.38 And, in so doing, and again in line with Awodey (2004),39

it should also be clear that when we characterize any EM organized axiom system as acategory, it should be understood in the Hilbertian schematic sense, that is, a category itselfis a schema used to provide “a system of conditions for what might be called a relationalstructure.” (Bernays, 1967, p. 497; italics added.)

Further witnessing the Hilbertian heritage of the use of the axiomatic method for theresulting category-theoretic consideration of an axiom system qua a relational structure, isMac Lane’s claim that

. . . a structure is essentially a list of operations and relations andtheir required properties, commonly given as axioms, and often soformulated as to be properties shared by a number of possibly quitedifferent specific mathematical objects. . . a mathematical object ‘has’a particular structure when specified aspects of the objects satisfy the(standard) list of axioms for the structure. This notion of ‘structure’ isclearly an outgrowth of the widespread use of the axiomatic methodin mathematics [as exemplified by Hilbert’s Grundlagen]. (Mac Lane,1996, pp. 174 & 176; italics added.)

37 I would like to thank an anonymous referee for pushing me to make this point clearer. Whatis doing the work here is neither the notion of a system nor the notion of an abstract system,rather it is the Hilbertian idea of a theory as a schema for concepts that, themselves, are implicitlydefined by the axioms. Thus, we do not need a “fixed domain” of elements (be these objects andmorphisms, or categories and functors), we do not need a system as a “collection” of elements (norany appeal to plural quantification to range over all such), and at the EM level we do not need forour system of objects and morphisms to be an abstract ‘object’ in a higher-order system of CCAFdefined categories. That is, we do not need for EM categories to be ‘objects’ in the CCAF senseand so do not need to appeal to “Lawvere’s functorial semantics. . . to give a purely categoricalunderstanding of what it means for such an object to satisfy the EM axioms.” As I argued in my[2011] paper, there are indeed many ways to provide a contentual analysis of logical structureto capture the needed meta-mathematical or logical (both semantic and syntactic) notions thatunderwrite the acceptability (here satisfiability) of a theory.

38 See Landry (2011), which works out the ways in which the category-theoretic structuralistposition can be rationally reconstructed along Hilbertian lines.

39 See, for example, Awodey, who, in responding to Hellman (2003), claims that “Neither G norC. . . are specific things here, they are schematic structures, as it were, specified or determined bythe configurations of objects and arrows and conditions on them. . . ” (Awodey, 2004, p. 62).


Now, of course to psychologically or heuristically get to these axioms we might havemade use of the genetic method and so appealed to other concepts; either set-theoreticallystructured concepts like ‘function’, ‘class’, or ‘set’ or even, pretheoretic unstructuredconcepts like ‘operation’ and ‘collection’. Indeed, we may, like Feferman, yet see theneed for an axiomatic theory of ‘operation’ and ‘collection’, as too we’ve come to seethe need for an axiomatic theory of ‘morphism’ and ‘object’. But, as should now be clear,this need is not borne out of some logical or epistemological priority of these notions withrespect to the concepts of ‘object’ and ‘morphism’, ‘set’ and ‘function’, or ‘category and‘functor’.

Again, we say in line with Hilbert: Despite the high pedagogic and heuristic value of thegenetic method, that is, the high pedagogic and heuristic value of the theory of operationsand collections (or the theory of functions, classes or sets), for the final presentation andcomplete logical grounding of our knowledge of category-theoretic concepts, the axiomaticmethod, which implicitly defines these concepts, deserves first rank.

Recall that there are four central claims of Feferman (1977): 1) that our aim is to give astructuralist account of abstract mathematics; 2) that category theory requires a prior theoryof operation and collection; 3) that we want a nonpluralist account of what a foundationis; and 4) that we want to avoid the Platonistic or extensional accounts and make room forconstructive or intensional accounts.

With respect to 1) and 2), I believe, that I have shown that we can provide a category-theoretic pure algebraic structuralist account of abstract mathematics, without needinga prior theory of operations and collections. Yet, in consideration of 1) and 3), if onetakes into account the philosophical literature as outlined above, we are still left askingwhich category-theoretic axioms are we to take as the foundation for a structuralistaccount of abstract mathematics; the EM, ETCS, or CCAF axioms? Against both Bell’s(1986) and (1988) topos-theoretic pluralism40 and McLarty’s (2004) category-theoreticpluralism,41 and in line with Awodey (2004), I take the EM axioms to be the foundationfor a structuralist account of abstract mathematics. Again, where I understand the termfoundation in the Hilbertian, organizational, sense.

As I explained in Landry (2011), and noted here on pages 18–19, the EM axiomsorganize what we say about the abstract structure of mathematics, both the abstractstructure of mathematical concepts, like the EM organized categories Set and Grp (wherewe take set and function, and group and homomorphism as ‘objects’ and ‘morphisms’),and the abstract structure of logical concepts, like the EM organized categories Lat andBool, (where we take lattices and Boolean algebras as ‘objects’, and structure preservinghomomorphisms as ‘morphisms’). The ETCS and CCAF axioms, in contrast, organizewhat we say about the structure of mathematics, insofar as we structure the mathematicalor logical concepts of ‘object’ and ‘morphism’ in terms of set and function or category andfunctor, respectively.42 That is, in line with McLarty’s (2005) claim, I too

40 Where, again, every topos represents a framework of a “possible world” for mathematical activity41 Where “each proposed specific categorical foundation [as given, for example, by the ETCS, SDG,

CCAF axioms] is a proposed answer” [to Hellman, 2003], because each is assertory. (McLarty,2004, p. 42)

42 Again, one can also use the EM axioms to organize logical structure as well. For example, we canconsider classical logical structure as given by the categories Lat and Bool, where we take latticesand Boolean algebras as ‘objects’, and ‘morphisms’ as structure preserving homomorphisms, thatis, (�, ⊥, ∧, ∨) homomorphisms. Or, and perhaps with reference to Feferman’s constructive aims,we can consider intuitionistic logical structure as given by the category Heyt, where we take

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. . . agree with Feferman that foundations of mathematics should lie ina general theory of operations and collections, only I say the currentbest general theory of those calls them arrows and objects. It is categorytheory. (McLarty, 2005, p. 49)

But again, against McLarty (2005), and in line with Awodey (2004), I feel no need, whenmaking such foundationalist claims, to read the EM axioms as assertory. Yet, neither, as isthe case with Awodey, do I feel the need to limit the term foundation to its constitutive, or“atomistic” meaning. Indeed, as already explained, when seen in the light of our rationalreconstruction of Hilbert, it is in virtue of their schematic character that the EM axiomscan be seen as providing the foundation, where foundation is here understood in theorganizational sense of the term.

Finally, with regards to Feferman’s claim 4), viz., that we want to avoid the Platonisticor extensional accounts and make room for constructive or intensional accounts, Ihave certainly shown that we can avoid Platonistic or extensional accounts. That is,I take it that as pure algebraic mathematical structuralists, we, like the non-Platonistor nonextensionalist Feferman, want to avoid the need for any “background theory,”including set theory and both Shapiro’s Platonist theory of structures and Hellman’snominalist modal theory of large domains, wherein our ‘objects’, like ‘collections’,would be considered “independent of any means of definition” and our ‘morphisms’, like‘operations’, would “identified with their graphs” (Feferman, 1977, p. 151).

To further demonstrate this, and in one last reply to Hellman (2003), what we simplynote is that if we take the CCAF axioms as organizing what we say about categoriesand functors, themselves structured as ‘objects’ and ‘morphisms’ then, as McLarty (2005,p. 52) notes, Hellman’s “home address” problem, insofar as it concerns the existence ofcategories, disappears. So in response to Hellman’s “existence” objection, I again turn toHilbert: . . . if it can be proved that the attributes assigned to the concept can never leadto a contradiction by the application of a finite number of logical inferences, I say that themathematical existence of the concept of a category. . . is thereby proved. In the case beforeus, where we are concerned with the CCAF axiom’s concepts of ‘category’ and ‘functor’the proof of the relative consistency43 of the axioms is at the same time the proof of themathematical existence of the complete system of categories.

Heyting algebras as ‘objects’ and ‘morphisms’ as (�, ⊥, ∧, ∨, →) homomorphisms. Finally,we can consider proof-theoretic logical structure more generally by taking a deductive systemitself as a category, where the ‘objects’ are formulas, ‘morphisms’ are proofs or deductions, andconditions on morphisms are taken as rules of inference.

43 Here I note that, as should be obvious, relative consistency need not be taken as relativeconsistency with respect to ZFC. For example, the consistency of the CCAF axioms can beproved with respect to the ETCS axioms plus one Grothendieck universe. More generally, ZFCis consistent if and only if ETCS plus the categorical reflection scheme is consistent. AndETCS is consistent if and only if the fragment of ZFC called “bounded Zermelo” is; wherebounded Zermelo is the fragment of ZFC where one drops the replacement axiom scheme, andallows the separation axiom scheme only with bounded quantifiers. More specifically, whateverproves the relative consistency of the ZFC axioms proves the relative consistency of ETCSaxioms. Moreover, the relative consistency proof between (the fragments and extensions of)ETCS and (the fragments and extension of) ZFC do not use either ETCS or ZFC. Rather,both are formalizable in weak fragments of second-order arithmetic by proof-theoretic means.Much thanks to Colin McLarty for providing me with the details need to make the abovepoints precise. Again, speaking to this last point, as noted here and in Landry (2011), forthe category-theoretic foundationalist, there are many meta-mathematical and logical options


Yet too this response seems to bring us up against even greater “size issues,” that is,what now about the status of the category of all categories?44 Again, however, we turn toHilbert, for our reply: Under the conception described above, the doubts which have beenraised against the existence of the totality of all categories lose all justification; for by thecategory of categories we do not have to imagine, say, the axioms as including the categoryof all categories, but rather—as just described—a system of things whose mutual relationsare given by the . . . axioms. That is, we need only assume, as the axioms do, that categoriesare ‘objects’ and functors are ‘morphisms’.

But, in our appeal to letting relative consistency imply existence, it would seem thatboth Hellman’s (2003) charge of ‘if-then-ism’ and Shapiro’s (2005) related claim thata meta-mathematical assertory background theory is needed to logically express claimsof “acceptability,” seem to lurk in the background, and so may yet push us into theextensionalist position of either the Platonist or modal nominalist. There are two repliesto offer here. One, in line with Awodey (2004), is that mathematics itself has an “if-then” structure, so we should not be either surprised or bothered that our foundation does.And two, as Bernays reminds us: the axiomatic method . . . consists in. . . understanding theassertions (theorems) of the axiomatized theory in a hypothetical sense, that is, as holdingtrue for any interpretation.

However, one may also read Hellman’s ‘if-then-ism’ charge as arising from the question,equally pointed at Hilbert and the category-theoretic structuralist, “How do you know thereare enough “things” to provide the models for the needed relative consistency proofs?”In reply, and again, as Hilbert explains: Here one customarily begins by assuming theexistence of all the elements. . . . As I have argued previously, in Landry (2011), thisassumption is no less problematic than is the Platonist assumption that there exists aset, or structure-theoretic universe, that makes our axioms true or the modal nominalistassumption that there possibly exits a large plurality of concrete parts that provides thelarge domains for our mathematical statements to range over.45

open for the contentual analysis of such proofs. Further, as I explain in endnote 37 (Landry,2011, p. 451), “[a]t the mathematical level, as algebraic structuralists, we [me, Shapiro, and theset-theoretic foundationalist] are [all] committed to some type of schema like “If theory T isacceptable (coherent or consistent, for example), then the objects over which it ranges exist”.With respect to the acceptability of T, we both agree that all we have is relative consistency (orrelative coherence). At the meta-mathematical level, however, as pure algebraic structuralists, thestatement of T’s acceptability (or consistency) is taken as internally assertory, i.e., is assertoryin some stronger theory TS [where by ‘stronger’ I mean whatever, in light of Godel results, wemean, for example, either computationally or information-theoretically stronger] which we takeas “acceptable” for the purpose of proving the relative consistency of the theory T. The statement‘T is consistent’ holds in TS, but it is not externally assertory in the sense that we need take thisstronger theory TS itself as true. Where we differ, then, is that I deny the claim that statementsof acceptability (or consistency, coherence etc.,) are externally assertory in the sense that at somepoint the “If . . . , then. . . ” dissolves because the statement of the acceptability of T is expressed insome true, assertory, meta-mathematical background, theory TT, that stops the possible regressof stronger theories, i.e., because for some TS, TS = TT.”

44 Note that Feferman’s theory of collections and operations faces the same problem. That is, as henotes of this theory T: “It is true that statements of results must now make distinction betweenpartial and total which previously were made between large and small.” (Feferman, 1977, p. 168)

45 Again, I would like to thank an anonymous referee for motivating me to make explicit thedifference between my approach and Awodey’s. That is, there is a distinction to be made herebetween my Hilbertian approach and Awodey’s “if-then” approach. Awodey clearly accepts thathis foundation too has an if-then structure. The Hilbertian, in contrast, as does the set-theoretic

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Again, put simply, the Hilbertian pure algebraic structuralist works top-down and sobegins by assuming the “things” that the Platonist, the nominalist, and even Feferman,feel the need to “structure” by working bottom-up, from either an extensional theoryof structures, sets, or large domains, or an intensional theory of collections. In anycase, however, none of these assumptions is need to “make up” the objects that foundsome constitutive notion of structure in mathematics. This point tells equally against thePlatonist/nominalist viewpoint as it does against Feferman’s constructivist viewpoint. Thatis, it tells equally against the need to appeal to any “background theory” as it does againstthe need to appeal to any “prior theory”; be it extensionally presupposed or intensionallyconstructed. Indeed, as I have used Hilbert and Awodey to show, there is no “making-up”or “constructing,” in terms of some atomistic notion of constitutive structure; there is only“organizing down,” in terms of the axiomatic notion of relational structure. And for thisfoundational task, category theory more than meets the mathematical structuralist’s needs.

In response to Feferman (1977), then, given that the pure algebraic structuralist workstop-down axiomatically, and only works bottom-up genetically, or heuristically, there is nosense of the talk of “construction” of concepts, and so no sense of the psychological priorityof any concepts. And given that the category axioms, the EM, the ETCS, and the CCAFaxioms, themselves provide implicit definitions of the concepts ‘object’ and ‘morphism’,‘set’ and ‘function’, and ‘category’ and ‘functor’, respectively, and so provide, again, toborrow from Hilbert, “the final presentation and the completely logical grounding of ourknowledge,” there is no sense of talk of either the logical or epistemological priority of anyconcepts.

Thus, given that a foundation, understood now in this Hilbertian sense, is that whichbest organizes what we say about the structure of mathematical concepts, and insofar asonly the axiomatic method structures our concepts, category theory as framed by the EMaxioms is currently the best foundation for a structuralist account of abstract mathematics.So, contra Feferman (1977), one can be a category-theoretic pure algebraic structuralistall the way up!

§5. Acknowledgements. For Aldo; who thinks it’s all abstract nonsense, but whosupports me nonetheless. I hope to have finally convinced you that, while it’s abstract,it’s not nonsense!

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The Genetic Versus the Axiomatic Method: Responding to Feferman 1977

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