Combinatorics (n+2+3)

Back to it. Alright.

The aim of my dissertation is to provide a helpful ontological explanation of the meaning of the term 'structure' as it is used within mathematics.

The first chapter involves a brief historical overview of geometry with particular attention to the developments leading up to the publication in 1899 of David Hilbert's 'Foundations of Geometry'. After this historical introduction, I turn to a discussion of the correspondence between Gottlob Frege and Hilbert regarding the way in which the latter employed the terms 'axioms' and 'definitions'. The point of this discussion is to illustrate that Hilbert's approach to the foundations of mathematics (which became almost universally accepted in the 20th century) is susceptible to Frege's criticisms, which imply a kind of conceptual confusion between the radically relational view of mathematics that Hilbert seems to want to offer and the actually limited 'absolute' sense in which his work does in fact operate. This basic confusion, [between what I will call the 'object-oriented' approach (of Frege and, confusedly, of Hilbert as well) and the 'structure-oriented' or 'relational' approach desired by Hilbert] will form the central conceptual focus of the remainder of the thesis.

In the second chapter I again return to some historical material, with a discussion of the successes of a Hilbert-style relational axiomatic approach and it's subsequent 'anchoring' in the reductive, foundational view of set theory championed by many mathematicians and philosophers in first half of the century. Of particular importance, given my earlier discussion of the Frege-Hilbert correspondence, will be Hilbert's own later, formalist views. I will argue that Hilbert's formalist position takes into account Frege's criticisms but that, even with this alteration, the combination of a 'relational' view with an 'object-oriented' view is subject to severe ontological problems. In particular, I highlight two instances in the history of mathematics and the philosophy of mathematics where this becomes evident.

The first of these, Skolem's 'Paradox' was first brought to the attention of the public in 1922. It's reception is remarkable, Ernst Zermelo initially believed it to be a hoax. The intricacies of the so-called paradox (actually a very interesting theorem and no paradox at all) are of course important, the main reason I draw attention to this incident, however, is to illustrate further the tension between a structural view of the nature of mathematics and an absolute, object oriented view. Many mathematicians at the time had come to see mathematics as ultimately grounded in (i.e., 'about') the universe of sets. This universe was considered to be a collection of objects of some kind (there are various dispensations explaining exactly which kind). What Skolem's paradox did was to call in to question the sense of the 'reduction' of mathematics to one particular universe, or, perhaps better, he called into question the 'obviousness' of the way in which the reduction was achieved by showing that there were, by set theory's own lights, very strange sorts of models which achieved the reduction in a way that no one found intuitively plausible. Though Skolem's Paradox was thoroughly discussed, relatively few mathematicians or logicians took a purely 'structural' approach in response to it. Nevertheless, as I will show, an increasingly structural approach in mathematics arose, largely from the more algebraically oriented sub-disciplines of mathematics (group theory, algebraic topology, and eventually category theory).

This brings us to the second instance in which the tension between the object oriented view and the structural view becomes apparent. Namely, Paul Benacerraf's paper "What Numbers Could Not Be", published in 1965. There Benacerraf poses a problem for set theoretic foundationalism which is similar in spirit, if not in precise content, to the problems posed by Skolem's Paradox. He points out the widely accepted fact that there are many different possible reductions of, e.g., the natural numbers to the universe of sets. We have, for instance the von Neumann ordinals and the Zermelo ordinals, both of which can fulfill the requisite number theoretic axioms. The problem posed by this, for Benacerraf, is that the acceptability of multiple different reductions to sets calls into question the object-based view of the reduction in the first place. If the numbers just are sets, but we cannot decide which sets, EXACTLY, they are, then what sense does it make to say that the numbers 'are' sets at all?

After having posed this question, Benacerraf suggests his own answer: numbers aren't really sets at all, numbers aren't any particular mathematical object but are positions in mathematical structures, which possess only structural or relational properties. His view, which he states somewhat loosely and tentatively, suggests a movement away from the absolute, referentially fixed conception of set theory as a foundation for mathematics and towards a relative or relational view somewhat akin to Hilbert's early approach to geometrical axiomatics. As noted, however, similar concerns about the referentially fixed foundational view of mathematics had arisen within mathematics itself much earlier (with, e.g., the publication of Skolem's Paradox), and Frege had also pointed out several pressing problems related to Hilbert's view. Nevertheless, it is Benacerraf's paper (combined with the widespread mathematical influence of Bourbaki and the rise of more obviously structural mathematical sub-disciplines like category theory) that first brought the structural approach into the philosophy of mathematics in a serious way. In the wake of Benacerraf's dilemma, several structuralist positions within the philosophy of mathematics have arisen. A debate of particular importance for the later portions of my dissertation has been between realist structuralists, who take structures to possess some sort of ontological importance, and nominalist structuralists, who view structures as a mere facon de parler. This debate obviously has much earlier roots in the philosophical positions of Plato, Aristotle, and many others regarding the nature of mathematical objects, but it begins to take on a highly specialized form within the philosophy of mathematics in the second half of the 20th century.

After establishing the historical links between Hilbert\s formalism, reductionist set-theoretic foundationalism, and the confusion between object- and structure-oriented views, the third chapter turns directly to more recent structuralist approaches. While there are quite a number of different approaches in the philosophy of mathematics which can be called 'structuralist', I am interested (initially at least) in two main varieties; namely, realist structuralism and nominalist structuralism. As my eventual aim in this dissertation is to establish a satisfactory view of 'structure' as a basic ontological category (a view in which 'structures' are accorded some kind of real ontological status), I begin my discussion of structuralist positions with opposed views in order to undermine both their motivations and their specific claims. There are several disparate 'nominalist' structuralisms, but to my mind the two most important advocates have been Geoffrey Hellman, with his modal, eliminative view (the non structures view) and Charles Chihara's constructive approach. While the specifics of both of these programmes are very interesting, my interest is to undermine the motivation for a nominalist view of the nature of mathematics. I attempt to do so by arguing against both the usefulness of a nominalist reduction (for mathematics and philosophy both), and also by suggesting that neither of these positions achieves a truly 'nominalist' view of mathematics, even by their own lights, precisely insofar as they rely on certain kinds of abstract entities that they forbid their realist counterparts access to.

After having seriously reduced the motivation for nominalism within the philosophy of mathematics, I turn to a discussion of it's broad alternative: realism.

In the last 25 years or so the two major advocates of structural realism have been Michael Resnik and Stewart Shapiro. Resnik views structures as 'patterns' which can be exemplified within or instantiated by particular systems of objects. Shapiro's view takes structures to be sui generis universals, kinds of abstract objects which exist independently of any possible or actual realization in a given system of objects. I explore these two realist positions in some detail in order to provide a sense of the problems facing such views more generally. Particularly important for my interests later in the dissertation are problems centering around notions of mathematical existence, ontological commitment, and the conceptual differentiation between mathematical structures (or patterns) and mathematical 'objects'.

The result of my dissection of the debate between realist and nominalist forms of structuralism should be the sense that the same ontological problems ultimately face both sorts of views, and also that neither strand has developed sufficiently clear ontological concepts capable of filling in the well-known gaps. At this point in the chapter I turn to a third form of structuralism which arose largely as an internal development within mathematics itself, namely, category-theoretic structuralism. Since it's early days (beginning with a famous 1945 paper on natural transformations by Saunders Mac Lane and Samuel Eilenberg) has been characterized as a possible alternative 'foundation' for mathematics which is conceptually and technically quite different from set theory. Just how the foundational nature of category theory is to be understood however is a question which has garnered much debate, beginning with F. William Lawvere's development of a category-theoretic treatment of axiomatic set theory.

Category theoretic structuralism within mathematics itself is neither definitively realist nor definitively nominalist in most of its incarnations, but is often treated simply as a technique to be used appropriately within a wide variety of mathematical contexts. My aim, in this and the next chapter, is to show that an appropriate ontological grasp of the concepts at work within the category-theoretic, foundational approach can help us avoid the confusion between object- and structure-oriented approaches that has plagued foundational views since at least Hilbert's 1899 work (and likely much earlier).

After having outlined my reasons for rejecting nominalism as a viable variety of mathematical structuralism in the previous chapter, I turn in the fourth chapter to my own positive project. This project consists, in large part, in combining a category-theoretic form of foundationalism within mathematics with my own ontological conception of structures. In this fourth chapter I focus on the problems facing a realist approach to category theory, as well as upon the problems we must face if we want to avoid the confusion between a referentially fixed, object-oriented approach and a relational, structure-oriented approach.

I discuss several varying conceptions of the nature of category theoretic structuralism. Of particular interest is an early (1977) paper by Solomon Feferman in which he argues that category theory, like set theory, requires a pre-theoretic grounding in concepts like 'collection' and thus does not offer us anything essentially new (in this regard). In other words, Feferman believes that the basic concepts of category theory, too, lead us to a non-structural ground and that, therefore, category theory does not offer an essentially 'structural' view any more than does set theory. By contrast, several later views either dismiss Feferman's view that category theory is or could be foundational for mathematics in the same way that set theory seems to have been, or they argue that category theory shows us that we can treat every notion as structural (in which case 'objects' themselves simply become structures when viewed from a particular mathematical position).

Taking Feferman's interesting paper as a foil, I argue both points. First I argue that category theory should not be treated as foundational for mathematics in the same sense that set theory has been (i.e., we should not see mathematical disciplines of greater complexity and specificity as supervening on the more fundamental mathematical/ontological universe in which categories are the only object). In this direction I follow a suggestion (made distinct by Elaine Landry) that we treat category theory as one, perhaps the best, language in which mathematics can be formulated. The benefit of this understanding is that the 'strictly mathematical' use of category theory is partially explained: category theory on this level simply provides us with an extraordinarily versatile and fruitful tool which can be applied in almost any mathematical context, as Mac Lane, Lawvere, and others have shown.

But, despite (or in addition to) this view of category theory as a framework for mathematics, I also want to combat Feferman's suggestion that category theory must ultimately rely upon non-structural conceptual elements in precisely the same way that set theory requires. What I suggest in the second half of this chapter is that the Feferman-type view that all foundational approaches require a ground in non-structural elements results from the 100 year old confusion between object-oriented views and structure-oriented views. At this point I begin my ontological focus on the twin concepts of 'object' and 'structure' and argue that the historical confusion has arisen (and continues) because mathematicians and philosophers, both, have had a conception of 'structure' which is insufficiently different, on an ontological level, than that a common notion of 'object'.

The sense derived from this chapter should be that a category theoretic, realist approach to structuralism in the philosophy of mathematics is possible, and also that it has run headlong into an ontological debate that has resisted clarification due to an unnecessary confusion between the ontological differences between structures and objects. In the next chapter I aim to tackle these ontological problems more directly.

In the fifth chapter I turn away from the specificities of the philosophy of mathematics and enter the more general forum of (formal) ontology.

I begin my discussion by returning to the work of Gottlob Frege, this time, however, to his fundamental distinction between concepts [or functions] and objects. Frege's distinction is interesting for a number of reasons. First, it is interesting insofar as it highlights the limits of rigorous systematization that Frege (and Hilbert) aspired to achieve.

The concepts of 'concept' and 'object' are for Frege indefinable (given his understanding of the nature of definition), but can become explicated though appropriate non-systematic, pre-scientific discussions.

Frege's explications seem to have the character of metaphors: for him objects are self-subsistent entities, fully saturated, and complete, requiring no supplementation. Concepts, by contrast, require supplementation, are unsaturated, cannot stand alone, etc.

In order to further explicate Frege's use of these terms, I turn to the famous view held by Frege that "the concept 'horse' is not a concept"

There does seem to be room within a Frege-style framework for a 'structural' view, if we treat concepts as structures. But it seems clear that for Frege this would have been an unsatisfying or plainly false view to take. For him, mathematics and logic, both, were grounded in the total system of thoughts, and this system was composed of objects (i.e., thoughts) which were the result of combinations of concepts and objects. The idea of a purely conceptual, i.e., relational, science made little sense to Frege: it would have no content, and could make no claims. We would have to turn the concepts into objects (by way of the 'concept horse' maneuver) in order to say anything meaningful, and in doing so the science would lose its structural/relational quality.

In order, then, to supplement Frege's view that science is non-structural (and hence rooted in objects), I turn to the more developed remarks of Alexius Meinong regarding the nature of objects (as elaborated in his 'Theory of Objects').

One of the most intriguing views of Meinong's work can be found in Gustav Bergmann's book 'Realism: A Critique of Meinong and Brentano'. There Bergmann characterizes Meinong's object theory as an attempt to treat relations purely on the basis of objects. Bergmann argues that this leads Meinong into denser and denser ontological problems, with the ultimate result that he broadens the concept of an object (improving our understanding of that notion considerably), while nevertheless failing in his ultimate goal. A proper ontological grounding of relations in a purely object-oriented view like Meinong's is, according to Bergmann, essentially impossible.

As will become clear, in order to pursue Frege's view of mathematics as rooted in a system of objects while also maintaining the importantly relational qualities of mathematics highlight by Hilbert and (later) by the category-theoretic structuralism I endorse, we are lead down Meinong's rich ontological path. Ultimately, though, even the richest object-based ontology comes up short in its ability to explain the radical relationality of mathematics as it is practiced.

Because Bergmann views Meinong's object theory to be the furthest end of object-based trajectory, he argues that its inability to adequately capture the nature of relations signals the sterility of any object-oriented approach. Accordingly, he embarks (in 'Realism' and many subsequent publications) to develop his own fundamental ontological picture in which a place is made for a fundamental, non-object category of ontological entity (he variously uses the expression 'nexus' and 'fundamental tie' to capture this notion).

I begin with the basic project of Bergmann but diverge heavily in specifics when I attempt to construct my own positive ontological view of the nature of structures. This view will be developed partly with the demands of category theory (understood as a framework or 'language' for mathematics) in mind, and partly with the Bergmann-inspired resistance to non-structural ontologies like those of Frege and Meinong in mind.