Chapter 1 and Chapter 2

Chapter One: Frege and Hilbert on the Nature of Axioms

Basic aim and outline of the chapter:

The aim of this chapter is to establish some elements of the ontological framework within which I wish to situate my later discussion of the concept of structure. In my research I have found the correspondence between Gottlob Frege and David Hilbert to be a crucial step in the historical development of structuralism within mathematics and the philosophy of mathematics. Accordingly, the chapter begins by recounting some of the conceptual elements of the history of geometry which led to the 1899 publication of David Hilbert’s Die Grundlagen der Geometrie [The Foundations of Geometry]. With this short historical portion I hope to illustrate the importance of Hilbert’s view for modern mathematics, as well as its radical divergence from a traditional view of the nature of geometry. From these historical remarks I then move on to a more detailed discussion of the content of Hilbert’s views regarding the nature and meaning of his axioms and definitions. I then move on to an evaluation of Gottlob Frege’s criticisms of Hilbert’s views, criticisms which have largely been ignored but which are, I believe, quite helpful for understanding both the historical development of ‘structuralism’ within mathematics as well as several of the conceptual problems involved in structural approaches in the philosophy of mathematics. The chapter concludes with the tentative suggestion that Hilbert’s later, formalist work began to take Frege-type criticisms more seriously as the philosophical difficulties of his initially radical position became clearer; I attempt to substantiate this suggestion in the second chapter.

Summary of the chapter:

After some very brief remarks regarding the historical status of Euclid’s Elements (with emphasis on problems with the parallel postulate), I move on to outline some of the conceptual shifts which occurred within geometry after the development of non-Euclidean, hyperbolic geometries by Lobachevsky and Bolyai between 1829 and 1832. My main claim here is that these new geometries, which were significantly extended and generalized a few decades later by Riemann and Gauss (among others), eventually created a widespread philosophical confusion about the nature of geometry itself. Prior to the development of non-Euclidean geometries, the generally accepted view was that geometry was a concrete science which helped to explicate the laws governing the actual, three-dimensional space in which we live. A commonplace epistemological adjunct of this view, emphasized and popularized by Kant, was the belief that the axioms and postulates of Euclidean geometry were truths that we can grasp immediately through our spatial intuition. These intuitive truths neither require nor admit of any form of proof, they are basic.

Beginning in the 1870s, when the work of Riemann, Helmholtz, Gauss, and others began to seem scientifically important in ways that the Lobachevsky-Bolyai geometries had not (or at least not immediately), the traditional philosophical account of the nature of geometry no longer seemed adequate. It is by no means intuitively obvious, for instance, that there are or can be points at infinity which guarantee the ‘truth’ of the axioms of projective geometry, nor is it obvious how to develop higher-dimensional differential geometries based solely on the deliverances of our spatial intuition. Perhaps least intuitively graspable among the new geometrical ideas was Riemann’s conception of a space whose curvature is different at each of its points. The full generality of the new forms of geometry was revealed by Riemann in a lecture (delivered in front of Gauss as his Habilitationsrede in 1854) entitled ‘On the Hypotheses which underlie Geometry’. In that lecture Riemann discusses five hypotheses which lie at the center of geometry, which he proceeds to treat with a level of abstraction and generality almost totally absent from the traditional Euclidean understanding of geometry. In this lecture, published only in 1868, Riemann provides clear and general formulations of metric and topological properties, with corresponding metric and topological ‘spaces’ determined by these properties. Despite the sophistication and facility with which Riemann and others employed their new geometrical concepts, there was no philosophical correlate to their work. Whereas the traditional, Euclidean view had a considerable philosophical edifice supporting it and linking it with the wider world, there was as yet nothing like this for the new approach to geometry. The conceptual shift remained an almost entirely internal development of mathematics itself. Of course, the lack of a firm philosophical understanding of the nature of their science did not dissuade geometers and other mathematicians from pursuing these deviant geometries, far from it. The second half of the nineteenth century saw an unparalleled explosion in geometry despite the lack of conceptual clarity regarding the nature of the various geometries themselves. Differing opinions as to what was going on within geometry eventually arose, however none of them met with much success until a general consensus became somewhat guaranteed by Hilbert’s 1899 work.

Following this portrait of the historical lead-up to Hilbert’s work, I will turn in earnest to the content of his views. My initial aim in analyzing Hilbert’s views on geometry and geometrical axioms is to illustrate how radically different his approach was from the traditional Euclidean/Kantian view of geometry. Though Hilbert paid lip service to the belief that his axioms, too, were grounded in intuition, in practice intuition plays a very minor role in his system. Instead what we find with Hilbert is a disconnection between the axioms and their fixed reference to an actual, concrete system like three-dimensional Euclidean space. While much of the terminology in Hilbert is familiar from Euclid’s Elements (he retains the terms ‘point’, ‘line, ‘plane’, etc), the meaning of this terminology is radically different. Hilbert’s intention, as he makes clear, is to describe an abstract structure which can then be interpreted in any number of appropriate concrete instances or systems. His view can be seen as an early version of modern day model theory, in which the uninterpreted elements of the abstract structure are assigned concrete elements by way of an interpretation. If all the axioms remain true on a particular interpretation, we call this interpretation a model for that set of axioms. (Hilbert’s terminology is of course slightly different, but the basic idea remains the same). In this way, Hilbert employs an abstract structure to prove claims about the concrete Euclidean system; in other words, he constructs a set of axioms for which the concrete Euclidean system, appropriately interpreted, is a model. He proves, for instance, the consistency of his axioms by showing that there is at least one model satisfying them. He also proves the independence of certain axioms from the others by providing interpretations on which the ‘independent’ axiom is false while the others are true. Especially interesting for my purposes is the fact that Hilbert takes the consistency and independence proofs for his set of axioms to imply the consistency of Euclidean geometry itself, and the independence of certain axioms of Euclidean geometry from certain others. This type of approach, as we shall see in the second chapter, was wildly successful, so much so that it is difficult to imagine the history of mathematics, logic, or even formal semantics in the twentieth century without the model-theoretic apparatus which is germinal in Hilbert’s work. It should also be clear, however, just how radically different this view is from the traditional Euclidean view of geometric axioms as intuitive truths about three-dimensional space.

From Hilbert’s views I move on to those of Gottlob Frege. Frege’s own views on the nature of geometry were largely ‘traditional’ in the sense outlined above. He was in several particulars a follower of Kant, and he considered geometry to be a concrete science whose basic truths were immediately apprehended by way of our spatial intuition. Yet, despite his very traditional view of geometry, Frege was perhaps the most perceptive and precise critic of Hilbert’s new approach to axiomatic. Frege’s criticisms began when he engaged Hilbert in correspondence after reading the latter’s Foundations of Geometry. Frege began by harshly criticizing Hilbert’s use of terms like ‘axiom’ and ‘definition’. Given Frege’s somewhat condescending tone, it is unsurprising that Hilbert’s response was lukewarm, nor is it surprising that the correspondence was a brief one. Nevertheless, in the sequence of letters (and in the subsequent exchange of papers between Frege and Hilbert’s defender, Alwin Korselt) there are quite a number of thorny conceptual issues which have yet to be satisfactorily sorted out.

Subsequent commentators have largely sided with Hilbert and tend to read Frege’s attack as the result of an outmoded conception of geometry which forbid any real connection between his criticisms and the actual content of Hilbert’s epoch-making work. To put it glibly, Frege’s criticisms of Hilbert have mostly been dismissed as the curmudgeonly musings of a defender of the old guard, the dying gasp of an ancient and obsolete point of view. But, as Paul Rusnock has convincingly argued, and as I will argue myself, there is no conceptual connection between Frege’s rejection of Hilbert and his own views on geometry. Frege’s criticisms are almost entirely methodological, and are more directly connected to his views on the nature of certain basic logical concepts. (Moreoever, we can find in Frege’s writings on arithmetic the elements of a criticism of his own Kantian views of geometry). His understanding of the nature of axioms and definitions, for instance, was well ahead of its time and, as I will argue, his views (or concerns related to his views) only began to be taken seriously by Hilbert and others within the mathematical community during the great foundational crises in mathematics of the 1920s.

After having dismissed the connection between Frege’s views on geometry and his views on axiomatics, I discuss the content of his criticisms of Hilbert in detail. In order to understand the correspondence, I think we require some prior grounding in Frege’s own work, which centered around the clarification and rigorization of basic elements of mathematics as well as logic. In order to provide this grounding, I begin another brief historical portrait which should, I hope, help to show the mathematical and philosophical context in which Frege’s work occurred. His work in mathematics and logic can be seen, at least partially, as a continuation of the project of rigorization and conceptual clarification begun much earlier in the nineteenth century by the Bohemian philosopher-mathematician Bernard Bolzano and developed later by the German mathematician Karl Weierstrass and his circle of students. Bolzano and Weierstrass (among others) were concerned to place the branch of mathematics known as analysis on a firmer foundation and to achieve this goal they presented far more rigorous proofs and definitions than had hitherto been required by most mathematicians. Frege took this approach and brought it to bear on some of the most fundamental concepts of mathematics, e.g., ‘number’ and ‘function’ in an attempt motivated by the desire to show that mathematical concepts were specifically reducible to logical concepts, that mathematics was ultimately a special branch of logic. He also made huge advances in logic, both in the technical symbolic apparatus and in its conceptual underpinnings, some of which are relevant to my subsequent discussion and will accordingly be highlighted.

Frege’s criticisms of Hilbert are continuous with his work on the logical foundations of mathematics and with his insistence on very strict standards of logical rigor within mathematics. By the time of the correspondence, Frege’s work for a decade and more had centered around the painstaking reconstruction of the most basic elements of mathematics, as well as an intensely inventive approach to the rigorization of logical and mathematical proof procedures. It is not suprising, then, that his reaction to Hilbert’s fast and loose employment of terms like ‘definition’ and ‘axiom’, concepts which Frege had spent years thinking very seriously about, would have seemed irritating at best. But Hilbert’s concerns were different than Frege’s, as were the concerns of the majority of the mathematical world (which is clear from a cursory glance at the virulent polemics Frege deployed against the positions offered by the mathematical community), and Frege’s criticisms of Hilbert were met with eventual silence on Hilbert’s part, though Frege continued to respond to Korselt.

Frege’s chief criticisms of Hilbert’s work have to do with how Hilbert himself understood the nature of his results. Hilbert’s view was that his consistency and independence proofs were directly applicable to Euclidean geometry. But for Frege this was by no means obvious. For Frege functions and objects had a determinate ontological status (which I will discuss at great length later in the dissertation). So, when Hilbert moves from the uninterpreted system to its various models, the proofs resulting from that move are either about the abstract, uninterpreted structure itself, or else about the concrete instantiations given via ‘intrepretations’. By Frege’s lights, it is not at all clear how we can make sense of the sorts of claims that Hilbert makes, particularly regarding his independence proofs.

In order to better understand what it is that Hilbert might mean, Frege begins by trying to reconstruct what it is that his axioms are supposed to do. It seems clear that they are meant to define the primitive terms of some system. This is acceptable, as far as it goes, though still radically different from Frege’s own conception of axioms as primitive, intuitively grasped truths. But, even after accepting the axioms-as-definition view, Frege has trouble understanding what Hilbert is up to. First of all, if the axioms are to act as definitions, they can only define the total system of concepts and not the individual concepts in isolation. This is so because Hilbert employs primitive terms to define other primitive terms in what appears initially to be a circular manner. For example, ‘line’ will be defined by reference to ‘point’, whereas ‘point’ will subsequently be defined by reference to ‘line’. Hilbert himself admitted as much and suggested that the whole collection of axioms (or at least of circumscribed groups of axioms) is to be seen as a kind of contextual definition of a system of primitive concepts. So we can talk only of the system of concepts which is defined by any group or subgroup of the axioms and not the individual, isolated concepts themselves. Though it is obviously not to his tastes, Frege is willing to allow this form of definition to Hilbert, and even offers his own account of how this might be done rigorously. But, still, problems remain.

In addition to the collective nature of Hilbert’s ‘defining axioms’, Frege also has problems with Hilbert’s independence and consistency proofs, which were at the heart of his success. In order for Hilbert’s independence proofs to go through, we have to treat sub-groups of the axioms as if they defined the same concepts as the complete group of axioms, otherwise the ‘independence’ of one axiom from the partial group really only amounts to the internal consistency of the concepts described by that partial group, and says nothing about the larger, complete group about which we are supposed to have proved something. Because Hilbert’s definitional axioms cannot define any of the primitive concepts in isolation, any alteration in the axioms (e.g., the removal of one or the addition of another) necessarily alters the actual primitive conceptual system at issue. In other words, the alteration of the axiom set alters the very content of the discussion, so it is not at all apparent how (or if) we can speak of independence when the procedure used to prove independence opens with one system of concepts and closes with another. Similar arguments apply to the procedures required for Hilbert’s consistency proofs.

Whatever Hilbert’s axioms are capable of doing, they are not, according to Frege, capable of showing the consistency or independence of Euclidean geometry, nor are they capable of providing isolated definitions of the primitive terms of Euclidean geometry. Even stronger, it is not clear that they can prove the consistency of a set of axioms or the independence of an axiom from such a set for any system. Despite his incisive critical remarks regarding the nature of Hilbert’s view of definition and axioms, Frege does not give up entirely on Hilbert’s project. Instead he offers him the benefit of his finely honed logical vocabulary in order to show Hilbert what it is he himself has done. The last section of the chapter is dedicated to a detailed account of Frege’s proposed alterations.

In the next chapter I will argue that Hilbert only began to take Frege’s criticisms seriously much later in his career, during his ‘formalist’ period. In that chapter I will attempt to draw out the crucial dichotomy between Frege’s ontologically grounded, object-oriented approach, and Hilbert’s (initially) radically relational vision of the nature of mathematics.


Chapter Two: From Set Theory to Structuralism: Skolem’s Paradox and Benacerraf’s Dilemma

Basic aim and outline of the chapter:

The primary goal of this chapter is to connect Frege’s criticisms of Hilbert’s early axiomatic approach to Hilbert’s later formalism and also to the widespread practice of set-theoretic foundationalism within mathematics in the first half of the twentieth century. And, following this, to argue that structuralism as a position within the philosophy of mathematics arose (in large part) in response to problems with set-theoretic foundationalism pointed out by Paul Benacerraf in his 1965 paper ‘What Numbers Could Not Be’. Pursuant to this goal, I begin with a discussion of Hilbert’s formalist position and argue that it shows evidence of having taken Frege’s criticisms (or at least the general drift of those criticisms) more seriously. From here I discuss some of the conceptual aspects of the development of set-theory as a foundation for mathematics, and argue that there is a close similarity between the set-theoretic approach to axiomatics and the tempered formalism Hilbert developed in response to Frege. After showing this similarity and establishing some of the details of set-theoretic foundationalism, I move on to a few criticisms and problems with this approach, which echo in more specific forms the criticisms of Frege. Of particular interest here is Thoralf Skolem’s so-called ‘paradox’, evidence of which was first published in 1922. I will argue later in the chapter that the uproar which this ‘paradox’ (actually an interesting theorem) caused in the 1920s within mathematics and logic was mirrored much later within the philosophy of mathematics when Paul Benacerraf posed a more directly philosophical dilemma for set-theoretic reductionism. The final portion of the chapter outlines Benacerraf’s own suggestion for a ‘structuralist’ position within the philosophy of mathematics, a suggestion which will segue into the third chapter dealing with a few of the more interesting varieties of contemporary mathematical structuralism, and the issues which separate them.


Summary of the chapter:

After having discussed Frege’s suggested remedies for Hilbert’s early axiomatic approach, I move on in this chapter to Hilbert’s later, more self-consciously philosophical position, formalism.

With the increasing popularity of reductions of more complex branches of mathematics to set theory, the appearance (beginning at the dawn of the twentieth century and continuing into the 1920s) of the infamous set theoretic paradoxes had a startling effect on the mathematical community. While the new attitudes toward geometry had encouraged wildly new developments in that sub-discipline and elsewhere, the ‘truth’ (however this was conceived) of the old theorems of Euclidean geometry was never called into question. But with the set theoretic paradoxes, and the nascent belief that mathematics was ‘really’ about the universe of sets, the certainty that had pervaded mathematics for over two millennia began to wane. In response to this crisis (or cluster of crises), many gifted mathematicians attempted to shore up their science by clarifying its nature and its foundations. Of particular interest for my discussion here is the debate between formalism (championed most notably by Hilbert) and the intuitionism begun by L. E. J. Brouwer and developed considerably by Hermann Weyl, Arendt Heyting, and others.

After a brief discussion of the collapse in certainty which provides the stage for the great debates of the 1920s, I turn directly to a discussion of Hilbert’s formalism. At the most basic level, Hilbert’s position is that mathematics can be grounded in a very simple system of intuitively acceptable stroke symbols and certain transformation rules. Now, much of the reasoning employed within the more interesting areas of mathematics goes beyond the limits of the stroke symbol system alone, it requires principles like transfinite induction (for instance) which have no analogue within the system of stroke symbols. Hilbert calls the mathematics we can achieve by means of the stroke symbol system alone ‘contentual’ mathematics, while everything else is non-contentual. This picture of the foundations of mathematics is, at first glance, starkly different from Hilbert’s early attempt to maintain a radically relational picture, in which reference to a fixed ontological ground (like that implied by the formalist’s stroke symbol system) was absent. But, when we look at the relationship between the non-contentual (but ‘interesting’) portions of mathematics and contentual mathematics, we can immediately see that Hilbert still wishes to maintain large parts of the relational character he gave such importance to in his earlier work.

Here the debate between Hilbert and Brouwer (and the other intuitionists/constructivists) is helpful, and so I take a brief detour into Brouwer’s views on mathematics. Brouwer, like Hilbert, was very concerned with the appearance of antinomies within mathematics and the effect these antimonies had on mathematical certainty. Brouwer, again like Hilbert, began by developing a form of mathematics which was epistemologically tractable and whose theorems would be in a certain sense ‘surveyable’. For Brouwer this entailed a large-scale rejection of considerable portions of classical mathematics due to their reliance on intuitively unacceptable modes of reasoning (this list, for Brouwer, notably included any reasoning employing the principle of the excluded middle). It is interesting that in spirit, at least, Hilbert’s contentual mathematics and Brouwer’s intuitively acceptable mathematics are very similar. The key difference between Hilbert and Brouwer, a difference which led to a sometimes bitter personal feud, was to be found in Hilbert’s view of the connection between contentual and ostensibly non-contentual mathematics.

For, unlike the intuitionists and other radical constructivists who dismissed large portions of mathematics as meaningless, Hilbert was interested in salvaging the bulk of mathematics while avoiding the disastrous consequences of a free-wheeling set theory which seemed to have called the whole edifice into question. His solution was to accept any foray into the non-contentual realm as long as it could eventually be re-connected, at the end of a chain of non-contentual reasoning, back to the contentual world of stroke-symbols. In other words, if the theorems we prove using ‘meaningless’ methods of reasoning can be shown to remain theorems within the stroke symbol system, then those theorems remain acceptable. The fact that we had to make a foray into the strange and technically meaningless realm of transfinite reasoning does not affect the fact that the results remain valid for the limited realm of contentual mathematics. We might think of the non-contentual portions of mathematical reasoning that result in contentual theorems as loops connected to a straight line. We might simply cut the loops and leave their end points on the straight lines; the loops themselves are dispensable and their removal leaves only acceptable, meaningful mathematics.

The specifics of Hilbert’s formalism and its divergences from intuitionism are of course interesting, but more interesting for my purposes is the shift from his earlier radically relational position to a relational structure that is ultimately grounded in an ontologically fixed system. In his earlier work, Hilbert eschewed Frege’s demands for a fixed referential system, and ignored also his criticisms that the independence and consistency proofs simply could not do what they were said to do without this kind of grounding in a fixed system. By the time he developed his formalist position, however, I argue that Hilbert had begun to take the problems pointed out in Frege’s critique far more seriously (by way of Russell and Whitehead’s Principia Mathematica and his problems with his own earlier attempts to get rid of the relativity of his consistency proof of geometry through an eventual grounding in the consistency of arithmetic). His turn to the problems pointed out by Frege resulted in a new position (his formalism) which resembles in a loose sense the very position Frege himself had outlined as a remedy much earlier. Hilbert in the meantime had accepted the relativity of his consistency proofs but began (beginning about 1917) to seek a deeper, absolute consistency proof which would ground the entire relational structure, proving once and for all what he had had previously to defer.

Hilbert’s type of view (which combined the relational quality of ‘higher level’ mathematics with a belief in the ultimate reducibility of the relational structures to some basic, fixed system) became popular throughout the 20s and 30s in various guises. Of particular importance for my purposes in this chapter is the increasing popularity of set-theoretic reductionism which was developed throughout the first third of the twentieth century and which remains popular in various forms up to the present day. The foundational, set theoretic view combines the sort of axiomatic approach developed by Hilbert in his geometrical work with an ontologically fixed, absolute universe (much as Hilbert attempted to do with his formalist work). Just as Hilbert’s stroke symbols were supposed to be the anchor that guaranteed the certainty of the more aerie heights of abstract mathematics, so, too, sets were thought to be the intuitively acceptable objects which could provide a ground for things like consistency proofs in more specific mathematical areas. The problems of set theory were of course well-known at this time but various systems were proposed to remedy the situation, and slightly limited axiomatizations of set theory seemed to avoid the complications at least to the satisfaction of mathematicians not overly concerned with the philosophical motivations upon which they rested.

After having discussed Hilbert’s formalism and its analogue in set theoretic foundationalism, I turn at this point in the chapter to problems facing those sorts of views. Two instances are of particular note for my purposes, namely Skolem’s so-called paradox and Benacerraf’s dilemma. These two problems strike directly at the very conception of set theory as a fixed point which grounds the vast relational edifice of mathematics. In 1922 the Norwegian mathematician and logician Thoralf Skolem delivered a lecture (subsequently published) which illustrated quite precisely the relativity (or ‘relationality’) of set theory itself.

At this point in the chapter I discuss the details of Skolem’s so-called paradox as well as its reception, particularly its reception by set theorists like Ernst Zermelo who were busy constructing the apparatus required to treat set theory in the foundational, absolute manner discussed above. When Skolem illustrated the relational character of set theory, he undermined (I argue) the attempt to stop the regress of relational structures in an absolute universe. What Skolem’s ‘paradox’ shows is that the universe of sets itself is a relational structure, and hence the regress does not stop, or at least not in the satisfactory manner envisioned by Hilbert’s demand for an absolute consistency proof. Perhaps because of the major conceptual problem that Skolem’s work posed for the widely popular practice of set-theoretic reductionism, its meaning was not immediately apparent, even less was it accepted and understood.

Following my discussion of Skolem’s paradox and its reception, I skip ahead, historically, to 1965 and the publication of Paul Benacerraf’s paper “What Numbers Could Not Be”. If Skolem’s 1922 paper brought the relativity of set theoretic notions to the attention of the mathematical world, then Benacerraf’s paper did much the same for the philosophy of mathematics.

I turn, then, to a discussion of the content and implications of Benacerraf’s paper. The paper constructs a thought experiment in which two young girls are taught alternative set-theoretic conceptions of the natural number sequence. The underlying set theory remains the same, but the way in which the natural numbers are reduced to sets differs. One girl learns the von Neumann ordinals, while the other employs Zermelo ordinals. Now, the dilemma that Benacerraf poses is this: if the natural numbers just are sets (as many forms of reductionism hold), then which sets precisely are they? If the natural numbers ‘are’ some sets, there should be a determinate answer to the question as to which set the number 2 is, for example. Benacerraf goes on to argue what any set theorist would accept, namely that there is no principled reason for employing one or the other reduction: both adequately fulfill the demands of number theory and both are equally well-grounded in the same sort of axiomatic set theory. None of this was terribly novel at the time, and it was widely accepted that a variety of different reductions were possible. The important part of Benacerraf’s paper (for my purposes) is that he points out the strange disconnect between the belief in set theory as the ground for mathematics and the belief that multiple different methods of reduction are possible. Equally important are the makings of a positive philosophical position which Benacerraf briefly develops at the end of the paper. There he argues that the numbers aren’t any ‘objects’ at all, hence it makes no sense to ask which sets the numbers are. Rather, the numbers should be construed as the abstract elements of an abstract structure, i.e., as elements which possess no qualities other than those which are determined by their position within that structure. Benacerraf’s dilemma leads him to a structuralist position within the philosophy of mathematics, and this position begins as a radically relational account of the nature of mathematics that is similar to Hilbert’s early work. In the following chapter I begin my discussion of structuralism within the philosophy of mathematics in earnest.