Towards a Philosophy of Real Mathematics

Although directed toward philosophy, Corfield’s book is of interest to mathematicians, especially now in a climate in which mathematics seems to be searching for new directions. The book’s title alludes to the British Campaign for Real Ale (CAMRA), a movement dedicated to maintaining traditional brewing techniques in the face of inundation of tasteless, fuzzy beers marketed by powerful, industrial-scale breweries” (p. 1). Corfield toyed with titling a talk he once gave “Campaign for Philosophy of Real Mathematics” until “it was sensibly suggested to me that I should moderate its provocative tone, and hence the present version” (p. 2). Nonetheless, he cannot be charged with pulling his punches. For example, his writes:

The intention of this term [(“philosophy of real mathematics”)] is to draw a line between work informed by the concerns of mathematicians past and present and that done on the basis of at best token contact with its history or practice. For example, having learned that contemporary mathematicians can be said to be dealing with structures, your writing on structure without any understanding of the range of kinds of structure they study does not constitute for me philosophy of real mathematics. (p.3)

A bit after this passage, he remarks: “By far the larger part of the activity which goes by the name of philosophy of mathematics is dead to what mathematicians think or have thought, aside from the unbalanced interest in the “foundational” idea of the 1880-1930 period, yielding too often a distorted picture of that time” (p. 5). He wants to tempt philosophers away from their concentration on foundational mathematics. (Corfield’s diagnosis that philosophy of mathematics is distant from mathematics is, I expect, uncontroversial.)

Why does this situation apply in philosophy of mathematics, but not in philosophy of physics, which, Corfield assesses, is governed by “a strong common belief that one should not stray away from past and present practices” (p. 5)? From his own experience with talking to philosophers of mathematics about this matter, Corfield suggests that there are two reasons given for this state of affairs, roughly as follows (p. 6). First, philosophers of mathematics feel that mathematics deals with longstanding truths, such as those from Euclidean geometry, unlike physics in which theories are continuously modified and rejected. “[E]ven if one wished to take a Lakatosian line by analysing the production of mathematical knowledge and the dialectical evolution of concepts, there is no need to pick cases studies from very recent times, since they will not differ qualitatively from earlier ones, but will be much harder to grasp” (p. 6). Second, philosophers of mathematics contend that relevant foundational mathematics was established by 1930, whereas physics “is still resolving its foundational issues: time, space, causality, etc.” (p. 6). In response to the first reason, Corfield observes that “the relevant constellation of absolute presuppositions, scene of inquiry, disciplinary matrix, or however you wish to phrase it, has simply changed” (p. 7). Furthermore, we have something to learn from studying modern mathematical tools, which have far eclipsed the crude implements of pre-twentieth century mathematics. In responding to the second reason, Corfield questions the assumption of a “foundationalist filter” (his term) on which the reason seems to rest.

Straight away, from simple inductive considerations, it should strike us as implausible that mathematicians dealing with number, function, and space have produced nothing of philosophical significant in the past seventy years, in view of their record in the previous three centuries. Implausible, that is, unless by some extraordinary event in the history of philosophy a way had been found to filter, so to speak, the findings of mathematicians working in core areas, so that even the transformations brought about by the development of category theory, which surfaced explicitly in the 1940s algebraic topology, or the rise of non-commutative geometry over the past seventy years, are not deemed to merit philosophical attention. The idea of a ‘filter’ is precisely what is fundamental to all neo-logicism. But it is an unhappy idea. Not only does the foundationalist filter fail to detect the pulse of contemporary mathematics, it also screens off the past to us as not-yet-achieved. Our job is to dismantle it, in the process demonstrating that philosophers, historians and sociologists working in pre-1900 mathematics are contributing to our understanding of mathematical thought, rather than acting as chronicles of proto-rigorous mathematics. (p. 8)

In line with his attempts to lure philosophers away from narrow foundationalist concerns, Corfield suggests that a variety of factors govern the course of mathematicians: “(a) logical and calculational correctness; (b) plausibility; (c) psychological factors; (d) technological factors; (e) sociological and institutional factors; (f) relation with others sciences; and (g) inherent structure” (p. 25). Along similar lines, he recommends “work[ing] out varied ways to liberate ourselves from the appeal of timelessness” (p.20). The significance and import of a piece of mathematics is not fixed in time. He writes:

[W]e might liken doing mathematics to kneading dough. If your time-scale allowed you only to describe the piece being flattened and stretched, you would miss the coming together of widely separated points during the folding process. As a piece is torn off and set aside, you might believe it looked finished. But had you waited a little longer you would have seen it under the knuckle to be reshaped into the rest of the dough. No mathematical concept has reached definitive form – everything is open to reinterpretation…(p. 21)

Corfield remarks, referring to the task of philosophers: “[w]e are faced with an enormous and daunting choice” (p. 21) about what field of mathematics to investigate. “[W]e need to make a systematic effort to engineer space for ourselves to work with a wide range of issues” (p. 20). Worthwhile philosophical observations can be, he suggests, drawn from recent developments in mathematics, including those within branches of mathematics, not just the whole of mathematics. His suggestions include the following three: mathematical projects to come up with a non-commutative geometry, probability and stochastic reasoning in mathematics, and higher-dimension algebras (beyond the linear) (p. 22).

Corfield’s philosophical notions arise from the direct experience of mathematicians. Take, e.g., the concept of inherent structure:

The idea of inherent structure, which may be chimera, answers to the sense that mathematics offers resistance to the mathematician well beyond that attributable to correctness within some universal calculus, and yet not just emerging from disciplinary training. For example, there are a number of ways to generalize or deform the concept of a group rigorously, but only very few have important properties. (p. 30).

He is also interested in realism in mathematics “but independent of the idea that mathematical entities really exist in some ontological realm or other” (p. 31). Instead,

[this realism] is about seeing things ‘correctly’, that is, how they ‘really’ are. This is the type of realism which underlies a suggestion that a mathematician may have ‘glimpsed’ something decades before another brings it into sharp focus. The key words here are ‘natural’ and ‘fundamental’. Mathematicians use them all the time to describe features of their work that appear to arise from the nature of the domain they are studying, rather than being externally imposed. (p. 31)

Corfield’s idea of the “natural” bears on his attitude toward set-theoretic underpinning of mathematics.

While set theory displays certain ‘foundational’ virtues, we must recognize that reformulating a piece of mathematics that way may run against its ‘grain’…. The price paid for universality is unnaturalness. Instead of seeing mathematical entities and constructions merely as ultimately composed of set theoretic dust, we should take into account structural considerations… (p. 31)

Titles of chapters yield a glimpse of their content: “Communicating with automated theorem provers”, “Automated conjecture formation”, “The role of analogy in mathematics”, “Baysianism in mathematics”, “Uncertainty in mathematics and science”, “Lakatos’s philosophy of mathematics¹ Beyond the methodology of mathematical research programs”, “The importance of mathematical conceptualization”,² and “Higher-dimensional algebra”.³ The chapters are all thoroughly informed by the content of advanced mathematics. This content makes it quite clear that practising Corfield’s version of philosophy of mathematics demands conversancy with advanced mathematics.

Mathematicians engaged in discussing the future of mathematics will likely appreciate Corfield’s book. There is some searching for new directions going on in mathematics. Borwein and Stanway (2005), e.g., propose a range of ways in which norms in mathematical practice should change. These include lending legitimacy to “almost certain” knowledge provided by mathematical software, recognizing the importance of computer visualization as an aid to discovery, and allowing the loosening of centralized mathematical authority in the face of easy availability of web-based publication. Seeking new directions is not confined to individuals. In October, 2004, a meeting representative of the Canadian mathematical community produced a paper entitled “Towards a global strategic plan for Canada’s Mathematical Sciences Community”. The paper includes this remark:

During the past decade, the Canadian mathematical sciences community has undergone a fundamental re‑evaluation and has developed a new mindset with a much greater appreciation of, and commitment to, interdisciplinary research and training, as well as industrial applications of mathematics. This transformation has been distinguished by a strong emphasis on innovation: emerging areas, areas at the interface of different branches of mathematics, areas that cross traditional discipline boundaries.

A climate of seeking new directions lends itself to looking to philosophy for conceptual resources and to engaging in philosophy. Along this line, Borwein and Stanway (2005) remark: “[W]hile a neglect of philosophical issues does not impede mathematical discussion, discussion about mathematics quickly becomes embroiled in philosophy…” (p. 7). With this observation in mind, Corfield’s book can be greeted as a welcome addition to an emerging discussion. As Harris (forthcoming) puts it, this book “is definitely part of the ‘conversation’”.

Notes:

1. Corfield’s account of Lakatos’s philosophy of mathematics clarifies and, in part, helps resolve some puzzling aspects of this philosophy, in particular, Lakatos’s view about axiomatization.

2. This chapter investigates the merit of groupoids (p. 211) as a basic algebraic structure. (Corfield uses the term “groupoids” in the sense of the structure first defined by Brandt in 1926.) The approach of Corfield can be partially glimpsed in this remark:

Philosophers should note that in the case treated here arguments for the ‘existence’ of groupoids did not figure in the array surveyed. This is through no oversight on my part –mathematicians make no use of the idea in their advocacy of the groupoid concept. On the other hand, turning to the arguments they do use, it is reasonable to wonder why so many different types are employed. I would explain this by pointing out that individual mathematicians weight the candidate criteria for progress idiosyncratically. (p. 230)

3. Michael Harris (forthcoming) writes that Corfield (in Towards a Philosophy of Real Mathematics) “is remarkably well-informed about trends in the most diverse branches of mathematics”.

References

“Towards a global strategic plan for Canada's Mathematical Sciences Community.” (2004). http://www.caims.ca/Society/Liaison.htm .

Borwein, J. and T. Stanway (2005). “Knowledge and Community in Mathematics.” The Mathematical Intelligencer 27(2): 7-16.

Harris, M. “Why mathematics?" you might ask”. http://www.math.jussieu.fr/~harris/ . To be published in the forthcoming book The Princeton Companion to Mathematics.

Dennis Lomas (dlomas@upei.ca) has studied computer science (MSc), mathematics (half dozen, or so, graduate courses), and philosophy (PhD). He resides in Prince Edward Island (Canada).