I want now to criticize, as briefly as I can, the philosophy of mathematical practice. I don’t want to criticize the numerous contributions which have been filed under this name recently, and are often interesting and serious studies, but rather the idea that this approach to the philosophy of mathematics should become a veritable “discipline” and the idea that the philosophy of mathematical practice is something new and young, as it is often claimed.
Philosophy of mathematical practice, whither now?
Around ten years ago a book appeared, edited by Paolo Mancosu, entitled “The philosophy of mathematical practice”. Soon after an Association was created, to stimulate:
research in philosophy of mathematics from the perspective of mathematical practice … [by bringing] together researchers that work on a variety of topics ranging from the way mathematics is done and evaluated to the study of its epistemology, its history and the educational strategies associated to it.
The passage just quoted is vaguely circular, but its message is clear: to understand what philosophy of mathematical practice is about one should look at epistemological or cognitive problems stemming from actual or past mathematics. This loose characterization is not obvious, even if it may seem so, especially if considered against what has been referred to as “mainstream” philosophy of mathematics. Paolo Mancosu contrasts the perspective of the mainstream philosophy of mathematics of analytical inspiration with the perspective brought aboyt by the philosophy of mathematical practice:
Contemporary philosophy of mathematics offers us an embarrassment of riches. Anyone even partially familiar with it is certainly aware of the recent work on neo-logicism, nominalism, indispensability arguments, structuralism, and so on. Much of this work can be seen as an attempt to address a set of epistemological and ontological problems that were raised with great lucidity in two classic articles by Paul Benacerraf. Benacerraf’s articles have been rightly quite influential, but their influence has also had the unwelcome consequence of crowding other important topics off the table. In particular, the agenda set by Benacerraf’s writings for philosophy of mathematics was that of explaining how, if there are abstract objects, we could have access to them. And this, by and large, has been the problem that philosophers of mathematics have been pursuing for the last fifty years.
As Mancosu put it, although “Contemporary philosophy of mathematics” (an overview of which can be found here) may have gained a considerable number of riches, it surely did so at the price of keeping references to history of mathematics and to mathematics itself to a minimum. This attitude resonates with a usual feature of analytic philosophy, which also disparages history in favour of fully-fledged rational arguments.
However, if such an operation might have the benefits of curtailing the subject matter of philosophy of mathematics in a way that allows one to clearly formulate an agenda of issues and questions, easily identify a “field” of research in order to get published in international journals and being recognised as a legitimate member of a group, community or party, this comes with the price of impoverishing the spectrum of interests and of cutting the connections with the discipline in flesh and bone (another unofficial “manifesto” of the movement is emphatically called: “Towards a philosophy of real mathematics”). Freed from historical encrustations but also from many of its connections with matters mathematical from the past and the present, mainstream philosophy of mathematics, even if addresses questions of general interest about the nature of mathematical knowledge and its objects, tends to fossilize into a vast amount of technical discussions whose interest for the working mathematician, the educator or the student is often next to nil. Too often the impression of philosophy of mathematics that one gathers from the outside is that of a self-contained discipline, focussed on discussions which end up having a life of their own, loosely connected or wholly separated from the actual body of mathematics, its history and its context. Rueben Hersch pinned down the difficult relationship between mathematics and its philosophy in terms which, although written more than 30 years ago, seem to be valid still today:
It has to be said that if a mathematician, uncomfortable with his philosophical confusion, looks for help in the books and journals in his library, he will be badly disappointed. Some philosophers who write about mathematics seem unacquainted with any mathematics more advanced than arithmetic and elementary geometry. Others are specialists in logic or axiomatic set theory; their work seems as narrowly technical as that in any other mathematical specialty.
I think that the difficulty in finding any mathematical insight in contemporary, mainstream philosophy of mathematics, i.e. to find any enlightment that philosophical debates may give to the mathematician or the historian of mathematics, should be regarded as a legitimate and serious concern, instead of being looked down as a naive question, when such a question is actually raised (has philosophy ever clarified mathematics?).
We can only welcome the philosophy of mathematical practice if, as its promoters often proclaim, will bring grist to the philosopher’s mill by reconciling existing philosophical debates about mathematics with the content of actual mathematics or its history, or by finding new themes worth of philosophical reflection out of the body of mathematics itself. One could imagine that philosophy fruitfully interacts with mathematics in at least three ways:
i) Mathematical practice may offer arguments, or raise problems in order to rediscuss classical issues in philosophy of mathematics (what are mathematical objects, how do we know them?).
ii) mathematical practice may provide new philosophical issues for the philosopher to reflect upon.
iii) philosophy of mathematics could be use in a normative sense, i.e. when it dictates its own agenda upon mathematical practice (what is good mathematics? whatnot?) or, more humbly, could be used as an instrument for cautious self-criticism, for instance by establishing which kind of philosophical stance can damage less the development of a certain area of mathematics.
I believe that all, or most of these points are already included in the agenda of the philosophy of mathematical practice. However, philosophers of mathematics (or mathematical practice?) are not happy with simple but clear descriptions, and show a tendency to discuss details to the extreme and to cut the hair into four, often only to achieve ridicolously sophisticated results. Thus Philip Kitcher has even formulated a very complicated attempt to define “mathematical practice”, and has come up with a mathematical structure – a quintuple, composed by Language, Metamathematical views, accepted Statements, Reasoning methods and Questions. All this is very sophisticated, but at the same time I wonder how it can illuminate mathematics itself and the way it developed, or the way its objects are apprehended and known.
Discussions like those around Kitcher’s “quintuple model” replicate the vice of analytic philosophy of mathematics: they pose esoteric problems for the sake of clarifying matter for themselves and a restricted number of people in the business of philosophy of mathematical practice. In short, these discussions show the dangerous inclination to turning a broad reflection on mathematics and its practice into a self-referring discipline with its rank of experts and selected problems of little interest beyond the discipline itself.
Philosophy of mathematical practice, whence now?
Together with the danger of reproducing the problems of analytic philosophy of mathematics, i see another danger in the philmath approach. Many have pointed out at the novelty brought about by the agenda of the Philosophy of Mathematical Practice, but few have paused to examine it critically. I find this attitude a bit puzzling, especially if it comes from people who profess an attention for history. Without a proper justification, the repeated claims that philosophy of mathematical practice is “new” or that it brings “fresh air to the philosophy of mathematics” (as stated here) , or that it is a “young discipline” , as stated by J. P. Van Bendegem (See van Bendegem, J.P. (2014). The Impact of the Philosophy of Mathematical Practice to the Philosophy of Mathematics. In Soler, L., Zwart, S., Lynch, M., & Israel-Jost, V. (Eds.), Science after the Practice Turn in the Philosophy, History, and Social Studies of Science. New York- London: Routledge, 215-226), sound almost like propaganda to me.
So let us go back to our steps and search where this fresh air comes from. Very simply the contrast between the contemporary analytic philosophy of mathematics and the philosophy of mathematical practice as dressed up by Mancosu considerations, closely reminds of an essay by Lakatos, Proofs and Refutations, and particularly of the preface of the book. Lakatos sternly criticised there a “formalist” or “dogmatic” component of mathematical philosophy:
An abstraction of mathematics in which mathematical theories are replaced by formal systems, proofs by certain sequences of well-formed formulas, definitions by ‘abbreviators devices’, which are ‘theoretically dispensable’ but ‘typographically convenient’.
Formalist philosophy of mathematics “disconnects philosophy from history”, and leaves problems related to aspects like growth of mathematics or the “situational logic of problem-solving” aside. Much of what the critics of the analytic philosophy of maths objet to the latter. So it looks as if the underlying and polemical motive of philosophy of mathematical practice – namely that philosophy of mathematics should include themes and questions from mathematical research and neighbouring disciplines, mainly history of mathematics and logic – is not such a novelty and it was already perfectly realised in the past. To become convinced of that, let us take a look at the index of Lakatos’ problems in philosophy of mathematics, published in 1967. The book contains indeed a collection of papers from a conference held in Summer 1965. Surprisingly, perhaps, the problems of philosophy of mathematics discussed in the book are the following (I am merely copying and pasting the index): Euclidean dialectics and Greek Axiomatics, the metaphysics of the calculus; problems in set theory; Godel and philosophy of maths; informal rigor; a discussion on logic and mathematical education and a section on Frege. The collection also contains a series of contributions with the encompassing title: “Foundations of mathematics, whither now?”
Ironically, I have titled this section “the philosophy of mathematical practice, whence now?”, as it seems to me that enthousiasts and supporters of this approach to the philosophy of maths have forgotten their recent past too quickly.
What then the “philosophy of mathematics” was supposed to be in 1965-67? Certainly it was also a gamut of questions from history, logic and foundations of mathematics. The impression that I got is that it was possible, also back at the time, in bringing together discussions in logic, Frege, Greek philosophy of mathematics or Leibniz under the encompassing name of Philosophy of Mathematics. Perhaps this was not a dominant view, but certainly an existing one.
On the other hand, and this is the main point that I want to make here, I believe that many of the authors to Lakatos’ volume would have had no qualms in praising the agenda of the philosophy of mathematical practice, and recognizing it as their own agenda. We might perhaps extend the scope of our subject matter, and consider scientists-philosophers such as Poincaré, Einstein, Mach or Hermann Weil. Would have they found that the only reasonable way to do philosophy of mathematics was by referring philosophical problems to their actual practice as scientists? Probably this was their only conceivable way of doing philosophy of mathematics.
Hence, should we call all of them philosophers of mathematical practice? Perhaps it is the case, and the philosophy of mathematical practice is not such a young discipline, after all. But then, why avoiding something simpler like “philosophy of science, or of mathematics” to denote philosophical reflection upon science or upon mathematics? Given that there seems to be no real qualitative difference between what some have been doing in the past, when doing philosophy of mathematics, and what they are doing today, when doing Philosophy of mathematical practice, why conjure up a new expression to denote an approach that is not that new, or even to claim that a new discipline is there, when it fact it has existed for a long time, even if no need was felt to turn it into a special discipline, properly speaking? After all what we have been calling “the philosophy of mathematical practice” in the recent years was simply “philosophy of mathematics” in 1967 or earlier, and why not continuing to do so?
I do not know how and when things precisely started to change, so that philosophy of mathematics had narrowed down its focus to the limited and esoteric array of problems that we know of today. Certainly the philosophy of mathematics needs not be confined within such esoteric problems, and was not so even in our recent past. To bring fresh air to the philosophy of mathematics, it may be perhaps more useful to admit that the philosophy of mathematical practice is, after all, nothing more and nothing less than philosophy of mathematics.
MAKE MATTERS WORSE 