Philosophia Mathematica (III), Vol. 13 No. 1 © Oxford University Press, 2005, all rights reserved
Book Review |
Martin H. Krieger. Doing Mathematics: Convention, Subject, Calculation, Analogy. Singapore: World Scientific Publishing, 2003. Pp. xviii + 454. ISBN 981-238-2003 (cloth); 981-238-2062 (paperback)
* Department of Philosophy, University of York York YO10 5DD England dc23{at}york.ac.uk
‘I want to provide a description of some of the work that mathematicians do, employing modern and sophisticated examples.’ Having read this opening statement of chapter 1, leafed through the text past terms such as ‘Langlands Program’, ‘Onsager phases’ and ‘Painlevé complex’, and taken a cursory glance at the index and bibliography, some readers of Philosophia Mathe-matica might have serious doubts as to whether Martin Krieger's book Doing Mathematics is a work of philosophy. No Frege, no Quine, no Boolos, no Benacerraf .... That the author takes himself to be very much engaged on a philosophical project is clear:
I take from Weyl the warrant for a philosophy of mathematics that is based on ‘what is really going on’, and on the connections with the historico-phi-loso-phical context. But, whatever else, such philosophizing is done through ma-the-ma-tical means, examining and explaining actual definitions, constructions, theorems, proofs, derivations and examples. Of course, one may prescribe how things ought to go, following an analogy or a program. But the facts of the matter—which definitions turn out to be the most fruitful, what you can actually prove—are in the end what counts. And ‘fruitful,’ ‘interesting,’ and ‘mathematics’ are not given a priori. They depend on the actual practice of mathematics, mostly by professional mathematicians. (p. 235)As someone who finds himself very much in agreement with these sentiments, I think this review may be of most service if I give some indications of how philosophy would have to reconfigure itself to allow Doing Mathematics even to be considered as a piece of philosophy, before the question of its merits is raised.
Let me begin, then, by sketching the topography of the land mass of contemporary Anglo-Saxon philosophy. Beginning near the Southern seaboard and stretching Northward across a wide swathe there lies a range of lofty peaks, bearing the names Epistemology, Metaphysics, Philosophy of Language, and Philosophy of Mind, where the high priests of philosophy dwell. Along the narrow coastal strip to the south of this range we find the philo-sophy of mathematics, cut off from the rest of philosophy by the mountain range, and connected by a very long and very thin bridge to the nation of Mathland. The existence of this bridge is both a source of pride and a source of concern to philosophers. Pride because the construction of the foundations for the bridge is believed to be largely the work of two philosophers, Frege and Russell, both now revered as demi-gods; concern because theories of knowledge, reference, and ontology have a nasty habit of sitting rather badly with mathematics.
Most of those living at the philosophical end of the bridge have now stopped visiting Mathland, relying instead on fading memories of brief childhood holidays, dusty old reports from Edwardian explorers and missionaries, and some occasional missives sent over by those inhabitants living closest to the bridge. The very few philosophers who have dared to cross the bridge and venture into the interior realise that the very small subpopulation of Mathlanders sending such missives are deemed by the remainder of the population to be unrepresentative of the inhabitants of the country as a whole. Philosophers who spend much time in deepest, darkest, Mathland, and who make the hazardous return journey over the bridge, arrive back amongst their compatriots with a wild look in their eyes, telling tales of unsurpassable beauty, and are generally ignored.
What Krieger and I share, along with a few others, is the conviction that this situation is untenable. If the set of questions philosophers of mathematics consider legitimate to ask precludes them from finding out about how mathematics is done and has been done, or even simply discourages them from doing so, then they are working with a wrong notion of legitimacy. It is simply a matter of a reductio. The issue then is how to redress the situation. Let us return, then, to our survey of Anglo-Saxon philosophy.
North of the range of the central peaks lie the provinces of moral and political philosophy, philosophy of social science, and aesthetics. In the regions of these provinces abutting the central range, much of the activity runs in accord with the strictures of the ruling paradigm, but towards the northern reaches we find communities operating under rather different principles. Many there show a keen interest in the history of the peoples of the land mass, which they interpret very differently from the official historians of the Lofty Peaks. The more anthropologically inclined discuss human existence in terms of language games and their associated forms of life, where the values operating in societies come under close scrutiny. Some speak dialects much influenced by languages of lands still further to the North, uttering mysterious terms such as ‘phenomenology’.
I believe there is enormous scope for philosophers of mathematics to forge alliances with the Northerners. Personally, I can read long stretches of a philosopher such as Charles Taylor discussing the wrongs of utilitarianism and arguing that we must focus on substantive values rather than procedural means in ethics, and with minimal modification transform them into something very relevant to mathematics. Unfortunately, many of the Northerners are none too scientifically literate, and do not realise that a satisfactory depiction of what mathematics is like is relevant to them, since this would provide insight into an activity of a very particular, yet very human nature, where the battle between freedom and constraint is especially closely poised.
Perhaps, our best hope is to turn to examine what is happening in the East, where we find situated the philosophy of science. Lines of communication with the neighbouring kingdom of science are so much richer than in the case of mathematics. Traffic is of a more variegated nature, and merchants have wandered more freely throughout philosophy. Some are also well versed in the practices of history and sociology. But, even amongst the established figures of this province dwelling closest to the central range, we find some interesting stirrings of new ways of thinking. Van Fraassen [2002]
in ‘The Empirical Stance' while criticising analytic metaphysics, invokes Sartre's theory of emotions to explain how it is possible for scientists trained in a tradition to contemplate radically new ideas. Meanwhile, Friedman [2001] appeals to Habermas to discuss communication within the scientific community.
Can we emulate the philosophy of science? Admittedly, at times the task seems overwhelmingly hard. You must understand the life of the indigenous people and translate their ways into a form comprehensible to those with only the slightest acquaintance of them. Forging strong ties between philosophy of mathematics and philosophy of science may be our best hope of acquiring an appreciative audience. The approach I adopt in my own book (Corfield [2003]) has been in part to revisit George Pólya's Bayesianism and Imre Lakatos's Research Programmes in the context of mathematics, languages I know philosophers of science will understand.
The more we bring philosophy of mathematics into contact with philosophy of science, the more we may be accused of not explaining what is distinctive about mathematics. But the only way to do this is to enter into the details of mathematical theories. The present time is especially auspicious as changes sufficiently profound to be called revolutionary are taking place in mathematics, and it is always useful to have a philosophical observer or two present when a revolution is underway, perhaps even to aid its passage. Of course this necessitates a considerable amount of exposition, leaving us open to the accusation that our work is merely descriptive. But there is no clear dichotomy between the descriptive and the normative, when one is describing something one considers well done.
Gaining the necessary experience to be able to depict contemporary mathematics exposes one to a risk of a kind that worries many in the philosophy of science—the risk of specialisation. That distinctive breed of philosopher of science known as the philosophers of physics may spend so much of their time in the land of physics that they have little inclination to return home, and so form expatriate communities, with the unfortunate consequence that the High Priests of the metaphysical heartland are not confronted with the fact that their ideas do not fit coherently with the discoveries of the physicists. If we are to emulate their level of sophistication, there is a considerable danger that those visitors wandering in the beautiful land of mathematics will feel tempted to stay, especially if when they return they are told that their discoveries are of no interest. Perhaps, we may be helped in mathematics by the fact that there are examples of contemporary rethinking of very basic concepts, e.g., reconsidering the integers gets you to knot theory surprisingly rapidly (cf. chapter 10 of my book).
Now, at last, we turn briefly to the book. As its title suggests, Krieger's foremost interest in mathematics is as an activity: mathematics is something some people do. Thus, he is continuing the anthropological work of his successful earlier book Doing Physics (Krieger [1992]). Where that book could find its place in the more freethinking philosophy of science, for the reasons explained above I imagine that this one will find life harder. The lack of any similar book in the philosophy of mathematics must surely have made it more difficult to write.
Krieger has set himself the remarkably ambitious target of conveying what a large slice of contemporary mathematics feels like. If he succeeds, a mathematician reading it would say, ‘That's the way it is. Just about right.’ (p. 1) The best word I can find for its style is ‘impressionistic’, and the correct impression he gives is of the staggering intricacy of modern theories. Contemporary mathematics is extraordinarily interconnected—number theory and statistical mechanics, algebraic topology and partial differential equations—every field relates to every other. But, perhaps for good reason, these connections take place through simple shared conceptions, such as composition, pasting, splitting, symmetry, sequence, re-presentation. The trouble is that these conceptions are worked over by mathematicians into a terrifically complex weave. That being so, given the ambitions of the author, I rather doubt that the reader will learn much mathematics through reading this book, but she will gain a sense of the range of theory she might strive to understand. The parts I enjoyed most were those where I already had a good idea of what was going on.
There is a tendency for those reacting to orthodox philosophy of mathematics to select aspects of algebraic topology to study. There are several good reasons for this: it is very representative of the twentieth century, it is well served by category theory, it provides a good counterpoint to point-set topology, it lies at the heart of much of the unificatory work of recent times, and it follows on from Lakatos's famous case study in Proofs and Refutations. Krieger could not help but treat this branch, but he treats an immense amount of other material as well, not only from mathema-tics, but also from (mathematical) physics and statistics. I was interested to note we had selected several references in common, e.g., André Weil's treatment of the role of analogy in mathematical research in a letter to his sister (translated in an appendix).
Krieger has the right idea that philosophy of mathematics must look beyond proof to take into consideration all of its aspects: analogies, diagrams, calculations, applications, etc. For the latter, he uses his experience of writing Constitutions of Matter (Krieger [1996]) to good effect to show what is at stake in a real example of calculation in mathematical physics, and how thoroughly intertwined its models can be with a surprising range of pure mathematical theories. Regarding the latter, there is plenty more scope for research. The revolution currently taking place in mathematics, which I mentioned above, has coincided with the drift of mathematicians from the former Soviet Union, e.g., Kontesevich, Manin, Drin'feld, Gromov, Kapranov, where the teaching of mathematics and physics was never so separate as in the West. String theory and quantum field theory are motivating an extraordinary amount of work at the forefront of contemporary mathematics.
As you will have gathered, we have a daunting task ahead of ourselves. It will take much more than Krieger's book and mine to get philosophy and mathematics back into sustained contact with each other. Krieger has driven a pile into the deep and wide waters that separate philosophy and mathematics. Bridge-builders of the future will have to decide whether and how to use it.
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REFERENCES |
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 Top References |
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Corfield, D. [2003]: Towards a Philosophy of Real Mathematics. Cambridge: Cambridge University Press.
Friedman, M. [2001]: Dynamics of Reason. Stanford: CSLI Publications.
Krieger, M. [1992]: Doing Physics: How Physicists Take Hold of the World. Bloomington, Indiana: Indiana University Press.
Krieger, M. [1996]: Constitutions of Matter: Mathematically Modeling the Most Everyday of Physical Phenomena. Chicago: University of Chicago Press.
Van Fraassen, B. [2002]: The Empirical Stance. New Haven: Yale University Press.
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