The n-Category Café

But, as John points out, I want to know “how you can do mathematics well”, and I take this to include the a study of the way the mathematical community operates. So I am interested in description and normitivity, an exercise R. G. Collingwood would have called criteriological. Let me re-quote MacIntyre on the tasks of philosophy:

…philosophy needs to be conceived as having at least a fourfold subject matter and a fourfold task. There is first of all that which has to be learned empirically: the rules and standards, concepts, judgments, and modes of argumentative justification, actually embodied in or presupposed by the modes of activity which constitute the life of the social order in which one is participating. Secondly, there are the dominant ways of understanding or misunderstanding those activities and the relevant rules and standards, concepts, judgments, and modes of argumentative justification. Thirdly, there is the relationship between these two in respect of how far the second is an adequate, and how far an inadequate and distorting representation of the first. And finally there is that of which a philosopher must give an account, if she or he is to vindicate the claim to have been able to transcend whatever limitations may have been imposed by her or his historical and social circumstances, at least to a sufficient extent to represent truly the first three and so to show not just how things appear to be from this or that historical and social point of view, but how things are.

Clearly this is meant to include ordinary political social orders, which it must be said are more chaotic in terms of its “rules and standards, concepts, judgments, and modes of argumentative justification” than a society such as that formed by mathematicians. But is there room for complacency here? I would say “No”, certainly not if Alexandre Borovik is even only half right when he writes about Mathematics in Crisis due to its failure to bring on the next generation. Well, perhaps this is not mathematics’ fault, you may say, internally we’re still doing fine, and externally what can one do in the current materialist culture? But is there not the faintest chance that what we on this blog perceive as lacking in mathematical communication, which we try to counteract by our discussions, is in some way connected with a failure to communicate outside the community to recruit the young? In the UK, for example, did academics allow private examination boards to gain control of the syllabuses of our teenagers in silence. (See the analysis of a typical examination question by Borovik in his Mathematical Abilities and Mathematical Skills.) Had there been a livelier ongoing discussion of the nature of mathematics, one which might have be heard outside of mathematics, could this have happened?

Perhaps mathematicians can lay some of the blame at the feet of philosophers. Philosophical representations of mathematics might be expected to play their role both internally to mathematics, as well as in its external relationships. Concerning the internal state of mathematics, certainly there have been voices - such as Lakatos’s - calling for a much greater openness, but there have not been many. Greek mathematicians were better served. Besides analyses of reasoning within the community, external considerations were also deemed important. Indeed Plato writes in The Republic Book VII:

After plane geometry, I said, we proceeded at once to solids in revolution, instead of taking solids in themselves; whereas after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have followed.

That is true, Socrates; but so little seems to be known as yet about these subjects.

Why, yes, I said, and for two reasons: –in the first place, no government patronises them; this leads to a want of energy in the pursuit of them, and they are difficult; in the second place, students cannot learn them unless they have a director. But then a director can hardly be found, and even if he could, as matters now stand, the students, who are very conceited, would not attend to him. That, however, would be otherwise if the whole State became the director of these studies and gave honour to them; then disciples would want to come, and there would be continuous and earnest search, and discoveries would be made; since even now, disregarded as they are by the world, and maimed of their fair proportions, and although none of their votaries can tell the use of them, still these studies force their way by their natural charm, and very likely, if they had the help of the State, they would some day emerge into light.

Think just how far we’ve come from our philosophical origins. Do the 856 pages of the The Oxford Handbook of Philosophy of Mathematics and Logic give even a hint that we should expect a quest for knowledge, such as mathematics, to have a political dimension? How many of these pages concern the lived experience of doing mathematics? Let’s recall MacIntyre’s warning:

Philosophers do in fact become irrelevant to others not only by making their utterances inaccessible, but also by losing sight of the often complex and indirect connections between their own specialized, detailed and piecemeal enquiries and those larger questions which give point and purpose to the philosophical enterprise, which rescue it from being no more than a set of intellectually engaging puzzles.

One philosophy/theology blog I very much enjoy is the always interesting Siris. This post pointed me to a discussion by Mary Midgley of the stance of her fellow moral philosopher Iris Murdoch. This extract begins with Midgley quoting Murdoch from The Sovereignty of Good:

We are anxiety-ridden animals. Our minds are continually active, fabricating an anxious, self-pre-occupied, often falsifying veil which partially conceals the world.

What chiefly pierces that veil is a sharp, direct perception of things which are no part of our own being. For instance:

I am looking out of my window in an anxious and resentful state of mind, oblivious of my surroundings, brooding perhaps on some damage done to my prestige. Then suddenly I observe a hovering kestrel. In a moment everything is altered. The brooding self with its hurt vanity has disappeared. There is nothing now but kestrel. And when I return to thinking of the other matter it seems less important … (page 84)

The veil, however, is persistent and terribly hard to detect. In every age it subtly provides new, unnoticed ways of evading reality. Detecting those new forms is a prime business of philosophy, but of course philosophers often find it no easier than other people.

To return to our health book, for which I have set up what I intend to be a fairly casual blog, Darian and I have tried to penetrate the veil formed by the way illness is represented by medical researchers and medical practitioners, and by the pharmaceutical industry. But what difference is there between veil-detection and the sociologists’ unmasking? It’s simply the belief that there is an objective notion of goodness as applicable to the way we conduct our affairs. Certainly, value judgement terms can be wielded merely to bolster positions of power, but then they are being applied incorrectly.