FOM: what sort of foundations
wtait at ix.netcom.com
Wed Nov 5 11:29:35 EST 1997
I like Stephen Ferguson¹s point that there are two prevailing ideas of foundations, that the first is epistemological and that the second is concerned with `conceptual analysis¹. But I think that what he says about the second is not entirely satisfactory. First of all, I note that he writes that it ``is driven solely by intellectual curiosity¹¹ and then, with some small inconsistency, that ``it was the first type of enquirey which got us started in this business¹¹. Neither is true of one sense of ``foundations¹¹ which would come under the heading of conceptual analysis. ``Foundations of¹¹ in this sense has somewhat the same meaning as ``principles of¹¹ or ``elements of¹¹; it is a matter of finding the first principles---primitive concepts, axioms, and definitions---of a science. It is a very old idea which can be found in Plato¹s Dialogues. Its motivation then, and I think now, is the realization that mathematics (exact science for Plato) is not literally true of what is given in experience----or better, does not even literally apply to it. Rather it is true of structures which may (but also very well might not ) be exemplified, albeit imperfectly, in experience, but which we can and must understand only intellectually. We do this by analyzing our idea to obtain the first principles (a process Plato calls ``dialectic¹¹): what is true of this structure then is what can be derived from the first principles. So it is not simply intellectual curiosity nor is it a question of establishing immunity to doubt that motivates foundations in this sense. Rather it is a matter of giving precise sense to propositions of the theory and so the criteria for their truth.
What is unique to our situation and clearly not contemplated by Plato is that, since Cantor¹s introduction of the transfinite numbers, it has followed that there can be no final (complete) foundations for mathematics. Whatever axioms we accept, closure of the transfinite numbers under these axioms leads to further axioms expressing the existence of still higher numbers.
I would like to drift from the subject for a moment and comment briefly on the bearing that this essential incompleteness has on a kind of Platonism¹ (having no more to do with Plato than the other kinds of Platonism¹ discussed in the present literature on philosophy of math). Much of what is taken to be a Platonist¹ position is really just a defense of ordinary mathematical grammar. E.g. if numbers are `created¹ then they exist in time; but it is simply grammatical nonsense to speak of their temporal location. Etc.
But there is one attitude towards the essential incompleteness of mathematics which I find hard to comprehend: namely that there is a fully determined totality of transfinite numbers or, equivalently, that the notion of a set of numbers is fully determined in its extension, in spite of the fact that no closure conditions---axioms---that we specify can ever fully determine that extension. I think that this is G"odel's position. But it leads to a kind of `Platonism' which goes beyond the simple defense of the ordinary grammar of mathematical discourse and affirms that there are mathematical `facts' which in principle escape the ordinary determinations in mathematics, e.g. by axioms, of its subject matter; and so mathematics becomes _speculative_. Moreover, on this view, there is no real solution of the paradoxes of set theory: one is left with the view that the numbers form a determinate totality in the same sense that sets are determinate, but it just doesn't happen to be a set. But now there is a genuine mystery: some determinate totalities are sets and others are not. I prefer the view that the concept of transfinite number is genuinely incomplete or open-ended, that there is no determination of it beyond what _we_ determine, by successivly stronger answers to the question: what properties of numbers determine sets? Our new answers may in some causal or psychological sense be contained already in the concept of transfinite number, but I would reject the view that they are _logically_ contained in it: logic begins after we have adopted the axioms. Before that, there is only dialectic: looking for the axioms.
Back to Ferguson¹s note: Although I agree with his evaluation of the first kind of foundations (viz. epistemological foundations), writing it off should not be thought to be too easy. It was, after all, foundations in _this_ sense that concerned Hilbert in his proof theory (though not in his _Foundations of Geometry_). It would seem also to be involved in the attempt to give a constructive foundation (in a less restrictive sense) for mathematics or parts of mathematics---or to restrict mathematics to what can be done constructively.
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