- Philosophers and logicians have been so busy trying to provide mathematics with a "foundation" in the past half-century that only rarely have a few timid voices dared to voice the suggestion that it does not need one. I wish here to urge with some seriousness the view of the timid voices. I don't think mathematics is unclear; I don't think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs "foundations." The much touted problems in the philosophy of mathematics seem to me, without exception, to be problems internal to the thought of various system builders. The systems are doubtless interesting as intellectual exercises; debate between the systems and research within the systems doubtless will and should continue; but I would like to convince you (of course I won't, but one can always hope) that the various systems of mathematical philosophy, without exception, need not be taken seriously.
- By way of comparison, it may be salutary to consider the various "crises" that philosophy has pretended to discover in the past. It is impressive to remember that at the turn of the century there was a large measure of agreement among philosophers — far more than there is now — on certain fundamentals. Virtually all philosophers were idealists of one sort or another. But even the non-idealists were in a large measure of agreement with the idealists. It was generally agreed any property of material objects — say, redness or length — could be ascribed to the object, if at all, only as a power to produce certain sorts of sensory experiences. When the man on the street thinks of a material object, according to this traditional view, he really thinks of a subjective object, not a real "external" object. If there are external objects, we cannot really imagine what they are like; we know and can conceive only their powers. Either there are no external objects at all (Berkeley) — i.e., no objects "external" to minds and their ideas — or there are, but they are Dinge an sich1. In sum, then, philosophy flattered itself to have discovered not just a crisis, but a fundamental mistake, not in some special science, but in our most common-sense convictions about material objects. To put it crudely, philosophy thought itself to have shown that no one has ever really perceived a material object and that, if material objects exist at all (which was thought to be highly problematical), then no one could perceive, or even imagine, one.
- Anyone maintaining at the turn of the century that the notions "red" and "hard" (or, more abstractly "material object") were reasonably clear notions; that redness and hardness are non-dispositional properties of material objects; that we see red things and see that they are red; and that of course we can imagine red objects, know what a red object is, etc., would have seemed unutterably foolish. After all, the most brilliant philosophers in the world all found difficulties with these notions. Clearly, the man is just too stupid to see the difficulties. Yet today this "stupid" view is the view of many sophisticated philosophers, and the increasingly prevalent opinion is that it was the arguments purporting to show a contradiction in the view, and not the view itself, that were profoundly wrong. Moral: not everything that passes — in philosophy anyway — as a difficulty with a concept is one. And second moral: the fact that philosophers all agree that a notion is "unclear" doesn't mean that it is unclear.
- More recently there was a large measure of agreement among philosophers of science — far more than there is now — that, in some sense, talk about theoretical entities and physical magnitudes is "highly derived talk" which, in the last analysis, reduces to talk about observables. Just a few years ago we were being told that 'electron' is a "partially interpreted" term, whereas 'red' is "completely interpreted." Today it is becoming increasingly clear that 'electron' is a term that has complete "meaning" in every sense in which 'red' has "meaning"; that the "purpose" of talk about electrons is not simply to make successful predictions in observation language any more than the "purpose" of talk about red things is to make true deductions about electrons; and that the whole question about how we "introduce" theoretical terms was a mare's nest. I refrain from drawing another moral.
- Today there is a large measure of agreement among philosophers of mathematics that the concept of a "set" is unclear. I hope the above short review of some history of philosophy will indicate why I am less than overawed by this agreement. When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need. And I include my own past efforts in this direction.
- I do not, however, mean to disparage the value of philosophical inquiry. If philosophy got itself into difficulties with the concept of a material object, it also got itself out; and the result is some modest but significant increase in our clarity about perception and knowledge. It is this sort of clarity about mathematical truth, mathematical "objects," and mathematical necessity that I should like to see us attain; but I do not think the famous "isms" in the philosophy of mathematics represent the road to that clarity. Let us therefore make a fresh start.
- A Sketch of My View. I think that the least mystifying way for me to discuss this topic is as follows: first to give a very cursory and superficial sketch of my own views, so that you will at least be able to guess at the positive position that underlies my criticism of others, and then to survey the alleged difficulties in set theory. Of course, any philosopher hates ever to say briefly, let alone superficially, what his own view on any topic is (although he is delighted to give such a statement to the view of any philosopher with whom he disagrees), because a superficial statement may make his view seem naive or even downright stupid. But such a statement is a great help to others, at least in getting an initial orientation, and for that reason I shall accept the risk involved.
- In my view the chief characteristic of mathematical propositions is the very wide variety of equivalent formulations that they possess. I don't mean this in the trivial sense of cardinality: of course, every proposition possesses infinitely many equivalent formulations; what I mean is rather that in mathematics the number of ways of expressing what is in some sense the same fact (if the proposition is true) while apparently not talking about the same objects is especially striking.
- The same situation does sometimes arise in empirical science, that is, the situation that what is in some sense the same fact can be expressed in two strikingly different ways, ….
Footnote 1: Dinge an sich: “things in themselves”.
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