Ebook: Philosophy of Mathematics: Selected Readings

This is a useful anthology. I shall not argue its merits or demerits, but rather submit the following thesis: the shortcomings of virtually all of these theories are traceable to their unwarranted identification of mathematics with 20th century axiomatic mathematics. Bernays asserts that since "the customary manner of doing mathematics ... consists in establishing theories detached as much as possible from the thinking subject," any contrary view is "extreme" (p. 267). "The customary manner of doing mathematics" is thus glorified: it is tacitly assumed that "the customary manner" will reign supreme for all future. Otherwise it would not be "extreme" to suggest that "the customary manner" may not be infallible. Now of course one does not infer the absolute truth of an economic theory from its successful account of one particular society. But precisely this is being done in the case of philosophy of mathematics.

Let us begin with the ever-foolish logical positivists. They define their position against Mill, who "maintained that [mathematical] propositions were inductive generalizations based on an extremely large number of instances" (Ayer, p. 317). Incidentally we later see Hempel making the exact same argument (p. 378), the positivist herd being predictable as always. But back to Ayer. The argument against Mill is that mathematical propositions are unfalsifiable: "Whatever instance we care to take, we shall always find that the situations in which a logical or mathematical principle might appear to be confuted are accounted for in such a way as to leave the principle unassailed" (p. 319). Therefore, the doctrine goes, mathematical propositions are analytic a priori, they are true "by virtue of definitions" (Hempel, p. 379), "none of them provide any information about any matter of fact" (p. 321), their truth do not depend on "facts about the world" (p. 316). Putting aside for the moment the trivial objection that some scientific laws (such as the law of inertia) are also analytic, this is still trivially false. Consider the discovery of the binomial series and other power series in the 17th century. These were obviously tested. There is a tacit appeal to "the customary manner of doing mathematics" here again. We now know that power series can be embedded in an axiomatic system that makes them true "by virtue of definitions." But this is a fact of experience. Only "facts about the world" tell us that mathematical propositions are susceptible to such treatment, i.e., that "the customary manner of doing mathematics" is all-pervasive. The claim that mathematics is analytic is the claim that another world would be impossible. Here is another passage in Ayer which plainly presupposes "the customary manner of doing mathematics": "Appeal to intuition [is] a source of danger to the geometer. It has, indeed, been shown that Euclid himself was guilty of ... make[ing] assumptions which are accidentally true of the particular figure he is using as an illustration" (p. 325). So? Were these theorems false? No. Were the proofs incomprehensible? No. To prove that intuition is dangerous it is enough to prove that the reasoning involved differs from "the customary manner of doing mathematics." There is no need to show that intuition has ever led to a single error (indeed, no such proof is offered).

The logisists are putting the cart before the horse in assuming logic to be prior to arithmetic and mathematics. Hempel (p. 380), for example, simply takes for granted that a=c is implied by a=b and b=c and that this is prior to arithmetic. The possibility is never considered that we may know the instances of the rule with greater certainty than the rule itself and that the latter depends on the former. Likewise the possibility is never considered whether the reduction of mathematics to logic may be vicious: if the rules of logic are abstracted from primitives (e.g., arithmetic) then it is not surprising that latter may be defined in terms of the former. This ignorance is due to the modern conception that there are some transcendent "rules of the game" fixed before any specific axioms have been chosen. Without this contingent fact about "the customary manner of doing mathematics" the logicist reduction would have no basis whatever. Indeed, looking at the matter with an open mind suggests that the reduction is absurd since "our logical intuitions ... are self-contradictory" (Gödel, p. 452) whereas our arithmetical ones are not.

Now let us look at the so-called "intuitionists." Their choice of "intuitions" has more to do with philosophical convenience than a serious study of intuition. For example: "However weak the position of intuitionism seemed after this period of mathematical development [i.e, non-Euclidean geometry], it has recovered by abandoning Kant's apriority of space but adhering the more resolutely to the apriority of time." (Brouwer, p. 80). It is obvious that they care more about "the customary manner of doing mathematics" than about intuition in their treatment of geometry, which they dismiss as "reduce[able] ... to arithmetic by means of the calculus of coordinates" (Brouwer, p. 80). To put it extremely mildly, "it seems a bit hasty to deny completely the existence of a geometrical intuition" (Bernays, p. 264). Anyone seriously interested in intuition should do what Poincaré did (not in this volume, of course), namely investigate the nature of geometrical intuition by means of detailed arguments. But Brouwer's attachment to "the customary manner of doing mathematics" and his eagerness to create an ism precluded this.

The third major philosophy of mathematics goes under the name of formalism. But this is a diverse crowd. On the one had there is Curry's terribly naive position that "According to formalism the central concept in mathematics is that of a formal system" (p. 203). There is little point in noting how this captures "the customary manner of doing mathematics" and nothing else. On the other had there is Hilbert, who was not really advocating an ism at all. He was simply suggesting a research programme to prove the consistency of mathematics. This proposal was a very poor one, and of course one based on "the customary manner of doing mathematics," but it was no ism. This is why Curry thinks that Hibert's position is "peculiar": "The peculiar position of Hilbert in regard to consistency is thus no part of the formalist conception of mathematics, and it is therefore unfortunate that many persons identify formalism with what should be called Hilbertism." (Curry, p. 206). Apparently it is "peculiar" not want to start an ism. "The part of mathematical activity concerned with a good choice of axioms had no place in Hilbert's 'official' conception of mathematics. If there is any real justification for calling Hilbert's approach 'formalist' it is certainly this deficiency of Hilbert's official conception of mathematics and not his use of syntactic formulation in the foundations of mathematics." (Kreisel, p. 226). In other words: the only justification for calling Hilbert's approach formalist is to interpret is as an ism, even though it was never intended to be one.