FOM: Physics and Math, Gardner and Hersh: some ambiguities wtait at
Wed Nov 12 23:07:41 EST 1997

JShipman (whom I am unfortunate enough not to know) writes ( FOM: Physics and Math; Gardner and Hersh)
>The old philosophers felt Euclid's
>parallel postulate was "necessary" but would have admitted they
>were wrong after reading Lobachevsky Š
There are some ambiguities. Amb 1. I am an old philosopher who thinks that the parallel postulate is necessary. Amb 2. It is necessary, not as a proposition about physical space (in those physical theories which are compatible with there being such a thing as space), but as a truth about Euclidean space---a mathematical object. Amb 3. It (or say, the statement P that every point is on some line in every pencil of parallel lines) is necessary also as an _axiom_ of the theory of Euclidean space in the sense that one can write down other propositions about Euclidean space which, together with P, axiomatize Euclidean geometry, but which do not imply P. Amb 4. The class of old philosophers who thought about such things as the parallel postulate divides up into different categories. There is, oldest of all, Plato who distinguished the question of truth in mathematics (idealized science) from empirical truth and never really felt that he had an adequate argument that the truths of the former really applied (as idealized truths) to the latter (although he thought that there must be such an argument). There is Kant, not quite so old, who made the same distinction but felt that he had proved that the truths of Euclidean geometry applied to physical space. Then there is Aristotle, quite old, who essentially made no distinction between truth in mathematics and truth in natural science.

I want to disagree also with Shipman's evaluation of the Hersh-Gardner exchange:

>Although I tend to sympathize with Hersh and feel that Gardner's
>criticism of him was confused Š.

I have read only the (I thought rather sad) line-by-line commentary on the review that  Martin Davis communicated to us and neither Hersh's book nor Gardner's review. Based on what I read, I thought that Gardner was more right than wrong; but that again there is an ambiguity. One may very well agree that mathematics is a social institution and that ideas like number, function, set, mathematical truth, and the like have meaning only within that institution. One may agree to all of this and, at the same time, believe these concepts and truths to be immutable and `necessary'. For, to say that 2+2=4 is not to say anything about what we would have said in the past, would say now or would say in the future, even though its truth is founded on the institution of mathematics as it exists and it is within that institution that it has meaning and is true. 

Now for the ambiguity: We might at some point change the institution of mathematics, so that sentences that were once true become false or meaningless. But this has two meanings---We may distinguish between changing the institution and changing institutions: the institution as it is now is immutable by definition (the institution de re, as the scholastics would have called it). What changes is not that; rather we may abandon that institution for another---we change institution_s_. (We change the institution de dicto). For example, number theory won't change; we may only abandon it for something else (which, of course, may come to be called `number theory'). Maybe, with this distinction, the strong feelings of rightness on the part of both Hersh and Gardner become explicable.

Hersh writes that there are three important realities in the world: physical, mental and social-cultural. He places mathematical objects in the latter reality, which includes "language, money, politics, war, shopping, families, etc."  Some of these, such as language and money may be susceptible to the de re/de dicto distinction made above: but to make that distinction would defeat Hersh's point. What is so of all of these cases, otherwise, and not true of mathematical reality, is that they are open to empirical investigation---what is true about the English language (for example), the currency exchange, etc. Isn't it better to just speak of mathematical `reality', independent of trying to make analogies? And at that point maybe we would agree that the question of what is real in mathematics is to be analyzed as, e.g., the question of giving content to the continuum hypothesis or the existence of large cardinals, rather than a question of trying to assign mathematical reality a place in some taxonomy of existence.

Bill Tait    

More information about the FOM mailing list