From Saturday's reading:
In the main, it is the applied scientists who call for numbers, numbers, numbers. Although the layman conceives mathematics as being the science of numbers, few mathematicians agree, and most of modern mathematics gives to numbers a role at best ancillary or illustrative in the development of concepts. Mathematics is the science of precise relations. Mathematicians habitually deal with properties that no measurement could verify or controvert, properties that can only be imagined. I refer to functions defined on infinite sets, continuity, passage to limits, relations of inclusion among infinite sets and in regard to membership in such sets, etc, etc. If the hard-headed empiricist replies,
I don't care about any such things, I just get numbers,in most instances the mathematician can easily see that he is deluding himself, for nearly all experimental results in physical science are explained and correlated by use of the operations of differential and integral calculus, in which the infinite is always present, and no finite set of rational numbers suffices for taking even the beginner's first step in that theory. Mathematics might have greater potential use if those who tried to apply it to the biologies and to social phenomena would learn that measurement provides only one of many possible kinds of quantification. The worst advocate for mathematics is the enthusiast who thinks he understands it but does not. For an informed, sober, and concise survey of the pitfalls in mathematical modelling, especially for the biologies, I refer the reader to Maynard Thompson'sMathematization in the Sciences.
-- C. Truesdell