In his March 10 communication, Ben Fitzpatrick mentions Philip J.Davis's "Of time and mathematics." Thanks, Ben, for bringing this stimulating essay to our attention. Professor Davis kindly sent me a copy, published in Southern Humanities Review 18 (Summer 1984) 193-202.
Davis is a well known author. His awards include the Washington Academy of Science Award, the Chauvenet Prize, and the American Book Award for 1983. With his consent, I'll quote from the essay, beginning at the beginning:
"Time, that mysterious something, that flow, that relation, that mediator, that arena for event, envelops us and confounds us all. What is time? The answer of St. Augustine has become famous: 'If no one asks me the question, I know; but if one should require me to tell, I cannot.' Two millenia later, two revolutions in physics later, we can still sympathize with this answer. Our shelves are filled with formulas and speculations, and we still cannot say what time is; we cannot agree whether there is one time or many times, cannot even agree whether time is an essential ingredient of the universe or whether it is the grand illusion of the human intellect.
"There are thus two conflicting opinions about time, and they have been around since antiquity. According to Archimedes (and to Parmenides earlier still, for whom ultimate reality is timeless), one must eliminate time, hide it, spirit it away, transform it, reduce it to something else, to geometry, perhaps. Time is an embarrassment. According to Aristotle (and to Heraclitus earlier still, for whom the world is a world of happenings), one must face time squarely, for the world is temporal in its very nature and its coming-into-being are real.
"Modern science follows the path of Archimedes rather than that of Aristotle..." Time is downplayed, ignored, transformed, eliminated. Cause and effect are replaced by description and relation: do not ask why, but how; and the successes of the Archimedean program characterize our scientific civilization."
******* continuing several paragraphs later:
"The common philosophy of mathematics says that personal, historical time is absolutely irrelevant for mathematics. Some authors have even expressed the opinion that mathematics is the ONE [I use caps for Davis's italics] subject in which time is irrelevant. The entities of mathematics are envisioned as timeless, existing perfected in a world of pure essences. The truths of mathematics are truths forever, outside of time, outside of mind and personality; the deductive dialogues take place atemporally in a world of pure logical transformation. Technical words in common mathematical discourse which seem to betray a temporal basis, words such as 'is,' 'exist,' 'let,' 'vary,' 'approach,' 'map,' 'construct,' and 'equip,' are held to be metaphorical expressions of a formalized time-free equivalent."
Davis then challenges the "common philosophy of mathematics" with reference to the history of mathematics. I won't quote much of his development because that might decrease your incentive to obtain a copy of the article and read all of it.
"One might here object, saying that the counting aspect is part of the primitive intuition of addition, and intuition of mathematics is not mathematics. It might be said that the embedding of the integers in systems of great subtlety is part of the history of mathematics, and the history of mathematics is not mathematics...[the list continues]...The assertion may then be made that there is still a formal meaning to 2+3=5 which can be abstracted from all of the above, and this formal meaning is out of the range of time. But in that case, I will invite anyone to tell me what that timeless meaning is and to tell me in a way and in a language or a metalanguage which itself is beyond the range of time. In my view, this cannot be done, because the meaning of 2+3=5 must be supplied as part of a wider, similar set of utterances, and this meaning is bound up in application, in intuition, in arrangement, in computation, in art, in mysticism - in short, the whole mathematical experience. The more of these elements that are stripped away to arrive at a pure, clear statement, the harder it becomes to communicate what remains and the closer we are to a formalism in which thought and action become separated from meaning.
"Mathematics embraces all these aspects, and even more. 'Pure mathematics doesn't exist," assets Didier Norden, and I agree.'"
**** End of quoting (but not the end of the article). I omit Davis's reference-notes and references except for Norden:
Would anyone care to defend pure mathematics? Is pure mathematics perhaps a formalistic system which IS separate from meaning? Can we (perhaps following Hilbert, and Cantor) think of [pure] mathematics as the collection of definitions and relationships - not necessarily depending on the difficult notions of "meaning," "existence," and "truth" except as part of the mathematical experience, as contrasted to mathematics itself? My key word and idea here is "definition". It seems to me that objects in mathematics, most especially infinite sets and sequences, are "defined" (or, in the case of the natural numbers, assumed) whether they are "real" or not, and that the definitions and relations are atemporal. Perhaps what I'm suggesting is best represented by the distinction between the meanings of "truth" and "consistency." Truth may be time-dependent, but is consistency? To put it another way, perhaps the first concern of a mathematician when broaching something new is whether it is "well-defined", not whether it has "meaning" or even "existence".
Davis mentions an "opinion that mathematics is the ONE subject in which time is irrelevant." I agree. But even if mathematics (again, the set of definitions and implications, not the mathematical experience) is time-dependent, it seems that mathematics is nonetheless VERY different from other subjects. Perhaps the "real" difference is that (pure) mathematics is the ONE subject that doesn't exist.