Wednesday, January 8, 2014

Why is Mathematics?

In an earlier essay, we considered “What is Mathematics?”   Whereas now, the question is:  Why.  -- We have deliberately given this post an awkward-sounding title, rather than the breezy and dismissive “Why math?” (the sort of thing Jughead might toss off with a shrug), aiming for a slight Entfremdungseffekt,   along the lines of Heidegger’s  Was heisst Denken? or Dedekind’s Was Sind und Was Sollen  die Zahlen?
Thus written, the sentence “Why is mathematics” sounds oddly as though translated from one of the more obscure dialects of the Carpathians, or perhaps an incomplete phrase of the sort that leads into a formulaic punning riddle -- “Why is mathematics like French plumbing?” “Because …”  (Prizes for best completion).

(Cf. Varro, De lingua latina V 2):  cur et unde sint verba 'Why and whence are words?')

Anyhow, here is a serious reply to the question:

Mathematics intrigues people for at least three different reasons:  because it is fun (the most important reason for inclusion in this book) because it is beautiful, or because it is useful.
-- Ian Stewart,  How to Cut a Cake (2006), p. 89

For myself, it is rather for a fourth reason:  because it is true.   And true, without contingency -- Necessary, like the Necessary Being.   From Whom, indeed, we receive it.
Like Christianity, were it not true, it would have no more inherent interest than sports or stamp-collecting.

Seen in that perspective, the ugling-duckling of a phrasing, “Why is mathematics?”, spreads the wings of a swan.   It becomes its own special philosophical language, like the innovations that the Pietists made to German.   What is Man, that Thou art mindful of him?  And whence cometh this “mathematics”, that it should contain things true before all worlds, before ever Man came into it, and that remain truths through all eternity:  yet are revealed, piecemeal, like Scripture itself, over the millennia?


There are some syntactic and semantic subtleties to that odd little phrase, “Why is mathematics.”   It is not quite idiomatic English, just as the epigrammatic questions of Heidegger and Dedekind are not quite German, though in a way that is difficult to put your finger on.  And yet we do (as linguists say, in a phrase that is not quite English either)  “get a reading” on them -- that is, an interpretation (albeit hazy) immediately suggests itself in each case.
Further, that interpretation is highly sensitive to syntactic and perhaps to lexical perturbation.  Thus,  Why mathematics?” would most readily be understood quite differently:  Why do mathematics?  Why go into mathematics?  (rather than biology or whatnot).  As for “Why maths?”, I don’t know British English well enough to know if that would be taken exactly  the same way or not.  (Ditto for “Pourquoi les mathémathiques?”)  As for “Why is maths?”  (“Why are maths?” ??), or “Pourquoi sont les mathématiques?”, I’ve no idea whether you could even say it.
(For the sense, if any, of “Warum ist die Mathematik?”, we must consult the shades of Dedekind and Heidegger.)

Since Spanish has two words translatable by ‘Why…?’, the question splits.  For the interpretation bzw. grammaticality  of things like “¿Por qué (0/es/son) la(s) matemática(s)?”, I defer to my hispanophone colleagues.  But the sense of “¿Para qué …” followed by any of these  seems pretty clear:  You are asking what maths are used for.
(There!  I just used “maths” spontaneously and non-metalinguistically in a sentence.  Time to toddle off to tea…)

As for Russian, here the case is different again, since it does not use a copula in present-tense equative sentences, and so cannot make the distinctions that English can in such cases.


In that earlier essay, we proceeded from the rather vaporous question “What is Mathematics?”  to the much more concrete series of questions,  “What is an affine connection?”, “What is a Lie group?” etc., the point being to seek out distillations of the essence of a subject, rather than opaquely motivated formal definitions, in a way that is both intuitive and mnemonic.    Now, likewise, we proceed from the formless blancmange of “Why is mathematics?” to characterizations of specific mathematical objects or topics, this time not in terms of what they “are” exactly, but what they are for (along the lines of:  A hammer is for driving in nails.)
Thus, take your friendly neighborhood Lie group, which in the definitions-oriented essay was thumbnailed as “Roughly speaking, a group in which one can meaningfully define the concept of a smooth curve.”    In the very next paragraph of the same article quoted there, the author provides a What-for motivation:

Lie groups were introduced in order to create an analogue of Galois theory for differential equations.
-- Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 230

Succinct and to the point!

Or (from the next page of the same essay), consider the group-theoretic notion of commutator.   The literal definition is simplicity itself: --   ABA-1B-1  -- but what does the thing do?  The answer is equally concise and revealing:  It “measures the extent to which A and B fail to commute.”   That fact established, we get a comparison with the related notion of the Lie bracket [X, Y]:

Informally, it represents the net direction of motion if one first moves an infinitesimal amount in the X direction, then in the Y direction, then back in the X direction and back in the Y direction.
-- Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 231


And, with reference to the discovery of p-adic numbers:

At first, most mathematicians seem to have found Hensel’s new numbers interesting in a formal way, but also to have wondered what the point of them was.  One does not adopt a new number system just for fun;  it needs to be useful for something.
-- Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 243

That faint-praise “interesting in a formal way” deserves to be highlighted and memorized;  it hints as well as anything at the appreciation for genuine depth and insight, as against mere symbol-shoving, among real mathematicians.  (Cf. further this Arnoldian epigram.)  One imagines that such is the bemused reaction of many mathematicians and physicists to the extravagent but perhaps inapplicable elaborations of String Theory.

Note too, that dismissive back-of-the-hand to mere “fun”,  which nonetheless figures first in the list of motivators, in the quotation from Ian Stewart  with which we began this essay.  Stewart was essentially writing for amateurs;  for full-time professionals, mathematics is just too darn hard to be nothing but fun.  You don’t devote your entire life to crossword-puzzles or sudoku.

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