Dedekind on ontology

[A footnote to this essay.]

Footnote re the irrationals:

Dedekind stressed the distinction of category  between cut and number  in 1888; against the view of his friend Heinrich Weber  that “the irrational number is nothing other than the cut itself”, he explained that “as I prefer it, to create something New distinct from the cut, to which the cut corresponds.  We have the right to grant ourselves such power of creation”,  and cuts corresponding to both rational and irrational numbers were examples.

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 87

A seemingly slight, even pedantic distinction;  but like many another such, it might have its point.   Cf. my astonished delight in junior high-school, upon meeting the distinction between  x (the thing itself) and ‘x’ (the name of x) -- already adequately foreshadowed in Alice in Wonderland, but encountered now in a new context.  Likewise the difference between  x and {x} (the singleton-set of x).

In the case of an algebraic number like √2, a simple number staring you in the face out of a hypotenuse  versus the infinite train of rational pilgrims (never quite arriving at their destination) of a Dedekind cut,  one is reminded of the variety of definitions of something so familiar as a tangent:  the slope of a curve (at a point); the closest linear approximation to the curve (at that point); versus the distressing definition in Loomis & Sternberg as an infinite equivalence-class of curves (through that point).

http://worldofdrjustice.blogspot.com/2012/02/on-rounding-out.html

More from the Mindscape

A footnote to this essay:

Dedekind … allowed his philosophy of mind  much reign, with a ‘proof’ that “there are infinite systems”;  for he gave  as evidence “the totality S of all things, which may be objects of my thought”, since  as well as any of its elements s,  it contained also “the thought s’ that can be the object of my thought …This ‘proof’ did not gain a good reception.”

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 105

Thoughts 'n' Things

  Rudy Rucker, Infinity and the Mind, p. 38:

(**) Just as a rock is already in the Universe, whether or not someone is handling it,  an idea is already in the Mindscape, whether or not someone is thinking it.

This is itself a pleasant thought, recalling the ditty about God-in-the-quad; but in actual fact – I don’t think so.

(So you see—I am not an uncritical Platonist.  Platonic heaven must be so gerrymandered, as to exclude such things as cheese doodles and Sponge Bob Squarepants.)

The actual universe has (for example) -- whatever geometry it has:  regardless of whether there are rational creatures capable of understanding it, let alone deriving it.  Likewise the landscape of math.  But particular formulations of physics, and perhaps even of math – matrix mechanics v. wave mechanics, Cauchy analysis vs. non-standard analysis – do not exist in complete independence from their proponents.  They are, one might say, propositions, not objects.  The objects (or patterns, or whatever they are)  exist  even in the absence of  a person to spout propositions about them; but the propositions require a proposer.  – Nothing specially abstract here; the same thing is true of rocks.  This rock exists independently of any finite mind, but: “There lies a rock” and “Behold that rock!” and “What a rock that is!” must come out of some actual someone’s mind or mouth.

            The unbridledly idealistic view in (**) conjures up a skyscape of untethered thought-balloons.  It is pleasant to contemplate, in a comic-strip sort of way, but not to be taken too seriously.  For one thing, unlike the situation with mathematical truths, where anyone at any place or time might discover them, there is no way for a rational creature in another galaxy or dimension to reach out and grab one of those thought-balloons by the tail;  he is required to blow his own bubbles.  Whereas the structures of mathematics are like fixed landmarks, which one encounters again and again, from different approaches.  For instance:  Yang-Mills gauge theories, discovered by the physics expedition; and connections on fibre-bundles, discovered by the math team; and lo, they meet in the middle.  Likewise group-theory.  Different body-parts of this have been grabbed onto by matrix theory, algebra (symmetries of solutions to equations), geometry (the Erlangen program), particle physics (glad you could get here; meet Sophus Lie), and in time it becomes clear that it’s all part of the same elephant.  Whether they come from physics, or mathematics, or computer science, two such explorers may not realise that they have come upon the same mountain, till they have circled around it a bit and compared notes.  And this happens repeatedly.  We may summarize in an epigram:  The mindscape of mathematics is a multidimensional torus:  whatever direction you set off in, you eventually wind up back at Hilbert’s Hotel.

It turns out that Shing-Tung Yau likes this montane metaphor as well.  Cf. The Shape of Inner Space (2010), p. 103:

A mathematical proof is a bit like climbing a mountain.

And he nicely outlines the Yang-Mills case (p. 290):

The physicist Chen Ning Yang was similarly astonished to find that the Yang-Mills equations, which describe the forces between particles, are rooted in gauge theories in physics  that bear striking resemblances to ideas in bundle theory, which mathematicians began developing three decades earlier, as Yang put it, “without reference to the physical world”.  When he asked the geometer S. S. Chern how it was possible that “mathematicians dream up these concepts out of nowhere,” Chern protested, “No, no.  These concepts were not dreamed up.  They were natural and real.”

            Contrast the case with “thoughts”.  Supposititious entities of the mindscape, even some popular thought-balloon, tethered to a billion different heads, need never be rediscoverable by another explorer, nor acknowledged as real should he simply be grabbed by the lapel by one of the thinkers, and treated to an exposition of same.  For example, the notion held dear by countless generations of schoolboys around the globe, of the uniquely funny nature of flatulence, will never appear among the gravely ellipsoidal thought-balloons of the solons of Fdrmrphlandia; even “funny”, for them, is not well-defined, and not particularly worth defining.

Now, probably Rucker meant to restrict the realm of “ideas” to just some of them.  Not, “Wouldn’t it be fun to dip Suzy’s pigtail into the inkwell!”, but things like “The square of the hypotenuse is equal to the sum of the squares on the other two sides.”  Fine; but careful, here.  The Pythagorean theorem has  as its basis  a fact about Euclidean geometry, in every possible world; just as Fermat’s Last Theorem expresses (in a possibly somewhat contingent and imperfect way) a fact about the natural numbers.   But a fact is not the same thing as an idea.  As a matter of fact, there is a coffee stain on this shirt; but “the idea of this coffee-stained shirt” is no strut or girder of God’s architectonics.  An idea concerning a fact of mathematics, in a finite mind,  may bear – must bear -- but an imperfect relation to the fact itself (‘fact’ here used broadly: it may refer to a wildly transfinite complexus of relations, some of them perhaps perceptible only to angels).   Most people’s ideas of mathematical truths bear as much relation to the truths themselves  as does a crayon scribble to the Sistine Chapel  which it might (based merely upon memory of a fleeting ill-lit glimpse) attempt to depict.  To posit that all truths of mathematics exist as Ideas in God’s mind, is logically allowable, but really adds nothing, and is in any case unknowable. To identify these truths with the neuronal states of the pitiful meat-wads sloshing around in our half-cracked crania, is to add nothing at all, but is rather to detract.

[Appendix]  Karl Kraus apparently entertained a notion of independent or pre-existent thoughts.  He speaks of someone being

von der Präformiertheit der Gedanken  überzeugt, und davon daß der schöpferische Mensch  nur ein erwähltes Gefäß ist; und davon, daß die Gedanken und die Gedichte da waren  vor den Dichtern und Denkern.

-- “Heine und die Folgen”, reprinted in J. Franzen, The Kraus Project, p. 88

The whole ‘meme’ idea (itself a meme) is similar -- not that the various Chiclet-thoughtlets were truly Platonically pre-existing, but that, once hatched, they lead a promiscuous existence, wandering into people’s minds  like pollen into our air-passages.

~

Footnotes from the 19^th century:

Dedekind … allowed his philosophy of mind  much reign, with a ‘proof’ that “there are infinite systems”;  for he gave  as evidence “the totality S of all things, which may be objects of my thought”, since  as well as any of its elements s,  it contained also “the thought s’ that can be the object of my thought …This ‘proof’ did not gain a good reception.”

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 105

For Frege,

In contrast to subjective ‘ideas’ (Vorstellungen), ‘thought’ was intended in an objective sense, rather like state of affairs, sharable among thinkers  and indeed independent of anyone thinking then.

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 190

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.