(**)
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 19th 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
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