(**) 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 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 gaugre theories in phhysics 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 dreams 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.