By extension, the debate goes to the issue of when to stop looking. For instance, Hawking is content when an experiment confirms the theory, but Penrose wants the theory to also make sense.
I am a Platonist - more like Penrose than Hawking. For instance, I perceive that geometry exists in reality and the mathematician comes along and discovers it, e.g. pi, Schwarzschild Geometry, Riemannian Geometry and so on. As a Platonist, I would ask “Why pi? Why not something else?”
It is important to know and/or pick a side because it has a lot to do with how this information (and other science information) will be understood. Here are the two sides:
According to the Aristotelian paradigm, physical reality is fundamental and mathematical language is merely a useful approximation. According to the Platonic paradigm, the mathematical structure is the true reality and observers perceive it imperfectly. In other words, the two paradigms disagree on which is more basic, the frog perspective of the observer or the bird perspective of the physical laws. The Aristotelian paradigm prefers the frog perspective, whereas the Platonic paradigm prefers the bird perspective....
A mathematical structure is an abstract, immutable entity existing outside of space and time. If history were a movie, the structure would correspond not to a single frame of it but to the entire videotape. Consider, for example, a world made up of pointlike particles moving around in three-dimensional space. In four-dimensional spacetime--the bird perspective--these particle trajectories resemble a tangle of spaghetti. If the frog sees a particle moving with constant velocity, the bird sees a straight strand of uncooked spaghetti. If the frog sees a pair of orbiting particles, the bird sees two spaghetti strands intertwined like a double helix. To the frog, the world is described by Newton's laws of motion and gravitation. To the bird, it is described by the geometry of the pasta--a mathematical structure. The frog itself is merely a thick bundle of pasta, whose highly complex intertwining corresponds to a cluster of particles that store and process information. Our universe is far more complicated than this example, and scientists do not yet know to what, if any, mathematical structure it corresponds.
The Platonic paradigm raises the question of why the universe is the way it is. To an Aristotelian, this is a meaningless question: the universe just is. But a Platonist cannot help but wonder why it could not have been different. If the universe is inherently mathematical, then why was only one of the many mathematical structures singled out to describe a universe? A fundamental asymmetry appears to be built into the very heart of reality.
As a way out of this conundrum, I have suggested that complete mathematical symmetry holds: that all mathematical structures exist physically as well. Every mathematical structure corresponds to a parallel universe. The elements of this multiverse do not reside in the same space but exist outside of space and time. Most of them are probably devoid of observers. This hypothesis can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato's realm of ideas or the "mindscape" of mathematician Rudy Rucker of San Jose State University exist in a physical sense. It is akin to what cosmologist John D. Barrow of the University of Cambridge refers to as "pi in the sky," what the late Harvard University philosopher Robert Nozick called the principle of fecundity and what the late Princeton philosopher David K. Lewis called modal realism. Level IV brings closure to the hierarchy of multiverses, because any self-consistent fundamental physical theory can be phrased as some kind of mathematical structure.
The view [Platonism] as pointed out earlier is this: Mathematics exists. It transcends the human creative process, and is out there to be discovered. Pi as the ratio of the circumference of a circle to its diameter is just as true and real here on Earth as it is on the other side of the galaxy. Hence the book's title Pi in the Sky. This is why it is thought that mathematics is the universal language of intelligent creatures everywhere....
Barrow goes on to discuss Platonic views in detail. The most interesting idea is what Platonist mathematics has to say about Artificial Intelligence (it does not think it is really possible). The final conclusion of Platonism is one of near mysticism. Barrow writes:
Do there exist mathematical theorems that our brains could never comprehend? If so, then Platonic mathematical realms may exist, if not then math is a human invention. We may as well ask, "Is there a God?" The answer for or against does not change our relationship to mathematics. Mathematics is something that we as humans can understand as far as we need...
Beyond the Doubting of a Shadow - Roger Penrose
9.2 Moreover, in the particular Gödelian arguments that are needed for Part 1 of Shadows, there is no need to consider as "unassailable", any mathematical proposition other than a P-sentence (or perhaps the negation of such a sentence). Even in the very weakest form of Platonism, the truth or falsity of P-sentences is an absolute matter. I should be surprised if even Moravec's robot could make much of a case for alternative attitudes with regard to P-sentences (though it is true that some strong intuitionists have troubles with unproved P-sentences). There is no problem of the type that Feferman is referring to, when he brings up the matter of whether, for example, Paul Cohen is or is not a Platonist. The issues that might raise doubts in the minds of people like Cohen - or Gödel, or Feferman, or myself, for that matter - have to do with questions as to the absolute nature of the truth of mathematical assertions which refer to large infinite sets. Such sets may be nebulously defined or have some other questionable aspect in relation to them. It is not very important to any of the arguments that are given in Shadows whether very large infinite sets of this nature actually exist or whether they do not or whether or not it is a conventional matter whether they exist or not. Feferman seems to be suggesting that the type of Platonism that I claimed for Cohen (or Gödel) would require that for no such set could its existence be a conventional matter. I am certainly not claiming that - at least my own form of Platonism does not demand that I need necessarily go to such extremes. (Incidentally, I was speaking to someone recently, who knows Cohen, and he told me that he would certainly describe him as a Platonist. I am not sure where that, in itself, would leave us; but it is my direct personal impression that the considerable majority of working mathematicians are at least "weak" Platonists - which is quite enough. I should also refer Feferman to the informal survey of mathematicians reported on by Davis and Hersch in their book The Mathematical Experience, 1982, which confirms this impression.)
Pi isn't axiomatic, it is derivative. And the axioms it is derived from are arbitrary; they are only selected because they seem to have utility as such.
LOL, I don't doubt that! But myself, I don't understand why only the Material or the Ideal must be thought of as real & the other the shadow.