Numbers exist as natural structure with intrinsic properties which happens to be suitable for purposes such as counting finite numbers of objects. M-theory exists in a similar way. In one of the approximately 10^500 classes of universe that exist within M-theory, it is possible for the patterns we know as humanity to exist. A particular view on one of the possible histories in such a universe is what we call our world. It is treating ourselves too seriously for us to worry about what it is possible for us to recognise, when we are merely a structure that happens to have the property of (somewhat imperfectly) considering such issues. Better to not introspectively (and self-referentially) think too much about what we can think, but to think about the things which have independent existence.
The Problem of Access
I have to disagree with some of what you said, and truth be told, I am left wondering how this adresses the paradox-my best guess is that you are adressing the less persuasive form of this paradox, which is more of just a question as to how we can have such access (which the Platonist replies to as being a red herring; Platonism does not have to give an account of how the access works). Although an interesting sub-question would be, how this access works, of course. But this is not what I was talking about-and forgive me if I misinterpret you-the objection called into question the logical possibility of any such type of access on Platonism being able to occur at all.
What I wanted to take issue with is "In one of the approximately 10^500 classes of universe that exist within M-theory, it is possible for the patterns we know as humanity to exist. A particular view on one of the possible histories in such a universe is what we call our world"-From what I know, m-theorists usually just invoke so many universe because m-theory does such a lousy job at describing our own universe...hence many such theorists are reduced to the weak anthropic principle, which is one of the many reasons that a lot of people have reservations about m-theory.
My point is that our understanding numbers is not much more mysterious than a computer "understanding" numbers (by being able to manipulate them or even to prove theorems about them). The difference is that we as humans have discovered the models that we use to work with numbers (or been taught them), but that is really a feature of our brain's ability to construct, store, use and communicate to others, models for what we perceive.
Do we ever know that we really "understand" something, rather than have a model that we are fairly sure does the job? If this seems unfair, bear in mind that there are many examples in science and even mathematics where most people have accepted a model that appears to be perfect and complete that later turns out to not be, showing that we do not always know how perfect our understanding of something is. Better to just acknowledge that we have a model that we believe is appropriate. Even in number theory the scope, power and perhaps the "meaning" of our models of the natural numbers were rather unclear until the crisis in logic last century. At present we have the situation where it is unknown if the Riemann hypothesis is true, false or undecideable. If the latter, we may be missing a vital axiom that is necessary for number theory, (such as the existence of some incomprehensibly big cardinal number to be used for transfinite induction). Godel's incompleteness theorem tells us that we need an infinite number of additional axioms to allow us to decide all theorems in number theory. Can we say we really understand numbers?
With regard to M-theory, I believe you are being unfair. The theory itself has a mathematical naturalness like many more familiar natural objects in mathematics (the natural numbers, the quaternions, the Fischer-Griess monster group ...). The multiple classes of universe it permits are a natural feature of M-theory. There are a finite (but extremely large, around 10^500) configurations, rather like the 3 ways it is possible to glue the edges of a square, giving three different topologies (the torus, the klein bottle and something else. These different configurations will apparently all give different values of the fundamental physical constants. I have for a long time resisted the anthropic principle, wanting the physical constants to be somehow derivable. Now I can see that this natural feature of M-theory allows the rich variety of universes that permits one in which we can arise, so the strong anthropic principle makes sense. (Until recently I didn't realise how unlikely the key sequences of events permitting intelligent life to arise were. This would make it an implausible co-incidence if the only possible set of physical constants also happened to be a set that permitted life to exist. This would be a co-incidence that deism could not solve, since the laws of mathematics cannot be changed even by a deity).
1. We need 3 space dimensions for orbits (and stars) to have gravitational stability. (The number of macroscopic space dimensions appears to vary between universes permitted by M-theory)
2. We need just enough inhomogeneity in the early universe
3. We need nuclear forces that permit the creation of light elements in supernovae
4. We need supernovae to explode rather than collapsing
5. We need stars to burn in such a controlled fashion that we have a few billions years of comfortable conditions on our planet for us to develop.
It turns out that these requirements depend on very delicately balanced physical constants. Most small shifts (as in others of the 10^500 classes of universe) result in cold dead universes.
A more knowledgeable and expert view about the strong anthropic principle and why it has gone from fringe to mainstream in the fundamental physics community in the last 10 years (not surprising that they are a bit quicker than me) is to be found in Enlightenment, Knowledge, Ignorance, Temptation by Frank Wilczek.
A very accessible view of these ideas can be found in Hawking and Mlodinov's 2010 book, "The Grand Design".
Fair enough...I suppose that your view of understanding "our understanding numbers is not much more mysterious than a computer "understanding" numbers " would assume that computers can manipulate any mathematical statement that humans can-but this is not something I can yet accept from certain logical statements (Godel senteces, etc), so I am skeptical about the handling of numbers themselves to a certain degree (I suppose some numbers could be more computable to computers than others). Another problem I would have is with the limited scope of equality computers have-one reason is because we have a very shifty notion of equality in some cases, currently, so I wouldn't expect a computer to do even better-and ultimately, I think that notions of equality are significant to being able to compute something, especially the more esoteric "somethings".
One of Penrose's claims was that computers were restricted to working in a predefined domain, for example a given axiomatic system, whereas humans were capable of changing the domain by some sort of paradigm shift. How we do this is obviously a tricky issue. But if you accept (as I do) that the brain is a complex electrochemical, analog (but also digital) machine that executes a process of association, induction and abstraction from things perceived in the real world (originally for the purpose of creating models of the real world in order to make plans for how to survive and thrive), it is virtually certain that a very sophisticated inorganic machine could do the same unless the brain does not obey the laws of physics.
Suppose the capabilities of humans exceed those of inorganic machines and that the brain obeys the laws of physics (i.e. the mind may be influenced by the brain but cannot influence by it, since if could, there would be a divergence from predicted behaviour of the brain, based on the laws of physics). This would be strange, as it would mean that the metaphysical mind could never influence any action by a person through motor nerves, or any thought pattern in the brain, etc. This is not what most people who believe in a metaphysical mind would expect, or require.
Are there subtle loopholes due to the non-deterministic nature of physics? Perhaps a metaphysical mind could select possibilities from those that physics would suggest were to be randomly selected, in a way which was not totally random, but indetectible by experiment? I don't see how, but I once came up with the idea that free will was a process by which selections were made by a person as to which of the possible futures predicted by physics would be selected. But surely such effects would be detectable, if a person was able to repeated make similar choices of future in a laboratory? (I am not at all sure what possible choices would be available, but I am aware of a theorem that says that if people have "free will", so do elementary particles).
" But if you accept (as I do) that the brain is a complex electrochemical, analog (but also digital) machine that executes a process of association...."-Of course, I agree this is part of the nature of the brain, but I never assumed the brain was responsible for all insight...your later statement, "the mind may be influenced by the brain but cannot influence [ ] it, since if [it] could, there would be a divergence from predicted behaviour of the brain, based on the laws of physics", is superficial, I think; I don't see why we have to assume at the beginnning that there would be a divergence from prediction at all. This is a statement to be proven, and would be very hard to do so, considering we don't immediately know what the predicted ability of the brain alone is, since there may be parts of the brain that play significant roles that are affected by quantum mechanics.
In my view, a function of the brain's behaviour is not necessarily deterministic, because a first choice cannot be defined by such a function (because there was no previous behaviour)-perhaps one free choice casts a partially deterministic shadow on the rest of a person's decisions, rather like "someone chosing freely to determine oneself", if I may put it that way. "I once came up with the idea that free will was a process by which selections were made by a person as to which of the possible futures predicted by physics would be selected"-Please expound on this more-it is interesting! What was the motivation for this? Would one person observing another person conclude that the person being observed is not making any free choices at all, or is the other person not governed by the laws of physics? My guess is that the first option would be picked, in which case why bother with saying that physics determines physics by an unnecessarily complicated way?
strangequark, either the behaviour of the brain is entirely explained by the laws of physics (as most scientists would assume without hesitation) or it is not. If it is then even if there is a metaphysical component to the mind it can have no influence on the behaviour of the brain (i.e. the behaviour of the physical brain is the same whether or not it exists). As more and more of the key functions of the brain (down to the precise things about which one is thinking) are being pinned down to the physical brain (not to the surprise of anyone working in the field), this means that these things are not influenced by the hypothetical metaphysical mind. I am sure purposes can be dreamed up for such a metaphysical object, but it is getting further from what I think of as my "mind" which involves my thoughts, including my consciousness. These phenomena are being more and more precisely pinned down to the physical brain.
As for my earlier musing about a way in which a metaphysical mind could make choices, it doesn't really escape this conflict any better than if the world was not quantum mechanical. Suppose a key neuron is put in a state where it may or may not fire according to physics, because of quantum mechanical uncertainty. Assume this neuron decides if I will blink or not. My idea was that maybe a metaphysical component could choose from the possibilities, but avoid doing anything that was not possible according to physics. But this situation could be repeated in an experiment, and we could generate results that were as statistically unlikely as we wish if free will could choose whether to fire the neuron (and blink), so physics would be broken to as high a probability as we wish, in a predictable way. So this merely replaces a situation where miracles are boolean breaches of physics, to where miracles are wildly unlikely statistical breaches of physics.
In summary, we either have a brain (and body) that operates according to the laws of physics without any influence from a non-physical mind, or we have a brain that breaks the laws of physics.
It is hard to prove that some effect is due to some particular cause. I think you have not taken into account the possibility of emergence: I have no problem saying that something physical can cause something metaphysical (the brain causing the mind). If this were so, we could have a purely physical account of all behaviour but still an acount that was due to the mind.
I disagree with nothing there.
Emergence is a good word for how I imagine the very high level functionality of a human being arises, made of electrons and nuclei, but with much richer structures at much higher levels. It is not the most useful model of a human being to describe it as a set of elementary particles interacting according to the laws of physics (worse than describing Microsoft Windows by giving the machine code into which it has been compiled 
).
My view is that emergence adds structure to the sum of its parts - if there is something created, it is the structure (which is not really a physical thing). I have come to the conclusion that the mind is such a structure on the physical material in a brain, which does not make it any the lesser. It seems to me the structure and its behaviour is much more important than the material out of which it is made.
A good example of emergence is Conway's Game of Life. The rules of this cellular automaton are simple enough to write in a few lines of code (the state of each cell depends only on the state of the 9 nearest cells in the previous generation), but its behaviour is very complex. It is interesting to trace how this complexity builds:
- There are structures that move in space while preserving their form perfectly (gliders).
- There are structures that produce gliders in a repeatable way.
- These structures can be used to create a computer that is Turing-complete (i.e. universal). [A bit like building a computer out of transistors, but with infinite addressable memory]
- There is a program for such a computer that is able to replicate the computer and its program (i.e. self replication)
It has been estimated that about a trillion cells are required to create a sell-replicating cellular automaton in this way, suggesting how demanding a behaviour self-replication is.
A question for fellow mathematical Platonists:
What is your "pet" rebuttal to the problem of access (numbers are causally inert which means that we shouldn't be able to recognize them (by causal theory of knowledge), but Platonism is committed to us knowing them therefore Platonism must be false)?
Is the causal theory of knowlege correct, in your opnion? Certainly, it is not necessary that the causal theory of knowledge is always correct (for example, the famous EPR thought experiment in physics), but is it not correct most of the time? Can we link the access problem to possible solutions to the mind-body problem? etc.