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Daniel, you make some good points which strike at the heart of part of what this article is about. But let me try to unpick a little of what you say to see if I understand you correctly.

You are worried about needing three realms. The two realms you seem to affirm are the mental and the physical. Even this is a leap, actually, since a pure materialist would say that the mental realm can be reduced to the physical, that our language for describing the mental realm is a kind of shorthand for what is "really" happening on a physical level. I disagree with this perspective, and you can find coherent book-length arguments against it in pop science books by Robert Laughlin (A Different Universe) and Stuart Kauffman (Reinventing the Sacred), and elsewhere. So you and I agree that there are at least two realms, the mental and the physical. While neither is reducible to the other, they are not independent either.

So the real question which I think you are asking is, "Do we need to posit the existence of a platonic realm." Well, to some extent, my article is all about this very question! But while I think that there is a lot of currency in your evolutionary argument, it doesn't seem to me to satisfactorily address all of the issues. As I said in the article, the evolutionary pressure would have been on (a) survival, and (b) mating. While understanding cause and effect, and being able to count distinct objects, and so on, would presumably have helped, it doesn't seem obvious to me that the leap you make to an evolutionary explanation for advanced mathematical ability is a necessary one, or one well-founded in the hypothesis of mathematics being somehow an encoding of the physical world in the mental. However, I think that John Barrow is a fan of this idea - I remember him discussing the work of Rosen in this context in his book "Impossibility". I'm afriad I can't get the reference for you now because I am in quake-shattered Christchurch, and I cannot get in to my office building!

As far as I can tell, an evolutionary approach to the question of the existence of applied mathematics does not address its ability to work in counter-intuitive realms, its apparent timelessness, its felt sense of independence, its coherence, its ability to explain the physical universe, or even our ability to find the universe comprehensible at all!

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