I just realized I didn't rate your essay so I am fixing that now. As you have a lot of votes, I hope mine is enough to make a difference.

Thank you for that. I voted for your essay a month ago or so. I don't remember the exact score I gave it. It was probably a 6 to 8 score.

Cheers LC

You give the impression that there is something wrong with the foundations of math, just because different formulations and axiomatizations are possible. But these possibilities make very little difference to the great majority of math.

    There are some questions concerning foundations of mathematics. I am not a great expert on this, but it does seem that as mathematical physics develops that it will embrace concepts that are not as tied to many aspects of point set topology with infinitesimals and the rest.

    I don't say there is something wrong with the foundations, and it appears that we are increasingly in a time where there are several such foundations. These things seem in some ways to be model dependent, with different proof methods and the rest.

    LC

    Dear Lawrence,

    Many points in your especially comprehensive essay are worthy of comment, but I find particularly intriguing the idea mentioned at the end. This is the suggestion that mathematical reality and physical existence have the same unusual organization. It might be that in both of them we find islands of order set amidst vast and encompassing chaos. If this is so, then perhaps, as you say, there might be no reason for this similarity between mathematics and physics. However, I think we would try to find some deeper reasons, though I am not sure how we would go about that.

    Thanks and best wishes,

    Laurence Hitterdale

      Dear Laurence,

      Thank you for taking interest in my essay. The idea is that the quantum vacuum as a set of qubits, say (0, 1) set to a|0> = 0, has a phase structure based on how qubits are transformed into each other. We normally think of the vacuum as invariant under a certain symmetry group, but underneath that it could just be a vast self-referential loop, where there are "accidents" that occur where the vacuum has a symmetrical structure. This means zones exist where there are dynamical structures, where symmetries are aspects of division algebras.

      These self-referential qubits, or loops of them, form a strange basis for the universe, or multi-verse, that can't be derived or computed. We can't then know what is not computable. It is similar to Chaitin's halting probability; we can know there are incomputable symbol strings in a set of them of length N, but we can't compute with certainty which are not computable (Turing's halting problem), we can't compute the number of them that are computable or not computable, or the probability for any of them to be incomputable or nonhalting. We are then faced with a bit of a conundrum; this would be a theory that tells us that this state of affairs exists, but we can't compute much of anything with it.

      If physics and cosmology reaches this state of knowledge it might be the end of these foundations. The end of scientific foundations might occur this way, though I suspect we have quite a ways to go before progress in physical foundations stops at this point.

      Cheers LC

      9 years later

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