Dear Lawrence,

I just finished reading your essay and let me tell you that it was one of the most original in this competition. I think that's justly reflected in your current position in the top.

What makes this paper special is your choice to treat computability instead of more vague questions. Surely this position is footed on more solid ground as it aims at describing potentially fruitful directions rather then simply focusing on the quirky side of the universe which brings forth coincidences and such. I found particularly striking your topological treatment of the wavefunction collapse due to a measurement needle state and I want to ask if you develop this treatment anywhere. I saw you are referencing a not yet published paper of yours on the topology of states on relativistic horizons, which is probably more to do with the equivalence you are drawing between a horizon and an N-slit (?). Anyway I'd like to get a better understanding of your work. I couldn't find it at arxiv - are your papers online somewhere where I can read them? I mean somewhere not behind a pay wall, since I have no affiliation and couldn't really afford it :)

Thank you for an engaging read! Should you have time to take a look at my essay, your comments are much appreciated.

Warm regards,

Alma

Dear Alma,

Thanks for the positive assessment of my essay. Fortunately a number of people seem to share your opinion. It has been near the top since the beginning, and I have been in #1 and 2 spot for nearly a week.

I see there being a sort of two fold system. Standard mathematics might be thought of as the "soul," or a "ghost," and mathematics that is restrained by concerns of Kolmogoroff complexity, types and so forth as the "body." It may not be possible to express all numbers between 10^{10^{10}^{10}}} and 10^{10^{10}^{10^{10}}}}, but this just means the body is not able to construct or contain the information space necessary to do so, but this still leaves room for the "soul." Mathematicians are then free to "pick their poison," where a pure mathematician may prefer to stay with the standard approaches to math, while a more practical minded analyst might prefer to stick with the "body."

I don't particularly get into the argument over whether the soul of mathematics exists or not. This involves things such as infinities, infinitesimal or even finite numbers that can't ever be computed. I am agnostic on the idea of there being a Platonic realm of ideals. The idea seems in one sense compelling, but it also seems to lead to some mystical notions that are not entirely comforting.

Cheers LC

    Dear Lc,

    I know what you mean by notions that are not entirely comforting and I appreciate that :) I am not a platonist myself because it feels - to me at least - a bit useless; I am more of an utilitarian. Math is what math is. Thank you for answering my comment and thank you even more for finding the time to read and give your thoughts on my essay. Wish you best of luck in the competition and I hope to see more of your ideas as they make a very good read.

    Cheers,

    Alma

    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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