There have been some ideas along these lines. There was axiomatic quantum field theory, but this never really seemed to catch on. There is also quantum logic, which has lead to some interesting developments. There is though Bohr's statement that quantum physics is best described by a system that permits discussion in ordinary language.

LC

Dear Adel,

I suddenly see that I got a fair number of posts and have risen considerably up now to the top. It is interesting to respond to two people, Alexey and Lev Burov above, who refute the MUH, and then within the same half hour discuss somebody who embraces it. Of course I will have to read your essay first. I will try to get to that as soon as I can.

Cheers LC

Jonathan,

Thanks for the positive assessment. Indeed, I seem to have popped up considerably in the last day or two. I will have to take a look at your essay as well to refresh my memory on it. I can't recall if I scored it as yet.

Cheers LC

I will have to respond in detail later. It is the morning before heading off to work. In many ways physics is coming around to the idea that information is at the foundation of what is important with respect to phenomenology.

LC

Excellent job Lawrence,

This essay is well-written and presents a Tour de Force of interesting Maths relevant to Physics. You have managed to work in a lot of topics that are very interesting to me, and about which I have much to learn, such as Homotopy Type Theory. The HoTT program is especially interesting to me, as it has a constructive geometric basis on the one hand, and a rigorous analytic procedure on the other. I also like that you wove in the Bott periodicity, which I was trying to find a way to fit into my own essay, because it is one of those invariant structures that one seems to bump into - as though it was there before you found it.

Being a constructivist, I think that perhaps numbers and counting are not the first Maths to arise, however. Having a Set of objects requires preexisting elements of geometric topology, so that objects with surfaces and containers to hold them are well-defined. Also, it is seen in young children that a sense of greater and lesser quantity is a kind of numeracy that exists apart from counting itself, and develops sooner. I would think that just as ontogeny recapitulates phylogeny, for developing organisms; so individual patterns of learning are reflected in the development of cultures. Perhaps counting is merely the earliest form of mathematical reckoning that could be written down.

I think you come out on the side of the formalists and logicists in the Brouwer Hilbert debate, while I am firmly in the intuitionist camp - and while this is sometimes termed anti-realist, I believe it is more realistic yo imagine that everything should be constructable for Physics. But at least you mention that there is a debate about this among mathematicians, which some might miss otherwise. It was a great effort overall, and you get high marks from me.

Regards,

Jonathan

    Jonathan聽

    I am rather agnostic on any of these ideas about mathematical foundations. I don't hold to any of them to much degree. For one thing these things are a bit removed from physical theory, which I am more interested in than pure mathematics. The other reason is there seems to be no way we can decide whether one is better than the other. In some sense maybe it is best to consider them as metaphysical tools that can be used or not depending on the situation.

    The homotopy and Bott periodicity involves my observation that groups involved with quantum information appear to have this period 8 structure to their topology. This seems to extend into the exceptional and sporadic groups as well. This means the quantum bits associated with a black hole event horizon have a type of degeneracy. This is the main reason why I think it is possible that this homotopy based mathematics with a correspondence to quantum bits might form the foundations of mathematical physics through this century. It would be curious to see what mathematical physics looks like in 75 years.

    Really I don't pretend to know the relationship between physics and mathematics. It is a completely mystery really. It may just come down to an instrumentalist argument that because physical science involves measuring things according to numbers that the subject must necessarily involve mathematical consistency.

    LC

    Dear Lawrence,

    Since the question of this contest is about universe theoretizability (using the word of our essay), the answer apparently cannot refer to such specific terms as 'vacua', 'energy', or 'quantum mechanics'. Such references are logical flaws, aren't they?

    As to Tegmark's MUH, we are refuting it on the factual ground, namely, on the grounds of the logical simplicity, large scale and high precision of the already discovered laws of nature.

    Cheers,

    Alexey Burov.

    Thanks for the thoughtful reply Lawrence..

    In his (non-contest) essay 'A View of Mathematics', Alain Connes speaks about Math as a single corpus, almost like a biological organism, that makes it hard to separate the parts or say what came first. I am of the opinion, however, that there must be some set of most elementary rudiments, from which that entire body of knowledge flows. Perhaps this does not mean it can be constructed deterministically, but there have to be some bones to hang the meat on somewhere.

    You might find interesting Andrei Rodin's book 'Axiomatic Methods and Category Theory' which relates strongly to the HoTT program, and is available as a download from arXiv. But I think it is wonderful that mathematicians are pursuing a better understanding of the underpinnings of Math, to expose its underlying simplicity, or inherent congruency, as well as probing the complexities and the details of that knowledge.

    All the Best,

    Jonathan

    I think you made reference to how mathematics by itself is not a minimal guide for physics. Many mathematical areas are vastly complex systems, which might not serve as an effective foundation to reality.

    Maybe a part of the problem is that we do not have a "unified theory of mathematics," assuming such a thing is possible. I think the foundation of the universe involves zeta functions, 8-fold periodicity related to Bott periodicity, homotopy indices, Langland number theoretic correspondences and so forth. This may involve some unification of some subjects in mathematics. I am not sure if this is comprehensive though. Physics in one sense involves working on a similar type of problem on deeper levels, while mathematics often involves pursuing the study of entirely different sorts of structures. These new structures can come into play with physical problems, but the method of thought is often very different between how a mathematician works out the consistency of some type of structure, and how a physicist frames a type of theory or solves phenomenology.

    I would agree that Tegmark's MUH does not appear to satisfy certain minimal conditions we would prefer. In that I would tend to agree with you. I am not sure if this is exactly a proof though. For all we know our requirement for simplicity could be a bias that is wrong. Maybe the universe is vastly complex at is foundation. There are areas of mathematics that have physical implications which involve a huge level of complexity. The universe might in fact have some extremal Kolmogoroff complexity condition, which means the foundations are not only bewilderingly complex, but unknowable. I am not saying I think this is the case, but on the other hand I do not know that this is not the case.

    Cheers LC

    Dear Lawrence,

    Following your last post, this is the type of application we can discuss. Until now, I focused on dessins due to their relationship to quantum geometries and contextuality as in my [12] and [17], now I mentioned in the essay the link to most sporadic groups, there are plenty of other applications, some have to be discovered. Cheers.

    Michel

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    • Post #114

    There are several statements in your comment, dear Lawrence, to which I'd like to respond.

    1. "I would agree that Tegmark's MUH does not appear to satisfy certain minimal conditions we would prefer." I cannot accept the verb "prefer" here. We are refuting the MUH not on the ground of preferences, but on the ground of facts.

    2. "For all we know our requirement for simplicity could be a bias that is wrong." In a sense, I share this caution. As Einstein used to say, "Subtle is the Lord but malicious He is not". We, humans, should not underestimate how subtle He might be. However, without a strong belief in the human ability to comprehend the yet unknown, this unknown would be unknown forever.

    Cheers,

    Alexey.

    The Rodin book looks to be a very long read. It could have some interesting insights into things. There are connections through groupoids to category theory and Grothendieck type of theory and cohomology.

    It is hard to know what the totality of mathematics is. It could be infinite in extent, which of course makes it difficult to know how this applies directly to physics. That would be difficult with respect to the Tegmark MUH conjecture. The one thing that is apparent is how many areas of mathematics are mapped into each other according to functors and categories.

    Cheers LC

    With respect to Rodin's manuscript, Voevodsky is a major developer of the HOTT.

    I think in one sense this will probably have to be simplified or made more applicable for it to be widely used in physics.

    LC

    Agreed,

    Rodin is not an easy read, but contains many useful insights. I also agree that category theory seems to subsume much of the structure in the rest of Math, and could be viewed as central or as essential to a full understanding of the rest.

    Best,

    Jonathan

    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.