Peter,

Thank you for the interesting remarks. Let me congratulate you as well for your place. Regarding your question "Do you agree that recursive higher orders should be able to model that formalism asymptotically?", I would say maybe.

> "One thing perhaps 'missing' was any reminder that a mathematical structure may not necessarily precisely model any physical process for which it's developed, though giving apparently correct outcomes. Do you agree this is possible?"

I think I addressed the question of models that are not "true" in a section of my essay named "Is mathematics merely a tool in physics?". If you refer to models that give all the correct outcomes, and can't be distinguished by other models by no matter what experiment, I think it is possible. But I think that if these can be related to anomalies and paradoxes, then they can be distinguished and falsified. Regarding "if we say ALL maths must ALL be right if the bottom line proves right then we can be misguided about how nature actually evolves in time", well, correct maths is right, but being right doesn't make it true about our physical world.

"You don't suggest a direction for any new formalism pointing towards a TOE". I think it is too early for such a suggestion.

I look forward to read your essay, which is on my todo list.

Best wishes,

Cristi

Hi Steve,

Thanks for the interesting comments and kind remarks. Indeed, some may qualify my essay as (too) optimistic, and I don't really think there are limitations that should stop us from trying to learn more. By the same criteria, any attempt to understand the universe has to be optimistic. In the same time, I advocate an agnostic and skeptical position regarding the belief in the absolute validity of our theories. And as you said, I indeed see no conflict between mathematics and causality. I think you have a keen eye and understood well my points. Your remarks about how changing the mathematical representation affects physical explanation, and the idea to take advantage from Gödel's incompleteness theorem seem to me very interesting. I look forward to read your essay. Good luck in the contest!

Best regards,

Cristi

Cristi,

Thanks for being engaged in my essay and for your kind words.

Jim

Christi,

I don't know how I missed your essay.

Your essay is delightful, like you have the essence of Socrates on your shoulder, distilling with questions and answers his method of inquiry and discussion. I think the potion you create works. It helps to clarify the mind-numbing. You also throw in helpful progressions of studies, GR beyond Newton, singularities.

Impressive essay.

Jim

    Thank you Jim, for reading and commenting my essay.

    Best regards,

    Cristi

    Dear Cristi,

    Since this comment relates to both yours and our essays, I am reproducing it here, in your blog, with a slight abbreviation.

    Thank you so much for your compliments; it is a true pleasure to be highly appreciated by one of the experts!

    You underline that your main take away from our essay "is the uniqueness of the laws." Maybe, our laws of nature are not exactly unique, but they definitely belong to a very special and narrow set of mathematical structures, much more narrow than Tegmark's multiverse suggests. In other words, our laws are truly beautiful in that deep meaning of mathematical beauty which was professed by Pythagoreans of all times, from Pythagoras and Euclid to Kepler and Newton and to Einstein and Dirac.

    In that light the statement of your essay that,

    "Mathematics is already there, eternal and unchanging. What we invent is the discovery of mathematics,"

    is revealed as having an even deeper meaning than it may at first seem.

    Cheers and good luck!

    Alexey and Lev

    I enjoyed your exploratory approach to the essay question, Cristi.

    You parse it into many questions. But I get the impression that you are not dogmatically attached to any of your answers. Rather you enjoy the dialectic struggle between opposing ideas.

    I would enjoy a joust over a beer with you sometime!

    ......David

      Thank you, David. I loved very much your essay. I am very pleased and honored that you liked mine. I have the feeling that you understand so well my position. I would be very happy to have a beer with you!

      Warm regards,

      Cristi

      Dear Christi Stoica,

      I enjoyed reading your essay. In particular, I liked your approach to include many questions. Most of the competitors (including me) are too eager to give answers, but I think it is an important goal to get the reader to wonder about these questions first.

      In addition, I would like to discuss two points.

      The idea of illustrating how you can code various texts as numbers in the [0,1] interval seems well chosen for the intended audience. On the other hand, I worry that you may have stretched it a bit too far when you write: "That line contains your entire life". After all, my life is not a text. ;-) And even if it would be narrated, a lot would depend on the wording - in particular: the first word! (If the narrater always starts with "Well,..." then all our lives end up close to 1.)

      You write at some that "the fact that mathematics is useful doesn't mean that the universe is mathematical". At a later point, you say that a full isomorphism (unknown so far) between the universe and a mathematical structure would mean that there is no difference between the two. This second part is not so clear to me: could there really be an isomorphism between the universe and a mathematical structure? In my opinion, at best we can find an isomorphism between _the structure of_ the universe and a mathematical structure. This makes all the difference: ascribing a structure to the universe leaves a lot of degrees of freedom, but more importantly, it would leave out the 'substance' of the universe, and it would avoid the conclusion that the universe is mathematical.

      Still, my general impression of your essay is a positive one. And the fact that it leaves topics for discussion is a sign of its quality, not a criticism.

      Best wishes,

      Sylvia Wenmackers - Essay Children of the Cosmos

        Dear Sylvia,

        Thank you for the comments. You raise interesting questions.

        One point you make is about my words "That line contains your entire life". You thoughtfully reply "After all, my life is not a text". Well, could you please tell me the part of your life which is not a text? I don't intend to be too curious, rather to ask you a trick question, since if you can tell this, you will make it into a text :) A quick reading of my essay may leave the reader with the feeling that I choose to ignore consciousness for example, but as I wrote, "I don't claim we can explain consciousness, with or without mathematics." What I wrote about this may clarify what I mean: "However, any feeling we may have, there are neural correlates associated to it, and hence, physical correlates. And these physical correlates are in the domain of known physics, which is strongly mathematized." About your excellent remark that you can choose to narrate the same life differently, and make it closer to one number or another, I fully agree, but I can't see why would this be a problem. As you know, there is a bijection between the points on a segment and those in a square, or the entire space, but that bijection can't be continuous. I understand that your remark reveals that representing everything on a segment is counterintuitive, and I like it. I just wanted to make a point regarding whether math is discovered or invented.

        Another point you make is "could there really be an isomorphism between the universe and a mathematical structure? In my opinion, at best we can find an isomorphism between _the structure of_ the universe and a mathematical structure." I agree, isomorphism is between structures. For example, there are more isomorphisms between the set of real numbers and other structures. The real line is isomorphic with a square, if we refer to the category of sets, with a line if we refer to the category of topological spaces, with a totally ordered set if we refer to the order, with a vector space, with a metric space, with a group, semigroup, ring, field, etc., it all depends on the structure we are interested in. As I explained in the essay, the structure is captured in the relations, all relations that can be described by propositions. So there's nothing that can be left outside the structure, if we take into account all the true propositions about the world. Saying "ascribing a structure to the universe leaves a lot of degrees of freedom" is not necessarily true, I mean, of course it is true if we leave outside some of the truths. About the substance, I don't know what you mean by this. Is it something that has effects? Then its properties are captured in the structure. If you think that there are properties of the substance that don't have effects to the structure, then I have nothing to say about it, and anything we would say would be out of our possibilities of verifications. I think the text in a book is what makes a book, and not the paper or the electronic memory used to keep a copy of that text.

        It was a deep pleasure to talk with you, and I wish your essay will do well, since I loved it.

        Best wishes,

        Cristi

        Dear Christi,

        Thank you for your (fast!) response.

        If I tell you an episode of my life, my life itself will not turn into words. To keep it simpler, let's talk about colours: mentioning a colour does not produce that colour (at best, it may trigger a memory of it). Even if I would be able to say everything about a particular colour (not just 'red', but the spectrum, possibly partial translucence, etc.) I would still not have reproduced the colour. Sure, I could use the information, in computer graphics for instance, to reproduce it. But I would need some hardware to run it on (part of the universe). With the universe as a whole, I don't see how a full description (for simplicity, let's rely on an outdated materialistic view: some mass here and some mass there) is enough (in the materialistic example: you would still need to get some mass in addition to a mere description thereof).

        On your view, can you destroy all copies of a book (including our memories of reading it etc.) without destroying the text?

        Best wishes,

        Sylvia

        Dear Sylvia,

        Thank you for the answer. I think I understand what you mean, but I don't think I made you understand what I mean.

        "If I tell you an episode of my life, my life itself will not turn into words." Sure it will not. And even if you give me the complete list of episodes, your life will not turn into words. What I mean is that its structure will be captured in those words. So, going back to my words "That line contains your entire life", I should have said perhaps "That line contains a complete description of your entire life". The main subject of my essay was about the relation between mathematics and physics, and I was careful to mention that I don't aim to explain unphysical stuff, perhaps consciousness being an example.

        You give a good example, that of a color. But you answer it yourself, you can reproduce it on a computer (and that information can be expressed if we wish as words, or as a point on a segment). Your worry that there will be not enough room for hardware in the universe to simulate the color is a new element you add. But the proof of principle remains: the possibility to describe that particular color exists. I am talking about the possibility of a description, not a real implementation of it, not of a hardware. I don't think that, in order to prove it, one should effectively build that description. For instance, the number π exists even if its decimals are nowhere written completely, and even if the entire universe is not enough to write it.

        "On your view, can you destroy all copies of a book (including our memories of reading it etc.) without destroying the text?" If you destroy all writings containing Pythagora's thereom, you will not destroy it, you will destroy the information about it that we have. It will be soon rediscovered. But if you destroy all books by Shakespeare, I don't expect that we will rewrite them soon. They will still be in that segment, but we will lose the address to retrieve them. Think for example at a computer. When you normally delete a file, you delete a reference to it, but the information remains on the hard drive, and can be recovered by special software, until you overwrite it with other information. When you lose the address, the file is not lost. A home is not lost when you lose the address. But you lose your access to it. In the case of a book, the address is the book.

        But when we talk about life, the things are different... I don't think that a complete description of your life values at least 0.000...01% of your life. You may think that this is a contradiction: on the one hand I claim that everything is isomorphic to a mathematical structure, on the other hand, I don't reject the possibility that consciousness or life is more than this. Maybe this is not obvious in my essay of this year (where I focused on the relation between mathematics and the physical world), but perhaps the one for last year is more on this topic.

        Best wishes,

        Cristi

        Dear Christi,

        Yes, it is easier for me to agree with the new formulation "That line contains a complete description of your entire life".

        Just a clarification: my point with the color example was not that there isn't enough room in the universe for the hardware. Simply that we need hardware, which presupposes that we already have the universe at our disposal* - and exactly this does not work when it is the entire universe we want to describe. If we have that mathematical description and delete the physical universe, it does not become reality. Our description will at best be isomorphic to the structure of the universe, but still something is missing to be able to bridge the gap from a mere description to reality.

        Thanks for your reply to my question about the book text! It was really nice to have this exchange with you.

        Best wishes and good luck,

        Sylvia

        *: It's a bit like the Sagan quote: "If you wish to make an apple pie from scratch, you must first invent the universe" ;-)

        Dear Cristi,

        A very enjoyable and easy to read essay, with interesting thoughts and observations. Especially noteworthy for us were your remarks on Godel's theorem.

        The duck: for us, a description of the duck is different from the duck; thus we said that mathematics lacks in material substance. Would you agree?

        And if pushed, what stance would you take: Platonism, or non-Platonism? :-)

        Our best wishes,

        Anshu, Tejinder

          Hi Cristi--

          I loved your essay! Incredibly well-written, well-structured, and thought-provoking. Your analysis of the applicability of Gödel's theorems to physics is spot on. Also, I think that you did an excellent job of rebutting Smolin's objections regarding time and particularities.

          Nonetheless, I do not share your view that "the universe is isomorphic to mathematics". Nor do I believe that "the universe is nothing but a mathematical structure". Let me ask just one "quick question" to sketch out my objection.

          If the universe is isomorphic to math, then why we are unable to analytically solve any of our PDEs in physics? You know what I'm talking about. We are forced to rely on analytical simplifications, numerical approximations, linearizations, and perturbations (to name just a few techniques), every day and in every way, to make progress in physics. Just how isomorphic can mathematics be to the physical world if physicists must typically rely on such mathematical techniques to get the job done? Put differently, if "A Supreme Something" had ordered me to design a physical world--and to do so in way isomorphic to mathematics--I'd like to think that I could have concocted a physical setup far more computationally efficacious than the one we now find ourselves in! And you say?

          I firmly believe that constructive criticism, including disagreement, is the engine that drives progress in physics. I may not agree with your overall position, but you did a great job setting it out and getting me to think about our points of agreement and disagreement. Accordingly, I have given you a high rating. (Not that you'll notice, as you ratings are already very high!). Congratulations.

          Very best regards,

          Bill.

            Dear Anshu and Tejinder,

            Thank you for the kind comments.

            "The duck: for us, a description of the duck is different from the duck; thus we said that mathematics lacks in material substance. Would you agree?"

            I don't really know what to answer, since I really know neither the nature of mathematics nor that of material substance.

            If we call material substance the thing from which things are made, what is that thing? Things are made of atoms, which seem hollow, and made of protons and neutrons, which seem hollow, which are made of quarks. Which are what? and why can't they have independent existence? Are particles waves of some stuff, or waves of probability? Are they fields of what? Can we know about these things more than mathematical abstractions? Are they more than clicks in detectors? You see, I am very confused about all this, and I must admit I don't know what is the substance making all this stuff. I don't want to sound like an ancient oriental wise pondering about the illusion of everything but nothingness, nor like a post modernist cubist surrealist, I just really don't know. What we know are operations from which we infer relations. We know propositions of the form "if we do this we obtain that outcome, with that probability". And we know that in the world there are some regularities, which look like the regularities we find in mathematical structures and nowhere else. So maybe these two things, matter and mathematics, are just one thing, the thread that connects these regularities. Call it the substance making the world, call it mathematical structure, in both cases is just a thread connecting the regularities. So if we say that mathematics lives on a material support, or that what we call material substance is in fact a collection of relations, or of propositions which are true about those relations, or mathematics, would it be a difference? And if there is a substance upon which the regularities are imposed in a way which looks like mathematics, then how can two so different things be so intimatelly connected, if they are not one and the same?

            So I agree that "a description of the duck is different from the duck", but we don't have ducks, we only have descriptions of them. We are talking here about two descriptions of ducks. If the descriptions are identical and there is no way to check how the real ducks are (or how real the ducks are), can they be different?

            "And if pushed, what stance would you take: Platonism, or non-Platonism? :-)"

            None. Should I pick one? If the Platonist view distinguishes between the ideal world and our world, which is just an approximate pale shadow of the former, I clearly am not satisfied with it. Why have an ideal world of shapes and live in the cave? What use would be for that ideal world? And, as you said, how can we know about it, other than by extra-sensorial perceptions, which I don't even know what can be? If non-Platonism means that mathematics is just a secretion of the thinking matter, then again I am not satisfied, because the very source of that secretion is subject to mathematical laws. To avoid circularity, I feel forced to admit identity between the two.

            But if I admit the identity of the two, this looks like a mathematical universe hypothesis, or mathematical monism. In this case, what brings the mathematical structure into existence? I don't know, but whatever we would consider to be the reality (including material substance), we face the same question: what brings it into existence? I don't have an answer for this, but I would rather have a single answer about the existence of the two ducks which are one, than two answers about what brings into existence two so different ducks, and another answer about what makes them so similar. Maybe all that there is is just mind, which contemplates an infinite diversity of propositions, which all arise from the principle of explosion (this essay page 9), and selects from these subworlds which are logically consistent, creating by this all mathematical structures, hence all possible worlds. Maybe. But I don't know :)

            Best wishes,

            Cristi

            Hi Bill,

            Thank you for the kind comments.

            You ask "If the universe is isomorphic to math, then why we are unable to analytically solve any of our PDEs in physics?"

            Why would we be able to analytically solve any of our PDEs in physics? You seem to state it as if it would be an inevitable consequence of the hypothesis that the universe is isomorphic to a mathematical structure.

            "if "A Supreme Something" had ordered me to design a physical world--and to do so in way isomorphic to mathematics--I'd like to think that I could have concocted a physical setup far more computationally efficacious than the one we now find ourselves in!"

            Why would you do it computationally efficacious? And could you do it like this, and in the same time allow the complexity we observe and we need to exist?

            Anyway, our current mathematical models of the physical world are very good approximations. Put it conversely, the universe seems to be able to approximate efficiently our mathematical models, which are indeed not so computationally efficacious. So even if the universe would not be isomorphic to math, it seems to be doing so well the job of a mathematical structure, including these computations.

            Thanks again for these interesting questions, and good luck in the contest!

            Best wishes,

            Cristi

            Hi Cristinel,

            Turing Machines, Game of Life, Free Will, Godel, Rule 110, etc.?! I think you'd like my Digital Physics essay, and the actual movie even more so.

            Are you fine with using all of mathematics to explain our universe? When using math to describe physical phenomenon, are you fine with incoroporating axioms that are merely known to be independent assumptions or would you prefer axioms to be self-evident? I think the concept of actual infinity, and its different guises (e.g. Axiom of Choice, Continuum, etc.) are the root of many paradoxes in both mathematics and physics. Have mathematicians and physicists forgot that a reductio ad absurdum means we should re-examine our assumptions?

            Please check out my essay if you get the chance.

            Jon

              Cristi,

              I haven't seen you announce it, though since I have Email confirmation from the Minkowski Institute Press that your PhD thesis has been published, please allow me to extend my congratulations. I note that your acknowledgments include David Finkelstein, one of the Minkowski Institute's founding members (along with Abhay Asktekar whom you also acknowledge, among other highly distinguished academics) -- Finkelstein is one of my favorite scientists.

              All best,

              Tom