Essay Abstract

We consider the unreasonable effectiveness of math in physical sciences by drawing clear differences between pure math and mathematical physics. We take into consideration the preferred methods of the fields, their historical background and examples from other fields. We build a theory of their similarities and divergences. We present some examples to support the hypothesis that the partial success that we investigate is based on the simplicity and the reducibility of the fields to their component parts, as well as naturalness and proximity to the fundamental parts of reality. Using our current knowledge of the directions and advancements of each field, we try to predict potential future directions.

Author Bio

Female, born 1983, Eastern Europe, life-long learner with a background in IT and business. Some formal math and physics background completed with courses and independent study during the past few years. I keep up to date with current research directions as well as theoretical developments and experimental results. Primary interests in high energy physics-theory, astrophysics, general relativity and quantum cosmology.

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Dear Alma,

Your beautiful essay is very fun and entertaining, and in the mean time so profound. You fill a gap and go at the root of the problem, when you discuss the differences between pure math and mathematical physics. Indeed, the simplicity and the reducibility of the fields to their component parts is responsible for the success of mathematics as a tool to investigate physics, or "Math works perfectly in physics because, as we will show, they are both simple and modular in their own way". As you well pointed out, a simple tool would be limited to the intention which led to its invention, while "when one makes the complex plane, fractals grow in it uninvited, without having been planned, placed there or even wanted, like leaves on a light bulb". This tool takes over and starts living its own life, leading us to new, unexpected destinations, "but when we lose the correct track we have no way to find it again without more new [physical] insights". You are right that we, as species, don't do math for that long time, and "there is still time for math". Despite the fact you mention, that "we haven't discovered anything fundamentally unexpected for years", I share your optimism, summarized in your conclusion that there is still time.

Best wishes,

Cristi Stoica (link to my essay)

    Dear Alma Ionescu,

    Your writing style is exquisite and you phrase the problem well: "It would be very nice to know once and for all if math is a human invention. ... If it is ... we could cross it out of our list of places to search for deeper meaning. If it isn't, and there hidden logical rules to the universe ... then we can see it as an intrinsic part of reality."

    I suggest that logical structure [the basis of AND and NOT 'gates'] is inherent at all scales, from telomeres, and RNA-DNA-protein operators, to crows counting, to microcomputers; and that counters [the essential basis of QFT's 'Number operator'] are ubiquitous. From there Kronecker says we do all the rest. I doubt very seriously that all of the physical counters and counting [and adding / subtracting] operations of the non-human portion of the universe should be ignored or written off. With Kronecker, I see the embellishment of these bases into "mathematics" as human invention and discovery. It is the logical structure [both static and dynamic] of physical reality that enables the relations we classify as 'math'.

    To draw a boundary between the physico-logico-mathematical structuring of physical reality and the abstracting and symbolic representation and manipulation of such seems not right. The only thing that man brings to math is the degree of consciousness that allows him to move beyond insect's and lower animal's direct use of math to formal representation and manipulation of symbols.

    Although "when one makes a complex plane, fractals grow in it uninvited, without having been planned, placed there, or even wanted, like leaves on a light bulb." Is this really any different than motion pictures growing from the light bulb, without having been planned, placed there, or even wanted? Actually, it takes quite a bit of purposefully created superstructure to bring out motion pictures or fractals. And it is not clear to me that either is more 'implicit' than the other. Both flow from the scale independent logical structure of physical reality, and I doubt that either has any meaning in any sense if one entirely removes physical reality from the picture.

    You note that "math lies dormant", but the "motion pictures" lay just as dormant. [This analogy is probably ruined by the existence of lanterns before light bulbs, but you get the picture.]

    You remark that "math made so much sense that it was worthy of worshiping..." If math is inherent in the self-consistent logical structure of physical reality, this should not really change that sentiment.

    I very much like your abstraction that "the square root of 2 emerged, and for all his efforts Pythagoras could not fix it." One can rearrange, but not alter the nature of physical reality.

    You ask so many cogent questions and make so many observations, one cannot respond to them all, but a very important observation you make is:

    "Mathematical physics is only as good as physical insight."

    I apply this reasoning to Bell's theorem, and hope you will read my essay and provide feedback.

    I very much enjoyed your many questions and astute observations.

    Best regards,

    Edwin Eugene Klingman

      Dear Edwin,

      Thank you for reading my essay and thank you for your very kind words!

      To be honest I was preparing to read your essay anyway. I will probably not be able to read all the essays, but I at least had in intention to read the highest rated works. I will gladly comment on it after your lovely invitation to dialogue.

      To respond at least partly to your comment, I think your counters are no different from what I had in mind when considering the self-assembling structure of math. The logical gates are meant for preserving (or are equivalent to) the quantity of information that's present in a system. Regardless of the manipulation of that information and the number and complexity of operations one applies to it, the information going into a structure (or calculation etc) needs to be equal to the quantity of information going out. It's the information preserving quality of math that makes it ultimately decomposable, because you can treat a collection as equivalent to the sum of the parts. However it is this insistence to preserve information that makes it different from other more fluid and unconstrained structures. I saw at least a movie that made me scream in pain for the time that was irreparably taken from my life and unfortunately I'm sure you can identify with that experience, regardless of the undoubtedly good intentions of the director and actors (intentions that were equally good to those of people who made films that we enjoyed). And yet surely one can find some fans of that excruciating film and no two people appreciate it for the same features. So it isn't only the emergence but also the constrained nature of math that makes it look (at least to me) more compelling that other emergent phenomena. Math has conservation laws, just like physical systems. (Thank you for helping me realize it. This is the first time I saw it clearly enough to be able to state it in unequivocal manner; otherwise I'd have put it in my essay). A good mathematical theory will be universally consistent for every (reasonable) being that observes it. (Now I am wondering where is the good will invested in making bad movies going and if it's conserved. I heard that it's paving the road to hell; so it is conserved).

      My comment about math sleeping for centuries was rather aimed to support Wigner's comments and stir the controversy. Also I stated in a rather matter-of-factual manner that math was worthy of worshiping to Platonists because it's interesting to note the aura that accompanied math through different stages of the human culture. As a tongue-in-cheek comment, I will mention that now we just say it's complicated and therefore accessible only to adepts, which is equivalent to worshiping it more than in the times when we considered everyone can understand it; only the chosen few can interpret the will of the god. It is perfectly true that math is discovered and then it takes ages to find an application for it (Galois' work), but that is probably because it's easier to find a certain logical theory - one only needs a brain - than build the LHC - one needs a bit more than just a brain. Some of my ideas are only half-formed since this wasn't an exhaustive attempt; I am open to discuss those ideas and comments but at this stage they are nothing more than the eerie music that accompanies the action hero venturing in a dark hallway while the camera peeks over his/her shoulder (I'm keeping your analogy because it's so rich). I am neither a Platonist nor a not Platonist. I am maybe an utilitarian. Evidently math has some special features and it's good to keep that in mind, eg, if math lies dormant and you need a special kind of math, maybe it's better to use google scholar instead of trying to make a theory.

      I know that I didn't respond to all your comments, but at least I hope to have made you laugh.

      Warm regards,

      Alma

      Dear Cristi,

      Thank you very much for your kind words!

      I am very glad you enjoyed reading my essay and I will be sure to read and comment on yours.

      Warm regards,

      Alma

      Dear Ms. Ionescu,

      You asked: "Why is a physical theory less sturdy than the math it's composed of?"

      Because physical theory and mathematics are both absurd abstractions.

      Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

      Warm regards,

      Joe Fisher

        Dear Alma,

        I'm very pleased that I could help you realize something enough to state it unequivocally. Sometimes all it takes is a slightly different way of seeing it.

        I really like your analogies, and completely agree with your view that math has become "complicated and therefore accessible only to adepts, which is equivalent to worshiping it more than in the times when we considered everyone can understand it; only the chosen few can interpret the will of the god." I suspect that's the way they like it. That is a sad truth.

        Also your brain versus LHC observation is very well expressed. I too am essentially utilitarian, but I tend to fall in the not Platonist camp. And yes, your colorful way of expressing yourself does make me laugh, or at least smile.

        Best,

        Edwin Eugene Klingman

        7 days later

        Dear Alma,

        Congratulations for a very interesting and well written essay. You have an enjoyable writing style, and some of your sentences have a literary feel - almost poetic! I really liked the spherical cow in the void, the light bulb growing leaves, and the list of ailments that befell our biggest mathematicians... I could almost feel their pain! You should start a blog or write a book...

        I found your section "The Shape of Things to Come" very interesting and original. It is interesting to try to evaluate how many mathematician-hours have been clocked over the course of civilization, and I share your optimism that "There is still time for math."

        One of the main ideas of your essay is, if I understand you correctly, that you hold on to the hope that we will one day be able to prove that "there aren't so many self-consistent recipes to build a universe from nothing" --- in other words, you hope that we will one day be able to answer by the negative the question that Einstein asked, "Did God have any choice when he created the Universe?" I agree that it would be quite an achievement --- but I also believe that it's very unlikely that we will ever reach that conclusion, because I don't see how the very specific ways that our universe is put together (all those families of particles (not to mention dark matter), all the particular values of the fundamental constants) is the only possible way a universe can be put together. I have a question for you: do you think that it's possible that we could show one day, from first principles, that the proton has to exist, and has to be 1860 times more massive than an electron?

        Near the beginning of your essay, I had trouble understanding what you mean when you state that it's possible that "there are hidden logical rules in the universe that make math obey a course and one course only." Are the rules of logic specific to a particular universe? In my opinion, there is only one logic, irrespective of the existence of any universe --- logic, like math, just is. I also don't believe that math obeys one course and one course only --- for me, math is the general study of structures, so math is infinite, even if our knowledge of math cannot be infinite and could very well be limited by our minds, even by the kind of universe we live in. If you have time to elaborate on this subject, I would appreciate it.

        So far, I have read over 70 essays, and in my opinion yours is clearly one of the best --- I hope you make it to the finals, and I have rated it accordingly. I wish you good luck in the contest.

        Cheers!

        Marc

        P.S. We do agree on one thing: "math does what math wants". I just believe that math wants to do it all!

          Alma,

          This is a well written and enjoyable essay.

          Sadly, it is true that many great scientists and mathematicians have died prematurely. Riemann and Maxwell for example. I understand that Euler blind? My memory may be fuzzy on that one. Emmy Noether did not even leave a textbook:-(

          But as you note, even a few 10's of thousands of man-years does produce results. And as long as it is written down, it will be remembered.

          You mention quaternions and their use for describing orbits ... You might be interested in my essay.

          Best Regards and Good Luck,

          Gary Simpson

            Dear Marc,

            Thank you for taking the time to read my essay and for your kind words!

            I'm not sure I can say I have an optimistic attitude or hold hope. Should there be a theory of everything, I realize it's not even granted that we will ever find it. Unlike winning the lottery (where someone has to win at some point), a TOE is not a guaranteed miracle. It's more that I am somewhat pointing to the possibilities of living in an open universe, with an infinite future, vastly larger than its past. Trapped in finite lives, it's easy for us to forget that others will follow us and that it's nearly impossible to predict the future of mankind (although it looks bleak at this point with ecological problems and whatnot).

            I don't think there is a way to show that a proton must exist, unless one takes quite a zen approach, eg. protons happened when time started, so in a sense they always existed and therefore must exist. This obviously doesn't produce a lot of helpful insight. But there is a way to create an infinite number of universes just like this one by keeping the ratios constant, as you said, the 1860 proton to electron mass. Thus the masses of the particles would be equivalent between universes, modulo something, along with all the other constants; all these universes would look the same. Take that something from modulo and replace it with all the real numbers, one at a time; suddenly the chance for our universe to be looks strangely different (the chance for this universe to be among an infinite number of universes made by varying the cosmological constant is 0; 0*aleph1, the cardinality of the set of reals, what's that? Should we multiply zero with the cardinality or with the value of the cardinality, and if we multiply with the value what do we get?). To make a toy model just for fun to explain this bizarre ratio, I'll choose the idea behind technicolor theory, where fundamental particles are made from a single type of constituent that's held together in bunches by a superstrong force; it's the only example I can find right now and I find it intuitive. Say this even more fundamental constituent is duplicated and configured inside each particle, so for an electron we have 10 pcs of it and for a proton we have 30 pcs, 10 for each quark. Then we take these configurations and plug them in some perturbation theory, the first few loops raising the difference in mass up to a factor of one thousand and then the next loops converging to some masses that share that ratio. Of course I'm not saying that there even is such a theory; I'm saying that we can't rule out the possibility that it exists or the possibility that we could one day find it. Even if the values that we have look weird.

            About the hidden logical rules. Our universe, the one that insists to make sense is also the one that influences our thinking. If we were to live in a level V universe, out math may have looked different; it would have been the math of the infinitely intelligent mathematician. He can understand ours, but we can't understand his (or maybe we can with proper training, but we're not inclined to use it from our own initiative). Our universe does have regularities and we were all born here, so our thinking seeks patterns. If we were to live in a universe with time swirls, we'd even have problems defining the number 4 in the way we do here. Natural numbers were born from the need to count so say that a prehistoric being is trying to count food urns, 4 of them, but when he finishes counting, a time swirl takes the 4th urn. Not just that only 3 urns remain, but only 3 urns were ever there because of the problem of the local direction of time. But the time swirl that took the urn left its memory unaffected so the notion of 4 will from now on be associated with 3 objects, in a superposition where you mean 4 and see 3. Of course this may not be a very good example but there's only so much one can do to make a definition have a meaning and the opposite. Depending on how that universe is constrained, it's maybe even possible to live in it (to keep it simple, say it has all our constants, but some small scale thermodynamic oddities), but the way that the inhabitants interpret the world should be different from ours. Perhaps they have a real-world number system that helps them count these awful disappearing objects and a second ideal number system that works like ours and is used only in philosophy classrooms, having no application to their world. A universe where the time direction is locally not constant would certainly generate minds that would seem odd to us. Our universe came with built-in math with conservation laws; why? Well that's a hidden rule, for now.

            I enjoyed your questions, by the way. Wish you all the best!

            Warm regards,

            Alma

            Dear Mr. Fisher,

            Thank you for reading my essay and for your comments! I appreciate your realistic view of the world.

            Wish you best of luck in the contest!

            Warm regards,

            Alma

            Thank you, Gary! Your kind words are much appreciated.

            Yes indeed, the one that went blind would be Euler. He tried to learn to write without looking while he still retained some sight, but after losing his sight his writing was too unclear to serve any purpose. I used that trick to encourage the readers' feeling of participation, because I thought you would recognize every case. I'm also glad you liked the man-years approximation.

            I downloaded your essay and took a look. It's certainly quite dense so it will take me a while to digest it, but it looks appetizing and I will certainly comment on it once I manage to understand it.

            Warm regards and best wishes,

            Alma

            Dear Alma,

            Fun speculations! I like the idea that beings living in a universe based on math that does not support conservation laws would have an appropriate mathematics that would work in their world, and also mathematics that resembles ours, but is only used in philosophy classrooms! It looks like something out of a Borges story, or something that could exist in some remote corner of Terry Pratchett's Discworld...

            Keep Maxiversing!

            Marc

            Dear Marc,

            Right, I forgot to say something; you first mentioned that my writing style is enjoyable. I think I can say I mostly learned English from Terry Pratchett's books, so even if I can't compare my writing to his, I think I retain some of the formulation quirks and comparisons he used to make.

            Cheers,

            Alma

            Dear Alma:

            A very nice and well written essay, I really like it! I agree that math can't work on it's own --- that's the biggest problem my "pragmatic physicist" has with Tegmark's mathematical universe. It isn't of much use knowing we live in a mathematical universe if we don't know where we live. We'd still have to go out and figure that out. It didn't really become clear to me though whether you are arguing for or against reductionism, or maybe the middle way in which reductionism works 'in principle' but not 'in practice'?

            -- Sophia

              Dear Sophia,

              Very nice to hear from you!

              Actually that's a very good question because I didn't consider reductionism when I wrote the essay. My thoughts on the topic are not as clear as they should be considering the difficulty of the topic - but to answer your question, I think that maybe I can say that reductionism can work in practice but not in principle. What I mean is that finding descriptions for emergent phenomena may be possible and they would probably come in handy, however it may not be completely satisfactory. To give the best example that can illustrate my case - some day we may find a great description of the human brain and it will have life-saving applications in medicine; that description however may not include qualia and even if it does and we can rationally argue that it's complete, I very much doubt that it can feel complete.

              Warm regards,

              Alma

              12 days later

              Alma,

              Interesting read. I like your chatty, almost poetic style, like a narrative we want to continue, and I like your Socratic inquisitions.

              When you say "We seem to come to a halt momentarily ..", do you mean nothing stunning has occurred, like finding the Higgs through LHC was expected -- but they now expect a larger-mass Higgs is being hidden as they increase energy.

              I think scientific approaches can be more directly looking for solutions to mysteries in the classical world like looking into the almost 100% efficiency of photosynthesis and tying it to superposition in the quantum world and quantum biology's look into the European robin's navigation N to S. The math follows.

              Enjoyed your essay.

              Jim

                Dear Jim,

                Thank you for reading my essay and for your kind words!

                I said that we seem to have come to a halt at the present time because some long sought after principles have failed to show, like SUSY or Higgs' brothers, dark matter signatures in LUX or any other number of different things we were hoping to find, but my main idea was to plead for patience. That we didn't find something in the last years does not mean much because we haven't been searching for long enough to despair.

                The examples that you give, new unexpected puzzles that may lead to new conceptions about the world or new valuable applications, are wonderful and help make my point.

                Warm regards,

                Alma

                Dear Alma,

                I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

                All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

                Joe Fisher

                  6 days later

                  Alma,

                  I revisited your essay and am thankful you checked out mine. I find that I did not rate yours, something I usually do to those I enjoy, so I am rectifying that. I like the bounciness, the hopefulness and the openness of your essay.

                  Jim