Essay Abstract

Just how "unreasonably effective" is mathematics, anyway? First, why does math accomplish so much for us in (apparently, in practice at least) understanding the "real world" around us? Then, what are the limits of what it can accomplish? Can it tell us why there is anything substantial at all, and why the world is the way it is? Here I argue related theses. First, mathematics is good at judging relative consistency (such as, whether a given model universe will be able to satisfy conservation of mass-energy.) This is demonstrated through a novel argument explaining why our universe must have three large spatial dimensions, in terms of the self-consistency of electromagnetic relations. Then, I argue that math has no ability to explain its own effectiveness in a "real world," or to explain the particular foundational properties of our world. Furthermore, math cannot explain why some possible worlds have "concrete" or "real existence" and others do not. Indeed, the very idea there is such a distinction is surprisingly problematic and questionable, as argued by Max Tegmark and others. Finally: any such existential distinction would be inaccessible to computational (AI-theoretical) intelligences.

Author Bio

I am an independent scholar interested in foundational questions. This is my fifth FQXi Essay Contest submission.

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Greetings. First, I realized I made a typo in submitting my little bio to the web form (also, if I write something I want to use first person to show that.) Then I appreciated, it was sort of cute - I do want to be "interesting" not just interested. Along with the others here, this essay delves into the fundamental question of how well, and why, does mathematics model reality.

Asking these deep and basic questions is a worthy enterprise. However, it is good to make actual direct contributions to physical knowledge, or at least come up with specific theoretical insights that might be true advances in knowledge. Here, I provide an argument showing why space has three (macroscopic) dimensions D. It builds on previous work, but apparently no one previously grasped how to put the parts together to show that space-times with analogous physics to ours, would not be internally consistent in all respects if D is not three. Hence we don't need to invoke anthropic considerations to explain our finding that feature, such as the difficulties life would have in universes with other numbers of large dimensions. If the universe originally had more dimensions, this could explain why the rest of them remained/compactified and literally could not expand along with the others.

Also, there is an erratum in the paper, which could be considered a graphical fail. In Equation (3) on page 4, there shows what appears as an absolute value of the vector p-dot (time derivative.) This would wrongly remove the negative sign which can be applicable. The correct representation should be, dot product of the magnitude of p. Alternatively, we could just simplify by restricting to a y-axis, and using y-components. BTW does anyone know how to improve equation setting in Open Office? It just doesn't work like MS Equation Editor.

Last but not least, I am very impressed by the quality of the entries to this latest Contests, and by the caliber of the participants. There are some "names" among us. I am not one of them, but I appreciate your interest and commentary. Thank you, and please have a good read.

Hi Neil -

Very interesting essay! I am in full agreement with several of your conclusions: "Math and logic don't have the tools to reach beyond their realms and characterize the status of another existential level." "We are free to decide what intuitions to trust in our quest for the horizon of what we and the universe are, and why."

I got to these conclusions with a very different approach and would love your thoughts!

Regards - George Gantz

    George, thanks for your thoughts. I will read your essay soon.

    Dear Neil,

    A logically strong presentation of views that I am in agreement with. Your essay is a vital contribution to this contest. Good luck.

    James

      Dear Neil Bates ,

      I agree with the essence of your essay: "math can tell us what works, but not what is real or why it is real". Then the next question arises: Why is it that math can tell us 'what works' and 'what does not work'? My essay is based on this question. The simple answer lies in the nature of 'working', which can be defined as a 'change'. For any change to happen, bodies or its constituents should physically move. Or it is motion that causes changes. Motion is a space- time relation that can be mathematically described. So all changes follow mathematical laws. So maths can tell us what works.

      As pointed out by you, maths cannot say why a body has mass, why a body has volume, why a body attracts, why bodies can move. In my opinion, we have to distinguish between properties and laws. Once the bodies have the given properties, it is mathematical laws that decide how bodies change with time. Please go through my essay: A physicalist interpretation of the relation between Physics and Mathematics.

        Thank you, James. Good luck to you as well. I read your essay and posted a comment. I hope I correctly understood your overall conceptual point (which seemed more important to me than the details of your reformulations of math concerning field theory.)

        Jose, I appreciate your interest. I'll take a look at your essay.

        A very closely reasoned essay.

        I think you might want to consider the implications of operationalism (P. W. Bridgman, The Logic of Modern Physics.)

        Best, TOm Phipps

          Dear Neil Bates,

          I enjoyed your essay, starting with your comment "math can tell us what works, but not what is real or why it is real", a statement I completely agree with. I liked discussion following "but our critical question here is: what does it tell us about "reality"? By reality, I mean what the person in the street means: "stuff" that actually "exists," not just mathematical models like the ones used to prove this point."

          I have taken a little different approach to this same problem my essay here. My goal was to find "real" objects (in the sense that they can be modeled visually) that match the mathematical models of the particles of the standard model. I hope you get a chance to have a look and offer a rating.

          A well done and well thought out essay, best of luck, you deserve a good rating.

          Regards,

          Ed Unverricht

            Thanks, Tom, it's good to hear from you again after so long. Yeah, operationalism is preferable, but has its limitations. We just can't experimentally test "other universes" - yet, at least. Your own essay made me think, as usual.

            Thank you, Ed. I will look at your essay. Best.

            Dear Neil Bates,

            I very much enjoyed your essay, and its support for the intuitively obvious fact that we live in a 3-dimensional world. I will have to carefully review your arguments for 'EMI equivalent mass', but, given that these arguments lead to your equations (2) and (5) you very nicely derive the result D - 2 = 1 / ( D - 2 ) => D = 3.

            Thank you for the simple demonstration

            Your discussion of 'existence' and 'reality' is also very cogent and well argued, and also that "math and logic ultimately constitute a self-referential universe", working on "structure not essence", and incapable of reaching the level of substantial reality.

            You say, "all that mathematics can do for us, is make relative judgment about model worlds in terms of various internal criteria."

            I fully agree with your conclusions about 'reality' and 'mind', so I would like to focus on your statement about "relative judgment about model worlds...". You say [p. 7] "all that math knows and can tell us in effect, is about math. When we think it is telling us something about "the world", we are just finding out about the model that we are using."

            In my essay I discuss the oversimplified physical model of Stern-Gerlach based on precession in a constant field, which leads to a null result; 0 not ±1. This contradiction is the basis upon which Bell builds his model, with the well-known "logical" consequences. I investigate a physical model of Stern-Gerlach that does not immediately lead to a contradiction, and attempt to see where that takes us. I invite you to read my essay and welcome your comments.

            Best,

            Edwin Eugene Klingman

              Edwin,

              Thanks for your comments. My argument about electromagnetic mass is somewhat complex but starts to come together for anyone with solid background who just follows along carefully. I appreciate that you are another of us, who realizes that math cannot just be glibly substituted as map for territory. There are many ways for the project to go wrong, both in terms of practical effect as well as deep questions of mind, determinism versus free action, etc. See my remarks about your essay at its link.

              13 days later

              Dear Neil,

              I have read your essay in the spirit of the Cartesian doubt with great interest. I totally agree with you:

              "Math and logic don't have the tools to reach beyond their realms and characterize the status of another existential level. Such systems can neither describe a more "substantial" realness per se, nor distinguish nor explain the high-level "accidents" of phenomenal existence - the apparent discriminatory actualizations of one (or more) mathematically-delineated possible worlds. Therefore, math cannot either describe what "concreteness" is, nor which if any model worlds should be manifested as "actual worlds." It cannot explain why any such transcendently "more real" world should be mathematically elegant or "simple," instead of messy and not effectively accessible through math."

              "All that mathematics can do for us, is make relative judgments about model worlds in terms of various internal criteria. It cannot answer existential questions, such as "why does anything exist (in the sense of being more than math)" or even if there is such a thing as "more." All that math knows and can tell us in effect, is about math."

              "Our world does not and cannot follow "from first principles" of any logical or mathematical sort."

              "We must transcend math and logic to grasp this."

              "If you do truly feel and know that the world is more than math, then it is, indeed - and so is your mind. I do not know what that "more" is, if it is so. We are free to decide what intuitions to trust in our quest for the horizon of what we and the universe are, and why. Mathematics cannot tell us about anything more than itself."

              Mathematics and Physics - the science without ontological justification (basification), as well as all "the fundamental knowledge". How to choose a path? Deeper limit ontology and dialectic in the spirit of Plato - Cusa - Hegel: "coincidence of opposites." We have now a lot of logic. To grasp the Universum as a whole (to connect Cartesian "res cogitans" and "res extensa") need the dialectical logic. Requires the deepest synthesis of all the accumulated knowledge, including Tradition. The "Occam's razor" should be extremely sharp.

              E.Husserl gave good tips in "Origin of Geometry": "Only to the extent, to which in case of idealization, the general content of spatio-temporal sphere is apodictically taken into account, which is invariant in all imaginable variations, ideal formation may arise , that will be clear in any future for all generations and in such form will be transferable by the tradition and reproducible in identical intersubjective sense. "

              I believe that only the deepest ontological turn of the fundamental science will provide an opportunity to get out of the "crisis of understanding", "crisis of interpretation and representation" to the new heuristics.

              I invite you to see my analysis of the philosophical foundations of mathematics and physics, the method of ontological constructing of the primordial generating structure, "La Structure mère" as the ontological framework, carcass and foundation of knowledge, the core of which - the ontological (structural, cosmic) memory and information - polyvalent phenomenon of the ontological (structural) memory of Universum as a whole. I believe that the scientific picture of the world should be the same rich senses of the "LifeWorld» (E.Husserl), as a picture of the world lyricists , poets and philosophers.

              Kind regards,

              Vladimir

                Vladimir,

                Thank you very much! Yes, we can't just put together an "explanation of the world" out of the thinking applicable to pure math itself (which does not tell us which of those should be picked out for further manifestation). We need to look at our experiences in the world, but it isn't clear yet just how all of this adds up. Since we experience the world through our minds, we have to acknowledge our own role in the way it seems to be. Naive realism is fundamentally wrong, and Kant was perceptive about the situation: we construct a lot of what we talk about as physical foundations. Phenomenologists realize this point, and how we must "bracket" out appearances instead of taking them for granted. This requires a lot of self-study and integration of physics, psychology, and philosophy.

                What I wrote might seem fanciful at first to more hard-nosed folks, but just consider how the intuitive concept of "flowing time" has been attacked lately by many physicists as a misrepresentation of what they claim is actually a fixed "block" instead of a traveling "now." So I talk both here and in my previous essay of the need to delve inward as well. However, unlike most of those considered the foundations, I don't think that any kind of mathematics can fully grasp the universe, because its special "realness" is not able to be described in terms of forms and quantity. I also argue that an AI (computational type) mentality could not grasp the distinction, and so our minds must be beyond such characterization.)

                In any case, we can certainly try to resolve more specific questions like, "why is our space three-dimensional"? Here I provided an argument based on self-consistency of EM rules in various dimensions, and show how things really only work out to conserve energy (for the electromagnetic mass) in three dimensions of space (taking time as granted.) Yes, this is a big question to grapple, and yet even this is separated by a great chasm from a question like, "why is there something instead of nothing?" The question about the nature of space is properly relative within the frame of some set of rules, where the ultimate question asks for where the framework comes from in the first place. We may or may not be able to answer such questions, but we can't even make an effort without deeply understanding ourselves and yes, our cultures and traditions first. Physics by itself cannot accomplish this. It must be a grand interdisciplinary effort and one that brings the world's various ways of doing, together.

                Vladimir, I will look at your essay soon in more detail and make some comments. Here is my preliminary observation: like the last time, I see you had some translation issues regarding which English word or phrase is best to state certain concepts. If I knew more Russian I could help, but alas I do not.

                PS: I apologize in general for my low activity in making comments etc. in this contest so far. More time will open up soon.

                Hi Neil--

                Your essay is excellent. I agree with your bottom-line: that math, being a closed set of rules, is best at handling self-consistency, but is unable to help us on matters "out there". For that, we need physics. Your analysis involving Eq. (2) was great--but why did you call the result "obscure"? Finally, I admit that I am a fan David K. Lewis and "Modal Realism". So, I was on your side from the start.

                Best regards,

                Bill.

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                  • Post #19

                  William,

                  Thank you very much. The paradox of math indeed is that it can tell us about anything else only through itself. If our world happens to parallel (is isomorphic to) some mathematical system, that same math cannot explain why it is the model for a world - unless the world literally is that mathematical system. Then it is a case of identity, not a mysterious echo. I don't believe the world is just math or that all possible worlds exist equivalently (as per David Lewis), but at least I appreciate that such viewpoints should be respected. Too many thinkers dismiss them as obviously wrong and absurd, able to be dismissed as "obviously wrong." Yet they don't even realizing the irony that believing in the specialness of concrete existence is akin to the "mysterian" idea of accepting conscious intuition for granted, and that the mind is beyond understanding in terms of matter. Ordinary physicalism is, ironically, a form of dualism itself - as recognized by critics like John Searle (and Lewis, in his way.)

                  Now, that doesn't mean we have to accept there is no physical "extra," just that if there is, it is transcendent. (I think the world and our minds are "extra" together, and form their own dualism vis a vis math - a twist on the usual divisions.) The ultimate answers to our universe being the way it is, won't fall out of equations a priori. So, we have to do experiments. Although I did not propose an experimental test this time, I did analyze a specific physical question instead of just ranging over basic issues in the abstract. (Nothing wrong with that, but we need facts and insights to fit into the pattern formed by our perspectives and methodologies.)

                  I said the result of Eq. (2) was obscure, because I see discussions of what physics might be like in 2-D flatlands (what atomic orbitals would be like, what 2-D life might be like as in the clever model-world fantasy The Planiverse" by A K Dewdney, etc.) These writers don't tell us that such a world would not "work" because of the infinite integral of potential energy between charges etc. I probably should have said, the implications of the integral are obscure (or shirked?) However, the shift of force exerted across a gravitational potential, really is obscure as best I can tell. You just don't see it discussed.

                  I will look at your essay and make a comment or more, when time opens up as it will.

                  Well, I got unwittingly logged out, and that happens to lots of us. The reply to William Parsons was from me.