Dear Ian,

Good essay. It shows the distinction between maths (the representational) and physics (the tangible) at an elementary level. Can you comment on the quantum reality? If one agrees that quantum measurements are contextual the quantum world is tangible, isn'it?

The cardinality of the Monster Group M (big but not infinite) is representational at the moment but it can become tangible (will be effectively counted with the computer), the cardinality of most sporadic groups became tangible.

All the best,

Michel

    Hi Lorraine,

    I think we're talking past each other here. I absolutely agree that we need to get away from Platonism, at least in physics. Several of my other FQXi essays have touched on this topic in one way or another. But I think the question of what tangible reality a number represents is a concept that really has little to do with number itself, with the exception that, as Russell pointed out, it is a way of creating a similarity class.

    Think about it this way. Imagine a toy universe that only has two objects in it. What purpose would the concept of number serve in such a universe?

    Ian

    Hi Ian,

    In the toy 2 particle universe you describe, only the following information can exist (from the point of view of each object):

    1. Categories of information (about "itself" and the other object) e.g. mass or a more fundamental category of information;

    2. Information interrelationships;

    3. Quantity information associated with the information categories.

    A physicist could represent this information and these information relationships with law-of-nature mathematical-type equations; and the quantities would be represented by numbers.

    It's clear that reality utilizes "categories of information" like mass, momentum, charge etc. And some of these categories are not fundamental, but are in effect built out of relationships between existing, more fundamental, information categories. Seemingly, at least one information category must be considered to be a "first principle" of reality.

    Similarly, it is clear that reality utilizes information interrelationships. We attempt to represent such interrelationships with symbols like "= + x", and at least some of these interrelationships must also be considered to be "first principles" of reality.

    When it comes to quantity, which we represent with numbers, we must also assume that there is an underlying simplicity. "Quantity" describes something that really exists in fundamental reality.

    But the objects in fundamental reality can make no distinctions relating to number of objects: there can potentially only be distinctions made relating to quantity. The reason for this is made clear by mathematician Louis H. Kauffman's article "What is a Number?":

    "Two classes are said to be similar if there is a one-to-one correspondence between the members of the one class and the members of the other class. . . [But] How do you know to take a given form as foreground against which to make the comparisons?. . . None of these questions can be addressed by a formal system alone. The ability [to] perform the relations indicated by these questions is a prerequisite to being able to make any formal system at all."

    Correspondence and comparison are complex many-step operations: clearly there can be no such ur-counting behind-the-scenes at the level of fundamental reality. Classes and sets have a hidden underlying complexity: sets have a complicated underlying infrastructure of axioms, defined symbols, defined equivalence relations, ordering etc. If one assumes that some sort of simplicity underlies physical reality, one must conclude that the quantities found to exist in fundamental physical reality cannot be modelled by the sets that represent the properties of number systems.

    On the other hand, a number seen as a ratio seems to be possible for fundamental reality. To give another toy example: assuming a toy law-of-nature relationship could be represented as "a+bc =d", then a (rational) number relationship could be represented as "(a+a+a)/a" . I'm saying that the substance of the reality underlying quantity could be as simple as the reality underlying laws-of-nature.

    (I attempt to explain in my essay how the complex numbers, algebraic irrational numbers, and non-algebraic numbers found in nature might be seen in the light of the above way of looking at numbers.)

    Cheers,

    Lorraine

    Hello Michel! Thank you for your comments. And your essay is on my list to read (I noticed that you wrote a dialogue as well). I absolutely believe that the quantum world is tangible. In my opinion, if it weren't, then nothing would be tangible.

    7 days later

    Ian,

    Very fun and imaginative dramatization. Math is a representational reality which means it must be peer reviewed by others (like BICEP2) to make sure it really represents tangible reality. Eddington's observation did help prove Einstein's theory but at the same time Eddington mistrusted mathematical derivations from relativity theory to explain "degenerate stars."

    Thanks for sharing an imaginative creation that helps clarify roles of math and physics.

    My "Connections" tries to show the relationships of math, the mind, and physics in new discoveries in quantum biology, LHC and DNA.

    Jim

      Dear Professor Durham,

      I am just a humble student but noticed a similarity of one moral your excellent story conveys to the main thesis presented in my little opera "Map = Territory" where I ponder the possibility of an actual merger of the description and the described in fundamental physics.

      Your message "the universe is nothing more than our description of it" reminded me of my beginner's take on the subject and I would very much appreciate an opinion of an accomplished professional like you (if some time can be found for that). I would be honoured by your feedback and advice.

      With deep respect and best wishes,

      Martin

      Hi Ian,

      I hope this isn't a re-post. I thought I commented on your essay but when I checked back to today I didn't see it. (Un)Luckily I had it saved in a word document...

      I totally agree with what you were trying to say via your dialogue... or maybe I completely disagree... or... Anyway, at the very least I liked the dialogue (reminded me of Gödel-Escher-Bach) and think the thoughts you provoked with it definitely relate to my Digital Physics movie essay. I think we're still trying to reach a consensus on infinity, probabilities, and mathematics in general, so I see little hope for achieving a Grand Unified Theory in physicists before some of these fundamental ideas are better addressed.

      I also tried to have a little fun with the essay format, so hopefully it makes it an easy read. I'd love for you to comment on my thoughts regarding real numbers in my essay. I think you might also like taking a crack at a few of the questions I have at the end of my essay. Here's a sample of one that might interest you:

      How quickly could a tape be processed through a Turing machine and is this constraint physical or informational in nature?

      Thanks again for making an interesting and entertaining read.

      Jon

      8 days later

      Dear Ian,

      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

      Dear Ian,

      Lovely! Here is a quote from G. H Hardy in his "Mathematician's Apology":

      (p. 70) " A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but '2' or '317' ha nothing to do with sensation, and its properties stand out more clearly the more closely we scrutinize it. ... 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."

      For Hardy, mathematics was more tangible than physical reality.

      Best,

      Lou Kauffman

      Ian,

      Time grows short, so I am revisiting essays I've read to assure I've rated them. I find that I rated yours on 3/28, rating it as one I could immediately relate to. I hope you get a chance to look at mine : http://fqxi.org/community/forum/topic/2345.

      Jim

      5 days later

      Dear Ian,

      I am particularly amused by and happy to have read you essay. I must say that I wanted to make mine a dialogue as well, but then reconsidered as I thought it might be deemed as not matching with the contest; I'm glad at least one person wouldn't have thought that ;) I decided in the end to just go with a light and funny writing tone.

      I like the symmetry of your essay. It begins and ends with the same action, somewhat like a Turing machine caught in an infinite loop. It must mean it's meant to be read and enjoyed by thinking beings. In this context, the eternal 6 o'clock made me consider what would happen if the March Hare would put on the table such a machine rummaging through an uncomputable calculation. I enjoyed very much how you are working your way to the argument of the representational and tangible realities and found very satisfying in this respect (and hilarious) the Hatter's remark that "A character in a dialogue has chosen tangible reality. How comforting."

      Profound and packed in dry humor. Great read! Wish you the best of luck in the competition and I'm rating your work accordingly, in the hope that it will make a difference.

      Warm regards,

      Alma

      Dear Ian,

      As others have commented, this is a wonderful choice of style. I wish you could of included one or two more topics, but this was fun and clear.

      All the best,

      Jeff

        Thank you all for the kind comments! I have been busy with work and have not been able to find the time to respond to each of you individually, but I appreciate the thoughts! I am very glad indeed that I was able to inject a bit of humor without taking away from the depth of the subject. I do wish I had had more space to elaborate on some points, but perhaps I will do so in the future elsewhere. Thank you all again!

        Ian,

        I am replying to my own thread, because I realized it was incomplete.

        In writing class (which you clearly do not need)we learned "show do not tell". This essay was showing, not telling. Is the idea of something the thing itself? Is the name of an idea the idea? Is an equation of the universe, the universe (at this point I clearly went beyond the text)? You let us make these decisions, which I do not know if I like or hate you for putting us in such a position.

        All the best,

        Jeff

        Dear Ian,

        Thank you for going to my essay. I expect that my essay will not be red as a tree but as a surface with punctures, it is non local in some sense. I also hope a big picture is emerging, some points are ongoing research (as those pointed out in the abstract), some technical aspects may not be familiar to quantum physicists (e.g. modular forms and characters).

        If you go to reference [17] just appearing in QIP, you can see that I cite a work of yours on the "order theoretic quantification of contextuality", meanwhile I also found another measure of geometrical contextuality that I am currently working on. A mathematician would say the Langlands program but I stay closer to physics.

        Best,

        Michel

          Dear Ian,

          Thanks for the very enjoyable essay. The distinction between representation and tangible realities is well articulated. One of the oddities of this dichotomy seems to be that some physical systems can characterize what may (or may not) be tangible using representation. So it seems these two modes of description are not entirely independent but that the representational influences the tangible and the tangible constrains what can be represented. You seem to suggest a starker line between the two, so I am curious on your thoughts where these two domains interact. In particular, my view is that addressing this aspect is essential to addressing why math seems to work for describing physical reality.

          Best,

          Sara

            Dear Ian,

            very nice essay. I still need to read it in full detail to appreciate all the sides of the conversation. For the moment let me say that I appreciate the choice of Lewis Carrol, which, btw, raised the famous logical paradox that proves that even the "modus ponens" must be formalized. If logic itself must be formalized, than you would agree with me that Physics must be formalized, otherwise would not be logical, whence not logical falsifiable, whence not scientific. As you know, this is the point raised in my essay.

            Hope to see you soon, and enjoy a thorough conversation about this crucial point about physical science.

            My best regards

            Mauro

              Hi Sara,

              Thanks for the comments. I think you raise a very important issue: where *is* the line between tangible and representational reality? Is the line, if it exists, well-defined or fuzzy? And if we err on the side of caution, should we err on the tangible side or the representational side?

              I'm torn on this issue, to be honest. I'm not a pure operationalist, but I'm not even sure operationalism, at least as it is practiced today, is even purely tangible. I do firmly believe that physics should be about the physical, but I also believe that well-founded theories are as broad as possible in their explanatory nature. So, to give you an example, if you've ever watched Susskind's videos on what you need to know about modern physics, he starts out talking about Newton's laws. Whenever I teach introductory material, I start with energy and momentum because Newton's laws are easily derived from conservation of momentum and energy. When I made this point to a friend of mine, he argued that introducing Newton's laws (and general laws of motion) first, made more sense from an operationalist view. So, in that sense, I am not an operationalist.

              On the other hand, as I said, I do think we need to default to physical reality. Feynman said something about this once. In fact Carl Caves said something to me once many years ago to this effect: we're physicists and we're doing physics so we should default on the side of physics.

              So, in short, I unfortunately don't have a neat and clean answer to your question. I think the line I painted in the essay is quite stark for that particular example. It may be that there is a stark line in *every* example, but that there is no single, unifying way of defining that line such that we can "know" it in a given case (if that makes any sense).

              Cheers,

              Ian

              Hi Mauro!

              Part of me completely agrees with you that physics needs to be formalized if it is built on logic (which it partially is). But another part of me says that it is ultimately built on what we can measure (see my reply to Sara above). In other words, it still has to explain what we see in a lab. It's not clear that we can formalize everything we see. Certainly people have tried ever since Hilbert's original proposal (and, to some extent, Newton and others in the 17th century tried as well, before physics took a more experimental turn). It seems telling to me that no one has fully succeeded in this endeavor yet, though of course that doesn't mean it can't be done.

              Hope to see you soon! Let me know if you will be in Boston at some point.

              Cheers,

              Ian