• apokrisis
    6.2k
    Many share your jaundiced view.

    So I'm reading Penrose, and all of sudden he explodes into excitement like a schoolgirl, fawning over complex numbers because they are "magical" and perform "miraculous" things, further spilling exclamation marks in the surrounding paragraphs about how he's only scratched the surface of "number magic!"

    https://www.physicsforums.com/threads/are-complex-numbers-magical.68277/
  • Agent Smith
    7.4k
    So I'm reading Penrose, and all of sudden he explodes into excitement like a schoolgirl, fawning over complex numbers because they are "magical" and perform "miraculous" things, further spilling exclamation marks in the surrounding paragraphs about how he's only scratched the surface of "number magic!"

    Alack, these be feelings I'll probably never experience!
  • jgill
    2.4k
    There's more magic in complex analysis than in complex arithmetic. :cool:
  • apokrisis
    6.2k
    There's more magic in complex analysis than in complex arithmetic.jgill

    It would help to familiarise yourself with the current metaphysical debate over Penrose’s argument. Here is one excellently clear paper….

    http://cejsh.icm.edu.pl/cejsh/element/bwmeta1.element.ojs-issn-2451-0602-year-2018-issue-65-article-439/c/439-462.pdf

    Penrose’s [sees complex numbers structures] as the primary fabric of the Universe. … Penrose uses this designation in regards to the mathematical structure of his main research focus in quantum gravity and twistor theory (Penrose, 1967).

    The main idea that Penrose articulates concerning the complex structures is that the results that require arduous computations with the use of the of the real structures are obtained “for free” as the complex structures come into play.

    Penrose does not hesitate to push this magic to the extreme as he openly states:
    Nature herself is as impressed by the scope and the consistency of the complex-number system as we ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales (Penrose, 2005, p. 73).

    The article then goes on to cover the gory mathematical details.

    What is key is that the complex numbers build in something essential to physical existence itself - something so basic that all more complex structure pops out of it “for free”, whereas getting the same outcomes from real numbers is arduous, as they lack the right dimensionality to describe the dimensionality that the Universe actually has.

    As a cosmic Darwinist, this tells me that we live in the Universe with the particular dimensionality that is the simplest way to produce richness, and that is why physics finds that complex number magic is matchingly the root description of Nature.

    The reals are too simple to generate a complex world. But the complex numbers have the minimal complexity that can then be the basis for generating all the complexity that ensues.

    The hypothesis I am pursuing is that it all ties together in rather obvious fashion with Noether’s theorem and Newton’s twin conservation principles - the dimension-defining dichotomies of rotation and translation.

    Dimensionality is a system defined by the local and global. And such a system can be broken towards its two dialectical extremes - the spin that defines localisation, the straight line or flat geometry that defines the globality of unbounded translations.

    If your number system has to have units that speak to both unit 1 rotation and unit 1 translation as their scale-anchoring identity operation, then what else would you expect your number system to look like as the simplest possible representation of such a unity of opposites?

    So this is why particle physics is cashed out in stories of spin symmetry, and why Penrose pushes so hard with with his hylomorphic/conformal view of spacetime geometry.
  • jgill
    2.4k
    Thanks for the link. Nice paper. I have Penrose's massive book, but have only read bits and pieces. The rotation/translation bit makes sense, although my knowledge of modern physics is weak. I have, in fact, done a fair amount of research involving Moebius (or linear fractional) transformations (that crop up in these physics discussions), but going the pure analysis direction in the complex plane rather than the geometries coming off the Riemann sphere. A non-terminating continued fraction - a kind of analogue of an infinite series - can be perceived as an infinite composition of such transformations (as can a series).

    (We were speaking of fixed points before. I introduced the idea of accelerating the convergence of limit-periodic CFs through the use of attracting fixed points of LFTs back some years ago.)
  • apokrisis
    6.2k
    I have, in fact, done a fair amount of research involving Moebius (or linear fractional) transformations (that crop up in these physics discussions), but going the pure analysis direction in the complex plane rather than the geometries coming off the Riemann sphere.jgill

    So what are your thoughts here when one direction looks to track the "deep maths" of Nature and the other choice may be just unphysical pattern spinning? What do we learn if this is the case?

    But am I right that you argue the complex plane has lessons in terms of the physics of chaos - patterns of convergence~divergence?

    I couldn't get much out of your 1970s paper, but I was thinking about how your fixed point paper illustrates my point about the dichotomy of rotation~translation in that the complex plane seemed marked by patterns of curl. Or convergence and divergence over all scales, as you would expect in a chaotic system.

    In a maximally turbulent system, you have the fixed points of vortexes appearing over all available physical scales. And the the deep naturalness of this scalefree or powerlaw behaviour was rather the point of the Franks paper I cited previously.

    So - in reference to the OP - it seems hard not to be interested in the maths that has some connection to reality, even if the ideology of maths is that is perfectly free to chase pure pattern for its own sake.

    And coming from the other direction, Nature itself has no choice but to be structured and fall into self-constraining patterns. Mathematical regularity of some kind must emerge.

    But I was reading this yesterday that warns the field of maths might indeed be shortchanging itself with some of the metaphysical choices it makes ... yet equally, one could argue that the power of maths lies in the fact it puts itself outside the reality it means to describe by making its "non-intuitionist" choices ... like the ones about infinities and empty sets that outrage the more metaphysically inclined.

    Gisin has traced the problem of the block universe to an unexpected source: mathematics itself. He notes that a century ago, mathematicians were split about how to describe numbers whose decimal digits trail off into infinity.

    On one side, led by David Hilbert, were those who thought every such real number was a “completed object” that exists in its entirety, timelessly, even though it has infinite digits. On the other side were ... “Intuitionistic mathematics.” In intuitionistic mathematics, numbers are created over time, with digits materializing in succession.

    Spoiler: Hilbert’s side won. “Time was expulsed from mathematics” and as a byproduct, from physics, too, writes Gisin (Gisin, 2020a). But, he wondered, what would happen if physics were re-written in the language of intuitionistic mathematics? Would time become “real” again?

    Gisin asks us to consider “chaotic” systems, in which two almost-but-not-quite-identical starting points evolve to wildly different end points. A classic example is the weather. ... whether you are talking about the weather, the evolution of the entire universe, or just your choice of what to have for dinner tonight, it is distressing to think that the future is already fully determined.

    Of course, physics is not required to make us comfortable. But Gisin points out that intuitionistic mathematics could offer a natural way out of the deterministic lockup. In the intuitionistic view, numbers—like the values of pressure, wind speed, humidity, and so on—do not have definite values from the get-go, but rather develop over time, with randomly-generated digits unscrolling as time passes. This mathematical treatment allows for a universe in which time actually flows, events truly happen, and randomness and chance are injected moment by moment.

    https://www.templeton.org/wp-content/uploads/2021/11/Time-by-Kate-Becker-1.pdf

    So there's a thought. Zoom in on your complex plane with its pattern of curl, and do you start to lose any sense of whether some infinitesimal part is diverging or converging? Is even a maximally curved part of the map now definitely one or the other. So same as the real number continuum. Zoom in and are your cuts a point on one side or the other? Bring in the observer (or observational scale) – as physics must - and the intuitionist position has a lot going for it.

    Seeing as I'm throwing out references, here is another that might interest you - a short, then longer version, of Peirce on the continuum debate. One can ask again whether maths made the right pragmatic choice even if Peirce is the metaphysically correct choice? And so everyone is right and wins a prize. :grin:

    https://cesfia.org.pe/villena/zalamea_peirce_translation.pdf
    https://uberty.org/wp-content/uploads/2015/07/Zalamea-Peirces-Continuum.pdf

    ... from the very beginning of their investigations, the pathways of Cantor and Peirce are opposite to one another; while Cantor and many of his successors in the 20th century try systematically to delimit the continuum, Peirce tries to unlimit it—to bring it nearer to a supermultitudinous continuum, not limited in size, truly generic in the transfinite, never totally actualizable.
  • Metaphysician Undercover
    10.4k
    Spoiler: Hilbert’s side won. “Time was expulsed from mathematics” and as a byproduct, from physics, too, writes Gisin (Gisin, 2020a). But, he wondered, what would happen if physics were re-written in the language of intuitionistic mathematics? Would time become “real” again?

    The physics and the mathematics go hand in hand, the problems of one are not the cause of the problems of the other, but they do amplify each other. Einsteinian relativity is what expulses time from physics. And since physicists choose to employ this principle, they also choose the mathematical axioms which facilitate the application of it. So as soon as the principle which expulsed time from physics was established, then the axioms which expulsed time from mathematics were applicable, and acceptable
  • apokrisis
    6.2k
    Einsteinian relativity is what expulses time from physics.Metaphysician Undercover

    Relativity united time and space in a way that made more general sense. And we now wait for quantum physics to catch up with the rest of the class.
  • Metaphysician Undercover
    10.4k
    Relativity united time and space in a way that made more general sense.apokrisis

    It really does not make "general sense". It's a counterintuitive principle which facilitates calculations when using electromagnetism as a measuring tool.

    Relativity is extremely counterintuitive because it describes time as passing at different 'rates', or 'speed', depending on one's physical circumstances. The intuitive way is to conceive of us human beings as being completely incapable of altering the passage of time. We experience time as passing in a way which is completely determined, or fixed by the physical universe, so that we are utterly helpless to slow down the arrival of the future, or alter the past. Relativity, if it is true, gives us the capacity to alter the flow of time through accelerations, and this is very counterintuitive.
  • jgill
    2.4k
    So what are your thoughts here when one direction looks to track the "deep maths" of Nature and the other choice may be just unphysical pattern spinning? What do we learn if this is the case?apokrisis

    Motivations vary in the mathematical community. What is common, however, is the drive to explore, sometimes with regard to the mysteries of nature, sometimes within the discipline itself. I'm a spinner.

    But am I right that you argue the complex plane has lessons in terms of the physics of chaos - patterns of convergence~divergence?apokrisis

    I don't know about the physics of chaos (other than my experiences as a meteorologist ages ago), but the patterns that appear as examples of weak emergence are fascinating.

    what would happen if physics were re-written in the language of intuitionistic mathematics? Would time become “real” again?

    Since computations are done on computers with computable numbers and functions, isn't that already the case? The notion that a non-intuitionistic approach damages the idea of time seems ridiculous. A purely philosophical tragedy.

    But Gisin points out that intuitionistic mathematics could offer a natural way out of the deterministic lockup.

    I've never known a fellow mathematician who claimed to be an intuitionist. The fact that all functions from [0,1] to R are continuous from that perspective is quite unappealing to someone who came up in classical analysis. (Yes, it depends on the definition of "function"). Equating the flow of time with adding more digits to a number seems a bit absurd, at least for me.

    Zoom in on your complex plane with its pattern of curl, and do you start to lose any sense of whether some infinitesimal part is diverging or converging?apokrisis

    The complex plane itself doesn't have a pattern of curl. It's a vector field based on a complex function that does the job. I have zoomed in up to 10,000X to display fascinating objects purely from curiosity. In the vicinity of an attracting fixed point, no matter the magnification, one sees convergence. It's just a matter of how one writes the computer program.


    One can ask again whether maths made the right pragmatic choice even if Peirce is the metaphysically correct choice?apokrisis

    I admire Peirce for his thoughts on nonstandard analysis. I once toyed with the idea of teaching a real analysis course from that perspective, but gave it up when a friend at a larger university did just that, with poor results. I find it a bit amusing that when one looks at the standard graphical depiction of a mathematical Category, one sees Peirce's Triangle.
  • Gnomon
    2.5k
    If somehow evolution has equipped us with mathematical minds, it is fair to hypothesize that the "book of nature is written in the language of mathematics" just because we see it that way.Doru B
    Some thinkers assume that Nature is Mathematical (abstract values, sans meaning) while others believe that Nature is Mental (logic plus meaning). So, I suspect that the replies to your topic will divide along those lines. A purely mathematical universe just is, and must be taken for granted. That's why many scientists assume that Energy (cause) & Laws (logic) exist eternally, and need no explanation. But some scientists observe that the universe, that began from abstract Cause & Laws, has evolved animated self-aware beings with personal values & meanings. How is that possible?

    I'll propose that, logically, the Potential for Awareness & Feelings must have existed, in potential, along with Causation & Organization. Over time, that Potential for both Material and Mental processes has been Actualized into physical Bodies with non-physical Minds. Why do I label Mind as non-physical (or metaphysical)? Because, the reductive sciences of humanity have not yet discovered an Atom of Mind. So, they label Mental processes as "epiphenomena", hence, not nearly as important (for practical purposes) as Physical phenomena.

    As a thought experiment, which would you prefer to be : a> a phenomenal Brain in a vat, without awareness & feeling, or b> an epiphenomenal Mind in a vacuum, with sentience & sensations. Your answer will reveal which you believe to be most essential to your being. Of course, a Mind without hands cannot do anything physical. But it can imagine doing anything imaginable. So, which is more important to you : Mechanism or Imagination? :joke:

    PS__How about option c> a physical body & brain plus a non-physical mind? Is that the best of all possible worlds?

    Epiphenomena :
    1. a secondary effect or byproduct that arises from but does not causally influence a process.
    2. mental state regarded as a byproduct of brain activity.


    Potential is Primary :
    In physics as in Logic, an Action Potential necessarily exists prior to the Action.
  • Gnomon
    2.5k
    The question seems a correspondent of the most popular question “Was mathematics invented or discovered?” and relates to the nature of mathematics as well as to the philosophical problem of applicability of mathematics. However, there are anthropocentric and evolutionary features that the philosophical investigations on this topic have not focused on much:Doru B
    Most of the proposed answers I've seen, to the "invented vs discovered" question, seem to conclude that it's a little of both. The "unreasonable effectiveness of mathematics" in science indicates that Nature is in some sense fundamentally mathematical. But the history of math shows that humans using abstract "pure" mathematical principles, eventually find concrete practical applications for many of them.

    That's why I have concluded that human Logic (Reason) is basically Geometry with words. Both manipulate abstract relationships (ratios ; relative values) in order to discover wider or narrower practical applications of those ratios. Ratios are inherent in Nature : for example, Energy is essentially a thermodynamic ratio between high & low, hot & cold, here & there. Likewise, Morality is basically an interactive ratio between me & you, us & them, good & evil. So, it seems that Evolution has programmed the human brain with a rudimentary sense of ratio, that we can expand on with education & practice. But, we can also lose that sense, when it is not exercised. Which is why I can no longer add 2 + 2, without the crutch of a non-human calculator.

    I just read an article that deals with a similar topic : Human mathematics and God’s mathematics, by Catalin Barboianu, mathematician & philosopher of science. For the purposes of this thread, you can interpret "God's Mathematics" as the logical structure of Nature, and "Human Mathematics" as the human talent for discovering and inventing applications for that knowledge. He seems to believe that "to mathematize is human", so to speak : "Whatever mathematics is (science, method, formal language or logical symbolism), we do mathematics without being mathematicians". Yet, he also offers an alternative theory, that humans project their own subjective values onto Nature.

    Speaking of subjective math, he adds that "The history of application of mathematics in the sciences also has a “mysterious” element. Being driven by their natural impulses of inquiring and generalizing, but also following some special criteria of beauty, symmetry and elegance specific to the mathematical creation". Although we can't yet describe the evolutionary mechanics that created the aesthetic sense in humans & animals, it's more taken for granted than viewed as mysterious. Without consciously thinking about it, we can immediately recognize the symmetry & elegance in a beautiful face. And some have even tried to reduce it to geometry. :yum:

    "Contemporary pioneering studies in what is called perceptual mathematics . . . . came to shape an interdisciplinary cognitive theory that claims that all mathematics is human, resides in the mind, and is not an external product of the mind. The human mind is endowed with innate primordial perceptions such as spatial (metric, linearity), numerical, and topological (proximity, relational structures), reflected by the common empirical concepts such as distance,motion, change, flow of time, and matter. It is further hypothesized that animals also hold such perceptions.Thus, the concepts of mathematics are not platonic, but are built in the brain from these primordial perceptions, and brain neurophysiology gives rise to the extremely precise and logical language of mathematics
    https://medium.com/@cb_67963/human-mathematics-and-gods-mathematics-682ac8e7bba

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