The quadrivium

The ancient Greeks considered mathematics to consist of four parts, usually translated as arithmetic, geometry, music and astronomy. The four-fold division reflects two dichotomies, between the discrete (arithmetic and music) and the continuous (geometry and astronomy) on the one hand, and the static (arithmetic and geometry) and the dynamic (music and astronomy) on the other.

In modern times, arithmetic has expanded to include algebra, number theory, combinatorics, and other parts of discrete mathematics. Geometry has expanded to include calculus, analysis, topology and other parts of continuous mathematics. Astronomy has expanded to include general relativity, cosmology and other parts of continuous physics. Music has expanded to include quantum mechanics, particle physics and other parts of discrete physics.

Unfortunately, the neat classification of the ancient Greeks has been destroyed: modern physicists try to describe music with geometry, instead of arithmetic. It can’t be done. Discrete physics requires discrete mathematics, and cannot be adequately described by continuous mathematics. The ancient Greeks understood the difference between the discrete and the continuous. Modern physicists clearly do not.


22 Responses to “The quadrivium”

  1. TJ Wence Says:

    Another remarkable discovery of the Ancient Greeks is the Platonic solids and Euclid’s proof that there are only five. Despite connections to Music Theory, this clearly stands in the domain of geometry. Plato is credited with the Platonic solids, although he rediscovered them after older civilizations which left behind historical pieces hinting they also knew about the Platonic solids.
    The rest of what I’m going to comment about, I have yet to distinguish the explicit domains that categorize them. I may have brought this stuff up earlier, though my computer has been placed aside for the summer, and I am on my phone. The tiny keyboard on it is something I don’t prefer, despite using now.
    Something that is a huge blow to Heim theory, and casts a strong shadow of doubt on faith in the model, is that there are no W and Z Bosons. The weak force, as empirically validated today, is not to be seen anywhere among what Heim originally proposed.
    My belief that Pauli was right about the motivation of both Heim theory, and also the entirety of Yang-Mills, as being fundamental flawed, has grown stronger. And the Yang-Mills skepticism is particulary important, because that theory is where modern physics stands today.
    Perhaps an error made by Eric Weinstein, who created Geometric Unity, is that he takes Yang-Mills theory far too seriously.
    Now for something entirely new, at least to me- Mark Hadley, after publishing his theory on time and QM in 2007, came out with another paper in 2008. He extends his unique line of thought to a new and more sophisticated conclusion.
    Titled “Quantum Fields as Gravitational Sources”, this paper describes “alternative ways to link classical GR with quantum theory using Bohm’s theory”
    He summarizes this development as: “linking GR to Bohm’s interpretation of quantum theory can work because Bohm’s theory is expressed in terms of an underlying realist theory”.
    I find that moderately fascinating in the regard that I’ve not yet encountered a unification scheme of this type. Unfortunately, I am not able to comment yet on whether or not I see it as fundamentally important. Potentially, however, that could be a possibility.

  2. Robert A. Wilson Says:

    The ancient Greeks understood. The modern geeks do not.

  3. Robert A. Wilson Says:

    BTW, my Clifford algebra paper, arXiv:2011.05171 has been accepted by Advances in Applied Clifford Algebras, after three revisions. But it still uses geometry to describe music, so I don’t consider it to be a particularly important paper.

    • TJ Wence Says:

      Eric Weinstein said he does not know anyone at all who has discussed the “connection between Einstein and Pati-Salam”. I immediately thought back to a certain blog post here that does. My strongest skepticism on Geometric Unity is the use of spin(6,4). This is the group he associates the Standard Model to, via Pati-Salam. Spin(6) x Spin(4) is, in Geometric Unity, a greater representation of Su(4) x Su(2) x Su(2). He uses Spin(6,4) as it naturally comes from the theory, motivated by interpreting Einstein’s work as derived from a higher structure of 14 dimensions that gives all the information of GR via pullback. Yet Spin(6,4) is a kind of So(10) theory. Going to higher gauge symmetry is as vague as higher Clifford algebras. Though So(10) isn’t as high as say, E8, or Cl(8). Yet Einstein used 4 dimensional spacetime. He chose to ignore Weinsteins 10 other dimensions, from his math, that apparently give spinors without a choice of metric. Its a matter of whether one thinks So(10) theory is physical, and I am rather certain that it is not.

      • Robert A. Wilson Says:

        Yes, I added a little bit more on the Einstein/Pati-Salam connection in response to one of the referees’ comments that revealed ignorance of the fact that there are three distinct conjugacy classes of SO(3,1) inside SO(3,3). One of these is called the Lorentz group in quantum mechanics, and a different one is called the Lorentz group in general relativity. This fundamental error is pointed out in Weinstein’s work, as it is in mine. It will take a long time for this fact to be understood by the mainstream, and all we can do is keep pushing.

      • TJ Wence Says:

        The error does not seem to be corrected in Garrett Lisi’s E8 theory as GraviGUT can be expressed as So(3,1) x So(10). The one Lorentz group is unified into the spin(3,11) that claims to describe interactions with gravity at the quantum level in a specific context. Perhaps it is corrected with his triality automorphisms from the split octonions, but that is for the three fermion generations and not for the different spacetimes. Eric Weinstein, on the other hand, uses So(3,1) but changes it with Sl(2,C) in the process. But he does know of the different spacetimes, though his theory, like Garrett Lisi’s, doesn’t apply them in any obvious way. Garrett Lisi simply uses DeSitter spacetime for the E8 model universe’s full spacetime symmetry. Going back to Eric Weinstein, I think he may have missed out on the following isomorphisms written below, which I have noted that he did not take advantage of:
        Spin(6,4) = Su(4) + Su(2) + Su(2) = So(6) + So(4) = So(3,3) + So(3,1) = So(3,4) + So(3).
        Although I suppose that in the end, it is all So(10). And he was very clear that the 6+4 splitting was right, and the 7+3 was wrong. Which means the 7+3 does not work for the mathematics of using So(10) from the *hidden* structure of GR, which he describes as 4+10. Geometric Unity has to be 4+6+4. Does that not look like Clifford Algebra?

  4. Robert A. Wilson Says:

    I rather like the idea of a “quadrivial” approach to the fundamental problems. It emphasises the fact that you have to consider discrete and continuous, mathematics and physics, all at the same time. And it contrasts with the standard “trivial” approach which is all about grammar, logic and rhetoric (the trivium) – mostly rhetoric, with a fair amount of grammar but not enough logic.

    • brodix Says:

      Doesn’t grammar contain both static and dynamic, as nouns and verbs? Not to mention that nouns tend to be discrete, while verbs, as actions, are continuous.
      My sense has been that energy, what radiates out, is continuous, as well as axiomatically dynamic, while form, as what coalesces in, is axiomatically both discrete and static.
      The problem then, is the assumption that energy is also essentially discrete, as in quantized, because that is the only way to measure it, whether as waves or particles. Thus this view of “it from bit,” where quantization is viewed as foundational.
      TJ mentions a paper linking quantization and gravity, which also arises from this, that both are ends of a spectrum of consolidation, of discrete form coalescing out of continuous energy fields.
      Which ties into the point I keep raising about time, that energy, as present, goes to the future, while the forms generated go to the past.
      That the feedback between these two dichotomies is endless, might create a lot of cross references.

      • TJ Wence Says:

        The 2008 Hadley paper I stumbled across appears to favor a world of non-locality where statistical aspects of the wave-function are not dependent on an observer. There are so many different interpretations of this thing we call the wave function, I find it trivial. Thus I hold an unconventional view that Schroedinger’s equation is an approximation too far removed from reality. The wave function is not the right way to go about figuring the universe out, and physicists have been trying repeatedly without success for many many years. Though I do not want to reject Hilbert space or Fock space. I just don’t see them as fundamental assumptions. Just mathematical tools to be used only when needed. For what is currently a large approximation. The mainstream always goes on and on and on about Schroedinger’s cat, and to me that is annoying. At the end of the day, whether Hadley’s calculations lead anywhere, although no follow-up is around nor is an explicit calculation that proves anything, the wave-function of QM is not it’s forte. And seemingly treated as far more relevant then I think it really is.

  5. Robert A. Wilson Says:

    There is a Chair in the University of Cambridge called the Lowndean (or Lowndes) Chair of Astronomy and Geometry which recalls this ancient association. For the first 150 years, most holders of the post were astronomers. For the next 100 years, they were all essentially geometers – in my Cambridge days, the Lowndes Professor was Frank Adams, whom you might prefer to call a topologist. The current holder is Mihalis Dafermos, who might be considered by some to span the two fields of astronomy and geometry. Sadly, there appears never to have been a corresponding Chair of Music and Arithmetic.

  6. brodix Says:

    I came across an interesting experiment and theory, called “loading theory” of light.
    To extrapolate from it, it seems that the only way to actually “observe” light is when it’s absorbed or interfered by some physical material. It’s not like we can use other light to measure light. So we can really only say it has wave or particle properties when it is interacting with some, more complex physical property.
    Now if it is quantized, that would seem to mean it is necessarily concentrated and focused by this process, ie. same energy confined to less space. Which is what gravity is, a centripetal effect, concentrating material in a smaller space.
    As with waves, such as a water wave, it isn’t just the energy, but the energy moving through and interacting with the medium, creating the ripple effect. Presumably light doesn’t need a medium to cross the vacuum of space, but then we can only really say the wave effects, the frequencies and amplitudes, exist as they are absorbed. How much is that the light and how much is the material used to capture or refract it?
    So it’s not so much the wave function collapses to a point, as the effect of it being absorbed focuses it in the absorbing material. Sort of like static electricity in the atmosphere creates a bolt of lightning when the connection with the ground becomes strong enough, while atmospheric lightning tends to be more diffuse.

  7. Robert A. Wilson Says:

    I agree entirely with your interpretation of the wave-function. I am told that Rovelli’s “Relational Quantum Mechanics” takes a similar view. What I am trying to do is take that physical insight and turn it into mathematics, in such a way that hopefully allows us in the end to deduce Schroedinger’s equation as an approximation.

  8. Robert A. Wilson Says:

    Zeno’s paradox, dating from almost 2500 years ago, is an ingenious argument in favour of a discrete rather than continuous universe. There are, of course, counter-arguments. But quantum mechanics essentially says that Zeno was right all along: continuous motion is not possible, and all change happens in discrete steps.

    • brodix Says:

      Yes, but Zeno’s paradox is nonsense, because they are not running relative to some abstracted fractional landscape, but otherwise continuous space.
      Our perception of reality is discrete, because it would be a whiteout otherwise. Like a movie camera takes a series of stills.
      Western logic also is fundamentally discrete, from atoms to individuals, while the evident reality is more a dichotomy of nodes and networks.
      In the West, we think of time as the future in front of us and the past behind, because we see ourselves as discrete entities, moving through our context. While the Eastern and Native American view is of the past in front and the future behind. Colloquially because the past and what is in front are known and the future and what is behind are unknown. Which is more physically accurate, because we do see events after they occur, then the energy transitions to other observers and events.
      This spacetime concept of time as some block dimension is based on our essentially narrative experience, as mobile organisms, navigating our environment, but the reality is that it is the the present that is real and the events emerge, like temperature, pressure, color and sound. Time is frequency, events are amplitude.
      Ideal gas laws correlate volume with temperature and pressure, but we don’t call them dimensions of space, because they are only foundational to our emotions and bodily functions, not the sequence of thought.
      It can’t even explain why time is asymmetric, short of entropy, but as a measure of action, action is inertial. The earth only turns one direction.
      Math is a useful cognitive construct, but it is a product of our perception. It formalizes and clarifies, it doesn’t create. It is emergent with the processes and patterns it models, not some metaphysically platonic ideal.
      Consider that in arithmetic, the operations are verbs, so it does model the dynamic. If you don’t actually add the two sets of one, you don’t have the one set of two. In reality, there is only the present, so the future is not determined, because it hasn’t been computed.

      • TJ Wence Says:

        I wonder about how the radiation from the sun makes its way through space and then illuminates the Earth. Space is dark and cold while Earth is bright and warm. Might apply to the loading light experiments, I would guess.

      • Robert A. Wilson Says:

        I disagree with your analysis of Zeno’s paradox. It is a reductio ad absurdum designed to show that the assumption of a continuous universe is inconsistent. It was not until the 19th century that an adequate mathematical description of a continuous universe was developed, sufficient to model a universe in which this paradox can be explained away.

        It is this same 19th century mathematics that is used in modern physics to describe everything, including quantum physics. But the mathematics relies on the process of taking limits of infinite processes, and this is something that cannot be done in the real universe. Hence there is no guarantee that the mathematics correctly describes the real universe. I would go further, and say that there is mounting evidence that it does not.

      • Robert A. Wilson Says:

        However, I do like your analogy (over on between money and mathematics. This 19th century mathematics is like money for physics, and if you cannot describe a physical transaction with this type of continuous money, then they won’t talk to you. But real wealth, unlike today’s electronic money, is not infinitely divisible. Real wealth is measured in cows.

  9. TJ Wence Says:

    One more isomorphism I forgot to mention regarding Eric Weinstein (as alternate pathways surrounding his original approach, which actually is based on a Clifford Algebra structure, Cl(7,7)- and I do not regard the work as complete or hold much faith in its “observerse”, the likes of which has structure for way more particles than ever observed. —
    So(3,3) = Sl(4,R) = Spin(6) = So(5,1) = Sl(2,H)
    By ignoring what mathematicians say about representation theory, or rather, the structure of each of these groups- the kind of thing physicists do all the time, ignoring the specifics- one can jump straight to the Quaternions, and therefore, the Pauli matrices. Kind of, anyway. But I suppose Eric Weinstein got spinors and particles and metric structures already wrapped up in the Cl(7,7) anyway, so this comment here is really meaningless…

    • Robert A. Wilson Says:

      I’d just like to point out that those five groups you say are equal are not: they are all different, five distinct non-isomorphic groups, with three distinct non-isomorphic Lie algebras.

  10. brodix Says:

    Probably an equally interesting question to ask, is what is space?
    The current assumption is that it’s “three dimensional.” Yet isn’t that just a mapping device, the xyz coordinate system. Is longitude, latitude and altitude fundamental to the biosphere of this planet, or is it a mapping device?
    The thing is that we seem to get hung up on these reductionist abstractions, like they are more elemental than what is being abstracted. It seems to me that space is both the equilibrium state and infinite. Equilibrium would seem implicit in Special Relativity, as the frame with the longest ruler and fastest clock would be closest to the equilibrium of the vacuum. Through which light travels.
    Now consider my point about time, That the energy goes to the future, while information goes to the past. Then consider galaxies, where the light radiates out, toward infinity, or at least until it is otherwise absorbed, or fades to ambient radiation. While the structure, aka, mass coalesces in, toward equilibrium. Or at least until it all cancels out, at the edge of the black hole/eye of the storm and the energy radiates back out.
    So the problem with trying to describe the light is that becomes information and effect, be it waves, photons, warming the earth.
    It’s like trying to explain the present, when all that can be described are the situations being manifest.
    So when there is some collective effort to understand, necessarily it settles onto some reasonably common denominator, because that’s what people need to communicate.
    Yet all that would seem to be just the information side of the situation. Even the intellectual process of distilling information is reductionist.

  11. brodix Says:

    I’m certainly not in the position to debate the history of mathematics.
    I would use the point about money to say wealth is not just personal possessions, but the strength of the network in which someone exists. In this node and network relationship, the node centers around equilibrium, while the network grinds out to infinity. Otherwise one’s valuables tend to become food for something larger and further up the food chain.
    I tend to take a contextual view to most propositions, otherwise it might just be some worm on a hook.
    As for cows, my father was a dairy auctioneer and horse trainer on the side. Up through the 70’s and early 80’s, the dairy industry managed to get some fairly large price supports lobbied into the market. The result became those famous warehouses of “government cheese,” that the Reagan administration started handing out, in leu of more welfare benefits. Then in 86, they pulled out all the price supports and literally half the dairy cattle in the country went to the butcher. Given the fears over butterfat at the time, these were largely those breeds with the higher cream ratios, like Guernseys and Jerseys. As my father was the largest Guernsey dealer in the country, he retired and went to dealing horses and land.
    Feedback and blowback.

  12. Łukasz Says:

    “The ancient Greeks understood the difference between the discrete and the continuous. Modern physicists clearly do not.”

    It’s a pity that many physicists do not understand a significance of mathematical logic.

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