What Are Numbers? Math is most abstracted physics!

German mathematician Richard Dedekind (1831–1916) published in 1888 a paper entitled Was sind und was sollen die Zahlen? What are numbers and what should they be? 

Here is my answer: forget what you know. 

Numbers are neural networks. Small numbers have small networks, big ones, big networks; so the nature of numbers, change, as they get bigger…(According to me, listening, delighted, to the indignant screams of distant mathematicians ).

Diagram Chasing all of them: not a coincidence. Instead of having “It from Bit”, one has it from action (arrow in Cat theory, action potential with neurons, fundamental process in physics…)

A few immediate applications of this master idea:

  1. Numbers are learned, because neural networks are learned.
  2. Advanced animals, having advanced neural networks, should be capable of having those neural networks we call numbers.
  3. Big numbers are different from small numbers, because big neural networks are different from small ones. Here again is the idea that energy should matter in mathematics (the conventional thinking being just the opposite: energy doesn’t matter).

***

Kronecker’s also quipped: “God made the natural numbers. Everything else is the work of man.

Kronecker proceeded to define numbers from Set Theory, invented for the purpose. Later Bertrand Russell found a problem with Set Theory, the set of sets which are not elements of themselves brought a contradiction. Russell tried to get out of that with a hyper complicated theory. In modern times, mathematicians prefer to use Category Theory. [1]

I go beast on how to construct numbers. Beasts have brains, and brains have neural networks.

Kronecker thought mathematics is the work of man. But, actually all advanced animals move in a way proving they are capable of differential calculus. Far from being the work of god, differential calculus is the “work” of dog. Without differential calculus, that dog can’t hunt. OK, dog is not conscious of god, or of the calculus it’s using. So what?   

Now for a few easy bits:

*** 

Let’s notice that numbers are definitely the work of the genus Homo: 

Consider the integer 152. 152 is the work of man. Just like “Yes” is the work of the Englishman. 

152 means: 100 + 5×10 + 2. But that’s only in base ten. In base 60, that would be: 60 x 60 + 5 x 60 + 2… Which converts to 3,902  back in base ten. 

So “152” is not an absolute notion. For that integer to make sense, the basis in which it lives has to be expressed (and what the notation means, such as 2 = 1+1…). The Babylonians invented base 60 to handle big numbers in astronomy. We still use base 60 to this day, for angles and time. So “152” is a cultural construction. In several ways. 

***

So how come Platonists claim that numbers live out there, in a special realm of their own, if there is so much human explanation and convention to provide, with just basic numbers? Most mathematicians also believe their are exploring that realm of Plato. But actually all they are exploring is the possible connections which can be built within the neural networks inside their brains. So they are exploring physics, a bit like a child on a beach explores which sand castle she can get away with. A difference with building sand castles is that the possibilities are few and are carefully recorded, becoming the body of that culture and language called “mathematics”. 

An example is the Archimedean axiom. The Greeks knew about it well: it’s in Euclid, and it says that, given two magnitudes, A and B, there is always an integer n so that: nA > B.

If one denies that axiom, one gets infinitesimals… That was made rigorous through Model Theory, in the 1950s, three centuries after Leibnitz first introduced infinitesimals, starting a fight with Newton.

No Plato universe of “forms”… or rather, they exist, but live as geometries inside brains…

Even more dramatic are hyperbolic and elliptic geometries: they were discovered at least a century before Euclid. Then they were forgotten, and a stupid debate occurred for 21 centuries about whether the parallel axiom (one parallel to a line, one only, through a point off the line) was independent of the others. Mathematicians, even the brightest, had forgotten that their ancestors had found geometries with many, or no, parallels…)

***

Let’s recapitulate: culture is composed of (vague, but good enough) descriptions of neural networks, which can be transmitted. Once contracted, those neural network templates modify brains in similar ways. Those similarly modified brains behave all similarly, mimicking innate characteristics.  

Language enables a transmission of neural geometries, topologies, logics, and categories. Language is primitive in most advanced animals, consisting in grunts, cooing, gestures, etc. But in Homo language became an advanced mental cultural duplication system (and some of the mentality passed is mathematical, but not only). 

True, advanced animals have a sort of pseudo-innate capability to evolve neurobiological mathematical structures: through trial and errors mimicking their relatives, or experimentation with what works, young animals brains learn to optimize trajectories: the brains of many predators in pursuit make subsets of themselves into differential calculus machines. 

So if Plato’s “forms” are real forms in (generalized) geometry and topology… what are the latter made of? Good question! Therein come our old friend, the Quantum Wave… 

Clearly, math is the most abstracted physics.

Patrice Ayme

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28 Responses to “What Are Numbers? Math is most abstracted physics!”

  1. brodix Says:

    Ants can count;
    https://www.ncbi.nlm.nih.gov/pubmed/17210957

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    • Patrice Ayme Says:

      Bees too. Apparently, according to French specialists, they even understand… zero.

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      • Patrice Ayme Says:

        And that means that numbers are neural networks… What else, if bees have them.

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        • brodix Says:

          I would say the left, logical hemisphere of the brain is a clock, while the right, emotional intuitive side is a thermostat.
          As mobile organisms, we need that sequential thought process, as a function of navigation. As the ants and bees need it.
          Plants don’t have a sequential cognitive process, as they don’t move intentionally. They are entirely thermostats. Heat expands/cold contracts.
          Of course, as I keep pointing out, we idolize time, as some metaphysical dimension of narrative, rather than understand it as the effect of change, by which the potential coalesces into the actual and recedes into the residual. Future to past.
          Meanwhile this dynamic churns along.
          We could use ideal gas laws to correlate temperature and pressure, with volume, but no one calls them the 5th and 6th dimensions of space, because they are only foundational to our emotions, bodily functions and environment, not the sequence of perception.

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          • Patrice Ayme Says:

            Are you saying that time shouldn’t be viewed as a dimension?
            Well, first, modulo C, time is another aspect of space, I would dare say… Locally… I still have to write that time essay…
            Second, as you yourself point out, dimensions are whatever parameters we need. That has allowed me to claim the brain to 50 dimensional or so (thanks to all the chemistry inside)…

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        • brodix Says:

          “time is another aspect of space”

          “dimensions are whatever parameters we need”

          Duration is correlated to distance, which is only one dimension of space. Ideal gas laws effectively correlate temperature and pressure with volume. Wouldn’t that make them dimensions of space, or it that starting to blur the lines a bit too much?
          The three dimensionality of space is the xyz coordinate system. Does that make it foundational to the nature of space, or a mapping device to describe space? Are longitude, latitude and altitude foundational to the biosphere of this planet, or are they a mapping device to relate locations in it?
          A coordinate system is build around a counterpoint of the axes, then there are directions of those axes. Say one coordinate system might have a different counterpoint, or different plane of the x and y and yet occupy the same space. Much as the Israelis and the Palestinians certainly use different maps to describe the same space. People in different locations on the surface of this globe necessarily have differently angled Euclidian coordinate systems. Wouldn’t that mean that space is infinitely dimensional? Especially when you include non-euclidian space.
          I would argue that when you remove all physical energy and form from space, it still has the non-physical qualities of infinity and equilibrium. Infinity, because it is unbound by any form. While equilibrium is implicit in GR, as the frame with the fastest clock and longest ruler is closest to the equilibrium of the vacuum. The unmoving space, aka, absolute zero.
          So space is the absolute and the infinite.
          Look at galaxies; The light radiates to infinity, while mass coalesces too……the other end of the number line, zero.
          Yet the effect is the feedback loop of a cosmic convection cycle, as the mass radiates away all its constituent energy, by the edge of the black hole, before reaching zero. Meanwhile energy eventually is absorbed back into mass, even it it has to travel billions upon billions of lightyears in doing so.

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          • Patrice Ayme Says:

            Space gets sort of bounded, circular, and terminal around Black Holes… Also speed of light is local, as and because coordinate systems are definitively local: this is the gist of differential manifold theory.

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        • brodix Says:

          Everything we measure, light and mass, is bounded, circular and terminal around black holes, but then there are those jets, quasars, shooting out the poles withs serious energy. Where did that come from, if not what had presumably fallen into them?
          In an accelerated frame, distance and the clock rate are dilated equally, so that the speed of light always measures C. For a frame moving at the speed of light, distance shrinks to zero and time stops. So what if we go back the other direction, to the frame with the longest ruler and fastest clock; Wouldn’t that imply a vacuum equilibrium against which light is moving and against which all these frames are relative?

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          • Patrice Ayme Says:

            Basic physics of jets is well understood.
            No frame goes at the speed of light.
            The question you brandish here, the absolute frame, is divisive.
            After a good while, Einstein himself, Mr. All Relative, was not sure anymore.
            Poincare seemed to be against. Died too early (1912)
            I am against.
            Because I believe in absolute speed TAU… 10^23 C at least.
            Independently of SQPR, there is at least one math reason for it: the embedding theorems (Whitney, Nash, etc.) Those give an absolute Euclidean or Pseudo-Euclidean space to embed the whole “spacetime” in.

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        • brodix Says:

          “Basic physics of jets is well understood.”

          So the energy comes from? Presumably most/all that is falling intot he black holes/eye of the storm.
          http://scitechdaily.com/ocean-eddies-mathematically-equivalent-black-holes/

          “No frame goes at the speed of light.”

          Only light. No rulers, no clocks.

          Because I believe in absolute speed TAU… 10^23 C at least.

          I thought you said it was local?

          “Those give an absolute Euclidean or Pseudo-Euclidean space to embed the whole “spacetime” in.”

          So it is absolute, relative to an inertial rest frame of space.

          Time is still an effect, aka frequencies.

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  2. Gmax Says:

    Numbers as neural networks? That sure is new. So the argument is that all the fundamentals look like neural networks, no? So if they look like it, they are it?

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    • Patrice Ayme Says:

      Well, sort of. Action potentials between neurons are exactly like arrows in Category Theory, and logic, and mathematics, can be built from the latter. A convergence between vastly separated sciences is appearing, and it’s bigger than expected…

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  3. ianmillerblog Says:

    Am I right in thinking you regard the quantal matter wave as a collection of numbers?

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    • Patrice Ayme Says:

      Hmmmmm… The quantal matter wave will have to be pre-numbers. Category Theory says numbers are are sets of arrows, basically… Themselves made of quantum waves… We need a foundation somewhere, and the Quantum Wave is it… The Quantum Wave is nonlinear, either imploding (collapse; making a particle, an interaction), or exploding (guided by its linear part expanding at ultra super luminal TAU)

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      • brodix Says:

        Patrice,
        Wouldn’t the “foundation” be the inertial state between the imploding and exploding?
        Empty space/absolute zero?

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        • Patrice Ayme Says:

          Nonlinear equations are a dilemma: they seem to either spread out into nothingness, or singularize. This is viewed as mysterious setback. But it may be a mathematical clue to the nature of the universe, the most fundamental process.

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          • brodix Says:

            Math is abstraction and in the reality from which it abstracts, light spreads out to nothingness and mass coalesces into the singularity of a black hole.
            Infinity and absolute.
            The reality we experience is a feedback loop between these two ends of the spectrum.

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          • Patrice Ayme Says:

            Math is also possible reasonings, thus, sometimes, imaginable, theoretical physics.

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          • brodix Says:

            It’s the map versus the territory. Obviously we need maps to understand the territory, but if they are taken to be anything more than what they are, we end up in ‘there be monsters’ land.
            Epicycles, as a modeling of our view of the cosmos, were very good math, but as they were somewhat biased by the fact we are not the center of the cosmos, or even the solar system, the physics of crystalline spheres were nonsense.
            As I keep pointing out, Relativity is very good math, but spacetime is lousy physics, as time is not the narrative flow, codified as measures of duration, but change turning future to past, with duration as this physical state, as events rise and fall.
            So while processes churn along, past to future, the patterns emerging go future to past.
            As our consciousness goes past to future, while the thoughts go future to past.
            Math is about patterns, rather than processes, so we get nonsense like ‘it from bit.’

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  4. Brodix Says:

    brodix Says:
    July 5, 2019 at 10:44 am
    It’s the map versus the territory. Obviously we need maps to understand the territory, but if they are taken to be anything more than what they are, we end up in ‘there be monsters’ land.
    Epicycles, as a modeling of our view of the cosmos, were very good math, but as they were somewhat biased by the fact we are not the center of the cosmos, or even the solar system, the physics of crystalline spheres were nonsense.
    As I keep pointing out, Relativity is very good math, but spacetime is lousy physics, as time is not the narrative flow, codified as measures of duration, but change turning future to past, with duration as this physical state, as events rise and fall.
    So while processes churn along, past to future, the patterns emerging go future to past.
    As our consciousness goes past to future, while the thoughts go future to past.
    Math is about patterns, rather than processes, so we get nonsense like ‘it from bit.’

    Like

    • Patrice Ayme Says:

      Epicycles is Fourier analysis (making a periodic function into a sum of its sine/cosine constituents).
      “Spacetime” is neither here nor there. Even Einstein didn’t like it. It’s good just as an abbreviation: spacetime gets wrapped around a Black Hole.
      Time is treated differently from space, even in relativity: see the signature of the pseudo-Riemannian manifold.
      Time has nothing to do with space in Foundational Quantum.
      However, in QFT, time is made, by fiat, into imaginary time, and used that way. Say I am 9K from Paris, and 10 hours flight. Then add: 9 + 10 = 19… That’s QFT for you… It works…

      You should have a look at Category Theory….

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      • brodix Says:

        It does seem to be apples and oranges, but I guess those are categories. I should look it up, though after work, my brain is thinking nap.
        Sorry about the double post. Thunderstorm passing through this morning and i thought the first one got lost.

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        • Patrice Ayme Says:

          Category Theory corresponds perhaps much towards where you want to go.
          I studied category Theory long ago… was spited for it, big time… Then it came back, carrying everything with it, from logic to physics. Superficially, CT looks like neural networks.

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          • brodix Says:

            Looking it up on wikipedia it does seem to be another useful mathematical mapping device for a complex reality. Yet as effective was it might be in the situations where it is useful, it still seems very simplistic, relative to the underlaying territory.
            As such it is patterns extracted from the process, with the intent of clarity, rather than understanding.
            Patterns describe. Processes explain.

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          • Patrice Ayme Says:

            Category Theory is immensely complex. Since its invention in the early 1950s, all what some research mathematicians have done is CT, and it gave tremendous results.

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        • brodix Says:

          Patrice,
          Reality is immensely complex and all people, throughout history have spent their lives studying it. All with various biases that distorted their objectivity.
          As such, math, as a study of patterns, is a tool for making sense of this reality, but as abstraction from reality, it is still a mapping device, not the basis of reality.
          The tendency is that as tools become over more effective, those using them become ever more wrapped up in them and they go from being tools to being gods. Think money, as one such tool of society that has become a god.
          As I keep pointing out, Physics, especially relativity, reduces time to measures of duration, in order to have a quantity, but still assumes the narrative flow, from past to future, with this quantity of duration being compared to measures of distance, as a measure of one dimension of space. Then assumes that all points on this dimension of time co-exist, like points on a spatial dimension. Yet while New York and London, circa 2019, can spatially co-exist, New York, 2000 and New York, 2019, do not temporally co-exist, out in the “fabric of spacetime,” because time is an effect of physical activity and the resulting change.
          So if you don’t quite appreciate the assumptions being built into the maps, it is possible there are biases being assumed.
          Much as epicycles were very good math, as a mapping of our perception of cosmic motions, yet the assumption that we are at the physical center and not just the center of our point of view, introduced biases and all the hands over the eyes and fingers stuck in the ears didn’t change that and its not going to change the fact that spacetime is nonsense.

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      • brodix Says:

        Then why it is such a problem to explain why time is asymmetric, as if it could be going either direction, of what is presumably a linear path?
        What is being measured is action and action is inertial. as in the earth turns one direction, not either.
        Saying the direction is only apparent with entropy is ridiculous. The relative order of the larger system is not what is measured.
        If there is a more coherent description, of why treating time as a dimension, than why isn’t it emphasized.
        It would seem dimensions, as conceptual devices to relate the properties of different phenomena, is similar to categories.

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