Not An Infinity Of Angels On Pinheads

Thomas Aquinas and other ludicrous pseudo-philosophers (in contradistinction with real philosophers such as Abelard) used to ponder questions about angels, such as whether they can interpenetrate (as bosons do).

Are today’s mathematicians just as ridiculous? The assumption of infinity has been “proven” by the simplest reasoning ever: if n is the largest number, clearly, (n+1) is larger. I have long disagreed with that hare-brained sort of certainty, and it’s not a matter of shooting the breeze. (My point of view has been spreading in recent years!) Just saying something exists, does not make it so (or then one would believe Hitler and Brexiters). If I say:”I am emperor of the galaxy known as the Milky Way!” that has a nice ring to it, but it does not make it so (too bad, that would be fun).

Given n symbols, each labelled by something, can one always find a new something to label (n+1) with? I say: no. Why? Because reality prevents it. Somebody (see below) objected that I confused “map” and “territory”. But I am a differential geometer, and the essential idea there, from the genius B. Riemann, is that maps allow to define “territory”:

Fundamental Idea Of Riemann: the Maps At the Bottom Are Differentiable

Fundamental Idea Of Riemann: the Maps At the Bottom Are Differentiable

The reason has to do with discoveries made between 1600 and 1923. Around 1600 Kepler tried to concretize that attraction of planets to the sun (with a 1/d law). Ishmael Boulliau (or Bullialdius) loved the eclipses (a top astronomer, a crater on the Moon is named after him). But Boulliau strongly disagreed with 1/d and gave a simple, but strong reasoning to explain it should be 1/dd, the famous inverse square law.

Newton later (supposedly) established the equivalence between the 1/dd law and Kepler’s three laws of orbital motion, thus demonstrating the former (there is some controversy as whether Newton fully demonstrated that he could assume planets were point-masses, what’s now known as Gauss’ law).

I insist upon the 1/dd law, because we have no better (roll over Einstein…), on a small-scale.

Laplace (and some British thinker) pointed out in the late 18C that this 1/dd law implied Black Holes.

In 1900, Jules Henri Poincaré demonstrated that energy had inertial mass. That’s the famous E = mcc.

So famous, it could only be attributed to a member of the superior Prussian race.

The third ingredient in the annihilation of infinity was De Broglie’s assertion that to every particle a wave should be associated. The simple fact that, in some sense a particle was a wave (or “wave-packet”), made the particle delocalized, thus attached to a neighborhood, not a point. At this point, points exited reality.

Moreover, the frequency of the wave is given by its momentum-energy, said De Broglie (and that was promptly demonstrated in various ways). That latter fact prevents to make a particle too much into a point. Because, to have short wave, it needs a high frequency, thus a high energy, and if that’s high enough, it becomes a Black Hole, and, even worse a Whole Hole (gravity falls out of sight, physics implodes).

To a variant of the preceding, in: Solution: ‘Is Infinity Real?’  Pradeep Mutalik says:

July 1, 2016 at 12:31 pm

@Patrice Ayme: It seems that you are making the exact same conflation of “the map” and “the territory” that I’ve recommended should be avoided. There is no such thing as the largest number in our conceptual model of numbers, but there is at any given point, a limit on the number of particles in the physical universe. If tomorrow we find that each fermion consists of a million vibrating strings, we can easily accommodate the new limit because of the flexible conceptual structure provided by the infinite assumption in our mathematics.


I know very well the difference between “maps” and territory: all of post-Riemann mathematics rests on it: abstract manifolds (the “territories”) are defined by “maps Fi” (such that, Fi composed with Fj is itself a differential map from an open set in Rx…xR to another, the number of Real lines R being the dimension… Instead of arrogantly pointing out that I have all the angles covered, I replied:

Dear Pradeep Mutalik:

Thanks for the answer. What limits the number of particles in a (small enough) neighborhood is density: if mass-energy density gets too high, according to (generally admitted) gravity theory, not even a graviton could come out (that’s even worse than having a Black Hole!)

According to Quantum Theory, to each particle is associated a wave, itself computed from, and expressing, the momentum-energy of said particle.

Each neighborhood could be of (barely more than) Planck radius. Tessellate the entire visible universe this way. If too each distinct wave one attaches an integer, it is clear that one will run out of waves, at some point, to label integers with. My view does not depend upon strings, super or not: I just incorporated the simplest model of strings.

Another mathematician just told me: ‘Ah, but the idea of infinity is like that of God’. Well, right. Precisely the point. Mathematics, ultimately, is abstract physics. We don’t need god in physics, as Laplace pointed out to Napoleon (“Sire, je n’ai pas besoin de cette hypothese”). (I know well that Plato and his elite, tyrant friendly friends and students replied to all of this, that they were not of this world, a view known as “Platonism”, generally embraced by mathematicians, especially if they are from plutocratic Harvard University… And I also know why this sort of self-serving, ludicrous opinion, similar to those of so-called “Saint” Thomas, a friend of the Inquisition, and various variants of Satanism, have been widely advocated for those who call for self-respect for their class of haughty persons…) 

The presence of God, aka infinity, in mathematics, is not innocuous. Many mathematical brain teasers become easier, or solvable if one assumes only a largest number (this is also how computers compute, nota bene). Assuming infinity, aka God, has diverted mathematical innovation away from the real world (say fluid flow, plasma physics, nonlinear PDEs, nonlinear waves, etc.) and into questions akin to assuming that an infinity of angels can hold on a pinhead. Well, sorry, but modern physics has an answer: only a finite number.

Patrice Ayme’


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30 Responses to “Not An Infinity Of Angels On Pinheads”

  1. Gmax Says:

    Away from breakshitters to tangle with infinity, alleluia! You argument seems pretty persuasive to me. If it doesn’t exist, it doesn’t exist, no?

  2. Alexi Helligar Says:

    To accept the idea that the Universe is “infinite” is to say that it is also “non-physical”. This is because infinity is a pure idea. No physical entity can be infinite. To say that the Universe is infinite is to say that the Universe is a pure idea.

  3. brodix Says:


    Units are finite, but is the context? It is the old dichotomy of nodes and networks. Even those who insist the entire universe is a singular unit are now talking about multiverses in some larger context, or network of universes.
    Every entity necessarily exists in some larger context, from which it coalesced and to which it disperses. To argue otherwise would be to argue against the conservation of energy. That the energy suddenly arose from nothing and will disappear back into nothing. That seems to me more problematic than infinity. To state it must be so would be to state the unprovable and that is a form of dictatorship. That there should be an infinity of the possible, than only one of the extant would be a more democratic conceptual basis.

    • Patrice Ayme Says:

      All I will say here is that the Multiverse/Many World critters deduce their belief from a PECULIAR INTERPRETATION of Quantum Mechanics. When first proposed, it was viewed as sheer insanity in Europe, althought the thesis adviser, Wheeler, and the institution, Princeton, were in high regards.
      Present day physics explains 4% of the observed universe. My own interpretation of Quantum Physics explains 100%, and does not use the fantastic hypothesis of zillions of universes which cannot be observed… Come to think of it the most absurd suggestion ever..

      • brodix Says:

        True. That’s why I think space, the context, not the unit, extends forever.

        As I’ve argued previously, using doppler effect to explain expansion presumes a stable vacuum, in which there are more lightyears between galaxies, not that light itself stretches. As Einstein did say, “Space is what you measure with a ruler” and the ruler of cosmic space would be the speed of light. If it takes light longer to cross, the ruler is not expanding, only the distance being measured.

        As for multiworlds, it is essentially that the past remains probabilisitic, as opposed to determinism, which assumes the future is pre-determined. As I keep arguing, we look at time backward. It’s not the point of the present moving past to future, but change causing future to become past. Probability coalesces to actuality and recedes into the residual. Events have to occur, in order to be determined. They are first in the present, then in the past. So neither cats or particles remain in multiple states, once they have been fixed in one.

        Nor is the future determined, as the information into any event only fully arrives with its occurrence and if you cannot know the input, you cannot know the output.

        • Patrice Ayme Says:

          The Doppler Effect is NOT a factor in the expansion of the universe. The Doppler Effect does not explain why a car goes away. A car, going away, stretches the sound wave it emits. A galaxy is not a car.
          The metric actually stretches. What is the metric? Light. It’s not Einstein who got that idea, but Lorentz and Poincare’, up to 20 years prior. Poincare’ expanded globally what he called Lorentz’s local time.

          I have no idea what you are trying to say about Multiverses. The problem is technical and originates in Hilbert Space theory, in conjunction with an error of Einstein (that is why I am so keen to demonstrate that Einstein was not so great!)

          Did the crazy, self-important prostitute philosopher ban you yet? (No, not an allusion to me, I am not paid to fake thinking, and self-importance.) Since his outrageous behavior, I did not even go to his site, once. Self-important whore philosophers are a big problem they tend to ban me. Although the subjects they claim to be interested by are important…

          • brodix Says:


            The doppler effect is being used to explain why galaxies are redshifted, not why they are moving away. That would be the BB and “dark energy.”
            Yet it assumes one metric based on the spectrum of intergalactic light and another, based on the speed of the very same light.
            If it is taking light longer to cross, that would make C the denominator and redshift the numerator.
            The only problem with galaxies moving away in a stable dimension, defined by C, is they are all redshifted proportional to distance. Which either means we are at the center of this bubble universe, in a stable dimension of space, or redshift is an optical effect, given we are at the center of our view of the universe.

            I don’t really see why this seems so difficult to understand. Everyone just seems to assume the speed of light as a constant, that is unrelated to the size of the universe. Is there any theory that light is relative to the size of the universe and speeds up as it grows? No.

            So what is this vacuum that is the basis of C, if it is not the expanding universe???? It’s just there, not a problem.

            Yes. It. Is. A. Problem.

            I’m afraid my interest in PF has flagged for the moment. Too much else to do, mostly work related.

      • brodix Says:

        My point being that even the argument that the universe is finite overlooks the fact that this makes it a singular unit and so the larger context of this particular unit is an open question, which some have tried to fill with other universes.

  4. Patrice Ayme Says:

    Brodix: I was ironical when I said what I said about what you said. And what you said was:
    ” As I’ve argued previously, using doppler effect to explain expansion presumes a stable vacuum, in which there are more lightyears between galaxies, not that light itself stretches.”

    Now what you say is:
    “The doppler effect is being used to explain why galaxies are redshifted, not why they are moving away. That would be the BB and “dark energy.”

    You do not understand. The Doppler Effect is NOT the reason for the redshift in conventional cosmology. OK, I agree that they talk as if it were. “They” being the officialdom of cosmology. They abuse the naive with their abuse of the language…

    The space is stretched. So the metric is stretched, So light is stretched. Should I copy and paste that 50 times?

    Also, once again, there is just one metric, locally:.
    dxdx + dydy + dzdz – cc dtdt

    All right, more exactly that was the square of the infinitesimal element of the metric… Your tale of two metrics is not cogent… Although the confusion comes from seeing a speed redshift where others see a COSMOLOGICAL redshift…

    • brodix Says:


      This is interesting!
      Light travels at 186,000 miles a second. So as the universe expands, does that mean miles are stretched? Such that when the universe started out small, miles were just little tiny distances? Or maybe seconds were really, really long?
      The problem is that it is completely meaningless to say something is stretched, if you have nothing to compare it to. So space is stretched. The reason we know this is because the light is redshifted. Why is the light redshifted? Because as space stretches, it takes light longer to cross and so the WAVES are stretched. Nothing about the SPEED. If the light speeded up, the waves wouldn’t be stretched.
      Again, the only way to know something is stretched, is to have a control to compare it to. They are using metric based the speed as the control to compare to the stretched waves.

      • Gmax Says:

        I think Patrice explained in the past that one could immerse space-time in a bigger space and measure distances there. She made fun of other physicists who never talk about that, because she suggested they simply don’t know the theorem

        • brodix Says:

          That might be one way to work it through. I’m not trying to start an argument. If someone presented a clear argument for why C isn’t being used as a control, I would try to understand the logic.
          Keep in mind the original premise of Relativity is that in a moving frame, both speed of light/time and distance dilate equally. So that you always measure light as C. So when they are arguing that intergalactic light is redshifted because it is taking light longer to cross, then distance and time are not dilating equally.

        • Patrice Ayme Says:

          Yes, there are two theorem, the NASH embedding theorem, and one older and simpler.

  5. De Brunet D'Ambiallet Says:

    Say you transform a black body into energy quanta. Are you saying there is a limit in the number of quanta? Why can’t we have an infinity of low frequency photons coming out?

  6. brodix Says:


    A simple question; Does the speed of light increase, as space expands?

    If yes, then why would light be redshifted, as the waves would still arrive at the same rate?

    If no, then what is this “vacuum,” through which light travels at C, if it is not space?

    “C” means “constant.” That light is always measured at the same rate, relative to the distance. If it is not constant to the distance, it is not relative to it. If it is constant to the distance and the distance is increasing, then the speed of light would have to increase as well, in order to remain constant.

  7. brodix Says:

    Let me try to make it even clearer;
    Light travels at 186.000 miles a second. Say that miles are a measure of space and since space is expanding, miles are too. So in order for light to still go at 186.000 miles a second, it would have to go faster, in order to remain constant.

    • Patrice Ayme Says:

      Dear Brodix: You are making the interesting confusion many do between globality and locality.The speed of light is an holonomic notion. It’s true only in the tangent bundle, to use fiber space semantics. It is not true globally in the differential manifold of space-time. In other words, the constant speed of light is just true in the limit of the ultrafilter of neighborhoods at a point. Einstein himself said that the speed of light was not constant in General Relativity (Einstein talked, debated and was instructed by the greatest mathematicians, such as David Hilbert, and Elie Cartan).

      • brodix Says:


        The point is that the two metrics are based on the spectrum and the speed of the very same intergalactic light.

        That it is redshifted, because it takes longer to cross. If it is taking longer to cross, it is not constant to the distance between galaxies, but some other frame, aka “the vacuum.”

        So there are more “vacuum units,” aka lightyears, between galaxies. What is the basis of this vacuum, if it is not “space?”

        • Patrice Ayme Says:

          I see that not only math, but astronomy is challenging to you. The distance is not FUNDAMENTALLY evaluated from redshift, which, as I said a zillion times, is not how distance is evaluated. Parallax, Cepheids, and Type Ia Super novas are the way. (Cosmological redshift can be used from distance, because we have determined what it depicted, from the other systems.)

          What you are trying to talk about is the integration of the infinitesimal metric element along a geodesic, according to the moving frame method of Elie Cartan. That’s learned typically around graduate school, majoring in math. Ever taken calculus? We are swimming in partial differential equations in many dimensions here…

          BTW, from the traditional way, there is an astronomical “distance ladder” (aka an astronomical metric in apparent space), as I said on top. Then there is the metric of the spacetime manifold, ds. Then, there is the metric of the nine dimensional pseudo-Riemannian manifold the spacetime manifold is embedded in (that’s the one I am the only one to look at, it seems…)

  8. brodix Says:


    Is it the premise that inter galactic light is redshifted, because the source is moving away, or not?

    (However it is explained, i.e., space expanding.)

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