the physicalization of metamathematics

Fri Aug 5 19:15:39 EDT 2022

If one is looking for a computational route to the cosmological constant that makes the cosmological constant time dependent (and so not a constant), it may be worth considering the following formula:

[cid:image001.png at 01D8A8FF.BB678480]

Where Harmonic(n) is the n’th number in the harmonic series and t is the number of Planck moments since the Big Bang. (Not the time since the Big Bang, the number of Planck moments since the Big Bang – the time divided by one Planck moment.)

According to Wiki, the age of the universe is 8.08*10^60 Planck moments.

https://en.wikipedia.org/wiki/Planck_units

Plugging 8.08*10^60 into the above formula yields 2.88844*10^(-122) [cid:image002.png at 01D8A8FF.BB678480] , which matches exactly the measured value of the cosmological constant. (What are the odds of that?)

https://en.wikipedia.org/wiki/Cosmological_constant

Of course, this is just a computation and doesn’t get at any of the underlying physics (if indeed there is any physics for it to get at). But one can give a very general physical interpretation of the above formula, which is consistent with the theory of entropic gravity put forward by Erik Verlinde.

Consider a process in which first a number is chosen from increasingly large initial sequences of the natural numbers greater than zero. The first choice is from the sequence containing just the number 1. The second choice is from the sequence containing 1 and 2. The third choice is from the sequence containing 1, 2, and 3. And so on. Suppose that at each step in the process, the number chosen from the sequence is assigned to a randomly chosen point on the surface of a sphere whose radius equals one Planck length, [cid:image003.png at 01D8A8FF.BB678480] . Let us suppose that one event includes both the choice of a number and an assignment of it to a point on the sphere.  After some number of events, n, suppose someone were to pick a region, r, on the sphere and randomly choose one number, N, from among the numbers that had been assigned to the points in r. (If no numbers are assigned to r, no number is chosen.) A question that can be now asked is: What is the square root of the probability that N=1?

Because of the structure of the choices, the expected number of 1’s that are chosen after n events equals Harmonic(n), where n is the nth number in the harmonic series. Hence, after n events, the probability that a randomly chosen number from among the numbers that have already been chosen equals 1 is given by: Harmonic(n)/n. And of course, [cid:image004.png at 01D8A8FF.BB678480]  is the square root of that probability. Letting n=t^4 and dividing this expression by the surface area of a sphere with radius equal to 1 Planck length yields an expression, which when multiplied by the area of the region on the sphere from which N is chosen, yields the desired square root of a probability.

If the cosmological constant is in fact constant, then the above result is a most remarkable coincidence. If, on the other hand, the expression above describes the evolution of the cosmological constant, then it may give some insight into the nature of the process that is the source of dark energy. If one were to suppose that the process is mathematically equivalent to the process described in this e-mail, then dark energy is the result of random processes whose possibility spaces become discretely larger over time (at a rate n=t^4) and whose outcomes are encoded by points on the surface of spheres whose radii equal one Planck length and that the number of outcomes encoded on the surface of a sphere at any cosmological time, t, equals the number of Planck moments between the Big Bang and t raised to the fourth power. Moreover, there must be some privileged bit of information, what in the example is the number 1, whose presence in a region (or rather, the probability that it is in a region) is linked to the expansion of space.

Paul Studtmann

From: FOM <fom-bounces at cs.nyu.edu> on behalf of José Manuel Rodríguez Caballero <josephcmac at gmail.com>
Date: Friday, August 5, 2022 at 5:39 PM
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Re: the physicalization of metamathematics
This email originated from outside Davidson College. Use caution, especially with links and attachments.

Vaughan Pratt asked:
I have a short list of criteria for what might count as a "testable
prediction".  At the top of my list is:
1.  Planck's constant h.  Does your combinatorial theory predict its value?
Second on my list is:
2.  Boltzmann's constant k.  Same question.

I will answer as an external affiliate of the Wolfram Physics Project, based on my own research, but my answer may differ from what S. Wolfram has in mind as there are various approaches in the group. So, I only speak for myself.

My starting point is to clarify that the goal of the Wolfram Physics Project is to find a fundamental theory of physics. Hence, we are only required to explain the fundamental constants (not other constants, which may be emergent or change of unit of measure). According to John Baez

https://math.ucr.edu/home/baez/constants.html<https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fconstants.html&data=05%7C01%7Cpastudtmann%40davidson.edu%7Ca56c93a849a74fe7262c08da772afa4e%7C35d8763cd2b14213b629f5df0af9e3c3%7C1%7C1%7C637953323743378308%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=Ck3%2Bbx2uN563YW7I1prJRW9OQ%2BN5isfMrLUBjkzhxMA%3D&reserved=0>

the fundamental constants are these 26 numbers:

  *   the mass of the up quark
  *   the mass of the down quark
  *   the mass of the charmed quark
  *   the mass of the strange quark
  *   the mass of the top quark
  *   the mass of the bottom quark
  *   4 numbers for the Kobayashi-Maskawa matrix
  *   the mass of the electron
  *   the mass of the electron neutrino
  *   the mass of the muon
  *   the mass of the mu neutrino
  *   the mass of the tau
  *   the mass of the tau neutrino
  *   4 numbers for the Pontecorvo-Maki-Nakagawa-Sakata matrix
  *   the mass of the Higgs boson
  *   the expectation value of the Higgs field
  *   the U(1) coupling constant
  *   the SU(2) coupling constant
  *   the strong coupling constant
  *   the cosmological constant

From the point of view of fundamental physics, the answer to this question is trivial: both Planck's constant and Boltzmann's constant are equal to 1 because they are used as units of measure in this field. In the same spirit, the speed of light in the vacuum c is 1 in fundamental physics.

There is a computational interpretation of the U(1) coupling constant (listed above), which is in the spirit of the Wolfram Physics Project but was developed independently of it by S. Lloyd. We can find it on page 17 of his preprint:

https://arxiv.org/pdf/quant-ph/9908043.pdf<https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fpdf%2Fquant-ph%2F9908043.pdf&data=05%7C01%7Cpastudtmann%40davidson.edu%7Ca56c93a849a74fe7262c08da772afa4e%7C35d8763cd2b14213b629f5df0af9e3c3%7C1%7C1%7C637953323743378308%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=%2F9lF0H3%2F%2BMrGIMlg1tBkGG0t8BrxxntK9nLap%2BJgKf4%3D&reserved=0>

Notice that S. Lloyd expressed his computational interpretation in terms of the fine structure constant, which, according to J. Baez's essay cited above,

The electromagnetic coupling constant [aka the U(1) coupling constant] is just another name for the fine structure constant

My research at the Wolfram Physics Project was in the direction of the computational interpretation of the cosmological constant (listed above). My idea was to try to link it to the growth of the descriptive complexity of the universe as a function of time, which I predicted was a logarithmic function for the whole universe (including all the many worlds of quantum mechanics). Nevertheless, each copy of us in a given world (in a quantum mechanical sense) will see that this function is linear instead of logarithmic This is for the same reason that, generically, a binary tree of height n can be described using log n bits, but each branch needs n bits to be described (of course, in come cases, there can be compression of information). Among all the fundamental constants, I think that this one is the closest to the physicalization of mathematics and the easier to adapt to math: the cosmological constant of the expansion of the mathematical space as a deductive system. My hypothesis is that an expanding space is linked to an increment in descriptive complexity as a function of time corresponding to the system inducing this space. I would like to clarify that this is not a result, but just an idea to motivate research in that direction.

Finally, I would like to conclude with a testable prediction of Wolfram's framework

Wolfram, Stephen. "Undecidability and intractability in theoretical physics." Physical Review Letters 54.8 (1985): 735.

which according to I. Pitowsky, is one version of the physical Church-Turing thesis

Pitowsky, Itamar. "The physical Church thesis and physical computational complexity." Iyyun: The Jerusalem Philosophical Quarterly/עיון: רבעון פילוסופי (1990): 81-99.‎

the other version is due to D. Deutsch,

Deutsch, David. "Quantum theory, the Church–Turing principle and the universal quantum computer." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 400.1818 (1985): 97-117.

The prediction is that everything in the universe (when including all the many worlds of quantum mechanics) is computational in the sense of computers based on classical physics. Some people argue that the existence of quantum algorithms that outperform the classical ones in a given task is a refutation of this thesis, but this is wrong: quantum mechanics can be simulated using a classical computer, and people inside the simulation may feel it as real. The point of the Wolfram model is to explain how an observer, inside a computer simulation, perceives the algorithm. This is why it is natural to consider the question of the mathematical observer: an intelligent entity that is embedded in the development of mathematics as a computational process. By observer, I mean a machine with memory, as it was defined by H. Everett III:

Everett, Hugh. "The theory of the universal wave function." The many-worlds interpretation of quantum mechanics. Princeton University Press, 2015. 1-140.

Kind regards,
Jose M.

Some essays that I wrote about the Wolfram model (these are essays, not scientific articles and they may have typos):

https://arxiv.org/pdf/2108.08300.pdf<https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fpdf%2F2108.08300.pdf&data=05%7C01%7Cpastudtmann%40davidson.edu%7Ca56c93a849a74fe7262c08da772afa4e%7C35d8763cd2b14213b629f5df0af9e3c3%7C1%7C1%7C637953323743378308%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=Xa8TWY049BtwgbgbByBdIXcZ1%2FNJwsoA%2FoSuuvS8DzQ%3D&reserved=0>

https://arxiv.org/pdf/2108.03751.pdf<https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fpdf%2F2108.03751.pdf&data=05%7C01%7Cpastudtmann%40davidson.edu%7Ca56c93a849a74fe7262c08da772afa4e%7C35d8763cd2b14213b629f5df0af9e3c3%7C1%7C1%7C637953323743378308%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=eL4VzoAYhb%2FDLB5LGzeDqUiV%2BVEUk7sJdW8KHT793Sk%3D&reserved=0>

https://arxiv.org/pdf/2006.01135.pdf<https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fpdf%2F2006.01135.pdf&data=05%7C01%7Cpastudtmann%40davidson.edu%7Ca56c93a849a74fe7262c08da772afa4e%7C35d8763cd2b14213b629f5df0af9e3c3%7C1%7C1%7C637953323743378308%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=xNoYKKDOAoW3JzbxWz2nn8dLscT71PDqn89Wp7Q3lRQ%3D&reserved=0>

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