Abstract: A new view is seen (“theo-ry”) for the relationship of mind and universe, and mathematics is central. The Mathematical Mind Hypothesis (MMH). The MMH contradicts, explains, and thus overrules Platonism (the ruling explanation for math, among mathematicians). The MMH is the true essence of what makes the Mathematical Universe Hypothesis alluring.


What’s the nature of mathematics? I wrote two essays already, but was told I was just showing off as a mathematician, and the subject was boring. So let me try another angle today.

The nature of mathematics is a particular case of the nature of thinking.

For a number of reasons, deep in today’s physics, as I have (partly) explained in “Einstein’s Error”, many physicists are obsessed with the “Multiverse”, an extreme version of which is the “Mathematical Universe Hypothesis” (MUH), exposed for example by Tegmark, a tenured cosmologist at MIT. Instead of telling people what happened in the first second of the universe, as if I considered myself to be god, I prefer to consider dog:

Dogs LEARN To Choose “y” According To Least Time

Dogs LEARN To Choose “y” According To Least Time

[Dogs can also learn to solve that Calculus of Variation problem in much more difficult circumstances, if the water is choppy, the ground too soft, etc. To have such a mathematical brain allowed the species to catch dinner, and survive.]

The “Multiverse” has its enemies, I am among them. Smolin, a physicist who writes general access books, has tried to say something (as described in Massimo’s Scientia Salon’ “Smolin and the Nature of Mathematics”).

“Smolin,” Massimo, a tenured philosophy professor also a biology PhD, told me “as a counter [to Platonism], offers his model of development of mathematics, which does begin to provide an account for why mathematical theorems are objective (the word he prefers to “true,” in my mind appropriately so).”

My reply:

Smolin is apparently unaware of a whole theory of “truth” in mathematical logic, and of the existence of the work of famous logicians such as Tarski. When Smolin was in the physics department of Berkeley, so was the very famous Tarski, in the mathematics department. Obviously, the young and unknown Smolin never met the elder logician and mathematician, as he is apparently still in no way aware of any of his work.

Thus, what does Smolin say? Nothing recent. Smolin says mathematics is axiomatic, and develops like games. That was at the heart of the efforts of Frege’s mathematical logic, more than 115 years ago. (Bertrand Russell shot Frege’s theory down, by applying the 24 centuries old Cretan Paradox to it; interestingly, Buridan had found a rather modern solution to the problem, in the 14C!) To help sort things out, it was discovered that one could depict Axiomatic Systems with sequences of numbers. Could not Axiomatics then be made rigorously described, strictly predictive?

Gödel showed that this approach could not work in any system containing arithmetic. Other logicians had proven even more general results in the same vein earlier than that (Löwenheim, Skolem and contemporaries). Smolin is now trying to reintroduce it, as if Löwenheim, Skolem, Gödel, and the most spectacular advances in logic of the first half of the Twentieth Century, never happened.

Does Mr. Smolin know this? Not necessarily: he is a physicist rather than a mathematician (like Tarski, or yours truly).

Smolin: “Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will.”

Smolin: Math brought into existence by HUMAN WILL. Mathematics as will and representation? (To parody Schopenhauer.)

So how come the minds of animals follow mathematical laws? Dogs, in particular, behave according to very complicated applications of calculus.

How come ellipses exist? Have ellipses been brought into existence by Smolin’s “human will”? When a planet follows (more or less) an ellipse, is that a “construction which has been brought into existence by exercises of human will”?

Some will perhaps say that the planet “constructs” nothing. That I misunderstood the planet.

Massimo’s quoted me, and asserted that there was no value whatsoever to the existence of mathematical objects:

I had said: “How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?”

Massimo replied: “They don’t.”

And that’s it. It reminded me the way God talked in the Qur’an. It is, what it is, says Allah, and his apparent emulator, Massimo. Massimo did not explain why he feels that the spiral of a nautilus does not exist (or maybe, he does not feel that way, because it clearly looks like a spiral). According to Smolin, the spiral is just a “construct of human will”.

If the spiral is a construct of human will, why not the mountains, and the ocean?

I am actually an old enemy of mathematical Platonism. However, I don’t throw the baby with the bath.

I agree that the “Mathematical Universe Hypothesis”, and Platonism in general are erroneous. However that does not mean they are deprived of any value whatsoever.

Ideas never stand alone. They are always part of theories. And idea such as Platonism is actually a vast theory.

MUH is: ‘Our external physical reality is a mathematical structure.’

I do not believe in the MUH. Because of my general sub-quantic theory, which predicts Dark Matter. In my theory, vast quantum interactions leave debris: Dark Matter. That process is essentially chaotic, and indescribable, except statistically (as the Quantum is). propose a completely different route: our mind are constructed by (still hidden) laws which rule the universe. Call that the MATHEMATICAL MIND HYPOTHESIS (MMD).

Here is the MMD: Our internal neurological reality constructs real physical structures we call “mathematics”.

This explains why a dog’s brain can construct the neurological structures it needs to find the solutions of complex problems in the calculus of variations.

Dogs did not learn calculus culturally, by reading books. Indeed. Still they learned, by interacting with the universe. (It’s unconscious learning, but still learning. Most learning we have arose unconsciously.)

From these interactions, dogs’ brains learn to construct structures which solve very complicated calculus of variations problems. As explained by the Mathematical Mind Hypothesis, (hidden) physics shows up in neurological constructions we call mathematics. And those structures, constructed with this yet-unrevealed, not even imagined, physics, are not just mathematical, but they are what we call mathematics, itself. That’s why dogs know mathematics: their brain contain mathematics.

Patrice Ayme’

Technical Note: Some may smirk, and object that my little theory ignores the variation in neurological structure from one creature to the next. Should not those variations mean that one beast’s math is not another beast’s math?

Not so.

Why? We need to go back to Cantor’s fundamental intuition about cardinals, and generalize (from Set Theory to General Topology). Cantor said that two sets had the same cardinal if they were in bijection. (Then he considered order, and introduced “ordinals”, by making the bijection respect order.)

I propose to say two neurological structure are mathematically the same if they produce the same math. (Some will say that’s obvious, but it’s not anymore obvious than, say, “Skolemization“.)

[Last point: I use “neurology” to designate much more than the set of all neurons, dendrites, synapses, axons and attached oligodendrocytes. I designate thus the entire part of the brain which contributes to mind and intelligence (so includes all glial cells, etc.). That ensemble is immensely complex, in dimensions and topologies.]


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  1. Ian Miller Says:

    Never mind the dog. Anyone who has ever played a ball game will know they have to determine where the ball will be in a very short time and take the appropriate action to send the ball back to somewhere else. I must confess when I used to play ball games, I just hit the ball. I most certainly did not sit down and solve a calculus problem in the way you do when studying. Most likely, in my opinion, the brain has evolved an analogue mechanism for doing these sort of things.

    • Patrice Ayme Says:

      The calculus problem input through reading about it is completely different from the natural one. We follow a number of formal, I call them axonal, rules, in the artificial case, that we learned by rote, from paper. In the natural case, our brain builds the neurology, the appropriate axonal system.

      For the latter, some may brandish a kind of min-max variational calculus involving a Hebbian mechanism, with a convergence towards the best solution by trial and error, thus ending with the appropriate neurological subset. But that does not contradict my vision of what happens

      Are you suggesting all moving animals have analogues centers for trajectory computing? Down to dragonflies? And wasps? The question is how do they grow to fit the world.

      My point, Ian, is that the brain IS made such elements. I don’t believe that brain of the dog has a calculus sub-machinery. I believe that the back and forth between brain of dog and environment takes the form of what we call mathematics.
      It’s like waves on a beach: the vertical back and forth creates an horizontal back and forth. Take a picture at any point, of waves or sand: what you see is math. The beach is math, the waves are math. Same here, except the math picture gets imprinted in a landscape we call neurology. And except we are not in three dimensions.

      I claim this is not trivial, as, first we are dealing with spaces (what is called in Quantum Mechanics CONFIGURATION, or PHASE spaces) of very high dimensions (more than 50, because of all the neurotransmitters). Second QM itself has got to be involved in the fine detail (we know memories are within neurons, latest research, and this sort of scale is definitively QM)

      • Ian Miller Says:

        Patrice, I don’t think we disagree on detail. In what I was saying, your “back and forth” is “analogue optimisation” but yes, i agree the brain will work at the quantum level, and at that level, because of the discreteness of QM, it presumably is digital. Maybe the brain does manipulate phase spaces; my view is that so far the exact mechanism is unclear. I agree completely with your last sentence, and that it is not trivial.

        My original point was intended to be more or less in line with your response, the analogue referring to, “that ball is going faster than the last one, and slightly to the right, so I must compensate by . . .” rather than solving a differential equation in the head.

  2. EugenR Says:

    I still don’t get it, why if neuron and all the rest of brain material is mathematically patterned and will be in the future perhaps even mathematically explained, out of it emerges mathematical knowledge of a dog, a human or even a professor of mathematics. Vice versus, doesn’t mathematical truth exist without its negative copy interwoven in the brain?

    • Patrice Ayme Says:

      Dear Eugen: it’s a chicken and egg problem. Mathematics as chicken, neurology as egg. The question: ‘can mathematical truth exist without the brain?’is quasi-metaphysical. Can the universe exists without the brain?
      Same for math.
      This is why Plato thought math had an independent existence.
      However, my angle is different.
      The universe exists. So do the laws of the universe. The universe is organized, and this organization we call laws. Physical laws.
      They show up in neurologies, not just in the orbs of planets.
      Is that more convincing?

      • EugenR Says:

        Dear Patrice, to me the idea of emerging property sounds more convincing. Maybe I have some block in my line of thought. Consciousness is an a emerging property or state, out of the physical activity of the brain. Mathematics is the emerging property raising from the human interaction, (inductive and deductive), with the physical world and its realities.

        • Patrice Ayme Says:

          Dear Eugen: there is no contradiction between my view and emergence. Emerging properties emerge from physical laws. I am just saying mathematics neurologically emerge from as yet hidden physics.

  3. Philip Thrift Says:

    On Patrice Ayme’s comment about mathematica truth, Tarski, and Gödel: What I think is closer to the Smolin principle is the mathematical pluralism of Joel David Hamkins (professor, CUNY):

    “Pluralism in mathematics: the multiverse view in set theory and the question of whether every mathematical statement has a definite truth value”

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