The MATHEMATICAL MIND HYPOTHESIS

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.

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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 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.

http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/do-dogs-know-calculus-of-variation

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.]

Tags: Mathematical Mind Hypothesis, Mathematical Universe Hypothesis, Mathematics, Mind, Platonism