There are two languages: common language, which is messy, and mathematics, which is much more precise, and contains basic physics. Indeed, all common languages are more or less isomorphic (same shape and preoccupations).

Mathematics is the part of common language which, given precise axioms, is the simplest and irreducibly deduced from those simplest notions (in physics, thus nature). “Physics” is a compendium of how nature looks, for sure, or how it works, de facto. How nature looks, as deduced from experiments, has varied in the last 100 million years… and that description is getting increasingly precise, as demonstrated by our ever greater power in making nature do as we wish.

But how nature works inside brains has become ever more powerful and precise ever since there are brains, and they have grown. Neurology is an emergent part of nature. Thus it is factual, being natural, and we also call its basic architecture mathematics, when we describe it. For example, basic category theory looks like the simplest abstraction of basic neurology restricted to the simplest axons…

Thus elucidated, counting becomes a matter of neural networks. 1 + 1 = 2 can be directly envisioned as a semantic description of a (very useful) neural network which has appeared in advanced species. That makes “2” a description of some neuronal architecture. There is no free will there. “2” is just the label for a particular type of neural network found in nature.

As a result of being the product of emerging neuronal networks, there is no more free will in “2” than in the Iron nucleus (Fe 56). And so on it goes: “pi” is the length of the circumference of a circle of radius 1. No free will there, either.

Nor is there for multiplication of real numbers. Even better: one gets in complex numbers by trying to build a multiplication in the plane which generalizes the multiplication of real numbers. There is a way to do this (multiplying distances to the origin, adding angles from the real axis): it enables us to get square roots of negative numbers… some numbers which multiplied by themselves, have a negative square. Not much freedom there. But then something spectacular happens: this gives the best description of light (including momentum, energy and polarization)… And as such becomes the basic language of Quantum Physics.

How could that all be?

Does that mean that our brain and how we build networks there, is not free from Quantum Physics? Indeed. Let’s inverse the question: how could the brain be free of Quantum Physics, considering, well, that Physics, Nature in Greek, is Quantum? Would that not be considering that brains are not natural?

If somehow there is no free will in the nature of the neural networks (and thus mathematics) we build, where could free will be? Well, in which kind of networks we decide to build, then? The networks themselves, at their simplest, are mathematics, and thus mathematics is digital… So is language. Being digital, and finite (in its mode of construction) make languages and mathematics, limited and pre-ordained. But Quantum Physics itself is based on a continuum, and that brings the freedom… of the butterfly effect. Free will is a subtle thing.

The famous mathematician Richard Dedekind said numbers were the work of God, and the rest of mathematics the work of man. It is probably wiser to acknowledge that we, or at least our mathematics, are the work of physics… self-describing…

Patrice Ayme

Tags: Axons, Foundations, Mathematics, Neurology

December 30, 2020 at 2:05 am |

Complex numbers solve light is an interesting way to put it. You mean, with enough light without looking out you get electromagnetism?

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December 31, 2020 at 9:37 pm |

Basically. This shows that mathematics is a form of extremely basic physics… What else? And the reasons are these networks, those ideas, which are physical objects…

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December 30, 2020 at 9:58 pm |

One could argue 2 is the term defined by adding 1 + 1. In my view, mathematics is essential for discussing and calculating physical effects, but it is not physics. Sorry, applied mathematicians that call themselves physicists.

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December 31, 2020 at 6:33 am |

Dear Ian, thanks for the comment: Long story. In the Theory of Types of Bertrand Russell & Whitehead, it takes 200 pages to get to 1 + 1 = 2. That TT was created because set theory is contradictory, as Russell showed with a variant of the Liar’s Paradox.

Recently CATEGORY THEORY came to the fore. It provides foundations without worrying about foundations. And has turned out extremely useful.

My point was that 1 + 1 = 2 is pretty obvious in Category Theory… and that’s exactly reflected by the simplest Axonal Networks… CT is AN!!!!!!!!!!!!!!!!!!!

THAT makes mathematics into the foundations of physics.Because surely you will agree that the brain is part of physics. This explains why, using only pure math, one can get ALL of electromagnetism… As the simplest possibility…. Among some increasing complications: multiplication on the plane to get a square root of negative number describes light… Maxwell equations come out of the simplest differential equations applied to this in 3+1 dimension…

I have made this point for a long time, it seems to be increasingly accepted by part of the foundational establishment… There is a whole cottage industry of it, although nobody is as clear as yours truly… I would dare to say…

Happy new year!

P

https://patriceayme.wordpress.com/

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January 5, 2021 at 7:16 pm |

From: Ian Miller

Sent: Thu, Dec 31, 2020 1:02 pm

Subject: Re: foundations math

Dear Patrice,

May you have a happy, prosperous and virus-free 2021.

I am not aware of the 200 pp getting to show 1 + 1 = 2, and in fairness I am not that interested. I am not sure how you show set theory to be contradictory. The problem with the usual form of the liar’s paradox is that the paradox truly only seems to arise from the use of a verb in place of a noun. The element of a set has to be a thing, not an action.

I disagree that using only pure math one can get all of electromagnetism. For example, apart from the fact that nature does not follow this, maths would permit div D + curl D = ρ. Why cannot an electric field spiral? You eliminate that by observation With gravity, you could construct a physics whereupon, putting F as the gravitational field, div F = m, whereupon mass screens gravity, and the gravity of a [planet is solely due to the outer shell. No, nature doesn’t permit that, but maths would. Back to electromagnetism, mathematics cannot require motion of the electric field to cause the magnetic field, and vice versa. Mathematics will work just as well otherwise. If we found a magnetic monopole, we would simply put further terms into the Maxwell equations. Mathematics could accommodate it, but they cannot determine it.

My view, anyway.

Best wishes for 2021

Ian

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January 5, 2021 at 8:43 pm |

Frege, the most famous German logician had written a book on the foundations of mathematics, inventing for so doing SET Theory. Russell got an early copy while the book was sent for printing. Russell considered the set U of all sets which are not elements of themselves. An element x of that set U is not an element of U yet it is also an element of U. Grege had to stop the press.

Russell’s fame was made. It helped that he belonged to The Crown or so…

Russell and Whitehead built the Theory Of Type, making Set Theory into a hierarchy. That, I believe is the key to the brain, and thinking, hierarchies. TOT was too cumbersome, and it’s not clear what it achieved.

In the 1920s, the famous Dutch topologist Brouwer (of the fixed point theorem), swept away all of math, and built INTUIONISTIC Math, where the infinity axiom was put in doubt. Mathematicians were aghast and ignored him. 40 years later, Intuitionism became central to Computer Programming.

Meanwhile the famous mathematician Hilbert (Hilbert spaces) had set up a program of conjectures to build mathematics on a firm basis.

[more laterrrr…]

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