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