What does mathematics really say about reality?

“This immense book which always remains open before our eyes, I mean the universe. […] “It is written in the language of mathematics,” wrote Galileo, one of the founders of modern science. The aim of this article is to question this stubborn assumption in science, according to which the concepts of reality, language and truth maintain strong and natural connections, to the point of merging. We will put it to the test with a mathematical theory that, from this point of view, seems extremely contradictory.

A century old paradox

Let's start with this theorem, whose centenary we will celebrate in 2024, known as the Banach-Tarski paradox, its authors. This statement is accompanied by a conclusive proof, according to the strictest mathematical standards, where truth is defined with extreme precision. It is therefore above all not a question of a paradox, but of a fact, and it is this fact itself that we will be interested in below. This theorem states the following:

“We can cut the ball into a finite number of pieces and rearrange them to form two identical balls.”

If he were speaking from an everyday experience, this statement would mean that we could cut an orange into carefully selected pieces and reassemble them (without distorting them) into two identical oranges (without a hole). Then repeat the process to get as many oranges as we want from one orange. The difference in this theory is:

“We can cut a given sphere into a finite number of parts and rearrange them to form a sphere of as large a radius as desired.”

We can then reassemble the orange pieces to form a piece the size of Jupiter.

false paradox

The paradox comes from the fact that this statement does not correspond to the stubborn experience of reality, where we intuitively assign to any object a volume in which the whole is the sum of the parts. On the other hand, each piece of division is practically unattainable, since it consists of an infinite number of points, while the total number of particles in the universe is finite.

Finally, this statement cannot be proven without accepting a premise called Axiom of choiceSo that its truth does not come from a natural principle but from the adoption of a rule of the linguistic game. This is the logical key to the false paradox: from this axiom flow the parts of the theory, to which the objection of the constancy of size cannot be invoked, because simply no mathematical notion of size can be applied to it.

But an embarrassing false paradox

The great advantage of this theory comes from the fact that the paradox persists, even after it has been technically clarified. To understand this psychological difficulty, we have to delve a little deeper into the history of ideas.

Sciences, such as physics, which study natural phenomena and are supposed to follow fixed rules, establish mathematical laws that offer three virtues. First of all, brevity, which allows the efficient dissemination of knowledge, open to scientific debate. Then, the prediction of the evolution of the phenomenon. Finally, fruitfulness is demonstrated: the emergence of new and non-intuitive concepts at the limits of the laws. Thus, black holes or some elementary particles were predicted for the first time through equations.

Under the influence of this mathematics of natural science, it is taken for granted that this “nature” is expressed in mathematical language. What is even more remarkable is the mutual model in which every researcher lives more or less consciously, according to which the mathematical interrogation of nature should reveal its secrets one by one, presenting a panorama of reality without shadows. Physicist Eugene Wigner calls this belief “the unreasonable effectiveness of mathematics in natural science.”

This belief carries with it the undeniable success of the mathematics of the natural sciences, since airplanes fly, as predicted by the laws of aerodynamics. We must add a powerful psychological spring: the idea that the whole truth of reality is revealed in (mathematical) language is attractive. Scientific progress, in its irrepressible advance, leaving nothing on the road to knowledge, will eventually exhaust this truth. It will all be a matter of patience.

Reality, Language, and Truth: A Far-Fetched Belief

Thus everything seems to happen as if it were reality, and truth as a concept, that is, the idea that we can assign a value to any discourse about nature (whether true or false) and that “in turn, every phenomenon is assignable to a law,” and finally, language, well calibrated in terms of grammar and logic, occupied the three vertices of a fixed, sometimes confused, conceptual triangle, which in the absence of one of the other two would slip away.

We can trace this concept, revitalized in the Renaissance, at least to Plato in the classical Greek period, who postulated the existence, certain but universal, of a reality existing in the inaccessible world of ideas, of which only the dark and crude projections of itself are shown (each representation here below of the circle is an incomplete image of the idea of ​​the circle). This existence so far from reality would make it necessary for us to walk towards it, because, guided by the principle of truth, we can be led to the paradise of ideas. It is Euclid (III^H BC) which under this system establishes mathematics as a hypothetical, deductive, model science: from the smallest number of axioms, logical reasoning, from truth to truth, raises us to the highest properties of geometry and numbers.

Previously, the distant Mesopotamian civilization had not formalized its relationship to reality, language, and truth in this way, nor had it linked the three together. Their texts show a performative and speculative use of language (reality corresponds to what is said, not vice versa). Mesopotamian mathematics, on the other hand, is not concerned with a system of axioms or with the generalization of formulas, which never go beyond practical problems. The civilization for which, as Richard Powers puts it in his novel astonishment :

“The world is an experiment in the invention of health, and contentment is its only guide.”

An illusion denied by mathematics itself.

It was Nietzsche who first worried about our Platonic belief in the holy trinity of reality-language-truth, where every utterance takes on the fantastic character of a veil lifted over the supposed truth of the world. This in turn gives a creative and satisfying appearance to (mathematical) language. He summed it up thus: Twilight of the Idols :

“I fear we will never get rid of God, because we still believe in grammar.”

Since the sciences of reality develop under this system of truth in an apparently satisfactory way, we would expect the directions taken by mathematics itself, as a paradigmatic science, to define more clearly this unshaded scheme of the truth of reality. However, a theory like the Banach-Tarski theory seems, on the contrary, to destroy these imaginary connections between reality, language, and truth.

Reality and mathematics are freed from their bonds.

What do we conclude? This example of unreal mathematical truth testifies to the autonomy of mathematics, which unfolds outside of sensible reality, in a language that is concerned only with its own internal truth, based on conscious choices of rules. Whereas the contradiction we feel is merely the effect of an illusion, the effect of the imaginary connections between real language and truth described above. Mathematics is far from what Wittgenstein suggests in his Philosophical Notes As a “phenomenological language, a grammar of facts on which physics builds its theories.”

This in turn calls into question the idea of ​​nature pre-written in mathematical language: when the mathematical arrow reaches the truth, it can be unreal. But we must equally take into account that its path ignores a set of phenomena that are inaccessible, because they are forever untranslatable into their language, and whose enormous potential configuration determines the inconceivable edges of expressible knowledge. This is the anti-Platonist hypothesis.

Certainly, breaking away from the reassuring view of a mathematical science that gradually translates all the truth of reality does not help to produce more knowledge, but it puts us in a much better position to think correctly about both reality and knowledge.