The development of mathematics, a key to the development of AI?

The history of computers and informatics is very rich. It is found at the crossroads of two communities if we want to make a simplified picture. On the one hand, let’s say between 1930 and 1950, there are mathematicians and logicians working on the foundations laid in the XX^e century by Bertrand Russell and David Hilbert. Two names will stand out very clearly: Alan Turing and John von Neumann. On the other hand, there are physicists and engineers who ask themselves very concrete questions about what computers can do, especially for numerical calculations applied to the resolution of computer problems. physique and engineering, and which are not necessarily concerned by very abstract questions, for example the notion of computability as it has been explored via the lambda-calculus of the mathematician Alonzo Church and the concept of the universal machine ofAlan Turing. We can consult on this subject clear explanations of Nobel Prize in Physics Roger Penrose.

However, in all cases, a certain level of competence in mathematics was necessary for all these actors of the computer revolution. The theory of information, itself, which has a significant role in the design and operation of computers as one can be convinced by reading computer lessons from Nobel laureate in physics Richard Feynman, was discovered and developed initially by a doctor of mathematics, also an engineer, namely the great Claude Shannon.

There is a lot of talk these days about the current revolution in artificial intelligence and the possibility of achieving an AI that not only is aware but has the same general intelligence as a human mind, to the point of passing the famous test. of Turing.

The question is an old one, of course it preoccupied Turing, but also von Neumann so much that he had started a short time before his death to write a famous work entitled The computer and the brain. His friend and collaborator during the design of the US A and H bombs, Stanislaw Ulam, pioneer like him of numerical simulations on a computer to explore in particular the nonlinear physics (not to mention their joint work on Monte Carlo method that we find in many fields such as those at the crossroads of Big Data and Bayesian inference that we use for machine learning in artificial intelligence), had also wondered about the subject with reflections drawn from the theory of general topology.

Finally, one of the pioneers of AI after WWII was none other than the mathematician Marvin Minsky, as recalled in the video above.

In short, as we will have understood, we cannot do without mathematics to seriously tackle AI problems (two excellent sources in English for entering the world of mathematics are Mathematics and logic by Ulam and Kac and What is mathematics ? by Richard Courant).

Of course, just as it is not necessary to be a professional mathematician, let alone a genius, to also be an excellent physicist, Einstein is a good example, it is not necessary to be a fairly gifted mathematician to be a good computer scientist and to advance the discipline. Moderate knowledge may be sufficient and one can approach concepts of Deep learning and networks of neurones without having the level of a degree in mathematics. We can be convinced of this by reading the excellent and very accessible book by Jean-Claude Heudin entitled Understanding Deep Learning: An Introduction to Neural Networks.

But, one can also see that to read completely the equally excellent course of information theory and machine learning of the late David JC MacKay (Information Theory, Inference, and Learning Algorithms), you still need significant mathematical skills, even if most of the necessary knowledge only calls for mathematics known towards the end of the 19th century.^e century for the most part (see also on this subject the book on Deep learning of the Turing Prize in Computer Science Yoshua Bengio).

However, it is well known that the brain human does not work like a simple computer and that the comparison with the neural networks of the Deep learning has limits, already because we know that it can accomplish certain tasks while consuming less energy. The mathematician and physicist Roger Penrose goes further by putting forward arguments resting the existence of consciousness on quantum processes in the brain some of which would be non-calculable and linked to a new formulation of quantum theory that remains to be discovered and more in accordance with the non-linearity of equations of the theory of general relativity

Without going that far, it seems possible that the general intelligence of the brain ofHomo sapiens requires to be understood a revolution in the theory of algorithms and this could well require the massive use of advanced mathematics discovered in XX^e century, for example in the rarefied fields of topology and modern algebraic geometry at the level of a Grothendieck. In fact, already today, some data scientists not only benefit from a solid training in statistical analysis, but also in geometry and topology, whether they are differential or algebraic.

More importantly, in recent years we have seen the emergence of a fascinating field which has been called the Geometric Deep Learning and as explained Gilles Babinet, in an article by The gallery, the exploitation of the increasingly massively available data in many fields seems to require increasingly sophisticated tools from branches of mathematics which are certainly not so recent, but whose development depends on the modern theory of graphs and some combinatorial topology initiated at the end of the 19th century^e century.

The more we think about what is happening now, the more the notion of noosphere as it has been explored by the paleontologist, geologist and philosopher Teilhard de Chardin (without going so far as to involve metaphysical speculations on this subject which he shared to a certain extent with a Sri Aurobindo) seems concrete with the development of a sort of collective psyche of humanity, made possible by constructions material, in this case the sources ofenergy, from matter first and now information that connects all Homo sapiens. We can also see that after the revolutions in writing and printing, we are still at the start of a much more fundamental revolution, that of digital and digital, which is having an increasingly major impact on the noosphere, both on its material bases and on its complexity itself.

However, it seems clear that this revolution cannot continue and will be mastered only by those who acquire and develop a high level of competence in mathematics and in research in this field. We can therefore only worry about the drop in the level observed in France in the general population with regard to mathematics.

All the articles of the special day: