The unreasonable effectiveness of algebra in representing cognition

In 1992 I was attending the Annual Conference on Supercomputing in Minneapolis, Minnesota. One of the keynote speakers was Larry Smarr, a leading pioneer in supercomputing. Behind him there were two screens: one showed a true tornado the other showed the simulation of a tornado made with a supercomputer. It was hard to distinguish the real from the simulated one. Larry commented that the computer image was obtained performing billions of elementary mathematical operations but still the result was a near perfect simulation of real observation. Larry concluded saying: “If you do not marvel at this result I do not know what will ever make you marvel.”

Eugene Wigner, a theoretical physicist who was awarded the 1963 Nobel Prize in Physics, did indeed marvel. In 1960 he wrote the famous article: The unreasonable effectiveness of mathematics in the physical science.

Now comes generative AI that exhibits the ability to answer questions, translate, summarize even invent texts. Generative AI exhibits cognitive abilities close to human cognitive abilities, at least for some specific tasks. It is truly surprising.

How does it work? The principle is in the end simple: it transforms words and sentences in numerical vectors/tensors and applies linear algebra operations. The “marvellous” fact is that linear algebra operations applied to vectors/tensors that represent sentences imitate superior human cognitive abilities such as summarizing a text.

The surprising ability of simple mathematics to represent human thinking deserves deep “human” thinking in itself.