Where is mathematics, physically?

 

Artificial Intelligence has solved math. At least, this is what is claimed on different social media. I don't know if this is true, and if the evidence is enough to claim such a strong statement, but probably it will be for sure in a few years (months, maybe?). And this leaves us (working mathematicians and theoretical physicist) in an uncomfortable position: what is our reason of being? what is our work for?

In my opinion, this is great news for us. Let me explain with an analogy: the geographic discovery. When human civilizations explored new lands, several specialized workers were needed, and they embarked in each expedition: road makers, home builders, defenders,... They had to accomplish fundamental auxiliary tasks that were needed for the project. But there was always a main character, a leading explorer, who brought the knowledge itself of the new land back, who integrated the new maps with the known maps, who told the other what was actually there.

In math research, the mathematicians themselves have to do al these different tasks (maybe split among several coauthors). The analogues, indeed. Writing down a pile of examples against which to check hypotheses, crafting proofs, solving some routine logical leap, coding, writing papers, tracing connections between different branches of mathematics. At the end, AI will be capable of doing all this, and integrate it in a paper, in a piece of new knowledge. Faster and better than us.

But this is good news! AI will clear the forests, build the roads and map the terrain at lightning speed. But at the end of the day, a real human being is needed to claim that we conquered the land. It will still be needed that the new knowledge is absorbed by the collective consciousness of humanity.

Suppose that we build super-robots, and we have sent them to colonize Mars and other planets, or even other solar systems. They are out there, building houses, clearing roads, and setting up infrastructure (artificial atmosphere, yielding foods,...). The galaxy is conquered, but we ourselves never actually visit these new worlds! A mathematics entirely solved by AI, but unread and unassimilated by humans (as a group), is a collection of empty houses on a distant planet.

We need to go to observe the new landscapes, to assimilate them, to communicate them, to teach others about them. It is not enough to have the knowledge encapsulated in books/papers, the real knowledge must be flowing from person to person. By the way, this is not new. Maybe the proof of Riemann hypothesis is already buried into a stack of preprints, written by some unknown mathematician. What is the point of having real knowledge locked in papers?

Mathematics, physically, is located in the mind of mathematicians. Therefore it must constantly be in motion, rewritten, re-discussed, re-taught, re-thought. Now more than ever.

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