Post by @antoinechambertloir@mathstodon.xyz

2026-09-10 08:22 UTC

#AI #Math 2/n Before going on specific matters, I need to go back to what math is and to what it does. TL-DR;-) As far as it goes about automatization, math is probably the easiest of all sciences. In 1994, the mathematician William Thurston published a paper with the title “On proof and progress in mathematics” in which starts with stating the question What is it that mathematicians accomplish? His answer — you might find it circular — is that mathematicians push forward the human understanding of mathematics. This means that mathematicians have under the eyes a whole range of (abstract) objects for which they use sometimes colloquial names (group, ring, field, memberhip, sheaf, relation, knot, crystal…) and sometimes more specific names (perfectoid, hypersolvable, function, proof, set…) and they try to better understand the properties that these objects possess, and the relations between them. A whole field of mathematics is motivated by the fact that some of these objects model phenomena of the actual world in which we live, with the idea that the behaviour of the abstract mathematical objects might help us make actual decisions. But a large part of mathematics is just motivated by these abstract objects themselves, possibly with the hope that, some day, they could be useful to the real world, and sometimes in the expectation that they will never be useful to it. For example when being useful means helping killing people. https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf

Replies (1)

@antoinechambertloir@mathstodon.xyz 2026-09-10 08:31

#Math #AI 3/n We have traces of mathematical activity for more than 3000 years now, so it seems that this human desire to understand, in an abstract way, our surrounding world is quite deeply ingrained in our minds. This quest has been parallel to the development of science, often prompted by questions of other sciences, but what happened in the modern era (beginning of 19th century, say), is more explicit desire for rigour in the argumentation. Not that mathematics lacked rigour, it always had a lot, like all science do, but mathematicians felt the need to have more precise definitions for their abstract objects. Mostly because standard reasoning often led mathematicians to oddities, if not mere absurdities. Let me quote just here the discovery by Bertrand Russell that if you wish to talk about collections of which things may, or may not belong, you need to be extremely careful. Russel's remark is simple: imagine you can create collections of anythings, for example collections of collections, and so on (after all, you can have a bag to keep all of your bags together…), then it might be possible that a collection be a member of itself (the idea of an idea is an idea…), and you could look at the collection R of all collections that are *not* a member of itself. This collection R is paradoxical : saying that the collection R is a member of R exactly means that it is not a member of R.

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