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Comment by pmoriarty | Hacker News Reader

As a non-mathematician, the last truly big mathematical project that caught my imagination was Hilbert's attempt to found all of mathematics on a firm foundation of logic.

That attempt famously and spectacularly failed with Gödel's incompleteness theorems.

Since then it seems that mathematicians have lost interest in foundations and are content to search for interesting results, structures, and systems, even if they don't have a solid foundation.

More recently I've heard some proposals to revisit the foundational project but with higher-order logics proving the consistency and completness of lower-order ones, which sounds interesting, but I'm not sure how much progress has been made, and to a non-mathematician/non-logician even that attempt sounds a bit like a house of cards.

Does anyone here know about this and if there are even any mathematicians around these days who are still interested in it?

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> Since then it seems that mathematicians have lost interest in foundations

This is very much not the case. What's closer to truth is that the discussion moved on from a framework laymen can seemingly understand conclusions, to one where conclusions (or their implications on mathematics proper) are a lot harder to explain to laymen. Foundational work is still a thing, but I don't think it affects the nature of mathematics in a way laymen can conceptualize.

Take Godel's incompleteness. People say it's something laymen can understand, and people attempt explaining it to masses every day in youtube, reddit etc. But if you truly get into the formal conclusion (i.e. with Rosser's trick, the conclusion is: "a theory cannot be an extension of Q, complete and consistent all 3 at the same time") you'll see that it's already pretty far away from what laymen thought they understood. And modern foundational work exponentially drifted away from this too.

I'm not a mathematician so everything in this comment should be taken with a grain of salt.

Noteworthy to me, is that it seems foundational work is to a significant degree, not considered mathematics or something people calling themselves “mathematicians” would spend much of their time on; but it’s left for logicians and more philosophically inclined people.

May I recommend https://sites.psu.edu/ehssan/wp-content/uploads/sites/7257/2..., a transcript of a 2000 talk by Michael Atiyah?

A tiny bit from section 9:

> The 20th century can be divided roughly into two halves. I would think the first half has been dominated by what I call the "era of specialization," the era in which Hilbert's approach, of trying to formalize things and define them carefully and then follow through on what you can do in each field, was very influential. As I said, Bourbaki's name is associated with this trend, where people focused attention on what you could get within particular algebraic or other systems at a given time. The second half of the 20th century has been much more what I would call the "era of unification", where borders are crossed, techniques have been moved from one field into the other, and things have become hybridized to an enormous extent. I think this is an oversimplification, but I think it does briefly summarize some of the aspects that you can see in 20th-century mathematics.

> As a non-mathematician, the last truly big mathematical project that caught my imagination was Hilbert's attempt to found all of mathematics on a firm foundation of logic.

The classification of finite simple groups aka The Enormous Theorem might also interest you. It is around 10,000 pages spread across 100s of journal articles by ~100 authors over a 50 year period.

https://en.wikipedia.org/wiki/Enormous_theorem

You could jump to the history section if you want to skip some technical parts.

There have been other huge programs but they might be harder to describe without a lot of jargon.

Mathematicians remain obsessed with the foundations.

Computer-checked mathematics is growing very fast.

Stuff like category theory ("Not real maths" - Kevin Buzzard) is also extremely popular.

Excellent book on just that subject, if you're not already familiar with it:

https://www.goodreads.com/book/show/748807.Mathematics

I'm not a mathematician or even very good at math but I love reading anything by Morris Kline. Even when I don't fully grasp what he's talking about, I have a great respect for his intelligence and grasp of the world.

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