In general there are two different kinds of methodology to produce...
François Chollet@fchollet
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Aug 22, 2025
~1 min read
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In general there are two different kinds of methodology to produce progress in any science or engineering field... both are important and can lead to transformative progress.
There's the "Edison way" where you brute-force a large predefined design space and you keep what works, without necessarily understanding why it works. This is akin to biological evolution. Nearly all of deep learning was built this way (despite the fancy math in papers, which is there to look nice 99% of the time).
And there's the "Einstein way" where you think up big ideas in a top-down fashion and derive precise results from them, before testing those results. This requires imagination, curiosity, high-abstraction thinking. This is, I suppose, the mode of progress favored by mathematicians.
There's the "Edison way" where you brute-force a large predefined design space and you keep what works, without necessarily understanding why it works. This is akin to biological evolution. Nearly all of deep learning was built this way (despite the fancy math in papers, which is there to look nice 99% of the time).
And there's the "Einstein way" where you think up big ideas in a top-down fashion and derive precise results from them, before testing those results. This requires imagination, curiosity, high-abstraction thinking. This is, I suppose, the mode of progress favored by mathematicians.
Edward Frenkel@edfrenkel
This is an unwise statement that can only make people confused about what LLMs can or cannot do. Let me tell you something: Math is NOT about solving this kind of ad hoc optimization problems. Yeah, by scraping available data and then clustering it, LLMs can sometimes solve some very minor math problems. It's an achievement, and I applaud you for that. But let's be honest: this is NOT the REAL Math. Not by 10,000 miles.
REAL Math is about concepts and ideas - things like "schemes" introduced by the great Alexander Grothendieck, who revolutionized algebraic geometry; the Atiyah-Singer Index Theorem; or the Langlands Program, tying together Number Theory, Analysis, Geometry, and Quantum Physics. That's the REAL Math. Can LLMs do that? Of course not.
So, please, STOP confusing people - especially, given the atrocious state of our math education.
LLMs give us great tools, which I appreciate very much. Useful stuff! Go ahead and use them AS TOOLS (just as we use calculators to crunch numbers or cameras to render portraits and landscapes), an enhancement of human abilities, and STOP pretending that LLMs are somehow capable of replicating everything that human beings can do.
In this one area, mathematics, LLMs are no match to human mathematicians. Period. Not to mention many other areas.
Calling on my friend @ericweinstein and @GaryMarcus, who has been one of the few sane expert voices on these matters lately. 🙏
h/t @hellheff
REAL Math is about concepts and ideas - things like "schemes" introduced by the great Alexander Grothendieck, who revolutionized algebraic geometry; the Atiyah-Singer Index Theorem; or the Langlands Program, tying together Number Theory, Analysis, Geometry, and Quantum Physics. That's the REAL Math. Can LLMs do that? Of course not.
So, please, STOP confusing people - especially, given the atrocious state of our math education.
LLMs give us great tools, which I appreciate very much. Useful stuff! Go ahead and use them AS TOOLS (just as we use calculators to crunch numbers or cameras to render portraits and landscapes), an enhancement of human abilities, and STOP pretending that LLMs are somehow capable of replicating everything that human beings can do.
In this one area, mathematics, LLMs are no match to human mathematicians. Period. Not to mention many other areas.
Calling on my friend @ericweinstein and @GaryMarcus, who has been one of the few sane expert voices on these matters lately. 🙏
h/t @hellheff
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AI (not LLMs, but search based AI) will be superhuman at the first type relatively quickly.