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> Of course computers can do arithmetic operations, but this is not the same as solving math problems, proving theorems, etc.
Computers can solve math problems and prove theorems; this remains a significant subfield of Computer Science with lots of industrial use cases. However, pure machine learning based approaches toward these problems remain subpar.
> Even mathematical objects are approximated up to an approximation error in a computer (like a differentiable manifold or a real number).
Only because it caught on (and in the case of non-computationally-intensive applications, for purely historical reasons). For example, Mathematica has Reals and even functionality for Reals that is literally impossible to implement for integers [1,2]. There are also precise characterizations of objects in differential geometry [3]. You could imagine applying LLMs to these types of programs a la Copilot, but when you do this you will find yourself agreeing with Paul Houle's observation that math is harder to fake than eg art, language, or even glue code for web apps (assuming you're not on a grant that economically incentivizes you to draw the opposite conclusion, ofc).
[1] https://reference.wolfram.com/language/ref/Reduce.html
[2] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...
[3] https://github.com/bollu/diffgeo