Maybe Hilbert's dream was not that crazy after all

woolion 19 hours ago | parent | next [-]

The dream in itself has been destroyed. The idea that you could just have a machine enumerate all valid theorems in a theory is part of it, but it's only a question of form. The point was that it was to prove "all theorems of Mathematic", not "theorems into a given axiomatic system that is useful in some contexts, e.g. ZFC".

You could even argue that it's the fundamental basis for post-modernism, since mathematics have destroyed the notion of absolute truth in any advanced domain. It's back to a form of "all models are wrong but some are useful" similar to what we have in physics. Sayonara, Plato.

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mettamage a day ago | parent | prev [-]

What is that dream? I don’t know much about it

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thrance 17 hours ago | parent | next [-]

The Entscheidungsproblem. Basically, is there an algorithm such that, taking an arbitrary statement as input, can output wether it is true or false? Turing and Church both independently proved that no such algorithm exists.

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

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ylliu a day ago | parent | prev [-]

[dead]

	▲	jrflo a day ago \| parent \| next [-]
		Mathematics not being axiomatically complete doesn't mean you can't have crazy progress from a formalized and mechanized systems. It just means that there are corners you can't reach mechanically, but we don't know if those corners are at all interesting or not. It could be the case that 99.99% of useful math can be found mechanically.
	▲	sigmoid10 a day ago \| parent \| prev [-]
		Mathematics in the sense of a complete set of axioms can't, but human research into mathematics apparently just needed enough compute to achieve the same output as a high-tier faculty.