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by nbouscal
4484 days ago
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I think it can in theory, but the reason it hasn't been done in practice is basically that mathematics is too damn big. Look at Whitehead and Russell's original attempt to do basically exactly what you're asking for: it took them a few hundred pages to build up enough machinery to prove that 1+1=2. |
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I only heard the same argument (this is the phrasing from Wikipedia but everybody seems to say something along these lines):
>"However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, there would in fact be some truths of mathematics which could not be deduced from them."
...so what?! The fact that there can be truths that can't be deduced from an "assembler language" just means that the system will sometimes just say something like "error: no proof in the current model database for 'fact x' found" and then the mathematician will just add "consider 'fact x' proven as in the defined in modelXYZ" (a model that can have a totally different logic than the current one - think of a model as a library written in a completely different programming language, in software analogy), taking responsibility for the fact the equivalence of the concepts 'fact x in current model' and 'fact x in modelXYZ'.
The long term goal would be unification of as many of the models as possible (even with, what I understand from Godel, as the impossibility of total unification - if something is proved to be impossible, it doesn't mean you can't get great benefits by always getting asimptotically closer to it) preventing such "forced equivalences", but it would still be a working system in the meantime. And more importantly, I guess, the system will make the "forced equivalences" obvious, and label them as problems for mathematicians to solve.