this post was submitted on 31 Jul 2024
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It's not pedantic. You can mathematically prove math.
You can't mathematically/algorithmically prove an event happened or did not happen.
You can prove mathematical logic and you can (not 1-to-1) tie that to symbolic logic, but since it's not 1-to-1, because of ambiguity of symbols, there will be much more complexity. I personally think that the future of various machine assistants lies there, and what LLM's now do is going to be used in auxiliary roles for that.
The problem is that mathematical proofs rely on the basic premise that the underlying assumptions are rock solid, and that the rules of the math are rock solid. It's rigorous logic rules, applied mathematically.
The real world is Bayesian. Even our hard sciences like physics are only "mostly" true, which is why stuff like relativity could throw a wrench in it. There's inherent uncertainty for everything, because it's all measurement based, with errors, and more importantly, the relationships all have uncertainty. There is no "we know a^2 and b^2, so c^2 must be this". It's "we think this news source is generally reliable and we think the sentiment of the article is that this crime was committed, so our logical assumption is that the crime was probably committed". But no link in the chain is 100%. "Rock solid" sources get corrupted, generally with a time lag before it's recognizable. Your interpretation of a simple article may be damn near 100%, but someone is still going to misread it, and a computer definitely can.
Uncertainty is central to reality, down to the fact that even quantum phenomena have to be talked about probabilistically because uncertainty is built in all the way down.
This is why many philosophers came to criticize metaphysical logic in the 1800s, viewing it as dealing with absolutes when reality does not actually exist in absolutes, stating that we need some other logical system which could deal with the "fuzziness" of reality more accurately. That was the origin of the notion of dialectical logic from philosophers like Hegel and Engels, which caught on with some popularity in the east but then was mostly forgotten in the west outside of some fringe sections of academia. Even long prior to Bell's theorem, the physicist Dmitry Blokhintsev, who adhered to this dialectical materialist mode of thought, wrote a whole book on quantum mechanics where the first part he discusses the need to abandon the false illusion of the rigidity and concreteness of reality and shows how this is an illusion even in the classical sciences where everything has uncertainty, all predictions eventually break down, nothing is never possible to actually fully separate something from its environment. These kinds of views heavily influenced the contemporary physicist Carlo Rovelli as well.