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JasonGross comments on Proofs, Implications, and Models - Less Wrong
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Mainstream status:
This is meant to present a completely standard view of semantic implication, syntactic implication, and the link between them, as understood in modern mathematical logic. All departures from the standard academic view are errors and should be flagged accordingly.
Although this view is standard among the professionals whose job it is to care, it is surprisingly poorly known outside that. Trying to make a function call to these concepts inside your math professor's head is likely to fail unless they have knowledge of "mathematical logic" or "model theory".
Beyond classical logic lie the exciting frontiers of weird logics such as intuitionistic logic, which doesn't have the theorem ¬¬P → P. These stranger syntaxes can imply entirely different views of semantics, such as a syntactic derivation of Y from X meaning, "If you hand me an example of X, I can construct an example of Y."
I can't actually recall where I've seen someone else say that e.g. "An algebraic proof is a series of steps that you can tell are locally licensed because they maintain balanced weights", but it seems like an obvious direct specialization of "syntactic implication should preserve semantic implication" (which is definitely standard). Similarly, I haven't seen the illustration of "Where does this first go from a true equation to a false equation?" used as a way of teaching the underlying concept, but that's because I've never seen the difference between semantic and syntactic implication taught at all outside of one rare subfield of mathematics. (AAAAAAAAAAAAHHHH!)
The idea that logic can't tell you anything with certainty about the physical universe, or that logic is only as sure as its premises, is very widely understood among Traditional Rationalists:
--Albert Einstein
Eliezer, I very much like your answer to the question of what makes a given proof valid, but I think your explanation of what proofs are is severely lacking. To quote Andrej Bauer in his answer to the question "Is rigour just a ritual that most mathematicians wish to get rid of if they could?", treating proofs as syntatic objects "puts logic where analysis used to be when mathematicians thought of functions as symbolic expressions, probably sometime before the 19th century." If the only important thing about the steps of a proof are that they preserve balanced weights (or semantic implication), then it shouldn't be important which steps you take, only that you take valid steps. Consequently, it should be either nonsensical or irrelevant to ask if two proofs are the same; the only important property of a proof, mathematically, validity (under this view). But this isn't the case. People care about whether or not proofs are new and original, and which methods they use, and whether or not they help give a person intuition. Furthermore, it is common to prove a theorem, and refer to a constant constructed in that proof. (If the only important property of a proof were validity, this should be an inadmissible practice.) Finally, as we have only recently discovered, it is fruitful to interpret proofs of equality as paths in topological spaces! If a proof is just a series of steps which preserve semantic implication, we should be wary of models which only permit continuous preservation of semantic implication, but this interpretation is precisely the one which leads to homotopy type theory.
So while I like your answer to the question "What exactly is the difference between a valid proof and an invalid one?", I think proof-relevant mathematics has a much better answer to the question you claimed to answer "What exactly is a proof?"
(By the way, I highly reccomend Andrej Bauer's answer to anyone interested in eloquent prose about why proof-relevant mathematics and homotopy type theory are interesting.)