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Have you followed Gödel's proof in detail? I know Fields medalists who like to speculate about Gödel, but haven't read Russell and Whiteheads Principia Mathemat

Have you followed Gödel's proof in detail? I know Fields medalists who like to speculate about Gödel, but haven't read Russell and Whiteheads Principia Mathematica (PM). Gödel's system is PM, but PM is quite an impossible read. Most of this discussed has been discussed over 30 years between 1900-1930. By the way Turing also builds on PM in his on computable numbers paper. PM is like a compiler before there were compilers. Today it would look more like metamath.org or https://github.com/vladimirias/Foundations https://github.com/vladimirias/Foundations > Formal mathematics has no concept of absolute truth; this is left for philosophy. No, it is called logic which is the intersection of both. And most mathematicians haven't done any logic whatsoever. They avoid it, because it means you actually have to think about things. You have to think about why greek symbols have special meanings. And as computer scientists/hackers know, computer code is an alternative medium. I remember when I asked a math professor about the "=" sign and how things can be equal at all, since if they are either the same (A=A is trivial), it is trivial, or if they are different they are not the same (A=B is false). About 10 years later I discovered this is why physics has the notion of symmetries, which is a statement of the form A - B = 0. Two things are equal iff their transformation leads to the origin. But I should have known this all along, because in computer programs "=" can stand for various kinds of operations. It is just that the language of mathematics in general is so inaccurate that one gets lost on the way.

asQuirreL 11y ago
I'm a bit dubious about your sweeping generalisation that "most mathematicians haven't don't any logic". I also agree with GP that logic also does not deal with absolute truth. Proofs in formal logic may be applied to models, but they only provide truths modulo the assumptions made in those models. As for your remark on equivalence. Your original description is one of syntactic identity, a sort of free equivalence. But any relation that is reflexive, transitive and symmetric can be considered an equivalence (indeed, there are infinitely many equivalence relations over the integers). Also note that this is not the same as how "=" is used in computer programs where for the most part it is asymmetric (a=b is rarely semantically equivalent to b=a).
__mbm__ 11y ago
This is the crankiest comment I've seen on HN in a while. You "know Fields medalists"? Right. You think that Pricipia Mathematica is the source of all truth in mathematics? Wrong. Most mathematicians working on foundational questions start out by learning Zermelo-Fraenkel set theory, which is well-understood, and avoids several difficulties that Russel had. While few people have read PM, it remains important because it goes through the hard task showing that high-level mathematics (calculus, &c.) can have rigorous, first-principle proofs. "PM is like a compiler before there were compilers." What? "I asked a math professor about the '=' sign and how things can be equal at all (...)" OMGWTFNO. I've seen comments like this before. Typically, it creates debate around a non-issue by sowing confusion and never nailing down precisely what we are talking about (hence the need for formal methods—it helps us call bullshit on comments like this). "Two things are equal iff their transformation leads to the origin. [A - B = 0]". Wrong. There are equality relations that do not require the existence of a zero or the concept of addition or subtraction.
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asQuirreL 11y ago

I'm a bit dubious about your sweeping generalisation that "most mathematicians haven't don't any logic". I also agree with GP that logic also does not deal with absolute truth. Proofs in formal logic may be applied to models, but they only provide truths modulo the assumptions made in those models. As for your remark on equivalence. Your original description is one of syntactic identity, a sort of free equivalence. But any relation that is reflexive, transitive and symmetric can be considered an equivalence (indeed, there are infinitely many equivalence relations over the integers). Also note that this is not the same as how "=" is used in computer programs where for the most part it is asymmetric (a=b is rarely semantically equivalent to b=a).

__mbm__ 11y ago

This is the crankiest comment I've seen on HN in a while. You "know Fields medalists"? Right. You think that Pricipia Mathematica is the source of all truth in mathematics? Wrong. Most mathematicians working on foundational questions start out by learning Zermelo-Fraenkel set theory, which is well-understood, and avoids several difficulties that Russel had. While few people have read PM, it remains important because it goes through the hard task showing that high-level mathematics (calculus, &c.) can have rigorous, first-principle proofs. "PM is like a compiler before there were compilers." What? "I asked a math professor about the '=' sign and how things can be equal at all (...)" OMGWTFNO. I've seen comments like this before. Typically, it creates debate around a non-issue by sowing confusion and never nailing down precisely what we are talking about (hence the need for formal methods—it helps us call bullshit on comments like this). "Two things are equal iff their transformation leads to the origin. [A - B = 0]". Wrong. There are equality relations that do not require the existence of a zero or the concept of addition or subtraction.

deleted 11y ago
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