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I broadly agree with your sentiment and am saddened (outraged?) to see the financially motivated cheapening of (destruction of?) what mathematics truly is. I ag

by pcfwik 20d ago

I broadly agree with your sentiment and am saddened (outraged?) to see the financially motivated cheapening of (destruction of?) what mathematics truly is. I agree that formal logic is merely a model of one aspect of what mathematicians do, in the same sense that a computer simulation of a roller coaster can never bring the same value to us as the real thing. However, I would be remiss if I didn't question your historical claim, which seems to me a bit too strong: > Mathematics predates the idea of formal proof by millenia [...] Formal proof only emerged early in the 20th century [...] You seem to associate the start of "Mathematics" with Euclid, but (as far as I know) he worked at approximately the same time as Aristotle. Aristotle's syllogisms are perhaps the most famous formal logic system: their correctness relates only to their form, not their content. All deductions of the form "All X are Y, All Z are X, hence All Z are Y" are valid (assuming the premises are), regardless of the meanings of X, Y, and Z. (Outside of Greece, my understanding is that a few hundred years earlier Panini had also developed a system of formal manipulations, but for representing grammars.) What, to my understanding, "emerged" only the 19th and 20th century was 'merely' a formal logic both expressive and sound enough to properly express modern mathematics (the Beggriffsschrift in the 19th century and FOL+ZFC in the 20th). Between Euclid and the 19th century the development of calculus was probably the biggest advance in mathematics, and my understanding is that Leibniz himself spent significant time working on formal logic. Perhaps I have the wrong definition in mind of 'formal logic' or 'mathematics,' but I do think the history of formal logic is much more closely tied to the history of mathematics than your post makes it seem on first glance. Though I certainly agree that "mainstream mathematics" has never felt it necessary (or necessarily that useful) to express proofs in a formal logic carefully enough that they could be checked by computers; this was a fringe focus of a minority group of mathematicians and computer scientists that was co-opted as a marketing stunt into 'what mathematics is' for major corporations trying to justify their money burn.

QuesnayJr 18d ago
The idea that logic is a branch of mathematics is itself a modern notion. Aristotle's logic was part of the trivium (grammar, rhetoric, and logic) in classical notions of education, while mathematics made up several parts of the quadrivium (arithmetic, geometry, music, and astronomy). Of course in retrospect we can see that the syllogistic part of Aristotle's logic can be formalized (as can grammar), but it was viewed as part of language or philosophy. I get the impression that a lot of more traditional philosophers of logic hated the formal turn. Leibniz anticipated the turn towards formalism, but he didn't publish any of it in his life and it wasn't rediscovered until the 20th century.
- pcfwik 15d ago
  Agreed that logic isn't a branch of mathematics (even today, my experience has been that logic classes tend to be offered by philosophy departments, not math). My only quibble was the implication that logic is somehow much newer than mathematics (/Euclid). It's not like people suddenly woke up one day in the 20th century and thought for the first time: "huh, I wonder what separates correct arguments from incorrect ones?" But point taken that Aristotle was not as formalist in his time. If you have some sources to read more about this: > I get the impression that a lot of more traditional philosophers of logic hated the formal turn. I would be very interested! (Don't doubt it at all, just sounds like an interesting period I don't know as much about as I'd like.)

by pcfwik 20d ago

QuesnayJr 18d ago

The idea that logic is a branch of mathematics is itself a modern notion. Aristotle's logic was part of the trivium (grammar, rhetoric, and logic) in classical notions of education, while mathematics made up several parts of the quadrivium (arithmetic, geometry, music, and astronomy). Of course in retrospect we can see that the syllogistic part of Aristotle's logic can be formalized (as can grammar), but it was viewed as part of language or philosophy. I get the impression that a lot of more traditional philosophers of logic hated the formal turn. Leibniz anticipated the turn towards formalism, but he didn't publish any of it in his life and it wasn't rediscovered until the 20th century.

pcfwik 15d ago
Agreed that logic isn't a branch of mathematics (even today, my experience has been that logic classes tend to be offered by philosophy departments, not math). My only quibble was the implication that logic is somehow much newer than mathematics (/Euclid). It's not like people suddenly woke up one day in the 20th century and thought for the first time: "huh, I wonder what separates correct arguments from incorrect ones?" But point taken that Aristotle was not as formalist in his time. If you have some sources to read more about this: > I get the impression that a lot of more traditional philosophers of logic hated the formal turn. I would be very interested! (Don't doubt it at all, just sounds like an interesting period I don't know as much about as I'd like.)