Controversies in field of mathematics | Hacker News Reader

Controversies in field of mathematics

mathoverflow.net

52 pointsnextInt6 comments

My favorite example of this sort of thing is in the preface of Sussman and Abelson's "Structure and Interpretation of Classical Mechanics":

https://mitpress.mit.edu/sites/default/files/titles/content/...

"Classical mechanics is deceptively simple. It is surprisingly easy to get the right answer with fallacious reasoning or without real understanding. Traditional mathematical notation contributes to this problem. Symbols have ambiguous meanings that depend on context, and often even change within a given context. For example, a fundamental result of mechanics is the Lagrange equations. ... The Lagrangian L must be interpreted as a function of the position and velocity components qi and i, so that the partial derivatives make sense, but then in order for the time derivative d/dt to make sense solution paths must have been inserted into the partial derivatives of the Lagrangian to make functions of time. The traditional use of ambiguous notation is convenient in simple situations, but in more complicated situations it can be a serious handicap to clear reasoning."

Footnote 2 is also fun to read.

I actually never really thought this was that big of an issue but your link is really interesting and brings up some great points. Hamiltonians I think solve the largest issues but I know from talking to people who learn mechanics that that becomes confusing to because dq/dt they think should be related to momentum simply by p=m*dq/dt but it's a tad more complicated than that because you're talking really about a manifold and fancy geometry talk not commonly know to physicists

sicm is great. another subject I would kill to have get this formalized treatment is statistics - so many people play fast and loose with conditional probabilities and expectations it drives me insane.

The submission title is editorialized. These aren't really foundational controversies. They are controversies about particular proofs or theorems. Even in the case of Fukaya's work, which is dramatized a bit by Quanta Magazine, there were decades of prior work in symplectic geometry that would have been unaffected by the incorrectness of Fukaya's proof.

[Edit: I see that the title has been changed to be more reasonable.]

One controversy that has settled into the foundations of math was over what it means to use infinite sets, and comparing their cardinalities. I was unsettled when I first encountered Cantor's diagonal argument to prove the reals are larger than the rationals, for example. Up until that point, infinite things were still just a kind of "and so on" instead of actual things.

https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_th...

Interesting question. As long as the controversies remain in the field of abstract/theoretical mathematics, I guess it would be hard to resolve them, but I would guess that if a controversial statement (not just a controversial proof) would be applied at some point to practical problems (in science or engineering) then reality would quickly point out who was right...

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