
Controversies in field of mathematics - nextInt
https://mathoverflow.net/q/282742
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lisper
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/...](https://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-
Z-H-5.html)

"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.

~~~
pducks32
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

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cwzwarich
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.]

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imh
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...](https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory)

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placebo
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...

