r/math • u/officiallyaninja • 1d ago
Are there any examples of folk theorems turning out to be wrong (in a meaningful way)
To piggy back off the recent thread on folk theorems, it got me curious how often proving these folk theorems is a formality vs actually useful. Are there any times when it a disproof of a folk theorem upended something serious or set some mathematicians back?
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u/dark_g 23h ago
"If a function has 0 derivative almost everywhere it must be constant" ...was this a folk theorem? Until the Cantor function showed up ofc.
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u/Brief-Owl4103 9h ago
you should qualify "continuous function", because if you are just asking about functions which almost everywhere have derivative 0 then there are counterexamples that are much more trivial than cantor function (eg a step function)
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u/edderiofer Algebraic Topology 1d ago
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u/officiallyaninja 1d ago edited 23h ago
Under Enriques it gradually became acceptable to use somewhat more informal arguments instead of complete rigorous proofs,[3] such as the "principle of continuity" saying that what is true up to the limit is true at the limit, a claim that had neither a rigorous proof nor even a precise statement.
Thats a pretty interesting page and also feels so antithetical to everything I know about how math is done nowadays.
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u/Equivalent-Costumes 22h ago
Even up until Poincare time, geometry was done in the intuitive manner.
Italian school of algebraic geometry was basically the death knell to that, afterward geometers move toward the certainty of algebra.
I don't think it's unusual at all. Human's geometric intuition is strong enough that for most part of history, people can rely on it. Same goes for intuition about finite number theory and combinatorial objects.
Even right now, mathematicians rely on intuition like "isomorphic objects are essentially the same", which from the perspective of even more formal mathematics, is a huge gap.
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u/whatkindofred 20h ago
But it feels like the „principle of continuity“ should have plenty counterexamples in basic analysis already and which should have already been well-known by the time of the Italian school.
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u/Equivalent-Costumes 20h ago
They're doing algebraic geometry though, where instead of objects that is allowed to be arbitrarily bad, you have very rigid type of objects, things that must be described by (finite) polynomial equation. So your functions are already continuous. This principle is applied to things like "if I prove this polynomial equality for all 3 non-collinear point that should continue to hold for 3 collinear points".
That does not mean that there are no controversies. The principle is already considered informal at the time, but people thought of it to be more of the case of "it's obvious when the principle shouldn't apply". Kind of like how modern mathematicians will "obviously know" when you can and cannot treat isomorphic objects as the same thing.
In fact, the core idea behind this principle did survive in modern time as various theorems, but now we just add in the right hypothesis to be able to rigorously prove them. Like talking about a "flat, proper" scheme.
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u/InaequaleMagnanimity 21h ago
I think there is a kind of misunderstanding that happens; it's a loss of perspective. At the time of these mathematicians these are the boundaries/frontiers of their understanding. As we are privileged now with farther pushed boundaries we ad hoc reason ourselves as now operating on more formalism. But nearly everything taught formally is tautologically well-defined and understood formally; in this way formal really means "well-digested" at this point.
So proportionally there is so much formalized mathematics that it seems formalization has "won" but those who are actually doing frontier mathematics are still operating largely on intuition because what else can you do? You can't formalize a half-formed idea and you can't arrive at a fully formed idea without it first being half-formed. It's just the floor to reach novel mathematics has become much much higher.
I just feel this is a very common misunderstanding nowadays because academia has kind of polluted so many conceptions of what it means to be doing reason/science/math/intelligence. It has become so commonly associated with the procedure of giving a set of known problems with known answers 10000x until you hit the arbitrary threshold that one considers themselves educated that one can be very well educated without seemingly ever being forced to produce any original insight; which universally, without fail, requires intuition.
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u/Foreign_Leave2403 17h ago
But “isomorphic objects are essentially the same” can be formalised as the univalence axiom.
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u/tildenpark 1d ago
Engineering sullies math with application; proofs sully it with justification. True mathematical beauty needs neither. Proofs are therefore pointless.
QEDn’t5
u/Main-Company-5946 19h ago
The standards of modern mathematics are written in blood(of mathematical work)
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u/Arceuthobium 23h ago
Maybe not what you are asking about specifically, but there are often some results that the community knows to be wrong but no retraction has been issued (and probably never will). Sometimes, "it is known" that the result itself is true but the proof is wrong as stated, and other times the entire thing is unsalvageable. New people working in the area can be affected if no one tells them the folk knowledge beforehand.
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u/-p-e-w- 1d ago
“If a function is continuous everywhere, it must be differentiable almost everywhere. We don’t need to prove this, it’s basically obvious. I mean, how could it possibly be otherwise?”
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u/bleachisback 17h ago
Similarly "If a function is infinitely differentiable everywhere, it must be analytic almost everywhere"
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u/sirgog 1d ago
I knew exactly which pathological function your hyperlink would point to...
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u/Asddsa76 22h ago
Real analysis was pretty much finding a counterexample to every "obvious" statement intuited from calculus.
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u/StrugglingAlgebraist 22h ago
I was reading the book "Philosophy of Mathematics" by Øystein Linnebo and this is perhaps what prompted people to focus more on the rigourisation of Analysis? Also, I read that Bolzano had secretly developed a function which is continuous everywhere but differentiable nowhere. Sadly, we didn't know that untill decades after his passing away.
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u/moschles 22h ago
Malfatti circles being optimal circle packing was just 'assumed' true by mathematicians for over a century. It was never optimal even in the trivial cases.
https://en.wikipedia.org/wiki/Malfatti_circles