# MATH HISTOTY: A very difficult question aboutthe history of mathematics

I have asked this question of seven mathematicians. From six I have received the same “answer”: “There must be one, but offhand I can’t specify it”. The seventh said “maybe it’s never happened”.

``````I know of a number of mathematical “proofs” that have been accepted for a good while only to be later shown to be false.
However, in every case I know of, it was the reasoning, not the conclusion, that was shown to be incorrect. I do not know of even one case in which the conclusion was not later proved to be correct.
For example, "Iron-clad proofs" of Euclid  contained unrecognized assumptions that made the “proofs” to be not proofs.   However, Hilbert later proved Euclid’s conclusions to be correct.
``````

Do you know of a “proof” that was long accepted and then demonstrated to have an INCORRECT conclusion?

NOTE: I know of a number of near misses: A book by Girolamo Saccheri published in 1773, was not widely accepted. Mersenne gave a list of exponents he believed generated Mersenne primes, but it was an incorrect list,not a proof. Riemann believed that his analysis of the frequency of prime numbers showed
that the integral of the logarithmic function remains larger than the prime number counting function for large n. This turns out not to be the case. But it was not a proof, but a speculation

I would think that the likelihood of proving something not true in mathematics is very low. Recall, to disprove a mathematical statement, all one needs is a single example where the premise is correct but the conclusion false. In the process of trying to complete a proof of a theorem, the proponent of the proposition would most likely come across at least one exception to the theorem if the theorem was false.

This post from your previous thread on this exact topic contains links to multiple examples of what you’re looking for.

Here is one of several…

Many people have “proven” Euclid’s Fifth Postulate, but it’s always turned out that the proofs rely on some added hidden axiom, which is equivalent to the Fifth.

Well there have certainly been theorems that have had to have additional hypotheses added to them but that is usually obvious. When I was an undergraduate (around 60 years ago) I was assigned to read the proof of a theorem of Frobenius (that the only real division rings are R, C, and H) in the German translation of Pontrjagin’s Topological Groups and it omitted the necessary hypothesis that R was central. A couple of my professors came up with a counter-example. But the missing hypothesis was obviously used in the proof. Note that the definition of algebra was a bit watery in those days and it was not clear that centrality was part of it. The chapter in question had not been included in the then extant English translation, but is in a more recent one.

But there was a real example I heard about from a colleague. There was something called, IIRC, Hamburg’s theorem that was accepted and used for a quarter century before a Danish student decided he didn’t trust the proof and eventually found an actual counter-example. What was notable (and unusual) about this example is that the theorem was accepted, cited, and used for a quarter century before the error was found. I imagine most false statements never get used or get used and show something patently false.

In the case of Cauchy’s theorem mentioned above, the problem was really that the distinction between convergence and uniform convergence was not clearly understood. Since Cauchy was struggling to find acceptable definitions of continuity, convergence, etc., it is not surprising that he missed this point.

I am unaware of any other example as striking as Hamburg’s theorem.

A theorem of Jan-Erik Roos in 1961 stated that in an AB4* abelian category, lim[sup]1[/sup] vanishes on Mittag-Leffler sequences. This theorem was used by a number of people, but it was disproved by counterexample in 2002 by Amnon Neeman.

How do I know this? Ok, ok, I cribbed it from this page on wikipedia: List of disproved mathematical ideas. This is the only theorem listed; it places the word theorem in scare quotes.

How many of these proofs were widely considered to be true, at least for a few years, like, for example, the four-color theorem was?

That’s definitely a mistake of Roos’, but the result does at least hold in nicer categories than just general abelian categories (e.g., for abelian sheaves, which is where a lot of this sort of thing takes place).

Do you remember any details about the theorem? I’m curious, but Google turns up absolutely nothing.

I don’t remember any details about the theorem in question, but here is a link to the student, Henrik Pedersen: https://www.math.ku.dk/english/staff/?pure=en/persons/132271. If you are seriously interested, there is an email address on the site. His thesis advisor at McGill was my colleague who told me the story.

Nifty, thanks.

Referring back to the old thread, did you ever decide whether or not 5 dimensions was enough for skew planes not to contain any parallel lines?

I guess I don’t understand what you’re talking about, despite thinking I did up until this point. I thought the idea was the only division rings that include a subring isomorphic to the real numbers are the real numbers, the complex numbers, and the quaternions. Whether this is true, I don’t know, but seems plausible. You then say a “necessary hypothesis” is that “R was central”. I don’t even know what that means, but it sounds like you’re trying to make some hypothesis on how the real numbers are structured, which is kinda not exactly possible unless it simply wasn’t known at the time whether the property “central” applied to the real numbers, and that the theorem was only true unless it was the case that the real numbers were in fact “central”.

Or wait…does “R” mean different things in each place? The first time it means the real numbers, the second time it means some abstract ring. So then the idea is that the theorem proves not that those three fields are the only real division rings, but they are the only “central” (again, whatever that means) division rings, where “central” is some property that was assumed for all division rings but actually was only true for a subset of them (the ones everyone studied to be sure, but not all of them).

If you start with a division algebra over the real numbers R, as in Frobenius’s theorem, then R is contained in the centre of the algebra more or less by definition, isn’t it?

Four dimensions is enough for two planes to not contain any lines parallel to each other, but such a pair of planes would still intersect at a point. Five dimensions is enough for them to not even intersect at a point.

Example: In a five-dimensional space with v, w, x, y, and z axes, the plane v=0, w=0, x=0 does not intersect with the plane v=1, y=0, z=0, and no line in the first plane is parallel to any line in the second plane.

R is central means that every real number commutes with every element of the division ring in question. And I forgot the crucial hypothesis that the division ring be finite dimensional over R. But the conclusion is false if R is not central. Whether centrality is part of the definition of algebra was not as clear 60 years ago as it is today where the usage has hardened. But this was especially unclear in Europe where the words translated as field (Koerper in German, corps in French) to this day do not include commutativity. And who knows about the original in Russian (well, somebody does, but not me)? All this illustrates the fact that definitions are not hard and fast and it is possible to trip over them. And I was an undergraduate student at the time and the book I was reading, in German, had not given the relevant definitions. But as I read the argument, it seemed that centrality of R was needed and not clearly stated. I now know a much simpler proof of the Frobenius theorem in which the role of centrality is even clearer.

I have a bit more information, from Pedersen’s thesis advisor. The theorem was called Hamburger’s theorem. It concerned something called the moment problem and was published around 1943. My colleague thinks it in some American journal, but he doesn’t recall which one.