Are there cases where a flawed proof seems correct?

Question

A mathematical proof is known to be wrong when either of the following is found:

1. a flaw in the logic (including perhaps unwarranted assumptions); or
2. a counterexample.

Wikipedia has an extensive list of incomplete proofs. Have there been any notable cases where a counterexample to a published proof has been found, yet the proof resisted attempts to find the logical flaw(s) that led to the erroneous result? By that, I don't mean that it takes a long time to fix the proof or adjust the statement of the result, but that nobody can identify where the proof went wrong.

Answer 1

A number of results in analysis were proven then counterexamples were later found. For example Ampere proved in 1806 that any continuous function from $\mathbb{R}$ to itself is differentiable everywhere except at some isolated points. However Weierstrass produced a counter-example in 1874 by constructing a continuous nowhere differentiable function.

Cauchy proved that an infinite convergent series of continuous functions was continuous however Abel produced a counter-example 5 years later using Fourier series, although definitions were still being created at the time so there is some debate as to if the result was actually wrong given the nature of the infinitesimal calculus Cauchy used.

I think the example I like the most was given by Euler who manipulated infinite series with complete abandon and came up with several absurd results. My favorite is the following $$s= 1 + 2 + 4 + 8 + \cdots = 1 + 2(1 + 2 + 4 + \cdots) = 1+2s$$ and so we have that $s=1+2s$ which gives us $s= -1$. So the infinite sum of the powers of $2$ is a negative number. Today we would simply say $s$ is undefined or converges to positive infinity by considering the partial sums.

I believe analysis was particularly ripe for this sort of error because when Descartes created the analytic plane because it bridged the gap between geometry and equations allowing for the rapid development of many highly practical results being available for well-behaved functions. This allowed a great deal of intuition to build up that would later prove problematic. For example there were many philosophical challenges to the use of infinitesimals and most famously George Berkeley said in The Analyst (1732), about fluxions which was the notion of an infinitesimal change as one might use in construction of the derivative or integral

And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

This would motivate a great deal of work towards putting analysis on a firm foundation over the following centuries, culminating with Cantor's work that has now become the paradise Hilbert predicted.

Even today peer review catches errors, such as with Andrew Wiles' proof of Fermat's Last Theorem. So even today it happens but the velocity of criticism is much faster due to our interconnectedness and the abundance of mathematicians.

Answer 2

It does seem to happen on a regular basis that a physics-y argument is given that purports to prove the Riemann Hypothesis... but, also, at the same time would seem to prove some other things which we know to be false.

This sort of argument does not at all directly show the flaw(s) in the original argument, but, in a way, only "proves existence" of a flaw.

(I think in most of these cases eventually someone has found the non-sequitur, etc., in the argument...)