Notable examples of “impossible” results ruled out by earlier barrier or no-go theorems or widespread beliefs

Question

there is a certain style of type of proof in mathematics something like a "barrier theorem" but which also relates to widespread mathematical beliefs/ "conventional wisdom".

• an example would be the proof that trisecting an angle is impossible.
• another was the historical belief (but not proof) that the parallel postulate was derivable from other geometric postulates, ie here the "barrier" is the previously widely-held idea that no other consistent geometry is possible without parallel lines.
• another case that comes to mind is Von Neumann's proof of the impossibility of local hidden variables in quantum mechanics, later overturned in its particulars but extended further by Bell.
• yet another old example is the discovery that $\sqrt{2}$ is irrational by Hippasus (supposedly leading to his sentence of drowning from a boat by Pythagoreans according to one legend).
• there was some shock at the time at the proof that the quintic polynomial is not solvable by algebra
• the discovery of the undecidability of Hilberts 10th problem after decades of investigation shattered Hilberts own program at the turn of the 20^th century of maximally extending provability.

etc! these barrier proofs or beliefs are sometimes very complex or deep seated in mathematical thinking. however, they make assumptions or impose conditions that sound reasonable but later are understood to be quite subtle and sometimes new discoveries are made that somewhat defy expectation wrt the "known barriers". barrier theorems are also important in TCS where several have been built wrt potential P vs NP proofs. (e.g. RJ Lipton summarizes some here, see also this cstheory question by Kintali.)

What are other key cases in mathematics of barrier theorems or beliefs that were later overturned, sometimes dramatically?

Are there references that study this phenomenon of reversal broadly/ in general?

Answer 1

The discoveries of "pathological" functions (e.g. functions that are continuous but nowhere differentiable) overturned conventional wisdom in the 19'th century. Previously, functions were generally thought to be well-behaved except at "exceptional" points. This was related to changes in the concept of "function". See e.g. The function concept.

Answer 2

Gerhard Gentzen's proof (1936) of the consistency of Peano arithmetic by "finitistic means" seems to contradict the commonly accepted conclusions from Kurt Gödel's second incompleteness theorem (1931). Even so Gödel himself warned in his paper that one should not draw these "commonly accepted conclusions", some people even today still doubt Gentzen's proof, claiming that he didn't show (beyond doubt by "finitistic means") that $\epsilon_0$ is well ordered. The "finitistic means" used by Gentzen to proof that $\epsilon_0$ is well ordered cannot be formalized in Peano arithmetic, but the proof is correct and uses only "finitistic means" nevertheless.

Answer 3

Grandi's series has Cesàro sum $\frac{1}{2}$. $$\sum_{n=1}^{\infty} (-1)^{n+1} = 1 - 1 + 1 - 1 + 1-\ldots = \color{#05f}{\frac{1}{2}} $$

This has been brought up by Grandi in $1703$, and was a term of great discussion for over a $150$ years.

It turns out that if you take the sequence $(s_n)$ of partial sums you get $\{1,0,1,,\ldots \}$ and then we have

$$1 ,\frac{1 + 0}{2} = \color{#05f}{\frac{1}{2}}, \frac{1 + 0 + 1}{3} = \color{#05f}{\frac{2}{3}}, \ldots, {\frac{1}{2}}, {\frac{1}{2} + \frac{1}{2n}} $$

which converges to $\frac{1}{2}$.

Grandi's series can be used to show that $$\sum_{n =1}^{\infty} n = 1 + 2 + 3 + 4 + 5 +\ldots= \color{#f05}{-\frac{1}{12}}$$

which at first makes absolutely no sense, since $\sum_{n=1}^{\infty} n $ is a divergent series, altough using the context of Riemann zeta function for $\color{red}{s = -1}$ this sum takes a different cloak and has this jewel number value $\color{#f05}{-\frac{1}{12}}$. This was first discussed by Euler who had proved it differently and then confirmed by Riemann several years later after he had developed a whole new theory.

This last sum appears in many different branches of mathematics and quantum physics.

Answer 4

The belief that there aren't any negative numbers. As late as the 16th century, mathematicians wouldn't write $x^3=5x-4$, preferring to write $x^3+4=5x$.

Answer 5

The belief that negative numbers don't have square roots. It took a long time for mathematics to come to terms with complex numbers --- but it was worth the effort!

Answer 6

Gauss's construction of the Heptadecagon, a regular 17-sided figure using only compass and straightedge, at age 19, is one of the striking discoveries in math at the time. from wikipedia

The regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge), as was shown by Carl Friedrich Gauss in 1796 at the age of 19.[1] This proof represented the first progress in regular polygon construction in over 2000 years.[1]

since no new regular polygon had been created in 2 millenia, no one thought it possible/ conceivable at the time. also re wikipedia entry on Gauss biography

This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle.[7])

the 17 sided case is actually a single initial case of infinite cases using a general law that allows creating even more complex polygonal figures based on Fermat primes.

... his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2) can be constructed by compass and straightedge.