In contemporary mathematics, ideas are supposedly not accepted absent
rigorous proof. Historically, some statements that were generally accepted were then shown to be false. • The idea of the
Pythagoreans that all numbers can be expressed as a ratio of two
whole numbers. This was disproved by one of
Pythagoras' own disciples,
Hippasus, who showed that the square root of two is what we today call an
irrational number. One story claims that he was thrown off the ship in which he and some other Pythagoreans were sailing because his discovery was too heretical. •
Euclid's
parallel postulate stated that if two lines cross a third in a
plane in such a way that the sum of the "interior angles" is not 180° then the two lines meet. Furthermore, he implicitly assumed that two separate intersecting lines meet at only one point. These assumptions were believed to be true for more than 2000 years, but in light of
General Relativity at least the second can no longer be considered true. In fact the very notion of a straight line in four-dimensional curved
space-time has to be redefined, which one can do as a
geodesic. (But the notion of a plane does not carry over.) It is now recognized that
Euclidean geometry can be studied as a mathematical abstraction, but that the
universe is
non-Euclidean. •
Fermat conjectured that all numbers of the form 2^{2^m}+1 (known as
Fermat numbers) were prime. However, this conjecture was disproved by
Euler, who found that 2^{2^5}+1=4,294,967,297 = 641 \times 6,700,417. • The idea that
transcendental numbers were unusual. Disproved by
Georg Cantor who
showed that there are so many transcendental numbers that it is impossible to make a one-to-one mapping between them and the
algebraic numbers. In other words, the
cardinality of the set of transcendentals (denoted \beth_1) is greater than that of the set of algebraic numbers (\aleph_0). •
Bernhard Riemann, at the end of his famous 1859 paper "
On the Number of Primes Less Than a Given Magnitude", stated (based on his results) that the
logarithmic integral gives a somewhat too high estimate of the
prime-counting function. The evidence also seemed to indicate this. However, in 1914
J. E. Littlewood proved that this was not always the case, and in fact it is now known that the first
x for which \pi(x) > \mathrm{li}(x) occurs somewhere before 10317. See
Skewes' number for more detail. • Naïvely it might be expected that a continuous function must have a derivative or else that the set of points where it is not differentiable should be "small" in some sense. This was disproved in 1872 by
Karl Weierstrass, and in fact examples had been found earlier of functions that were nowhere differentiable (see
Weierstrass function). According to Weierstrass in his paper, earlier mathematicians including
Gauss had often assumed that such functions did not exist. • It was conjectured in 1919 by
George Pólya, based on the evidence, that most numbers less than any particular limit have an odd number of
prime factors. However, this
Pólya conjecture was disproved in 1958. It turns out that for some values of the limit (such as values a bit more than 906 million),{{cite journal |last=Tanaka |first=M. |date=1980 |title=A Numerical Investigation on Cumulative Sum of the Liouville Function |journal=
Tokyo Journal of Mathematics |doi=10.3836/tjm/1270216093 |mr=0584557 |volume=3 |issue=1 |pages=187–189 ==See also==