Georg Ferdinand Ludwig Philipp Cantor (December 3, 1845, St. Petersburg, Russia – January 6, 1918, Halle, Germany) was a German mathematician who is best known as the creator of set theory. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities." He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
Cantor's work encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré, and later from Hermann Weyl and L.E.J. Brouwer. Ludwig Wittgenstein raised philosophical objections. Nowadays, the vast majority of mathematicians who are neither constructivists nor finitists accept Cantor's work on transfinite sets and arithmetic, recognizing it as a major paradigm shift. (Full article...)
An animated geometric proof
of the Pythagorean theorem
, which states that among the three sides of a right triangle
, the square of the hypotenuse
is equal to the sum of the squares of the other two sides, written as a2 + b2 = c2.
A large square is formed with area c2
, from four identical right triangles with sides a
, fitted around a small central square (of side length b − a
). Then two rectangles are formed with sides a
by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a2
, which together must have the same area as the initial large square. This is a somewhat subtle example of a proof without words