Category Archives: A Matter Of Importance
There are two kinds of people who are ridiculed in this world; imbeciles and visionaries.
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician who lived from around the middle of the 19th century to the turn.
Of Georg Cantor’s mathematical discoveries, he is most famous for developing set theory – an approach to solving proofs related to the span of all mathematical studies and wildly popular in the mathematics circles. His notation was so popular, that for a brief spell, pre-Gödel, some believed Cantor had discovered a means of proving or disproving any statement or proposition one could make.
But while Cantor’s set theory was popular, one of his greatest discoveries was so poorly received that the details of its reception are a matter of notoriety amongst mathematicians.
The subject was on what today is called transfinite numbers; and Cantor’s conclusion was something of the following:
Suppose I take all the positive real numbers 1,2,3…n and lay them in a set unto themselves and denote this R. And suppose that next to that set I take all the numbers between 1 and 2 and put them into a set and denote that S.
Which of these two sets of numbers is bigger?
After much deliberation, Cantor finally came to the conclusion that both had to be the same size. Thus, he developed the concept of
finitely countably infinite sets and infinitely uncountably infinite sets. Between any two numbers there exist an infinite number of other numbers.
A second of Cantor’s controversies was around the following problem. Which is bigger, the set of all real numbers from (…-3,-2,-1,0,1,2,3…), or the set of only the positive real numbers (1,2,3…)?
The widely accepted answer was that the first collection of numbers must be bigger than the second, because the second set is contained within the first.
Cantor proved otherwise.
Cantor argued that for any of these sets, there exists an endless number of other numbers to make up the set. As both sets extends to infinity, it is formally impossible to reason that the first set is bigger than the second.
The first set contains the second set. Yet they are the same size.
This caused an uproar. Poincare and Kronecker (two extremely accomplished mathematicians in their own right) launched an assault on Cantor that crossed from merely objective to the highly personal. Other mathematicians eventually entered the attack also.
Then the Christians got involved. They seemed to feel that Cantor’s work was a direct assault on the omnipotence of God, a charge Cantor would spend his entire life fighting.
Cantor suffered a nervous breakdown in 1884 and was hospitalized. Upon release, he spent the next few years trying to piece together his reputation and make amends with his detractors. In 1899, he was checked into a sanatorium. Then again in 1903. In 1904, a paper from the mathematician Konig attempting to disprove his theorem, read out loud in front of his family, left him so thoroughly demoralized, he is said to have questioned the existence of God over his treatment. This set off a string of stays in mental hospitals that spanned every two to three years.
He retired from mathematics in 1913, in a state of abysmal poverty. He suffered a final mental breakdown and died in 1918. The effort of contemplating the infinite cost him his career, his reputation, and his very mind.
Shortly thereafter, his detractors finally admitted they could find no means of disproving his central theorem of transfinite sets. Today, it is so readily accepted that it is taught in undergraduate courses and occasionally introduced as early as grade school.