*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.

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As a physicist, I really enjoyed reading your post

Thank you for sharing the story

zzz

So one may be correct longer than one may remain solvent– er, sane.

Haters got to hate.

Awesome piece. Set theory was a footnote thrown into my linear algebra class…an afterthought.

The concept that there exists an infinite string of numbers between any two numbers in an infinite string is some “heavy shit”.

Fractals come to mind.

Thanks for sharing.

Nice piece. I never really understood the reasoning of infinitesimal increments between semantics. Numbers, for lack of a better term, are symbols used for aggregation to illicit simplicity. Simplicity begets laws unto which discovery is synthesized. It is much easier to work in a world of approximation.

There were two Soviet mathematicians/physicists in the mid 80’s who were working on a problem that involved lubrication flowing across a ball bearing.

The first mathematician, who was a recognized doctorate (highest level, can’t quite remember what the SS term for it was), used theory and reached the conclusion that, as liquid flows around the ball bearing the pressure differential generally follows a normal distribution.

EXCEPT, for one location. The theory revealed to him that about an eighth revolution around the bearing, an anomoly occurs; a massive pressure spike that has almost no width but immense force.

The second, who was applying for his doctorate, simply approximated the normal distribution from the get go, and missed the pressure spike entirely.

I cannot remember these two men’s names – but I can tell you that the first man, who was on the second’s review process, black balled him for several years…

CT – nice esoteric numbers piece, to go along with the criticism lesson. In a former life, in the early 80’s (I was about 9 yrs old), I wrote some code for a DTM application using the Triangular Irregular Network algo — fancy arithmetic given we were using an 8 bit processor = thousands of lines of code. With the state of screen resolution at the time, I became a bit obsessed with the relationship between map scale and screen pix count. Found myself contemplating very small increments between at 1/n! — hurt my head. Now I am old+stupid, — uh, colorful, oh well.

BTW – sold IMN.tsx = Cdn copper (will buy it back under $45); and bought POT, MEOH. Will look to enter CCJ soon.

Excellent post.

great piece. got me to read his entire wiki. turns out his son died right before his second hospitalization, tough life.