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The ‘Transfinite’ and the Limits of Mind


Madness is not a consequence of uncertainty but of certainty.  
--F. Nietzsche

Georg Cantor (1845-1918), German mathematician and inventor of ‘set theory’, pushed the boundaries of infinity well beyond its implications for mathematics—into metaphysics; theology; and perhaps to the limits of his sanity as well.

His revolutionary theories also produced criticism, both procedural and personal. Cantor was moving beyond the bounds of what had been defined as the proper realm of mathematics. Cantor proposed there is an “infinity of infinities” and defined transfinite numbers as the set of numbers representing different levels of infinity.

“… [Cantor’s work] between 1874 and 1884 marks the real origin of set theory, which has since become a fundamental part of modern mathematics. Although the concept of a ‘set’ had been used implicitly since the beginnings of mathematics, dating back to Aristotle, this was limited to everyday finite sets. In contradistinction, the “infinite” was kept quite separate, and was largely considered a topic for philosophical, rather than mathematical discussion. Cantor, however, showed that, just as there were different finite sets, there could be infinite sets of different sizes, some of which are countable and some of which are uncountable.”*

Later, Cantor expanded his theories to embrace what became known as the continuum hypothesis, a theory regarding the potential sizes of infinite sets. Later, in the 20th century, Kurt Gödel and Paul Cohen built upon Cantor’s theories. Yet it was in his frustration in satisfactorily proving many of his theories that led to the erosion of Cantor’s emotional state.

His expositions moved beyond the established realms of mathematics into the theological, claiming that transfinite sets reinforce the view of God as the only true ‘Absolute Infinity’—a claim he believed had been communicated to him by God Himself.

Cantor spent much of his later years in and out of mental institutions. The precise cause of his institutionalization remains unclear. Possible causes put forward were extraordinary stress, brought on, in part at least, by his ‘unprovable’ theory; the rejection by colleagues; or, as some have suggested, a manifestation of what is now diagnosed as bipolar disorder.

Whatever the cause, the ‘infinite’ had a hold on Cantor. To maintain his bearings, he’d embark regularly on scholarly diversions. Principal among these was losing himself in the works of Shakespeare. His text-analysis of Shakespeare’s works produced a variety of commentaries, including two pamphlets purporting to prove that Shakespeare’s true identity was Francis Bacon.

In 1904, The Royal Society, sponsors of the monumental Catalogue of Scientific Papers, awarded Cantor the Sylvester Medal, the highest honor for work in mathematics. Sadly, as Cantor’s work on ‘transfinite’ numbers emerged as settled doctrine in the field, his mental health was in steady decline.

He spent his final year in a sanitorium in Halle and died in 1918, as The Great War came to an end.

To read more about the monumental Catalogue of Scientific Papers, visit the Resource Description


*The Story of Mathematics,


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