Infinity

I'd like to highlight two fascinating minds that have graced the pages of history: Rene Descartes and Georg Cantor

Rene Descartes was a French physicist in the 17th century (1600s, as I always have trouble matching them). He developed a technique for systematically breaking down and interpreting problems in physics. His most noted contributions are to geometry (a normal x-y plane is the epitome of Cartesian coordinates) and optics. While many of his discoveries were moot within a century or so, I have described before the change in approach his physics embodied.

One of the problems he encountered was that his new system didn't rely heavily on God and, used carefully, implied a heliocentric solar system. Having seen the impact of Galileo's Dialogue Concerning the Two Chief World Systems on that astronomer's research (forced self-repudiation and house arrest), he spent the bulk of his energy couching his physics in a philosophy that he thought the Catholic Church would like. Among these arguments was the following, summarized by Tom Sorrell in "Descartes: A Very Short Introduction":

"Descartes thinks an infinite substance is more real than a finite substance[...]. In the case of an idea of God, the idea represents an infinite substance, something whose category or degree of reality cannot be exceeded. According to Descartes' principle, therefore, the cause of the idea has to belong to the same category as the thing it is an idea of. More explicitly, the idea of God has to be caused by an infinite substance. But there is only one infinite subtance, namely God." (Kindle Edition, location 688)

I will not deal with his argument here; you can imagine the difficulties ahead. Instead, I would like to highlight the idea of infinity here. The finite and the infinite are different categories, with that which is infinite being the greatest degree of reality (and paradoxically, filled with exactly one thing).

Enter Georg Cantor in the 1800s.

Beginning as an engineering student at university in Germany. He soon took up mathematics instead and gained a name as an analytical theorist. In 1873, he published one of his first breakthroughs on the topic of infinity.

To explain, I'd like to briefly describe the concepts involved.

Countable Infinity
Take a box of an infinite number of blocks. If you can pick them up, one by one, and count them, all the way to infinity, this is a countable infinity. The most familiar countable infinity is the set of counting numbers, one to infinity.

Uncountable Infinity
Given a similar box of, say, all (real) numbers, start picking them up again. Count them. No matter how careful you are at picking numbers as close together as possible, I can find one in between them that you missed. This is an uncountable infinity, as they cannot be counted using the counting numbers, e.g. you can't pair each real number up with a counting number, as there are too many.

To Cantor, this ballooned rapidly. If you could have one infinity be bigger than another infinity, you could find a third bigger than they all. After those three, a fourth, larger infinity can be found.

In his work, he describes God as the biggest infinity of them all.
(This is an impossibility according to later advances in transfinite theory, but this fits with an entity who is everywhere but nowhere...)

His peers had a hard time with this. Up to his seminal article in 1874, "infinity" was merely some incomprehensibly gargantuan behemoth that had one size: infinity. It was like in elementary school, when you said "You're stupid times infinity!!!" and the argument was over because Nothing(TM) is bigger than infinity.

I do apologize I haven't been able to finish this arc of thought; I hope you liked this much!