# Alex Peek blog

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Saturday, August 25, 2018

## Georg Cantor and infinity

Photo source: Wikimedia Commons

Georg Cantor (1845-1918) was a German mathematician best known for pioneering work on set theory and transfinite numbers. Mathematics historian Eli Maor said,

"In 1874 the German mathematician Georg Cantor made the startling discovery that there are more irrational numbers than rational ones, and more transcendental numbers than algebraic ones." (The Story of a Number, 1994)

The rest of this post is some quotes from Cantor.

### Infinity

"The actual infinite arises in 3 contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the absolute infinity or simply absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type." (Quoted in Mind Tools by Rudy Rucker)

"Thus I believe that there is no part of matter which is not - I do not say divisible - but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures." (Quoted in Out of the Mouths of Mathematicians by Rosemary Schmalz)

"This view [of the infinite] which I to be the sole correct one, is held by only a few. While possibly I am the very first in history to take the position so explicitly, with all its logical consequences, I know for sure that I shall not be the last!" (Quoted in Journey Through Genius by William Dunham)

### Mathematical freedom

"Mathematics is in its development entirely free and is only bound in the self evident respect that its concepts must both be consistent with eachother and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already established." (Quoted in From Kant to Hilbert by William Bragg Ewald)

"The essence of mathematics lies entirely in its freedom." (Quoted in From Kant to Hilbert by William Bragg Ewald)

### Advancement of knowledge

"Great innovation only happens when people aren't afraid to do things differently." (AZquotes.com)

"A false conclusion once arrived at and widely accepted is not easily dislodged and the less it is understood the more tenaciously it is held." (AZquotes.com)