If you have been following my recent blogs, you see that I have been referring to a few dead geometers. I am finished for the time being discussing Axioms and definitions. There were of course many other things I could have talked about but, for the time being I am satisfied with what I have written.

However, before I continue reading Euclid, for my own benefit, I would like to make a few quick notes regarding the authors of the books that I have referring to. I have already talked a little about Euclid, who is believed to have died approximately 1500 years before Playfair was chair of mathematics at the University of Edinburgh. So, now only wish to make a few short comments about Hilbert and Playfair. I am not sure I want to read Hilbert or Playfair's book at the present- but will probably continue to refer to them occasionally.

John Playfair: Playfair was an educator who held the chair of mathematics at the University of Edinburgh from 1785 to 1805. He put together a popular rendition of Euclid's first 6 books, about 1500 years after Euclid. His book "Elements of Geometry" included a commentary and a discussion of a few additional topics in geometry. In his appendix, he proposed replacing Euclid's 5th axiom regarding parallel lines with the axiom you find in most modern textbooks. You can find both of those axioms in my last post that included a comparison chart. Playfair did not invent the axiom, in fact it said to have been first proposed by the Greek mathematician Proclus perhaps 200 years after Euclid. Playfair himself attributed the suggestion to one of his contemporaries.

David Hilbert: Nearly a hundred years after Playfair was in his prime, David Hilbert was at his. Hilbert was a very influential member of the mathematical community in his day, helping to organize mathematical conventions and the like. Hilbert believed in axiomatizing all of mathematics. He wanted to unite all of mathematics under a universal and complete system of axioms. He favored Playfair's, "more rigorous" suggestion for the parallel axiom. Of course, the axioms have been shown to be logically equivalent. To this end he helped inspire logicians Alfred North Whitehead and Bertrand Russell to collaborate on Principia Mathematica. Principia was an attempt to create the most logically complete treatise of mathematics every written. If famously took 300 pages to lay sufficiently rigorous foundations to establish 1 + 1 = 2. Hilbert's dream of a logically complete and closed system of mathematics was met with disappointment when the mathematician Kurt Godel proved the famous incompleteness theorem. Hilbert's idea of completeness was to be based on building a foundation on basic arithmetic. Reminiscent of Euclid's proof that there are infinitely many prime numbers, Godel showed that there is no such thing as a finite number of axioms that can cover the needs of every possible mathematical proof. Godel did have a chance to talk to Hilbert about his proof, after Hilbert had read it. Hilbert accepted it, but found it a bitter pill to swallow in light of his life long dream.

While I normally don't list sources for my blogs. I am immensely grateful for the enrichment me little books have provided me. I decided to run a quick internet search and found a cool blog as well.

Sources:
The History of Mathematics by David M. Burton
Men of Mathematics by E.T. Bell
Incompleteness: The Proof and Paradox of Kurt Godel by Rebecca Stein
http://blog.computationalcomplexity.org/2011/07/why-did-112-take-russell-and-whitehead.html