... Anaximander of Miletus, who probably lived from 610 to 540 B.C.E. ... described the ultimate material principle as apeiron, "the Infinite" or indeterminate; "something without bound, form, or quality."
This is a quote from Naming Infinity: A Crisis in Mathematics (Harvard) by Loren Graham and Jean-Michel Kantor. I just started reading it last night and expect I may finish it by tomorrow. Briefly put, it tells the story of how a group of Russian intellectuals who were adherents to a mystical form of Russian Orthodoxy (having to do with the Philokalia and the Jesus prayer) were able, by virtue of their mystical take on things, to make one of the great mathematical breakthroughs (having to do with infinity).
The description of the Infinite in the passage I have quoted bears a striking resemblance to the Tao. Anaximander and Lao Tzu were roughly contemporary. What I am interested in is the light the book may shed on how the way we think affects the outcome of our thought. Specifically, if the Russians' mystical approach was in fact more effective in arriving at a breakthrough necessary to advance mathematics, then maybe that approach is sounder than a purely rational one.
The assumption that math could proceed without intuition was destroyed by Kurt Godel, who said that 1+1=2 couldn't be "proven" as Russell and Whitehead had tried at the beginning of the 20th century in their Principa Mathematica. Rather it was only shown by intuition, and any attempt to remove intuition from math was doomed:
ReplyDeleteBy removing all intuitions so that points and lines are no different from beer mugs and bar tables, the [Russel-type] reasoning went, mathematics would be forever free from the dangers of unrecognized and possibly misleading assumptions. It would, in principle, be possible to design mechanical devices--which, of course, have no intuitions--to follow the rules and deduce all the truths for you. (This, remember, was before computers were invented.)
This approach to mathematics, to formulate all the axioms you need to deduce all the truths in a particular branch of mathematics mechanistically, became known as the Hilbert Program. For many, the search for axioms became something of a Holy Grail, although Hilbert, who had great respect for the role played by human intuition in the practice of mathematics, was not one of them, and never himself proposed that the program to which others attached his name should be carried out.
One of the most sustained efforts to carry out the Hilbert Program was made by the English philosophers Bertrand Russell and Alfred North Whitehead. Their mammoth, three-volume work, Principia Mathematica, published from 1910 to 1913, was an attempt to develop basic arithmetic and logical reasoning itself from axioms.
It was the axiom system in Principia Mathematica that Gödel took, by way of an exemplar, to demonstrate beyond any doubt that the goal of the Hilbert Program was unattainable.
http://www.sciencemag.org/cgi/content/full/298/5600/1899
What mathematical breakthrough was it that these russians managed?
ReplyDelete(Sanskrit writing on infinity is a bit more interesting than Greek writing in my opinion...)
Hi David,
ReplyDeleteI haven't got that far, but I presume it has something to do with naming infinity. I will update when I finish the book.
Hi Joe,
Thanks for a thoughtful and informative comment.
I rather think that it is the wedding of mysticism (intuition) and rational logic by which mathematics proceeds. Examples abound of insights achieved via intuition, even revelation (although defining revelation is an issue), leading to mathematical advances.
ReplyDeleteI don't think mysticism can replace rational logic, it can only guide it. I say that as a mystic AND as someone science-trained.