Sunday, September 21, 2014
Sunday View (the nature of proof)
"What mathematics accomplishes with its reasoning may be more evident from an analogy. Suppose a farmer takes over a tract of land in a wilderness with a view to farming it. He clears a piece of ground but notices wild beasts lurking in the wooded area surrounding the clearing who may attack him at any time. He decides therefore to clear that area. He does so but the beasts move to another area. He therefore clears this one. And the beasts move to still another spot just outside the new clearing. The process goes on indefinitely. The farmer clears away more and more land but the beasts remain on the fringe. What has the farmer gained? As the cleared area gets larger the beasts are compelled to move farther back and the farmer becomes more and more secure at least as long as he works in the interior of his cleared area. The beasts are always there and one day they may surprise and destroy him but the farmer's relative security increases as he clears more land. So, too, the security with which we use the central body of mathematics increases as logic is applied to clear up one or another of the foundational problems. Proof, in other words, gives us relative assurance."
-- Morris Kline in "Mathematics: The Loss of Certainty"
[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]