Saturday, March 5, 2011
- Many seem to agree that the epsilon-delta definitions are annoying, This can be soon both at Karl's calculus page or on mathcs.org
Does that delta-epsilon method that we have been through a bunch of times now seem like a pain in the butt? Mathematicians are, by their nature, a lazy lot. If they can get away with doing some pain-in-the-butt operation just once and thereby come up with some easy and useful rule, then that's what they'll do. And once they know the rule to be true, they'll use it every time instead of the pain-in-the-butt method. Not only does the rule make life easier, but without such rules, mathematics would be so thick with undergrowth as to make it virtually impossible to understand."
"This, like many epsilon-delta definitions and arguments, is not easy to understand."
- The fact that infinitesimal calculus was heavily supported by Newton and Leibniz, tells me again, that the classical definition of limits simply does feel annoying and people tried to find 'better' ways.
- At the same time, based on Bishop, they are common sense, so there must be a way to understand why they are the best way to describe what they do describe.
- They are a relationship between domain and range
- The relationship is TOTALLY dependent on the function, the algorithm that maps domain to range
Those are 2 uncomfortable categories, they do not allow for laziness, impatience, or lack of intuition and very solid understanding. It is a game...
Not only does the rule make life easier, but without such rules, mathematics would be so thick with undergrowth as to make it virtually impossible to understand.""