When is one thing equal to some other thing? (pdf)

posted by   prashant (270)   26 months ago (www.math.harvard.edu)

One can’t do mathematics for more than ten minutes without grappling, in some way or other, with the slippery notion of equality. Slippery, because the way in which objects are presented to us hardly ever, perhaps never, immediately tells us–without further commentary–when two of them are to be considered equal. We even see this, for example, if we try to define real numbers as decimals, and then have to mention aliases like 20 = 19.999 . . . , a fact not unknown to the merchants who price their items $19.99.
The heart and soul of much mathematics consists of the fact that the “same” object can be presented to us in different ways.