@MISC{Mazur07whenis, author = {Barry Mazur}, title = {When is one thing equal to some other thing?}, year = {2007} }

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Abstract

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. Even if we are faced with the simpleseeming task of “giving ” a large number, there is no way of doing this without also, at the same time, “giving ” a hefty amount of extra structure that comes as a result of the way we pin down—or the way we present—our large number. If we write our number as 1729 we are, sotto voce, offering a preferred way of “computing it ” (add one thousand to seven hundreds to two tens to nine). If we present it as 1 + 12 3 we are recommending another mode of computation, and if we pin it down—as