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Why are you permitted to define $1 − 1 + 1 − 1 + . . .$?

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Please see the embold phrase below. Why doesn't $1 − 1 + 1 − 1 + . . .$ possess an intrinsic, Platonic objective meaning? The best way to showcase my confusion, is to burlesque the Riemann Hypothesis. If Hardy and humans can simply define $1 − 1 + 1 − 1 + . . .$, why not just define $\zeta(s)=\sum_{n=1}^{\infty}\dfrac{1}{n^s} := 1/2$? Then problem solved! LOL!

      The tradition is called “formalism.” It’s what G. H. Hardy was talking about when he remarked, admiringly, that mathematicians of the nineteenth century finally began to ask what things like

$1 − 1 + 1 − 1 + . . .$

should be defined to be, rather than what they were. In this way they avoided the “unnecessary perplexities” that had dogged the mathematicians of earlier times. In the purest version of this view, mathematics becomes a kind of game played with symbols and words. A statement is a theorem precisely if it follows by logical steps from the axioms. But what the axioms and theorems refer to, what they mean, is up for grabs. What is a Point, or a Line, or a frog, or a kumquat? It can be anything that behaves the way the axioms demand, and the meaning we should choose is whichever one suits our present needs. A purely formal geometry is a geometry you can in principle do without ever having seen or imagined a point or a line; it is a geometry in which it’s irrelevant what points and lines, understood in the usual way, are actually like.

Ellenberg, How Not to Be Wrong (2014), page 400.

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