Q&A

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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By definition, a positive real number is a real number greater than zero. That statement cannot be proved to be right; it cannot be proved to be wrong. We either reject it and use a different definition, or we accept it and move on.

Consider the statement "$1-1+1-1+... = 0$."

Here is a "proof" that it is right: $1-1+1-1+... = (1-1)+(1-1)+... = 0+0+... = 0$

Here is a "proof" that it is wrong: $1-1+1-1+... = 1-(1-1)-(1-1)-... = 1-0-0-... = 1 \ne 0$

But that is absurd; a statement cannot be right and wrong at the same time. Either we give up and say that the expression is meaningless, or we can define it be equal to some value. If the statement is a definition, then it cannot be proved to be right and it cannot be proved to be wrong. We either reject it and use a different definition, or we accept it and move on.

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