Communities

Writing
Writing
Codidact Meta
Codidact Meta
The Great Outdoors
The Great Outdoors
Photography & Video
Photography & Video
Scientific Speculation
Scientific Speculation
Cooking
Cooking
Electrical Engineering
Electrical Engineering
Judaism
Judaism
Languages & Linguistics
Languages & Linguistics
Software Development
Software Development
Mathematics
Mathematics
Christianity
Christianity
Code Golf
Code Golf
Music
Music
Physics
Physics
Linux Systems
Linux Systems
Power Users
Power Users
Tabletop RPGs
Tabletop RPGs
Community Proposals
Community Proposals
tag:snake search within a tag
answers:0 unanswered questions
user:xxxx search by author id
score:0.5 posts with 0.5+ score
"snake oil" exact phrase
votes:4 posts with 4+ votes
created:<1w created < 1 week ago
post_type:xxxx type of post
Search help
Notifications
Mark all as read See all your notifications »
Q&A

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

+3
−1

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.

History
Why does this post require moderator attention?
You might want to add some details to your flag.
Why should this post be closed?

0 comment threads

1 answer

+0
−0

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.

History
Why does this post require moderator attention?
You might want to add some details to your flag.

0 comment threads

Sign up to answer this question »