Post History
#3: Post edited
by
Chgg Clou
·
2021-06-05T07:54:43Z (almost 3 years ago)
Why are you permitted to define $1 − 1 + 1 − 1 + . . .$, rather than unearth its intrinsic meaning?
- Why are you permitted to define $1 − 1 + 1 − 1 + . . .$?
Please see the embold phrase below. 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.
- 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.
#2: Post edited
by
Chgg Clou
·
2021-06-05T07:50:52Z (almost 3 years ago)
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.
- Please see the embold phrase below. 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.
#1: Initial revision
by
Chgg Clou
·
2021-06-05T07:50:25Z (almost 3 years ago)
Why are you permitted to define $1 − 1 + 1 − 1 + . . .$, rather than unearth its intrinsic 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.