Should the third term in equation (3.11.18) be multiplied by half?

[mad mathematician]

https://arxiv.org/pdf/hep-th/0108200
on page 135 equation (3.11.18) shouldn't the third term on the RHS of the equation be multiplied by half?

 

[martinbn]

https://arxiv.org/pdf/hep-th/0108200
on page 135 equation (3.11.18) shouldn't the third term on the RHS of the equation be multiplied by half?

It seems that it should. By the way you mean page135 of the file, which is page 125.

 

[mad mathematician]

Another question from the same text. On page 136, after equation (3.11.20), I don't understand how did they get: $\Pi_n= \Pi_n^2$, unless equation (3.11.20) should be: $\Pi_m\Pi_n=\delta_{mn}\Pi_m^2$?

 

[Demystifier]

Any projector is equal to its square, that's a definition of a projector. But their explanation is perhaps a bit confusing.

 

[mad mathematician]

Any projector is equal to its square, that's a definition of a projector. But their explanation is perhaps a bit confusing.

Their derivation that their $\Pi_n$ satisfies that it equals $\Pi_n^2$ seems wrong, as I said we should have $\Pi_n\Pi_m= \delta_{mn}\Pi_n^2$, since then: $\Pi_n=\Pi_n \sum_m \Pi_m=\sum_m \Pi_n\Pi_m=\sum_m \Pi_n\Pi_m=\sum_m \delta_{mn}\Pi^2_m=\Pi^2_n$.

BTW, I glossed over Perleman's papers on the arxive, it's written there something that looks wishy washy to me:

For an arbitrary metric gij on a closed manifold M, the function
μ(gij , τ ) is negative for small τ > 0 and tends to zero as τ tends to zero.

No logical quantifiers... psk psk.

 

[martinbn]

BTW, I glossed over Perleman's papers on the arxive, it's written there something that looks wishy washy to me:

No logical quantifiers... psk psk.

Looks perfectly clear! Not sure how it relates to this thread!

 

[mad mathematician]

Looks perfectly clear! Not sure how it relates to this thread!

small compared to what?

 

[martinbn]

small compared to what?

If this confuses you, may be you shouldn't read maths papers.

 

[Demystifier]

If this confuses you, may be you shouldn't read maths papers.

Or perhaps he wants everything expressed strictly mathematically with $\epsilon$ and $\delta$.

 

[Demystifier]

"For arbitrary metric ..."
No logical quantifiers... psk psk.

For logical purists I see an even bigger problem. The metric itself is an infinite set, because it is a function defined on an uncountably infinite number of points. Hence the phrase "for arbitrary metric" involves a quantification over infinite sets, which involves a second order logic. But second order logic has many problems, which is why logicians usually try to avoid it. And Perelman didn't say how he copes with these problems, for example does he use standard or Henkin semantics?

I'm half-kidding of course, but the point is that mathematical purism that makes sense in one branch of math doesn't make sense in another. A good mathematician should be able to understand a hand-waving mathematical statement in the context where it should be obvious how to make it more precise.

 

[weirdoguy]

A good mathematician should be able to understand a hand-waving mathematical statement in the context where it should be obvious how to make it more precise.

This. Usually scientific papers are written for people working in the same field. Writing everything out would even make this papers hard to read.

 

[mad mathematician]

For logical purists I see an even bigger problem. The metric itself is an infinite set, because it is a function defined on an uncountably infinite number of points. Hence the phrase "for arbitrary metric" involves a quantification over infinite sets, which involves a second order logic. But second order logic has many problems, which is why logicians usually try to avoid it. And Perelman didn't say how he copes with these problems, for example does he use standard or Henkin semantics?

I'm half-kidding of course, but the point is that mathematical purism that makes sense in one branch of math doesn't make sense in another. A good mathematician should be able to understand a hand-waving mathematical statement in the context where it should be obvious how to make it more precise.

I see your point. I guess there's a lot of background reading before actually understanding the papers.
But eventually how should one be sure if a proof is legit? proof by Authority?

 

[weirdoguy]

But eventually how should one be sure if a proof is legit?

By checking if it is legit? If one is a mathematician working in the field, it should not be problematic.

 

[martinbn]

By checking if it is legit? If one is a mathematician working in the field, it should not be problematic.

And just to add, if one is not a mathematician working in the field, the one shouldn't worry about such things.

 

[mad mathematician]

By checking if it is legit? If one is a mathematician working in the field, it should not be problematic.

That's correct. Then how come there's no consensus for the alleged proof of ABC conjecture?

 

[martinbn]

That's correct. Then how come there's no consensus for the alleged proof of ABC conjecture?

There is consensus, the proof has a gap and therefore is not a proof.

 

[mad mathematician]

BTW, may I remind you of Veovodosky's work on higher homotopy theories where he said that his peers didn't see his mistakes but he eventually seen them...
Even if something is peer reviewed it doesn't mean there are no mistakes.

 

[weirdoguy]

Even if something is peer reviewed it doesn't mean there are no mistakes.

Yes, but probability is very low. What is your point?

 

[weirdoguy]

where he said that his peers didn't see his mistakes

They eventually did. Papers are not revieved by hundreds of people, but by 2 or 3. Then, after it is published, it takes time for others to look at it.

 

[mad mathematician]

Yes, but probability is very low. What is your point?

One should judge a book by its content and not its cover...

i.e just because someone has such and such reputation, doesn't mean we shouldn't be critical of his work, regardless of his or hers stature.

 

[weirdoguy]

i.e just because someone has such and such reputation, doesn't mean we shouldn't be critical of his work, regardless of his or hers stature.

Yes, that is how science work. And all scientists know that, and no one here said otherwise. What is your point then?

But still, criticism of someone who is not a mathematician rarely can count, because it takes YEARS of study to be knowledgable in the field. Non-mathematicians don't have proper background. And if they do, they are probably physicists.

 

[mad mathematician]

Yes, that is how science work. And all scientists know that, and no one here said otherwise. What is your point then?

Well it depends. I've seen Susskind interview in YT that he says that sometimes you need to adhere to the majority consensus even if you feel that they are wrong.

Anyway, mistakes may happen. From my point let's end the conversation here.

If I'll have other questions from SUPERSPACE I'll sure to ask here in beyond forum.