1 + 2 + 3 + ... = -1/12

[Albert Chan]

Is this a joke ?

[Thomas Klemm]

(10-01-2018 08:26 PM)Albert Chan Wrote:  Is this a joke ?

A response by the Mathologer:

[SlideRule]

from the source document
[attachment=6392]
for the curious

BEST!
SlideRule

[Albert Chan]

Hi, SlideRule

You are amazing !
That was the same page from the video !

Are you by any chance a historian ? Books collector ?

I "know" the sum does not add up to -1/12, but the guys were so serious ...

[SlideRule]

(10-01-2018 11:34 PM)Albert Chan Wrote:  Hi, SlideRule … Are you by any chance a historian ? Books collector ? …

Glad the posting was appreciated!
I was a Doctoral Candidate as well as a prior instructor in technical mathematics … but mostly now a retired 30 yr, combat engineer, mil veteran. One of the former allows me continuing academic access from multiple venues (I share when permissible).

ENJOY!
SlideRule

[Maximilian Hohmann]

Hello,

thanks for sharing this! Very interesting. I watched the original video and the whole 40 minute reply of Burkard Polster both of which are worth seeing. Incredible what can be deducted from such a simple summation. Between the lines it confirms two points that I have always defended in our weekly amateur astronomer's meetings:

- There is no such thing as infinity in physics or engineering. If a (mathematical) equation yields infinity as result then it is the wrong equation to describe the problem.
- String theory is just a paper excercise juggling with equations that has zero connection to reality.

Regards
Max

[Albert Chan]

After some investigation, -1/12 may not be wrong ...
It depends on what "Equal" meant: Does 1+2+3... Really Equal -1/12 ?

Above article included a link to Numberphile more convincing second proof.

Here is a good explanation from Professor Frenkel:

[pier4r]

(10-02-2018 05:52 PM)Maximilian Hohmann Wrote:  - String theory is just a paper excercise juggling with equations that has zero connection to reality.

I often find such statements self disqualifying.

They sound like "I don't understand it, and I belittle it and all the people that work in the field".

As an analogy one can say the same of whatever discipline, for example engineering "what do engineers do? They take formulae from math and physics, actually never producing anything new themselves, they plug two numbers and they get the result. Whoever person with a calculator can do the same".
Self disqualifying, that's it.

[Maximilian Hohmann]

Hello!

(10-03-2018 03:42 PM)pier4r Wrote:  As an analogy one can say the same of whatever discipline, for example engineering ...

Until not so long ago physicists and engineers tried to shape their models according to reality. Real world observations usually are the basis, the model tries to explain it in terms of equations, and the extrapolations and ramifications of that model point the experienters in new directions, where they can either confirm or discard the model. A very prominent example is the trheory of relativity, many of whose predicted effects could be veryfied by experiments, some of these experiments even required dedicated satellites to be built and launched into space. The same holds for many aspects of quantum physics. Even exotic and incredible stuff like quantum entanglement can be demonstrated in experiments.

But string theory is different because most of it is way outside the physical world which is accessible to experimentation. We can not probe beyond trhe 3 or 4 dimensions of our world. We can neither verify nor falsify the claims of that theory. And because of that I call it paper science. But maybe history proves me wrong and it is actually string theory that will show us how to build a warp drive...

Saluti
Max

[pier4r]

(10-03-2018 04:15 PM)Maximilian Hohmann Wrote:  Hello!


But string theory is different because most of it is way outside the physical world which is accessible to experimentation. We can not probe beyond trhe 3 or 4 dimensions of our world. We can neither verify nor falsify the claims of that theory. And because of that I call it paper science. But maybe history proves me wrong and it is actually string theory that will show us how to build a warp drive...

Thanks for the explanation of your viewpoint. What I am saying is that I don't like, in general (not only because string theory), that plenty of people that are evidently not mad and put a lot of effort in their work are easily dismissed. This because the same can happen for all the fields that I think are non trivial.

I don't know enough about string theory to contest or confirm your view, but I would think that if enough smart people work on it, I trust them that it has some sort of meaning. Even if the string approach is completely wrong, maybe some byproduct of it will be useful.

As long as at least the theory stands (that is a way to refute hypotheses, although little for physics) we will have to wait.

Anyway I'd like to emphasize that the point of my comment is the dismissal of a subject without being expert enough in the subject itself.

[Thomas Okken]

(10-03-2018 05:13 PM)pier4r Wrote:  

(10-03-2018 04:15 PM)Maximilian Hohmann Wrote:  Hello!


But string theory is different because most of it is way outside the physical world which is accessible to experimentation. We can not probe beyond trhe 3 or 4 dimensions of our world. We can neither verify nor falsify the claims of that theory. And because of that I call it paper science. But maybe history proves me wrong and it is actually string theory that will show us how to build a warp drive...

Thanks for the explanation of your viewpoint. What I am saying is that I don't like, in general (not only because string theory), that plenty of people that are evidently not mad and put a lot of effort in their work are easily dismissed. This because the same can happen for all the fields that I think are non trivial.

I don't know enough about string theory to contest or confirm your view, but I would think that if enough smart people work on it, I trust them that it has some sort of meaning. Even if the string approach is completely wrong, maybe some byproduct of it will be useful.

As long as at least the theory stands (that is a way to refute hypotheses, although little for physics) we will have to wait.

Anyway I'd like to emphasize that the point of my comment is the dismissal of a subject without being expert enough in the subject itself.

The problem with string theory is that there are two too many parameters whose values are unknown. By tweaking those parameters, the model can be made to fit anything, which makes it useless in terms of offering testable predictions about reality.

https://en.wikipedia.org/wiki/String_theory#Criticism

It may be elegant, but as it stands, it is not science.

[Maximilian Hohmann]

Hello!

(10-03-2018 06:22 PM)Thomas Okken Wrote:  It may be elegant, but as it stands, it is not science.

This is what I also hear from some of my astronomer colleagues - I am only an amateur astronomer, but some of them are really physicists specialised in astronomy and astrophysics. The problem for them is that quite a few of these guys have to earn their money in IT and in the automotive industry because the physics departments of many universities devote a lot of resources and manpower to stuff like string theory. Leaving no place for "real world" physicists. But again, maybe people with more insight than the rest of us know the purpose behind arranging things that way.

Regards
Max

[mfleming]

I bought it, but then didn't bother to read it

~Mark

[SlideRule]

A modest quote from the first paragraph of chapter 18, Other Points of View, on page 250 of Not Even Wrong Copyright © Peter Woit 2006.
[attachment=6412]
This book has surveyed the current state of fundamental particle
physics from a very particular point of view, that of a
mathematically minded particle physicist. The emphasis has been
on the standard model, the mathematics behind it, and the accelerator-
based experimental techniques that led to its discovery and
whose limitations now make further progress difficult. There are
other different points of view on the problems of particle physics
and this chapter will consider some of them.
emphasis in bold mine
BEST!
SlideRule

[Albert Chan]

I finally see the logic of assignment a divergent sum to a value.

Example: find \(\sum _{10}^{25} 2^x x^2 \) with divergent "sum"

t = \(\sum _{1}^{\infty} 2^x → \frac{2}{1-2} → -2 \)

Getting coefficients of polynomial of t, to calculate the "sum"

Code:

x  x^2 Forward Difference Table
10 100
11 121 21
12 144 23 2

26 676
27 729 53
28 784 55 2

\(\sum _{10}^{\infty} 2^x x^2 = 2^9 \sum _{1}^{\infty} 2^x (x+9)^2 → 2^9 (100t + 21t^2 +2 t^3) → -67584\)

\(\sum _{26}^{\infty} 2^x x^2 = 2^{25} \sum _{1}^{\infty} 2^x (x+25)^2 → 2^{25} (676t + 53t^2 +2 t^3) → -38788923392\)

\(\sum _{10}^{25} 2^x x^2 = \sum _{10}^{\infty} 2^x x^2 - \sum _{26}^{\infty} 2^x x^2 → 38788855808\)

Amazingly, it match true result


Update: t polynomial trick always work. It is based on summation formula:

\(F(x) = (\frac{-a^x}{1-a}) \{1 + (\frac{aΔ}{1-a}) +
(\frac{aΔ}{1-a})^2 + \cdots \} u_x \)

\(\sum_{x=0}^{n-1} a^x u_x = F(n) - F(0) \)

Above formula, being a finite sum, does not require convergence.

t polynomial trick assumed x from 1 to ∞, and F(∞) = 0, and simplify to T = -F(1):

Let \(t = \frac{a}{1-a}\), then \(T= \sum_{x=1}^{\infty} a^x u_x = t \{1 + (tΔ) + (tΔ)^2 + \cdots\}u_x \)

For divergent sum, assumption of F(∞) = 0 is wrong, but finite sum T_a - T_b still correct.

[toml_12953]

(10-01-2018 08:26 PM)Albert Chan Wrote:  Is this a joke ?

From a purely non-mathematical POV:

1 > -1/12

and if you're adding positive integers to 1 you are going to get numbers that are even greater than 1.

If your sum is S then

S > 1 > -1/12

So the sum, S, can never be equal to -1/12.

In fact it can never be equal to a fraction since the set of natural numbers is closed under addition and multiplication. That is to say that any two natural numbers added (or multiplied) together produce a natural number. Now let's try three numbers, A, B and C. Let A+B=M. M must be a natural number. M+C must also be a natural number. So A + B + C is a natural number. By induction (whatever the hell that is), it can be shown that the sum of ANY number of natural numbers is a natural number.

The very first sum he forms is incorrect.

At least that's how I (as a non-mathematician) see it.

[ijabbott]

I must sit down and watch this again sometime:

[jebem]

(10-01-2018 08:26 PM)Albert Chan Wrote:  Is this a joke ?

Thank you for sharing this.
As an engineer, I don't have the mathematical silks shown in these videos, but I reaaly enjoy to follow them and learn that this subject is as old as Euler time at least, and studied by other mathematical minds after him.

Apparently these outlaw mathematicians are the ones that added new dimensions to classic mathematica.
As one mathematician said in one of the above videos, imaginary numbers that we can not live without today, once were outlaw mathematicians thinking out of the stablished system.

Fantastic stuff indeed.

[Gerson W. Barbosa]

That’s just Zeta[-1]

https://blog.stephenwolfram.com/2016/04/...ramanujan/

[Albert Chan]

10 years later, The Return of -1/12 - Numberphile