Dear SteeleRedux,

Let me first reiterate (also as a comment on warmair’s post) that you can “do” mathematcs (at any level) without knowing anything about quantum physics or string theory, but not the other way around. Physicists can talk about the APPLICABILITY to their problem of this or that part of mathematics, but PHYSICS as such cannot decide what is correct and what not in mathematics. As quoted in my paper http://www.onlineopinion.com.au/view.asp?article=15928 , “while our perception of the physical world can always be distorted, our perception of the mathematical truths can’t be”. Apparently, string theorists found the zeta function (which is not easy to define at this level) useful, and I think the Wikipedia article I linked to is a good explanation of why zeta(-1) = -1/12 can be associated with 1+2+3+4+ ...

I looked at the links you gave and I admit I did not understand how they relate the picture and the calculations to the zeta function. True, the two regions A and B both have an infinite area, actually they are of the same shape.

>>why can't you subtract area A from area B and get left with area C which equals -1/12th?<<

Because both these areas are infinite numbers, and you cannot subtract one infinite number from another, more precisely, you could reasonably assign anything to the result:

For instance, there are infinitely many natural numbers (1,2,3,4 ...) and there are infinitely many even numbers; if you “subtract” (i.e. remove) the latter from the former you get the set of all odd numbers, and there are again infinitely many of them. On the other hand, if you “subtract” from all natural numbers all those greater than 2014, again infinitely many of them, you get the set of all the natural numbers less or equal 2014, which is finite, its size being exactly 2014.

So in the first case we “derived” infinity - infinity = infinity, in the second case infinity - infinity = 2014.

Dear SteeleRedux
Frankly I am out of my depth on this topic, anyway thank you for bringing it up, I found it fascinating. I mentioned it to my nephew who is a mathematical whiz kid, I expect in about 6 weeks I will get a very detailed response which will be way over my head.

I did in the end however come to the same conclusion as George that infinity - infinity can equal whatever you choose.

Thanks George for providing a good deal of clarity on the issue.

Good afternoon to you STEELEREDUX...

I've carefully examined the conundrum you've kindly posed for us all. My only answer can be best described by my candid admission that having passed, six subjects in the (1957) NSW Leaving Certificate, general mathematics was not amongst them. Though I managed to scrape through in chemistry, much to the incredulity of Mr Potts, our Chem. teacher ?

In my former occupation, it was required that I perform a couple of years penance in the Academy. In those days, government thought it better they attract more undergraduates, ostensibly lifting the basic educational standards of police recruits. I recall on one occasion, listening in to a bunch of these recruit's arguing over the efficacy of adopting a modified type of anemometer (wind speed measuring device) as a traffic calming strategy ? They proposed all manner of mathematical formulae, some supporting the device, and others strenuously opposing such a strategy ! All based on these complicated math. type diagrams ? They all appeared, on the face of it at least, to be brilliant mathematicians ?

Interestingly though, later on while marking some of their law theory examination papers, many of these same individuals had made some truly awful (simple) spelling errors, together with an almost non-existent ability to structure their narrative style answers, using correct syntax and basic punctuation ?

Upon enquiring later on why this was so, I was duly informed, at university little attention was/is paid to either correct spelling or correct grammar, as long as their responses were clear and comprehensible, and naturally correct ?

I suppose on reflection, their mathematical adroitness is a much more important and desirable expertise, rather than spelling an answer incorrectly, with badly structured grammar ?

STEELEREDUX, how things have changed since many of us went to school ? I agree with FOXY, there's a whole new world out there. Consisting of formulae, theorems, symbols, reasoning, abstract concepts, etc etc ? A really good topic I reckon ! However, it would be a pity, if you allowed it to further inflate that hauteur of yours now ?

Dear George,

Firstly I want to thank you for taking the time to deliver expanded answers to a pleb like myself. I am grateful.

You wrote;

“I looked at the links you gave and I admit I did not understand how they relate the picture and the calculations to the zeta function.”

That's good since neither do I. But plugging the formula G(n)=n(n+1)/2 into a spreadsheet (using .1 increments) then graphing it directly delivers the result shown in my earlier link.

This is where I am struggling. If area C was moved anywhere to the left it would retain the value of -1/12th. But apparently if we move area A to the right by 1 then it is not allowed because it is infinity?

I had imagined that if we keep our calculations strictly to the areas and not the figures contained within then a sum of those areas; (-)A + B + (-)C then why wouldn't it leave us with (-)C?

The answer again it seems is because it is infinity. But for a pleb the question becomes if we can change the rules one way for infinity what prevents us changing them in other ways?

I just keep getting the feeling I have all these roads leading to Rome, the video, Ramanujan, Bender, Physics Central but I kept being told I'm heading in the wrong direction.

Cont...

Cont...

You also wrote;

“”For instance, there are infinitely many natural numbers (1,2,3,4 ...) and there are infinitely many even numbers; if you “subtract” (i.e. remove) the latter from the former you get the set of all odd numbers, and there are again infinitely many of them. On the other hand, if you “subtract” from all natural numbers all those greater than 2014, again infinitely many of them, you get the set of all the natural numbers less or equal 2014, which is finite, its size being exactly 2014.”

I get what you are saying yet I'm not sure it applies here. I would have assumed if you take away all the natural numbers (1,2,3,4...) from all the natural numbers (1,2,3,4...) that you got zero. From that doesn't it follow if you add all the positive natural numbers plus all their corresponding negative numbers you would again have zero?

Thanks again for the effort you are putting in to slap me into shape. Also I have just realised I have been using 'illiteracy' instead of 'innumeracy'. Sorry folks.

Dear warmair,

I'm glad you got a buzz out of it and believe me there would be a hell of a lot more depth under my feet than yours. My nephew is also a mathematical whizz kid. 17, already a double degree under his belt and about to embark on his masters. My usual retort when they get a bit dismissive on this one is to ask them to explain why I should be taking their word over that of Ramanujan, widely considered the most naturally gifted of mathematicians over the last 200 years.

Dear o sung wu,

I will endeavour not to let your appreciation of the topic to over-inflate your impression of my impression of myself.

Dear SteeleRedux,

I am still curious how the author of http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html got the value of zeta(n) for odd integers n other than -1, i.e. where he got his generating function for those n. (If you are just after the numerical results, the online calculator http://keisan.casio.com/exec/system/1180573438 will give them, though I have no idea what algorithm or programme they used.)

As to your question, first a clarification in terminology: You know the formula which gives the AREA (a number) of a triangle (a REGION). So the picture denotes the REGIONS A and B, which both do not have a finite AREA but it is reasonable to denote their area as infinity in the following sense:

In the picture, the part of B going from - n to -1 is the same (only flipped over) as the part of the region A going from 0 to n-1, hence both have the same area which you can calculate directly (or using Wolfram which I do not have access to) as (1/12)(2n+1)(n-1)^2 which clearly tends to infinity as n tends to infinity, i.e. as the partial region is becoming the whole of A. This is what one understands when saying that the area of A is infinity; the same for the area of B.
(ctd)