What.. No. This sum diverges. It does not equal -1/12.
Now if you plug -1 into the Riemann zeta function, you get -1/12. One could interpret that to mean the sum 1 + 2 + 3 + .. can be mapped to -1/12.
But the sum has never, and will never equal -1/12. It diverges, simple as that.
And, there are some results in physics that actually measure close to the number -1/12, from something that looks like a sum of 1+2+3... and that's kind of telling us that the whole construction is not just some math sleight-of-hand, it actually has some meaning in the real world.
Example: Casimir effect
So my point is... that kind of suggests the smoothness, or continuity, or differentiability, or whatever we want to call it, of the underlying function. The opposite of discrete. Is what my point was.
There are many ways of making the sum, including limit of partial-sums/summation by parts (the one we use most of the time), but there is also Abel summation, Borel summation, Ramanujan, Cesaro, and more. And frankly there is no reason to think that "summation by parts" is the "right" way. It is surely not "right" in physics. Example: why do we start summation from 1? Is the first element somehow more important than others? No, that is just our (ie human) arbitrary pick.
Essentially, it consists of text pulled from exactly the same source that you originally cited.
(Cue sound of hands washing in sink.)
You cannot argue that with high-school math. There is more to it.
This may be useful: https://motls.blogspot.ca/2014/01/sum-of-integers-and-overso...
" The real problem is that the definition of the sum involving the limit of partial sums – limits that way too often "diverge" or "refuse to exist" – isn't the only definition or the best definition or the most natural definition that may be connected to the sum. There exist better definitions of the infinite sum – numerous definitions that turn out to be more natural in physics applications – and they generally produce the result −1/12−1/12. It is no trick or sleight-of-hand. The value −1/12−1/12 is really the right one and the rightness may be experimentally verified (using the Casimir effect). "
If we take the set { 1, 2, 3, ... 4 } the summation or partial summation of no subset of this set, finite or not, converges on any fraction or negative number. No matter what order we choose for traversing that set for generating a series, we never see anything resembling -1/12 as a partial sum or limit.
The ordinary arithmetic sum of any collection of positive integers is an integer which is strictly greater than each the integers.
The pages you're referencing are all crackpottery.
So... I can trust Ramanujan and Abel, that published results on these things, and Terrence Tao that has a nice writeup, and a bunch of others, or I can trust HN user "Kazinator", who... published some middle-school algebra "proof" on HN.
Guess who is crackpot here.
I can see how overlapping notation can be confusing for laymen (me included).
But that does not mean that people that came up with those are crackpots, just because they reused the "=" symbol.
Unfortunately, the syntax chosen for representing the sequence is that of an additive series, where a binary + operator is interposed between terms.
The semantics being shown does not seem to follow from a redefinition of that operator per se as stand-alone binary operator.
You really want to show this as, say
fun([1 2 3 4 ... ])
n fun([x0 x1 ... ]) = fun([nx0 nx1 ...])
fun(v0 + v1) = fun(v0) + fun(v1)
Basically, none of this means that the natural numbers add to -1/12; only that the sequence of natural numbers can be fed into some decimating calculation which ends up with -1/12.
Well, no kidding; the sequence of natural numbers can be fed into a decimating calculation which converges on any value you want, if you can freely choose the decimating calculation, and that calculation can be chosen to be linear operator.
Note that we don't have:
fun([x0 x1 x3 ... x42 ...]) = fun([x3 x42 x0 ... x1 ...])
Only through a derivation which involves some cute but unsound pseudo-algebra on infinite series.
See, in the same page, the remark "Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums".
It is very well-known and long-recognized that our natural intuition is very wrong when it comes to infinities and infinite objects, and mathematics had a crisis in trying to come to grips with it.
Sure it looks like that, but, amazingly, there is a physical meaning to that number.
I replied here: https://news.ycombinator.com/edit?id=16164915
Terence Tao has a nice lecture published... trying to find it, brb.
In any case, the gist of it was: long as we treat the numbers in the sum as integers (ie discrete), we will get ill-defined sums, contradictions, infinities, and so on.
Once we switch to real numbers (and there is an underlying function that is differentiable), things "can" be made to work and converge. To -1/12 always.
So that's it. In particular, if we decided that our smallest unit of measure is... the size of atom, or whatever finite value, many of these techniques will just fail to work, and they won't match what we measured. Which means something is wrong. Maybe things really are continuous. Or maybe they are not, it is just that our math is not sophisticated enough.
Obviously this is an open question.
Atoms have structure, they consist of... whatever parts. And these parts have structure (like... quarks and whatnots). And... are we sure that quarks do not have structure themselves? Whatever the case is, it is not something that one should declare in a HN thread.
Even more interesting... 1+2+3+4... = -1/12
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...
And that result actually pops up in physics.
That kind of indicates continuity/infinite divisibility.