1 + 2 + 3 + 4 + 5 + … = -1/12

“I find this confusing.”

You ain’t kidding. These sort of things always make my brain hurt. Probably why I dropped calculus in my first year of university.

Belle and I agree that the sleight of hand comes at 3:11 when he ‘shifts it along a little bit’. (My cat is looking at me, skeptically.)

When he says that 1-1+1-1… = 0.5, he’s applying a particular definition of the value of the infinite series that is different from ‘what limit does it approach’. ‘What limit does it approach’ is the more common definition of the value of a series like this. And 1-1+1… doesn’t approach any limit, neither does 1+2+3+4…

It’s like those proofs where they prove 2+2=5 by dividing by 0. Except that while it never makes sense to divide by 0, apparently the assumption that 1-1+1-…=0.5 actually takes you someplace interesting, so the video isn’t bogus, but it is sneaky.

Or, it’s a bit like if this were a geometry problem, and they used a non-Euclidean geometry without telling you (we tend to assume things are Euclidean…).

Somebody who knows more math could give more details.

Oh! I never thought that CT would one day post on my area of research! You guys taking guest post (who wouldn’t die to know how one could have guessed the result beforehand)?

http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation

The arguments are *not* sleight of hand. They aren’t entirely rigorous either, to be fair, but they could be made rigorous with a bit of effort. It would just alienate the viewer it’s meant to entertain.

Anyway, they use a different notion of a ‘convergent series’ from the standard analysis text. Whenever standard analysis says a series is convergent, this definition will agree and give you the same limit. But this definition is more general and allows you to define a limit for a larger class of series.

I found that by shifting the sets along a different number of places when adding the two identical sets I got very different results. Say, none, such that it was:
+1 -2 +3 -4 +5 -6…
+1 -2 +3 -4 +5 -6…
+___________
2…-4…6…-8…10…-12
If I then choose to group these together variously it seems I start with 1 and can imagine everything else to be -2? Oder, two places rather than one:
+1 -2 +3 -4 +5 -6 +7 -8…
+1 -2 +3 -4 +5 -6
+_______________
1…-2…4…-6…8…-10…12
Now, if I fancy, we will begin with -1, but go on with -2, hypothetically? But what if I should decide to group them the other way around, such that the larger positive number befriended its adjacent lower negative buddy? And how did we ever get to 1/2?

What I got out of it was that you can ask a simple question: Does a set converge? And if it does you might be able to compare it to another set, if it converges also. If one set doesn’t converge, then the right side of the equation is not useful for that application.

Likewise, you don’t get from increasing positive numbers to a negative sum.

dn – that link was helpful. An exchange in the comments says that it is possible, w/ important assumptions that also weren’t stated in the video (i.e. if the series is in the complex plane), to assign a Cesaro sum to a series that ‘goes to infinity’, but even if that is correct, they still go wrong with the ‘shift’.

http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/#comment-33103

Sort of like the way the writers at Slate would teach math, if they were to try.

Hah. That’ll leave a mark (or should, but yeah, no).

The point is that

1+2+3+…

is a string of numbers, and you can choose how to assign a value to the ‘sum’. The most common way is to take the series of partial sums, i.e.

s1 = 1, s2 = 1+2 = 3, s3= 1+2+3 = 6, …

and see if that sequence has a limit in the traditional sense. In this case, the sequence clearly doesn’t have a limit, so the sum does not exist.

However, there are many other ways you could assign a ‘sum’ to the original sequence. One might also require that this assignment satisfy a number of nice properties. Examples of this are things like Cesaro summation and Abelian summation. Those two, for example, assign a value of 1/2 to the ‘sum’

1-1+1-1+1 …

Even more, you can show that any assignment that has some nice properties must give a value equal to 1/2. These two don’t work for the ‘sum’

1+2+3+4+…

however, but you can do zeta-function regularization or Ramanujan summation to get the value -1/12.

The cool thing is that this isn’t just mathematical sleight of hand. These values are telling you something about the original series. One (very) mathematical explanation is given by Terry Tao here. Hardy wrote an entire book about this stuff called, “Divergent Series”. But, being a physicist, the cool thing for me is that you can actually see this in the laboratory as it’s related to the Casimir effect.

Kaveh @12 – ah, that comment makes sense. I, too, was slightly wrong. Thanks for that.

Math is challenging, but it’s not out to tear down all your intuitions.

Speaking from ignorance here but aren’t imaginary/irrational numbers extremely useful in physics, and aren’t they fundamentally inconceivable?

mattski @ 19:

Speaking from ignorance here but aren’t imaginary/irrational numbers extremely useful in physics, and aren’t they fundamentally inconceivable?

Yes, they are extremely useful, no they are not fundamentally inconceivable.

Imaginary numbers are a straightforward extension of real numbers, motivated by including the element i (the square root of -1). Irrational numbers (and hence the real numbers) are necessary as soon as you want to define the square root of 2 and in a bunch of other operations.

Both are a bit counter intuitive and involve some weirdness, but they are still approachable.

mattski,
I see what you mean about imaginary numbers but I don’t think irrational numbers are inconceivable. Draw a square with sides of one inch and the lengths of that square’s diagonals will be irrational.

Yes, I was aware that irrational and imaginary numbers not the same category. But isn’t the square root of -1 fundamentally inconceivable? Seems to me that it is.

Speaking as a very visual person – who did maths up to 18 mostly by imagining pictures in my head – yes, imaginary numbers are difficult, nay, impossible to visualise. So you don’t. Inconceivable? No. Like the name says – it’s imaginary and used to work other things out.

Katherine @26, they’re actually not that hard to visualize at all, once you get past just the basic concept of sqrt(-1) = i and start building up structures with them, like the complex plane.

Inconceivable? You keep using that word. I do not think it means what you think it means.

@belle #8

Regarding regrouping and shifting series, there is a very interesting result called Riemann series theorem [1] that states that given a convergent series that is not absolutely convergent [2] you can get to sum to any number you want by rearranging the terms.

[1] https://en.wikipedia.org/wiki/Riemann_series_theorem

[2] a_1+a_2+…< ∞, but |a_1|+|a_2|+… = ∞

I appreciate the responses. Thanks all. I do find this sort of thing fascinating, and lament the deterioration in my gray-matter.

dn, depends what you mean by “inconceivable”

Yes. So, I’m skeptical of the idea that simply slapping a label on something makes it conceivable. The word “god” makes for a great example. For my money, that is an attempt to conceive the inconceivable and the result is quite a bit of confusion.

As far as the square root of minus 1, doesn’t it violate what we understand as the rules of multiplication? How is this hurdle overcome by the pasting on of a label? The question of whether simple negative numbers are conceivable is very interesting. I’m not sure about it, but visualizing a negative number as a process seems to satisfy my intuition… (it might also be a helpful reminder to think of positive numbers in a similar, instrumental way.)

The main reason this confuses people is that, as stated, it is bullshit, and there is nothing to understand. A smart mind that knows basic math cannot see much but gobbledygook in that. It is not impossible for it to make sense, but then you have to redefine the meaning of things (like equality and sum) rather fundamentally and do so explicitly. I hope the guys in the video are aware of that, otherwise that would be quite shocking.

People tend to get immediately better at maths once they realize that it is entirely legitimate to NOT understand things, and to insist on the details.

I think that these guys are trying to popularize mathematics with videos like this, but they probably harm that cause more than they help it.

They could have used similar processes to “prove” that 1=0. A conclusion like that in a direct proof means that you made a mistake somewhere. In a proof by contradiction, it may mean that you made a mistake or may mean that your proof by contradiction was successful.

I’m also curious to know what explanatory text appeared around that equation in their string theory textbook. One would think that the accompanying text would go a long way to explaining the intricacies of that non-standard result along with cautions about where and how to apply such things.

You all know about heat rays. But have you seen my proof of the existence of coolth rays?

In addition to what MattF says: everything in math gets more complicated when you start throwing in such notions as “infinity” or “infinitesimal”, which are intuitively conceived as indefinite. This is the very reason why the differential and integral calculus were somewhat controversial when first introduced; the concept of the “limit” was developed after the fact as a way to get the infinitesimals out of the infinitesimal calculus. More rigorous that way. (Abraham Robinson in the 1960s did take a shot at a non-standard calculus which revived the use of infinitesimals themselves in a more rigorous way; his method has its admirers but has never really caught on in the mainstream.)

“It suppose” s.b. “I suppose it”

I get that the square root of -1 is an extremely useful concept that lets you do all sorts of mathematics, and I heartily endorse it.

But I’ve never understood what kind of a number we’re imagining it is. We have rules that say the square root of a number is a number which multiplied by itself gives that number. We have rules that say that two positives or two negatives multiplied by one another give a positive number. So are we imagining that the imaginary unit is neither positive nor negative, or maybe both at the same time? Or is the answer just that its useful, and we don’t need to worry about it?

OCS: You’re correct to say that the imaginary unit is neither positive nor negative. This just means that ‘positive’ and ‘negative’ are properties that have limited usefulness. C’est la vie.

dn: I’m not sure that anyone who has not reached the age of ‘mathematical maturity’ should actually try to read this, but here’s Terry Tao’s attempt to make nonstandard analysis seem reasonable to working analysts:

http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/

but it just seems so totally confusing that I don’t know at all what to think.

Well, Belle, as Barbie famously (but apocryphally) proclaimed: Math is hard. It took mathematicians two millennia to understand how to compute with infinite sums so you shouldn’t feel diminished if your are unable to rediscover all that lore starting from scratch. Nevertheless, the identity ζ(-1)=-1/12 which forms the title of this post and its close cousin ζ(0)=-1/2 (so 1+1+1+1+1+…=-1/2), an even superior one in fact, are such gems of human knowledge that their propagation in popular culture can only be deemed a good thing.

Imagine you had a relay that switched between 1 and 0 volts. So you measure the voltage at the output. As it increase how fast the relay switches, the needle on your meter can’t move fast enough and settles on .5 volts, quivering slightly. Faster and faster: eventually the capacitance of the output wire acts like a “weight” that prevents the actual voltage from changing very fast, and the real instantaneous voltage stays close to .5 volts.

I don’t know whether this has anything to do with physics. I am an engineer. It does occur to me, tho, that the observable universe isn’t big enough to contain any infinite sequences, so if such things are important, we need them to fold up into something definite. And if you are trying to describe continuous chaotic functions accurately, you need infinite sequences like the Taylor series. The Universe is a stranger place than you thot, bro.

Belle: it doesn’t matter how you pair up the numbers. Arithmetic says all those sums must be the same because “commutativity”. He picked that particular pairing because he knew it was going to work, is all.

There is a geometric interpretation of imaginary and complex numbers. This interpretation is important historically because it made it clear that complex numbers were a well-defined thing.

At this point, not only are the complex numbers are well-understood, but many other systems with more exotic multiplications, such as the quaternions (where the order of multiplication matters) octonions (where multiplication is not even associative), or the p-adics (which contain rational numbers, but not all real numbers, and where many weird infinite sums hold, such as 1 + 2 + 4 + 8 + … = -1).

mud man: Finite and infinite sequences obey different sorts of rules. Procedures that are acceptable with finite sequences are not necessarily allowable with infinite sequences. That’s even more true with non-convergent sequences, or even with certain classes of convergent sequences (http://en.wikipedia.org/wiki/Riemann_series_theorem).

Commutativity does not excuse his particular choice of pairings, the presenters operations with 1 – 1 + 1 -1 … are unjustifiable.

@mattski, you ask “doesn’t [root -1] violate what we understand as the rules of multiplication?”

No! The whole point is that it doesn’t; if you assume it is a number, then everything works out algebraically. That is its only meaning – there is nothing else to it.

Of course it is true, as you note, that no one can show you what root -1 of an apple (say) looks like. But why get bent out of shape about “imaginary” numbers? No one can show you an example of a typical real number either; that is, if you randomly select with uniform probability a number from a finite real interval, then with probability one you will select a number that cannot be identified numerically with finite information. Numbers like 3 or 1/2 or pi or root 2, which all can be quantified with finite information (i.e. a computer program), are vanishingly rare.

Real numbers are only needed because it is convenient to say that the limit of a sequence, if it exists, is a number (this is how real numbers are constructed.) But once you assume they exist, they follow all the rules. You can’t visualize a typical real number, but you were still happy to accept them; imaginary numbers are just an extension of this principle.

It should be pointed out that the first computation of the value of the Zeta function at -1 […] is due to Leonhard Euler; and that his calculation (which is heuristic, as no complex analysis was available at the time)

Because a CT thread should include controversy and because Mao found it in himself to troll divergent series at 43, I will note that I disagree with the statement that Euler’s proof was heuristic. It was correct to the standard of proofs of the time and Euler could have, with no doubt, proved it to today’s standard if he had been required to. To be more specify (and for Christian benefit), the computation of negative zeta values does not require any non-trivial complex analysis, just a clever real analytic manipulation of the Taylor series of tdlog(1-t)/dt, it is the meromorphic continuation to the whole of C, which is much harder, which had to await Riemann and the assorted technology.

Whoops, and of course, even better, you may then recognize you are already familiar with this rotational procedure in the case of multiplying by a negative number, which does entail a 180 degree rotation. So with real numbers, the rotation is either 0 degrees or 180, with purely imaginary numbers, 90 degrees, or 270, and for complex numbers in general, any possible rotation.

Seeing all this way back when translated i, for me, from something that looked like a definitional trick to a part of an expanded conceptual framework.

This may be the first time CT has been facilitator of trolling an entire profession.

It’s too damn bad that mathematicians were responsible for picking annoying and misleading terms for concepts that are neither irrational nor imaginary, but the mathematical sense of humor is as difficult to appreciate as math itself. Physicists aren’t much better, with their quarks and gluons and strings and half-dead cats.

But you do have to give entities names or you can’t talk about them.

@61

Counterpoint. One of the most overused adjectives in the field is normal.

On a tangential note, any else here read Mathematics Made Difficult ? I feel it is very apropos.

mathbabe has a cool blog post on this at http://mathbabe.org/2014/01/21/if-its-hocus-pocus-then-its-not-math/

“If it’s hocus pocus then it’s not math.”

The topic is one of series, not sequences. They are different things.

Mathworld’s definition of divergent series.

Additionally, the site “vitutor.com” is not a place that I’d look for mathematical definitions.

#71 and “half dead cats,” meet #68.

What mud man said is correct, but applying the same reasoning to an infinite stream of pulses -1,1,-1,1… gives an integrated value of 0, not 0.5. That’s a different type of analysis than whether a series converges, though.

Complex numbers are very useful in electrical engineering for calculating both the magnitude and phase of a signal. Nothing really imaginary about it. Just a conversion from polar to cartesian coordinates. However the cartesian coordinates happen to be in the complex plane.

Anyway, as any fule kno, all numbers are equal to 47.

Phil Koop,

Thanks for the response, I appreciate it!

But why get bent out of shape about “imaginary” numbers?

Who said I was bent out of shape? :^)

No one can show you an example of a typical real number either; that is, if you randomly select with uniform probability a number from a finite real interval, then with probability one you will select a number that cannot be identified numerically with finite information.

OK. I think I see what you’re getting at here. But if I understand you correctly what you are claiming seems problematic.

I think you are saying that if we randomly put our “stylus” on a “line” representing the real numbers we are going to have a problem of infinite or unknowable decimal places. But that would raise an objection. In what sense have you pointed to a number if you cannot identify it? Wouldn’t it be reasonable to say, “you have not identified any specific number.”

At least with numbers like Pi we have a means of identifying the number (as a ratio of two numbers) and the ability to produce the decimal series.

But additionally, we can intuitively understand fractional numbers as volumes or weights. How many pounds does this bowling ball weigh? Well, how many decimal places do you want to go? How sensitive is your equipment? And as a benefit (?) thinking this way diminishes the importance of “excessive” decimal places, which might be useful when calculating atomic weights but for mundane purposes we don’t need them.

Now, back to the root of -1. Yes, I can understand it as a “stipulation” or a “rule”. But I can’t map it onto my experience the way I can map regular multiplication. So it seems to me that there is an important sense in which I can never conceive it.

??

Mattski @88:

“Now, back to the root of -1. Yes, I can understand it as a “stipulation” or a “rule”. But I can’t map it onto my experience the way I can map regular multiplication. So it seems to me that there is an important sense in which I can never conceive it.

??”

I attempted to answer this before, but if you see complex numbers as points on the plane and multiplication as multiplication of the magnitudes (distance of the point from the origin) and *addition* of the angles from the positive number line, then multiplication of complex numbers makes sense. It completely encompasses multiplication of real numbers you have an intuitive sense for, but includes another dimension of multiplication, quite literally. With a little “playing around” with this, you can develop an intuition for it, too: it is not just a definition, but has a geometric representation that can come to seem fairly natural: add the angles, and multiply the magnitudes.

mattski: Have a look at http://www.mathsisfun.com/algebra/complex-number-multiply.html and scroll down to “Now For Some More Multiplication”. Once you get this, you have the central insight that makes complex multiplication intuitive.

The Youtube video is amateur hour. They need not just one, but several errors to obtain the result.

Instead, I can show you a correct proof that 1=0, using geometry. The proof works very well on napkins and steamy windows, but there is an online version too.

First we prove that all triangles are equilateral. The proof is here .

Then, using a triangle with two sides of length 3 and 4 respectively, which we just proved to be equilateral, we have that 3=4. Subtracting 3 we get our result, along with the result that all numbers are equal to zero.

I always liked both irrational numbers and imaginary numbers I was happy when they came along in math. I was talking about the video last night with John just after he posted it, and I was thinking that the claim, “this is important for string theory! This is how we know there are 27 dimensions rolled up inside all the regular stuff!” is not of necessity compelling. I mean, may people seem to believe that etc. but is it the case that they know about the various additional dimensions? I am unsure.

Alex K

Just because a chosen angle is bisected – does not mean that AP = PB for all triangles.

In fact,

AP = PB if and only if the triangle is isoceles.

In other words, it is proving something by assuming it in the first place.

Just like Arrow and Debreu.

Another way to spot the sleight of hand is to consider what is happening out at the infinite end of the series. infinities are being subtracted from infinities. Which can produce any result you like.

It seems that the reason the astronomers have fallen for it is that they have a tendency to confuse models with reality. Superstring theory is not reality, nor does reality have 24 dimensions (or whatever the number is this week). At best the physicists have a MODEL that has 24 dimensions that is consistent with empirical observation. Most times they just have a model…

Basically it is maybe possible to create a calculus of divergent series that is consistent if you restrict the transformations on the series that are permitted and the sum of the series is not the ‘total’ it is more like a characteristic index. And the resulting indexes are the sort of thing that can be then given meaning in some other calculus… So its not necessarily complete nonsense but showing that it means anything requires more effort than the shell game version…

“If you draw a exaggerated scalene triangle you will see that any bisector is always biased towards the sides with the greater angles.”

As I said, the proof works very well on napkins.

Alex K

Previous post did not accept mu greater than and less than symbols.

Try:

… if angle A less than < angle B.

… if angle A greater than > angle B.

Try drawing an triangle with sides 1, 8, 10. and playing with angle bisectors.

I can’t believe Chris Warren is wasting time arguing this – it is one of the most famous fallacies there is.

http://en.wikipedia.org/wiki/Mathematical_fallacy

I think everybody should start dividing by zero. What the hell!

@OP and Belle, #9, and others

Nah, the “shift” is very useful, but you must be careful about what is contained in the “…”. Here are two simple examples of the utility of shifting and grouping:

What is the sum of all of the first 100 numerals, i.e. 1+2+3+…97+98+99+100? If you rearrange them in a convenient way, you have 100+(1+99)+(2+98)+…+(49+51)+50 which is now a much easier problem (5050).

When trying to convert infinitely repeating decimals to fractions, i.e. 0.7777…:
10*0.7777…=7.777…
– 1*0.7777…=0.777…
—————————-
(10-1)*0.7777…=7
9*0.7777…=7.000000 = 7
divide both sides by 9 –>
0.7777…=7/9
Even this is a little dubious since you will get a result that 0.9999…=9/9=1.

In this case, the sleight is not the shift, it is the 1-1+1-1… = 1/2.

Heh … i saw the exchange just now, work and all.

After scanning the days comments, I’ve to admit to a tremendous increase in respect for Mao’s trolling skillz … how does on do that on a math thread ?!!!

dn @ 18: “Math is challenging, but it’s not out to tear down all your intuitions.”

I find it strange that this evolved into a discussion of complex numbers.

Math has plenty of counterintuitive results. The bit about a randomly selected real number is good. What about the existence of non-measurable sets, given the axiom of choice? That shows that one cannot extend what seems obvious to the infinite sets without question. Or how about the existence of, at least, a countably infinite number of infinities? Or the Banach Tarski paradox?

Even this is a little dubious since you will get a result that 0.9999…=9/9=1.

Sir, I have the unshakeable belief that 0.9999… does indeed equal 1.
I just can’t remember why anymore.

One interesting thing about the “inconceivable” imaginary numbers is that real, physical things behave in ways which can be represented by imaginary numbers, especially things with frequencies.

For example, the electrical thingy often called Resistance in simple physics texts is characterized by a real number, but when dealing with alternating current (like what comes out of a plughole/outlet), the corresponding thingy is Impedance, which is a complex value with both a magnitude (real) and a phase (imaginary).

If you put an AC voltage across a load with a certain impedance, the current you get is given by dividing two complex numbers.

And because of wave/particle duality, everything has a frequency, and the basic equations of quantum theory (such as the Schrödinger equation) are written in complex numbers with imaginary parts.

Mattski @88 – nope, 47

http://www.47.net/47society/47story/

I was wondering about taking on the Barbie comment, but I figured you’d do it so much better Belle. And boy, I was right.

I need, like, something to go on besides ‘apparently sexist math dick who grants a few comments later that my grouping in the sets in question was as good as any other, which is to say, all options can be employed for evil in the manner in the video.’ Maybe fill me in on some person details? Or get a more ludicrous nym? What are you, Zorro?

Belle, I can see you’re angry. Please please take in account that English is not my native language when I say I’m not 100% sure about what exactly. Math is hard. That’s a fact. So you shouldn’t feel bad if you find computations about divergent series difficult to make sense of, especially when they are performed randomly as in the video, seeing it took centuries to the best mathematical mind to make sense of it. I wrote this as a word of consolation. Not how you took it, most clearly. Also, I did not exactly grant that your ways of doing it was equally good a few comments later; I always thought so: you watched a confusing video about hard math, you felt confused. Nothing wrong about it. The contrary would have suggested some lack of appreciation of what math is. That’s what I meant.

Now I believe that the source of your wrath is the Barbie quotation. OK, I guess I should have selected the other synonymous quote I enjoy about this topic (to recall, the topic of being confused about math): “Young man, in mathematics you don’t understand things. You just get used to them” from Von Neumann. Perhaps this one can also be interpreted in a sexist way, so I apologize in advance if it is, as it is not my intention.

You ask for personal details. Rhetorically, I guess. Nevertheless, I will point out that if you follow the link on my name and delve a bit, you should find relatively ample evidence of the fact that I’m quite (like just a tiny little bit) involved in the promotion of mathematics (you know, to the tune of several hundred of hours a year) among young students who desires to start again scientific studies after having chosen a different path earlier. I’ll let you guess what is the overwhelming gender of these students (I mean, I recognize fully your right to write anything you like about anyone on your own blog, but I won’t hide that “sexist math dick” hurt a bit).

More accurately, I thought Henry had told you when he told Hector.

How on earth is starting out with “as Barbie said ‘math is hard'” in a discussion with a woman who has just said, “I think the way they added these two sets looks like bullshit, also I don’t see how on earth the first set equaled 1/2 in the first place” going to equal anything other than ‘sexist math dick’, in any language? You know how badly math studies are skewed, gender ratio-wise. Do you want to know how badly ‘political blogger’ gender ratios are skewed? Do you want to know how badly ‘active commenters on this blog’ gender ratios are skewed? 20%, 20%, 20%, fuck all, ain’t nobody, and fuck all, stated variously. Why piss me off like that when there’s no cause?

You are talking to a woman whom you do not know, who is confused about a genuinely confusing thing. You choose a gender-specific statement, that originally (if apocryphally in your view) came from a plastic, highly sexualized doll intended for 10-year-olds. Can you imagine any possible world in which that is not sexist or offensive?

You don’t actually need to shut up after that one, I’m genuinely curious to hear your response and I’ll read it in the morning.

Then I should have been paying closer attention to your non-assholishness, probably. It’s maybe the single letter problem! It’s much harder to remember a single letter as a pseudonym I find. And the reason I asked, “do you want me to get mad at you or what?” was because you otherwise seemed totally reasonable. In your subsequent comment also. OK, fine, you didn’t pull a jar of wasps onto your head with all the wasps inside. Nonetheless I’d have to say that unless your interlocutor is a personal friend who is also a set theorist, I’d just never, ever, ever…(now we may imagine)… bust out the Barbie thing. Ever.

I think that math should not be that hard. I conclude that it is usually poorly taught!

Math has only a few basic types of actions: collecting, counting, comparing, computing (add-subtract-multiply-divide), ordering, rearranging, closeness and estimating, etc. Teachers ought to find a way to present them all at the beginning of study, in a single simple symbolic format. I do not think that “category theory” does the trick, but it is close. (That list is a rearrangement of the very great Saunders MacLane, Mathematics: Form and Function, p.35. Almost a unique book; there are a couple of books in the same vein by Martin H. Krieger.)

One pedagogic problem is in the types of students. The smart students who immediately understand math are already locked into the context of this list of basic actions with simple objects (like numbers). They are “mathematically inclined”.

The smart students who don’t get math easily are usually more facile in changing to other contexts of life, in which these basic actions do not easily apply, or can be found ridiculous. They may be better at emotions, of at formulating higher logical types of concepts, or at assimilating humor or paradox.

One thing teachers could have done for me personally would have been to start by explaining the areas where math can never apply, or is highly unlikely to.

I was interested in the health of the ecosystem, and in the sources of new ideas and innovations. If teachers had explained to me at the beginning that math will never precisely predict complex systems, and that math will never precisely predict the emergence of new properties in biological and social systems, then I wouldn’t have found the constant limitations of mathematics in these areas to make the subject of math so repulsive in my student years.

At the risk of beating a dead horse, I would restate the necessary insight here as: Mathematical concepts may have alternative definitions.

For example, suppose you put two dots on a piece of graph paper such that one dot is three over and four up from the other dot. What is the distance between the two dots? You could compute that distance as “taxi cab” distance (3 blocks over plus 4 blocks up = 7 blocks, as a taxi would drive between two locations in Manhattan) or “as the crow flies” distance (5 blocks, which you can measure with a ruler or calculate with the Pythagorean theorem). Both concepts of distance are sensible and valid if used consistently, and indeed there are many other meaningful definitions of distance. We only get into trouble if we mix and match definitions in the same discussion. (“Alice says the two dots are 7 blocks apart, but Bob says they are 5 blocks apart. IT’S A LOGICAL PARADOX! Mathematics is disproved!” No, it isn’t.)

Likewise, the concept of “sum of an infinite series” can have multiple definitions. One of those definitions (Ramanujan summation) gives 1 + 2 + 3 + … = -1/12. This result is unexpected in the context of this video because this definition is being slipped in under the radar. If the video had said, “Here is a highly specialized and counterintuitive definition of how the sum of an infinite series should be calculated, and that counterintuitive definition produces this counterintuitive result,” no one would have been surprised, and we wouldn’t be having this discussion.

Katherine @ 117

Well, I am temporarily stymied. But give me some time.

alkali @ 135: That touches on one of the other annoying parts of the video (which gets to a bit of mathematical shorthand as well).

(The notation of this post will be a bit lumpy, hopefully it’s not too confusing)

Outside of specialized contexts, when a someone with a moderate math education sees notation like:

a[1] + a[2] + a[3] + a[4] + a[5] + … = b

It’s typical to assume that this is shorthand for:

Limit(n->infinity) { Sum{m in 1..n} (a[m]) } = b

Evaluating this (commonly-assumed) convention for the given series (1+2+3+4+5+…) shows that the limit is +infinity — exactly as a non-mathematically-inclined person would expect.

So one sleight of hand in their result is substituting the Riemann zeta function (evaluated at -1) for the series (1+2+3+4+5…) and/or substituting Ramanujan summation for the more-typical limit process.

Another very large issue, of course, is that the process they use to present that result is bogus and much more likely to confuse than to enlighten.

It’s somewhat like the following well-known “proof”:

16/64 = 1/4

Because:

Start with: 16/64

Cancel the 6s: 16 / 64

Result: 1/4

The video is not just an ordinary bit of sleight of hand. It exploits the seeming paradox that arises when you treat infinity as if it were just a very big number that acts like any other number. It isn’t. Infinity behaves in ways that appear to be paradoxical, as here, but in fact work out very neatly once you accept that infinity is not just another number.

The best known version of the paradox is “The Paradox of the Grand Hotel,” or “Hilbert’s Paradox,” after the mathematician David Hilbert, who devised it to help explain the theory of transfinite numbers, discovered by Georg Cantor.

See
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

But before we get to the Grand Hotel, let’s take a look at the series.

The series 1 – 1 + 1 – 1 + 1 … would have a definite value if it were not infinite – that is, if it stopped at any point. If it stopped 100 billion trillion gazillion, the value would be zero. And if it stopped at 100 billion trillion gazillion and one, the value would be one. But because the series is infinite and does not approach a limit, it’s value is neither zero nor one, nor anything in between. We can make the value appear to be as big or small as we like, positive or negative, all the way up to infinity.

The video tells us that the value of the series is 1/2. But by the same logic we can make the value 1. Here’s how:

Write the series as 1 + -1 + 1 + -1 + 1 …

Now, put in some parens: 1 + (-1 + 1) + (-1 + 1) …

Reverse the order of the numbers in the parens: 1 + (1+ – 1) + [(1+ -1) …

Drop the parens: 1+1 + -1 + 1 + -1 …

Add the first two 1’s together:

2 + -1 + 1 + -1 …

Put the minus signs back:

2 -1+1 -1 …

Which by the video’s logic is 1.

And you can do this again and again, and move another 1, and then another 1, and then another 1, up to the front an infinity number of times, and there will always be an infinity number of 1’s left to pair up with the infinity number of -1’s that you didn’t move up. So the series can be made to appear to have any value, all the way up to infinity. That can’t be right, can it?

And you could do the reverse, and move a -1 to the front, and then another and another, and there will still be an infinity number of -1’s to pair up with the infinity number of 1’s that you didn’t move. So you can make the value appear to be a large a negative number as you’d like, up to – infinity. That can’t be right either, can it?

That’s what infinity means. It’s not the same as “a really big number,” it’s an inexhaustible supply of numbers. To our minds, it’s genuinely inconceivable, hence it can be used to trick us, as the video does. Yet it turns out that there’s whole branch of mathematics devoted to it.

When I was a wee boy, I read George Gamow’s wonderful little book called “1,2,3 … Infinity” (published 1947), which has a chapter on the mathematician Georg Cantor’s insights into the nature of infinity and his discovery of transfinite numbers – that is, that there is an orderly series of infinities, from smallest to largest. There are an infinity of them, and that they can be the subject of operations like any other numbers – except that they behave rather oddly based on our intuitions.

In explaining the very first, smallest infinity – the one that’s being exploited in the video – Gamow makes use of Hilbert’s Paradox of The Grand Hotel. He asks you to imagine an ordinary hotel, with 100 rooms and 100 guests. A traveler looking for a place to stay is turned away – no vacancy.

But now imagine The Grand Hotel, with infinity rooms and infinity guests. A traveler arrives. Every single room is occupied. No problem, says the manager, you can stay in Room No. 1. We will ask the guest in room No. 1 to move to room No. 2, and we move the guest from No. 2 to No. 3, and from No. 3 to No. 4, and ….

And if two travelers arrive separately, we move the guests in Nos. 1 and 2 to Nos. 3 and 4, and the guests from 2 and 4 to 5 and 6 ….

So how many vacant rooms are there at the Grand Hotel? Well, there are none. But how many rooms does the hotel have for additional guests ? Well, an infinity of rooms.

And this works because infinity plus one is infinity. So is infinity plus 2. That’s pretty obvious. But what is infinity plus infinity? That’s also infinity. And since infinity plus infinity is infinity, what is infinity minus infinity? It must be infinity.

And that’s the value of the series in the video – an infinity of ones, minus an infinity of ones, which equals infinity. Not zero, not one – infinity.

If this is at all interesting, I do recommend Gamow’s 65-year old book – the relevant part is the first chapter, and it is available as a pdf is here.

http://selfdefinition.org/science/25-greatest-science-books-of-all-time/10.%20George%20Gamow%20-%20One%20Two%20Three…%20Infinty%20(1947)(1988).pdf

There’s a little bit of literally accurate but objectionable discussion of the mathematical ability of a Hottentot, but otherwise it’s as fun and informative as I remember it.

I think Belle’s initial reaction demonstrates why this video is the worst thing ever to happen. An intelligent non-mathematician like Belle can see that the argument in the video lends itself to all kinds of answers. But of course any non-megalomaniac will think not that the argument in the video is gibberish but they must be confused about something. The effect of the video is actually remove mathematical knowledge from the world. Math is hard, but it’s not that hard.

And it’s obviously ridiculous to say that 1 + 2 + 3 + 4 + = -1/12 is “the” correct answer, the hallowed names of Euler, Riemann, and Ramanujan notwithstanding. The obvious non-mathematician objections to this equation are all correct. The partial sums are all positive numbers and steadily increase, so the limit sure seems like it should be positive. It’s not Cesaro summable: even if you buy the idea that if the series is oscillating you should take the average, here the average is +infinity. Intuitively, -1/12 sounds like the wrong answer, and that intuition is perfectly sound.

Here’s another ridiculous consequence (inspired by robotslave’s comment above). According to the zeta function argument, the sum 1 + 1/2 + 1/3 + … diverges: the zeta function doesn’t have an analytic continuation to s=1. But using the analytic continuation argument, the obviously larger sum 1 + 2 + 3 + 4 + … magically converges. But it’s not magic. It’s misapplying totally comprehensible mathematics to produce mystical stoner mathematics. In the hands of once-in-a-generation geniuses like Euler and Ramanujan mystical stoner mathematics can produce surprising patterns that future generations will want to explain, but for the rest of us all we’ll get out of mystical stoner mathematics is the munchies.

This thread is kinda awesome (if often over my head), but wanted to note that dn’s link waaaaay up at #4 is super helpful for anyone still scratching their heads.

The one to listen to here is Steinsaltz. The Scientific American explains it fully at http://blogs.scientificamerican.com/roots-of-unity/2014/01/20/is-the-sum-of-positive-integers-negative/, and Terry Tao has a more general treatment using something called “smoothed sums” at http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/.

I know George Gamow only from a nice edition of Mr. Tompkins stories…and an anecdote about a science demonstration gone awry.

It’s unfortunate that you find page after page of results in Google when you look for ‘Carl Sagan overrated’ (and that nasty theme had started even back at the height of Sagan’s fame in the early ’80s), and you also quickly find out that people think Stephen Hawking and Neil DeGrasse Tyson are overrated too. I bet Bill Nye is overrated, also! Somebody thinks Albert Einstein is overrated. When you get down to it, though, many people have learned over the years from the popularizers of science, and I hope there will be many more men and women to follow in their footsteps.

Some people have criticized Neil DeGrasse Tyson should be spending his time fighting theism…I don’t see that. He’s been wise not just to avoid that, but I think he’s also been wise not to go far from his core competency just in the way the Bad Astronomer has here: We want more people talking up science, but we don’t want them crowding out the best people and the best information with quick and easy contrarianisms. That annoys people here on CT and it’s just as useless in the public sphere at large.

It was a fun video, with links in the comments to deep and rigorous explanations. As the presenter said in the video, he tried to think of a semi-intuitive way to present a mathematical result which has actual, real-world applications, and what he did in five minutes was the best he could think of. It raises, or should raise, questions in people’s minds for deeper study and for some humility in appreciation that the universe is a very strange place; and maybe human beings aren’t as smart as we would like to think. I for one enjoyed it.

No, man. Numbers are real and would exist even if humans never existed. Facts about squares are just true facts about squares. I’m a Platonist about math. Possibly a stoner Platonist about math, but nonetheless. IME lots of mathematicians are, though of course many (most) are not. My friend who did algebraic topology was like, ‘naw, we pretty clearly just made this shit up.’ Set theorist friend by contrast? Secret Platonist but embarrassed to tell other mathematicians.

The concepts (or is it objects?) of mathematics are pure formal operations. They are neither empirical objects, nor substances: when one counts such, one doesn’t find the numbers inside them. (One can add apples and oranges; just reclassify them as fruit). That’s why Bertrand Russell, an empiricist, who thought that “knowledge by acquaintance” is the primary form of knowledge, remarked that when we do math, we don’t know what we’re talking about. The only requirement is that systems or sets of mathematical inferences be self-consistent.

So the old question is: are mathematical “truths” invented or discovered? My sense of this is that when a new domain of mathematics is opened up, (because mathematicians have somehow intuited that it is not just operationally doable, but makes mathematical “sense”), then the basic rules that establish that domain also function as meta-rules, i.e. rules for the generation of further rules. Which gives to the development of further implications and operations in that domain the feel of something being discovered “out there”.

But what exactly does mathematical “sense” consist in? Obviously, it depends on the evolved/inherited state of mathematical problematics. My sense of the matter, (and, mind, I only got to the level of basic calculus and statistics and that was IIRC a billion years ago), is that beyond the formal axiomatics of “proof”, (which famously have now been shown formally to have its limits), it tends to boil down, as others have indicated above, to the projection of point systems in n dimensions. And curiously, the development of new mathematics has mostly, though not always, preceded any (thought of) empirical application.

An interesting, if speculative, question is to what extent the intuitions of mathematical “sense” are influenced by general cultural forms of sense-making, the “metaphysics” of different cultures. Greco-Roman math had no concept of zero and indeed the culture also conceived of being as substance, as what is unchanged in all change, thus “eterrnal”. It was the Indians, whose meditative practices strove to remove attachment from the sensory world and achieve “oneness”, who invented the concept of zero and the Arabs who believed in a creation ex nihilo who transmitted and further develop its implications. And then there is the story about a medieval Chinese mathematician who wrote the most advanced algebraic treatise of that time, demonstrating a wide variety of methods all of which resulted in the same answer or solution, (rather than, as a presumably Western mindset, reducing a wide variety of solutions to a single method).

Another curious question is how are brains, which evolved largely as analog pattern-matching devices, and are not at all like digital computers, nevertheless can develop the ability to do advanced mathematics. Some of the mysteriousness and confusion attaching to the business and its status might derive from its “unnaturalness”.

But I don’t think a foundationalist account could be considered at all credible anymore.

Belle Waring, for what it’s worth, I just came to this page and read through the comments, and when I saw that Barbie reference I had no idea that the author was addressing a woman, or that it could easily be taken as sniping rather than something that had been pastiched in the Simpsons. I only realised otherwise when I saw your reaction. It’s quite possible that he (should I assume he?) was simply doing it recklessly and negligently rather than wilfully, either not knowing who you were or (if he had bumped into you in other contexts), just not making the connection, not putting two and two together as it were. You’d have to ask him (?) to find out if he (?) was even giving it a moment’s thought, at the time. Me, when I see “Barbie” (in any spelling), I tend to think “Klaus Barbie” and not “symbol of sexism in popular culture”, simply because I’m not a product of U.S. culture (apart from some influence at my very first school, which had a U.S. teacher – in Iraq).

Belle Waring, you’ve just jumped in and accused me of offending you. Clearly I must really have offended you, but you appear to have proceeded on the basis of jumping to conclusions:-

– You accused me of not actually reading any of this. Actually, I read all the comments in sequence, in full, but without memorising the cast of characters, rather paying attention to the matters raised. When I saw the Barbie reference, I literally did not recall that the person being addressed was female. For some reason, I hadn’t been paying much attention to whether readers were male or female until then; I misguidedly thought that the subject matter was more important.

– Since I hadn’t been paying any attention to that until then, and only then looked back to get the details straight, it occurred to me that the original writer might also not have been paying attention to that specific issue; attention, that is, to whether it would push your buttons, rather than being the very pastiche reference that I took it for, that was used in the Simpsons (Lisa versus Malibu Stacy, or something like that). I drew that possibility to readers’ attention, further suggesting that the only way to be sure was to ask.

– You accused me of being deliberately insulting, on the basis that you don’t need any fact checking to just know the truth of the matter, and that I knew it too. Well, I may have succeeded in offending you, but I can assure readers that I sincerely believed that my intention was only ever to raise another possibility, based on my no doubt faulty impression of how it had struck me. But having been told in no uncertain terms that it is offensive enough to warrant insulting me for just suggesting asking if it could possibly have been the inadvertent result of oversight, I see that that also is proof enough that I cannot have meant to advance enquiry but can only have been deliberately provocative; it seems I do not know my own intentions as well as others do.

And that is my Apologia pro sua. If it serves to condemn me yet further, for venturing to reply to righteous indignation, then I will know that the only acceptable reply to rage is to validate it by confirming its righteousness, and I will let others learn from my fate at your tongue and pen.

But there aren’t any squares! Outside of our minds that is.

Once someone drew a square it was there, surely; and with its own telos?

P.M.Lawrence: The exchange you’re commenting on was an actual conversation occurring between actual people. It also featured moderately-heavy, poor-quality trolling, which has since been removed.

At the time of the conversation, the participants were paying attention to each others’ identities.

It’s not surprising that you, as a non-participant who is reading the history of that conversation, take it differently. You can afford an ignorance of who-said-what.

Additionally, it’s an issue that has since been resolved and your posts look like ordinary trolling.

Lawrence@159 – “and I will let others learn from my fate at your tongue and pen.”

My goodness, that sounds dire !
I can picture it verse –

Some, from fallen Lawrence learning,
lashed themselves to the mast
while others succumbed insane
to Belles’s soft siren song …

Sorry, I just couldn’t resist – Lawrence’s response was too funny.

P.M. Lawrence: women who are insulted in any way on the internet are nearly always called upon to prove, not only that the insult occurred (and one gets plenty of pushback there) but also that the person intended to insult her and in precisely that way. Since this is, absent confession, almost impossible to obtain, women’s complaints can nearly always be dismissed with what has a surface appeal to neutral standards. “Maybe you misunderstood him?” “Who knows what he was thinking?” and so on. The important thing to remember is that, for the purpose of driving women out of public fora, it doesn’t matter what the various people intend by their comments. If the net effect is that after 50 dismissive comments along the Barbie lines a woman gives up, and stops asking questions about math, while an equally ignorant man will go on asking those same questions, it is a pernicious, awful thing–even if those 50 negative questions are the incredible result of a chain of improbable, totally innocent, unintended consequences and forgettings. It is also self-reinforcing: the fewer women there are who are willing to participate in comments on blogs, the fewer women will want to join up and start doing so as a n00b.

Here at CT it is difficult to forget I am a woman. I am not a random commenter of whom one might easily lose track but in fact an active front page poster. I am additionally mentioned by name IN THE POST. Your claims to be an open-minded, helpful skeptic whose assistance is being spurned by your poisonous hostess fail in various ways.
1) you say I “have” to do something. To review, a woman has been told not to feel stupid about not rediscovering set theory from scratch (sure, who would). But she has also been, as I noted, addressed with terms used by a highly-sexualized doll intended for 10-year-olds, which had to be recalled by the company after public outcry. This woman doesn’t have to do anything before taking offense. She can just plain take offense. You cannot require me to do anything further, such as get into a conversation with the other person and learn exactly what he was thinking at the time, before deeming it a situation in which it was acceptable to feel pique. I was named in the comment, with my name, Belle, which close to zero human males have as a given name, BTW. Hard to imagine how the whole “who was being addressed?” thing kept going so, so wrong for you.
2) Had you actually read the thread, as I noted above, you would have seen that Z was genuinely upset to think that–even for a moment–I might imagine he was saying that to me in all seriousness. He took it to be so offensive a joke that it was obviously, of necessity satirical, given that I probably knew who he was due to his long tenure as a commenter. He also apologized, and I said, ‘OK fine, I understand and am not angry with you, but I recommend that you only ever say this to women who are both personal close friends and set theorists.’ This was an exaggeration. They need not be set theorists. It’s not so cutting edge anymore, is it?

Now, after the discussion, his explanation, his apology, and my acceptance of his apology, you come along and decide to call me on the carpet because I am making assumptions about his thought processes, and tell me I ‘have’ to check what he intended. a) I don’t. b) Per impossible, if I did, I would just re-read Z’s comment 130, in which Z explains in considerable detail exactly what was going through his mind when he wrote his comments. Is there some reason that this step-by-step list of intentions fails to satisfy you, if I may ask?

It’s a fair accusation that I am easily provoked to anger and when provoked I say things that are more vehement and unpleasant than they need to be. This is a personal failing on my part and I should try to be a better person. However, it is also true that I am deliberately insulted all the time, on purpose, by men who are sexist. I know this sounds like really meager motivation, but, for real. So much of the happens. Now, of people who actively comment at our site, I have made a number of random polls at various times (I do know people’s gender in most cases) and I’ve never come up with anything more than 20% of comments by women in a given thread. Usually it’s worse. If I sit around and let random commenters employ the classic “just askin’ questions. Why are feminists so afraid of rational inquiry?” strategy I will doom our threads to worse. Do you see how Katherine, above, says she sort of wanted to respond to the Barbie comment, but didn’t? Maybe there were 20 Katherines reading the thread who wanted to respond to the Barbie comment too, but then never commented at all. This is the tragedy of the commons. Bad commenters will drive out good ones. If someone says something racist, we don’t need to “look into their heart” and see if they are “a racist” or “truly hate brown people.” We just need to acknowledge that that thing they said was racist. So when people say something sexist, it also doesn’t matter what they intended. In this case, I asked Z “what the fuck?” and he said, “oh God, I can’t believe you thought I really meant that; I’m sorry,” and I said, “OK.” Why did you then feel the need to muddy the waters, exactly, when they had been distilled in a way so unusual and pleasant and clear, if I may ask precisely what you intended, P.M. Lawrence? Since a full discussion had gone on and everything had been settled and everyone’s motives and misunderstandings had been made crystal clear, what made you want to tell me that I ‘had’ to do something before getting offended? If you did read the thread, why did you pretend not to know whether Z was a man, when he himself said that he was? Do you think someone named “Belle” is likely to be a man, given that you know it is not a pseudonym of any kind? If you think this is plausible, could you please produce some evidence for the assertion? If not, why did you pretend to have been unable to discern whether the Barbie comment was addressed to a man or a woman? I have more, more detailed questions about your motivations, and whether I may have misunderstood you, and why you said precisely the things you did, but if you would just answer all these questions it would be very elucidating. I know you only commented from the start in a helpful spirit, so I don’t imagine you’ll mind answering? I really can’t imagine what you were thinking, so give me a hand, please.

If I sit around and let random commenters employ the classic “just askin’ questions. Why are feminists so afraid of rational inquiry?” strategy I will doom our threads to worse.

Thank you, thank you, thank you for going after this kind of shit, swiftly, precisely, and without fail. It must be exhausting having to shoulder this kind of niggling burden each and every time you post or comment, but it really is much appreciated and makes this space safer for other women.

Saurs: I appreciate the support. It is a pain in the ass sometimes. All the times.
Z: It’s cool man. There were plenty of good comments. Don’t apologize to P.M. Lawrence, though, he’s trolling like a mofo.

Sorry Lawrence, I am just a long time lurker, but I am embarrassed at how folks are getting treated here lately. Good luck.