# 1 + 2 + 3 + 4 + 5 + … = -1/12

by on January 21, 2014

I find this confusing. (via Gizmodo.)

For the first time I’m going to write the following sentence. The top YouTube comment link is helpful. Or would be, if I were good at math. I trust.

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.)

This earlier video provides a nice introduction as well.

1

MPAVictoria 01.21.14 at 3:27 pm

“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.

2

dn 01.21.14 at 3:32 pm

This has been making the rounds on Facebook, and it’s silly. The sleight of hand is more fundamental than any “step” in the math – they’re pretending that the sum in question is convergent when it isn’t. 1-1+1-1+1-1+… doesn’t add up to 1/2 either. It adds up to nothing, not even zero. Not convergent.

3

Kaveh 01.21.14 at 3:35 pm

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.

4

dn 01.21.14 at 3:37 pm

Good explanation here.

5

Z 01.21.14 at 3:42 pm

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)?

6

David Steinsaltz 01.21.14 at 3:50 pm

There is no such thing as 1+2+3+… This is the kind of nonsense that physicists love, taking some formal rules and giving it a name that sounds like something people are familiar with (e.g., “strings”).

What they are talking about is that you can define a function that is equal to (1/1)^k+(1/2)^k+(1/3)^k+… for k bigger than 1, but also has values defined for other k. If you look at what it is for k=-1, the result is -1/12. But this has nothing to do with anything a reasonable person might call 1+2+3+… Even if that person were a physicist.

It’s not that they did anything wrong — though you could make an equally convincing video to prove that it’s a different number — but it’s basically just mystification.

7

Petar 01.21.14 at 3:54 pm

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.

8

dn 01.21.14 at 3:56 pm

Wikipedia shows that the video is in fact even wronger than Kaveh is saying; if you scroll down, it indicates that the Cesaro sum of 1+2+3+4+5+… is also not convergent, i.e. it equals nothing. The creators of the video pretended that you could equate the Cesaro sum with the actual sum and then do algebra with it, which is not the case. (Again, 1-1+1-1+… does not equal 1/2 either. You can’t equivocate between “does not exist” and “is 1/2”.)

9

Belle Waring 01.21.14 at 3:57 pm

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?

10

Belle Waring 01.21.14 at 3:58 pm

I mean, I know this is intended to be some sort of helpful illustration, and not anything like a proof, but it just seems so totally confusing that I don’t know at all what to think.

11

Ed Herdman 01.21.14 at 4:04 pm

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.

12

Bloggers have been all over this. In addition to the link in 4, this is pretty helpful.

I think part of what irritated people is that the reason the folks in the original video ended up misrepresenting things is that they were overly eager to arrive at a facile “Ha! Look how crazy and counter intuitive math is!” moment. Sort of like the way the writers at Slate would teach math, if they were to try.

13

Kaveh 01.21.14 at 4:07 pm

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’.

14

elm 01.21.14 at 4:08 pm

This presentation is pretty deeply misleading.

The misdirection starts when they arrive at “1 – 1 + 1 – 1 + 1 …” = 1/2

That is not accurate. It’s wholly inappropriate to use equality for the process that they employed. My own two semesters of university calculus tells me that the series “1 – 1 + 1 – 1 + 1…” does not converge, so its sum has no particular value.

Any process that builds on a misstep like that in mathematics is simply incorrect and flawed.

Belle @8,9: You’re absolutely correct. Shifting terms and adding series in that way is a tricky business and the presenters in the video play very fast and loose with how they do it.

There is a sense in which you can get the result they present — that the sum of natural numbers is -1/12 — but it involves analytic continuation and taking a series outside of its circle of convergence. It’s also true that some elements of quantum physics, like QED, rely on similar mathematically-nonsensical operations.

However, if they had arranged their presentation somewhat differently, they could have arrived at any answer whatsoever, so their presentation has nothing to do with that bit of useful nonsense.

15

Rob in CT 01.21.14 at 4:09 pm

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).

16

David Steinsaltz 01.21.14 at 4:13 pm

#9 is exactly right. There is a true mathematical fact at issue here, but this is the sort of thing that gives “formal proof” a bad reputation. It’s not a procedure that can reliably give the right answer, and it doesn’t clarify any important principles. On the contrary, it seems intentionally obfuscatory. The intention seems to be primarily to provoke a reaction like that of #1 here: You must be pretty smart if you can make sense of this kind of weird shit.

17

Aaron 01.21.14 at 4:15 pm

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.

18

dn 01.21.14 at 4:15 pm

What AcademicLurker said. I’m not a mathematician, but I do enjoy math. It frustrates me when people come up with misleading sh*t like this to try and portray math as some deep mystery beyond the ken of mere mortals. Math is challenging, but it’s not out to tear down all your intuitions.

19

dn 01.21.14 at 4:19 pm

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

20

Warren 01.21.14 at 4:19 pm

There is a theorem in calculus/real analysis, the Riemann Series Theorem or Riemann Rearrangement Theorem.

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

If you take one of these series with infinitely many positive terms and infinitely many negative terms, strictly speaking a “conditionally convergent” series, you can rearrange them to get any real number which you want. Good for paradox but not good mathematics.

21

mattski 01.21.14 at 4:23 pm

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?

22

Walt 01.21.14 at 4:37 pm

This video is the worst thing ever to happen. Yes, worse than the Iraq war or the series finale of Battlestar Galactica. The argument in the video is 100% bullshit. There are settings where you can assign a value to divergent series, but there is no unique universal way of doing so. For example, in some settings if you assign a value to 1 + 2 + 3 +…, it’s -1/12. In other settings it’s obviously +infinity. (What’s weird about the video is that they use Cesaro summability for 1 – 1 + 1 – 1…, and if you apply Cesaro summability to 1 + 2 +3 +…, you get +infinity.)

23

elm 01.21.14 at 4:43 pm

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.

24

dn 01.21.14 at 4:44 pm

mattski – Imaginary numbers (complex numbers) and irrational numbers are not the same. Irrational numbers, at least, are very conceivable and you can see them every day, not just in weird physics. The square root of 2, for example, or pi, both of which can be easily illustrated geometrically. Complex numbers are a little more tricky, but they don’t really replace your intuitions about real numbers; they’re just a different class of number that you can also work with in well-defined and often useful ways. This demonstration, on the other hand, plays fast and loose with definitions/assumptions to produce a result that conflicts with our intuitions about natural numbers.

I vaguely recall reading of an exchange between Wittgenstein and Russell, in which Wittgenstein essentially criticized Russell and Whitehead’s attempt to ground arithmetic in logic by arguing that they were reasoning the wrong way: if the logic produced an arithmetic that conflicted with our ordinary understanding of how natural numbers behave, we would take this not as a demonstration that we were doing the math wrong, but as an indication that there was something wrong with the logic. (I may be mangling Wittgenstein here. I’m not any more a philosopher than I am a mathematician.)

25

Anonymous 01.21.14 at 4:47 pm

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.

26

MPAVictoria 01.21.14 at 4:50 pm

I love that there are people out there who actually understand this. I am quite jealous.

27

mattski 01.21.14 at 4:50 pm

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.

28

elm 01.21.14 at 4:56 pm

mattski @ 24:

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.

To paraphrase Richard Feynman — How can you say it is inconceivable when you have already conceived the idea? It may be counter intuitive, or you may not like it, or it may be hard to visualize it, but you have already conceived it.

29

Katherine 01.21.14 at 5:02 pm

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.

30

Kaveh 01.21.14 at 5:04 pm

@21 Ditto that, I found complex numbers more approachable than a lot of other things I studied in math, back in the day. I was really sold when I learned that you can use them to get the mathematical function for the bell curve.

31

Kaveh 01.21.14 at 5:05 pm

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.

32

dn 01.21.14 at 5:08 pm

mattski – depends what you mean by “inconceivable”. A person might not, in ordinary life, know where to look for a complex number, or how to “picture” it. But “the square root of -1” is, in another sense, entirely conceivable; It’s just a number that, when multiplied by itself, yields -1. (For that matter: can you conceive of -1 itself? How would you draw me a picture of it? Even the concept of zero took hundreds of years for mathematicians to arrive at.) The imaginary unit may strike us as a strange number, but as I see it, it behaves as consistently as any other number and it doesn’t contradict my intuition about what 2+2 equals.

33

Futility 01.21.14 at 5:09 pm

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

34

dn 01.21.14 at 5:16 pm

In another sense you CAN picture a complex number; you just reimagine the “number line” as a plane, a two-dimensional space. You can draw geometric figures in this space and use them to illustrate some conclusions about complex numbers that would seem odd if simply notated symbolically. A unit circle drawn in the complex plane, for example, provides a visualization for the close relationship between exponential and trigonometric functions which can initially blow your mind if you’re only shown the equations.

35

Another lurker 01.21.14 at 5:19 pm

@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.

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

36

Another lurker 01.21.14 at 5:21 pm

An example of a non absolutely convergent series is:
1-1/2+1/3-1/4+…

37

mattski 01.21.14 at 5:23 pm

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.)

38

kent 01.21.14 at 5:26 pm

I am not a mathematician, but:

Take 1+2+3+4+5+6+… , and call it S1
Then subtract it from itself, but just “move it along a little bit,” as follows: 0+1+2+3+4+5+…

1+2+3+4+5+6 ….
MINUS
0+1+2+3+4+5 ….

1-0 = 1
2-1 = 1
3-2 = 1

Result: S1 – S1 = 1+1+1+1+1 …

As a result, either S1 does not equal S1 [a clear contradiction] — or else 1+1+1+1… = 0 [which is stupid].

I conclude that if you are allowed to “just shift this along,” you can get pretty much any result you want. Thus “just shift this along” is not a valid mathematical procedure in this type of case. And thus the whole thing is bogus.

39

Mario 01.21.14 at 5:38 pm

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.

40

MattF 01.21.14 at 5:47 pm

Here’s the important point– what you get when you ‘sum’ an infinite series is a matter of definition. One definition may give you the answer “You can’t do that”. But another definition may give a different answer, e.g., -1/12.

One way to look at it is to ask “What do those three little dots mean?” They cannot mean “Do some operation an infinite number of times”, since that would take forever– and we don’t have forever to wait for an answer. One better possibility is that the three little dots mean “Seek professional advice”, e.g., ask a mathematician. In this case, a mathematician may think something technical like “Well, maybe analytic continuation could make this meaningful” and then attempt to explain that in a non-technical manner.

41

elm 01.21.14 at 5:48 pm

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.

42

mattski 01.21.14 at 5:54 pm

Also, back to the OP. The following is impressionistic and strictly amateurish but, isn’t there a symmetry between the distance from 0 to 1, and the distance from 1 to infinity? So, can we visualize numbers in general as a sort of oscillation on between these two “regions?” And does that take some sting out of the paradox?

43

bianca steele 01.21.14 at 6:07 pm

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

44

Nine 01.21.14 at 6:12 pm

dn@30 – “A unit circle drawn in the complex plane, for example, provides a visualization for the close relationship between exponential and trigonometric functions which can initially blow your mind if you’re only shown the equations.”

Not sure if this will convince mattski but equation is
Euler’s identity

45

dn 01.21.14 at 6:18 pm

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.)

46

bianca steele 01.21.14 at 6:32 pm

It supposed could be read as a satire on the difference between literary and scientific cultures. Except it reminds me more English physicist piss-taking in the vein of the coolth ray experiment (which I’ve never heard mentioned by an American but actually don’t remember well enough to repeat here).

47

bianca steele 01.21.14 at 6:33 pm

“It suppose” s.b. “I suppose it”

48

Billikin 01.21.14 at 6:34 pm

1/2 = 0.

Proof:

S = 1 – 1 + 1 – 1 + . . . = 1/2 (Already proven.)

S = 1 – 1 + 1 – 1 + . . .
= (1 – 1) + (1 – 1) + (1 – 1) + . . .
= 0 + 0 + 0 + 0 + . . . = 0

S = 1/2 = 0 QED.

49

OCS 01.21.14 at 6:44 pm

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?

50

Billikin 01.21.14 at 6:49 pm

Belle Waring: “I know this is intended to be some sort of helpful illustration”

No, it is not. It is intended, as Castaneda’s Don Juan put it, to astound the Indians.

I suppose that there is some interpretation of 1 + 2 + 3 + . . . = -1/12 which makes sense in physics. They did show a text in which that equation appears. However, they did not then provide any such interpretation. Instead, they went off into Fallacy Land.

51

MattF 01.21.14 at 6:54 pm

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/

52

christian_h 01.21.14 at 7:12 pm

It should be pointed out that the first computation of the value of the Zeta function at -1 (which is really what these guys mean when they say “1 + 2 + 3 + … = -1/12, as others have pointed out before) is due to Leonhard Euler; and that his calculation (which is heuristic, as no complex analysis was available at the time) can on the one hand be turned into nonsense as in the video – and on the other can be made into a proof by re-thinking it using 19th century methods. It would have been educational to make a video explaining this – without the context it is misleading and as has been said, a mystification of maths. I have to run, but I will try to find Euler’s original argument later today. Here is a link sketching it in modern terms:

http://people.reed.edu/~jerry/311/euler.pdf

53

Z 01.21.14 at 7:19 pm

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.

54

dn 01.21.14 at 7:19 pm

OCS @44: The imaginary unit is, in a sense, “positive” in that there is also such a thing as -i, whose relationship to i is analogous to the relationship of -1 to 1. [More specifically, -i=(-1)i.] The mistake is to try to locate both 1 and i on the same axis, because 1 is real and i is not. They’re perpendicular.

When people say imaginary numbers are unintuitive, their intuitions would best be characterized as “not even wrong”. Your intuition is fine; it’s just not precise, because it ordinarily doesn’t have to deal with such objects as complex numbers. Eventually you train yourself not to privilege the real axis and the distinction no longer worries you. They are both just axes, and they both behave pretty much exactly the same, except that one has an extra symbol attached to it and the other doesn’t.

55

mud man 01.21.14 at 7:41 pm

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.

56

mbw 01.21.14 at 7:42 pm

@49 It’s pretty misleading to say that the real and imaginary axes are “pretty much exactly the same”. The real axis forms a field, you get a real number when you multiply two reals. The imaginary axis doesn’t. They’re very different.

That particular side discussion reminds me of when mathematicians in the early 60’s wanted to set a whole new curriculum. IIRC ordinary 12 year olds were supposed to learn the complex numbers as the ring of polynomials mod (x^2+1). Slower kids would just have to learn them as the set of all ordered pairs with a particular multiplication rule.

57

Walt 01.21.14 at 7:50 pm

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).

58

Odm 01.21.14 at 7:54 pm

mud man: As people have pointed out upthread, when an infinite series is divergent (the partial sums do not approach a number), commutativity no longer holds.

From what I remember, when a series is convergent, then you can rearrange as you please.

59

elm 01.21.14 at 7:58 pm

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.

60

Z 01.21.14 at 7:58 pm

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.

I’m afraid that’s not true, as has been pointed out before in the thread already: commutativity does not extend to infinite sums. He picked this particular pairing because he wanted to arrive at the correct result, and he did, though by an entirely wrong derivation, but Belle’s pairing was as justified (and in fact it is not hard to prove Walt’s assertion above that one could have arrived at any real value by choosing carefully a rearranging).

As an aside, though it is true that the fact that the result is correct has a (not so easy) physical interpretation, I want to insist that this is a result of pure math, that it predates the physical interpretation by centuries and that the investigations of such results is ongoing and still considered very much central (or so says someone who earns his living doing it anyway). If I were to make a comparison, I would say this identity is to math what Newton’s apple is to physics (in particular, it is much more important than the more well-known e^(iπ)+1=0 which, though cute, is completely trivial).

61

Phil Koop 01.21.14 at 8:14 pm

@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.

62

bianca steele 01.21.14 at 8:51 pm

Really, all that happened is that someone read a technical book under the assumption that it’s unnecessary to have any theoretical or technical vocabulary, or to read books in one field differently from books in another. The person of technical bent runs up against some confusion, and stops to figure out what he’s missing. The non-technical person feels the text should be written in plain language and interprets everything in that light regardless of context. The results are very different. When the latter type of person finds a contradiction, he blames the text, and that’s all.

But I’ve found another proof that 1-1+1…=0.5. S is the sequence. Rewrite as 1-(1-1+1…). Then S=1-S, and S=0.5.

63

Z 01.21.14 at 8:53 pm

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.

64

Just some commenter 01.21.14 at 9:00 pm

@mattski, @Phil Koop:

For me, the key to understanding i was to see a geometric explanation of addition and multiplication on the complex plane. I can see how if you’re told only that there’s a defined quantity called i which equals the root of -1, but you’ve never grasped the mechanics of how it works to do arithmetic with it, it might seem incomprehensible that one can do arithmetic with it.

The key is to understand (as others have explained) that i exists on a second axis, defined to be at right angles to the real number line, with both axes together defining the complex plane. Any point on that plane is a complex number, with a real and imaginary part, and multiplication of the numbers in this plane proceeds exactly as before for the real numbers, all lying on the real number line, but via a *rotational* procedure for complex numbers (even better, the real number multiplication is just the trivial case of this rotation — a rotation of zero degrees). Working through some examples, and learning how you can do the multiplication either through a strictly algebraic procedure, or though an exactly equivalent rotational procedure based on the angles and magnitudes of vectors can make this somewhat more intuitive again.

Once you understand how that works, and you can see that it’s an expansion of the concept of multiplication that’s wholly consistent with everything you learned about operations on real numbers, it seems like a brilliant and useful — but not mysterious or imponderable — expansion of the concept of numbers, because it allows for arithmetic with previously undefined quantities, and solutions to previously unsolvable equations.

65

Just some commenter 01.21.14 at 9:07 pm

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.

66

elm 01.21.14 at 9:14 pm

I’m not familiar with the technical lingo in this language, but colloquially, is 1-1… really ‘divergent’? it seems more like ‘oscillating’

It’s not appropriate to intermix colloquial and technical language. In the technical sense, that series is divergent.

And if some value is oscillating, then the center of it seems like a reasonable approximation of its value.

Mathematical reasoning does not stand on how things seem. This is especially so when working with infinite or infinitesimal quantities.

Furthermore, there’s a world of difference between equivalence and approximation.

67

P O'Neill 01.21.14 at 9:16 pm

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

68

Bruce Wilder 01.21.14 at 9:29 pm

bianca steele @ 55

Really, all that happened is that someone read a technical book under the assumption that it’s unnecessary to have any theoretical or technical vocabulary, or to read books in one field differently from books in another. The person of technical bent runs up against some confusion, and stops to figure out what he’s missing. The non-technical person feels the text should be written in plain language and interprets everything in that light regardless of context. The results are very different. When the latter type of person finds a contradiction, he blames the text, and that’s all.

I just flashed on CT comment threads being replaced by a programmed loop spiralling robotically into an infinite future, to end only with the final dimming of the sun . . .

69

Theophylact 01.21.14 at 9:35 pm

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.

70

dn 01.21.14 at 9:52 pm

Re: terminology, I think calculus would be a lot more fun if we’d stuck with Newton’s terminology: not differentials, but “fluxions”.

71

bourbaki 01.21.14 at 9:53 pm

@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.

72

Collin Street 01.21.14 at 9:55 pm

@theophylact.

Rational <- ratio, no? a number that isn't a ratio, then, a non-ratio number, must be…?

And "imaginary" came in contrast to the label "real".

What else would you suggest calling either of them?

73

Billikin 01.21.14 at 10:01 pm

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.”

74

Just some commenter 01.21.14 at 10:06 pm

@Collin Street, hmm, what about “second dimensional”? Of course, that might seem to include all complex numbers, not just the imaginaries, but perhaps the reals could be called the “first dimensional” numbers, the imaginaries could be called the “second dimensional” numbers, and complex numbers generally could be “two dimensional”, or planar numbers. Fairly descriptive, I think.

Of course, “surreal numbers” and “hyperreal numbers” are already taken.

75

elm 01.21.14 at 10:22 pm

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

Mathworld’s definition of divergent series.

76

mud man 01.21.14 at 10:22 pm

Thank you all, I am corrected.

77

elm 01.21.14 at 10:25 pm

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

78

dn 01.21.14 at 10:35 pm

elm @78: I love that quote from Abel from your link: “The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.”

79

Ed Herdman 01.21.14 at 10:38 pm

#71 and “half dead cats,” meet #68.

80

Ed Herdman 01.21.14 at 10:39 pm

elm: I would also stipulate the attitude “Mao Cheng Ji in a math discussion? HA! HA! HA!” *places monocle*

81

maidhc 01.21.14 at 11:01 pm

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.

82

ezra abrams 01.21.14 at 11:09 pm

1) assume the universe is infinite

2) in any given volume (say our solar system) there are so many ways of arranging all of the matter and energy – a large, but finite number; call in X (I’m neglecting transport in and out of hte volume – perhaps a fatal error)
X is the number of ways all the matter and energy in our solar system can be arranged.

3) If (1) is true, then there exists another volume of space with X identical to our own.

4) infact there are an infinite number of places where x is identical – an infinite number of solar systems *exactly* like this one

5) let Xa be a slightly different arrangement of matter and energy; say where in English the letter E is written as square C
6) there are an infinite number of places where there is a solar system exactly like our own, except E is written as C

Infinity is so much fun

83

Katherine 01.21.14 at 11:34 pm

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

84

christian_h 01.22.14 at 12:37 am

Z @64 (8:53): I do not disagree with anything you write there – just expressed myself badly being in a hurry and all… so no controversy, sorry ;) The point I was trying to make is that while the video gives an incorrect argument, it is an argument based on a bad understanding of Euler’s proof obtained by substituting 1 in for t in the logarithmic differentials. So the video is a chance missed, not a random assault on reason.

85

mattski 01.22.14 at 1:03 am

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.

??

86

mattski 01.22.14 at 1:09 am

Also, Katherine. I think you fucked up. It is 17.

You will find the proof encrypted here.

87

Just some commenter 01.22.14 at 1:50 am

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.

88

bourbaki 01.22.14 at 1:57 am

@88

One thing that might help (if you are interested) is to recognize that numbers have many different way of being thought of that preserve there underlying algebraic structure. I actually think that saying i is the square root of -1 obscures things and a more geometric perspective helps (though I study geometry so am biased).

[warning this is a bit pedantic]
For instance, one way to think of the real numbers is as an “action” on the set of points on the real line itself (this may seem pointless at first but I think the analogy is useful). Under this action multiplication by a positive number x acts on a point p on the real axis at distance |p| from 0 by stretching it out so it is at distance x|p| from 0. Multiplication by a negative number does the same thing, only it also reflects across the origin. Multiplication by 0 sends everything to 0. What one observes that acting by x*y is the same as acting by y and then by x.

Now you could just as easily imagine this action occurring on the plane. In this case multiplying a point p in the plane at distance |p| from the origin by a real number x gives you a point that is distance x|p| from the origin and lives on the line through the point and the origin (negative x again reflects while positive does not and 0 maps to the origin). I.E. we again scale. Now in the plane we also can think about rotations. In this picture the action by i the squareroot of -1 is by rotation by 90 degrees about the origin. The reason for this is if we act twice then we get rotation by 180 degrees about the origin which is the same as multiplying by -1 that is i*i =-1. In general, any transformation of the plane that is a composition of a scaling and a rotaiton about the origin can be thought of as a complex number — i.e. a number of the form x+i y where x and y are real. The algebraic multiplication of complex numbers then corresponds to composing the transformations. Obviously one has to do some work to make sure this is completely justified but it is not too hard.

What is interesting is that something along the same lines happens also in dimension 3 and one can construct an “extension” of the complex numbers to something 4 dimensional (but where one no longer has commutativity). These are called the quaternions.

89

Just some commenter 01.22.14 at 1:58 am

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.

90

mattski 01.22.14 at 2:15 am

@ 90, 91, 92

Many thanks!

91

Alex K. 01.22.14 at 2:16 am

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.

92

Jason 01.22.14 at 2:16 am

Terry Tao is awesome.

To the proof that 1/2 = 0 by Billikin above: you are assuming associativity in infinite series, which is false.

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

Also the -1/12 is the reason bosonic string theory is only consistent in 26 dimensions; the 12 becomes the 24 in the condition (D – 2)/24 = 1 [or something like that].

If you want pretend you can add an infinite number of things, weirdness is what you get.

93

Belle Waring 01.22.14 at 3:00 am

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.

94

Belle Waring 01.22.14 at 3:15 am

Ed Herdman, elm. No need for monocles. We have banned Mao, but unlike Hector who has quite politely excused himself when asked…

95

Chris Warren 01.22.14 at 3:34 am

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.

96

Alex K. 01.22.14 at 3:56 am

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

Actually, P is chosen precisely as the midpoint of AB — that’s how P is defined. There is no trickery there.

97

PHB 01.22.14 at 4:01 am

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…

98

Chris Warren 01.22.14 at 4:04 am

Alex K

If P is the midpoint, then the perpendicular will not intersect with the ray from the angle opposite AB.

The perpendicular bisector of AB will pass to the left of the ray if angle A angle B.

The bisector of angle C can cut a perpendicular to AB, but there is no reason why this should pass through the midpoint of AB.

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

99

Alex K. 01.22.14 at 4:08 am

“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.

100

Alex K. 01.22.14 at 4:09 am

There is also no mistake in reasoning, given the drawing.

101

Chris Warren 01.22.14 at 4:13 am

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.

102

Alex K. 01.22.14 at 4:15 am

“Try drawing an triangle with sides 1, 8, 10.”

An unfortunate choice of sides surely?

But, you don’t need to overexplain the thing.

103

Nine 01.22.14 at 4:24 am

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

104

Chris Warren 01.22.14 at 4:35 am

Alex K.

There is also no mistake in reasoning, given the drawing.

Yea – and that is how they teach economics at university to students who then go on to run the country!!!!

105

Lee A. Arnold 01.22.14 at 4:36 am

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

106

Chris Warren 01.22.14 at 4:37 am

Nine

I blame you – why didn’t you mention this earlier ????????

107

Eric H 01.22.14 at 4:44 am

@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.

108

Lucie Rie Mann 01.22.14 at 4:44 am

When I was a budding string theorist this used to be the Riemann Zeta function, and Zeta of -1 was indeed -1/12. Has that ceased to be the case? Is analytical continuation now considered Bad Math?

109

Nine 01.22.14 at 4:47 am

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 ?!!!

110

Alex K. 01.22.14 at 5:15 am

“Yea – and that is how they teach economics at university to students who then go on to run the country!!!!”

Maybe it’s about time that this thread turns into a discussion of socialism.

111

Robert 01.22.14 at 6:10 am

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?

112

Lee A. Arnold 01.22.14 at 6:23 am

I think there is a conceptual problem in the step of adding an infinite sequence to itself, calling it 2S, then dividing though by the 2 later. Because if you add infinity to itself, I don’t think you get 2x infinity. If you think you can do that, then it seems to me that you are claiming to have disproved Cantor’s continuum hypothesis.

113

peter ramus 01.22.14 at 6:38 am

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.

114

The Raven 01.22.14 at 7:27 am

peter ramus@115: indeed that series converges to 1. This can be seen by noting that, for any partial sum of the series, there will always be another partial sum further on in the series closer to 1, and no matter how small a number one chooses (denote that number by δ) there will always be a partial sum in that series that will be closer to 1 than 1-δ. The “trick,” if one can call it that, is to never make a direct statement about infinite values, but instead to observe that the partial values of the sums, always the sum of a finite number of terms, get ever closer to the limit value.

For the rest, this is in fact very difficult mathematics, and it kept brilliant mathematicians happily occupied for around two centuries. The best mathematicians were able intuitively to arrive at valid conclusions, but it took a long time for them to justify those conclusions with rigorous logic.

115

Niall McAuley 01.22.14 at 9:21 am

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.

116

dk 01.22.14 at 11:02 am

Mattski @ 87, you might be interested in the idea of ” computable numbers”.

117

Katherine 01.22.14 at 11:28 am

Mattski @88 – nope, 47

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

118

Belle Waring 01.22.14 at 2:17 pm

I meant to say, Z, with regard to your comment above (“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”), what in the ever-loving blue-eyed fuck? Are you trying to provoke my righteous wrath for some reason? Why, when you can just wait around ten or fifteen minutes? You ain’t got to tip the jar onto your own head. The jar that’s full of wasps, all of them shiny new, glistening yellow and black, just crawled out of their paper hexagons.

119

Katherine 01.22.14 at 2:19 pm

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.

120

Mao Cheng Ji 01.22.14 at 2:23 pm

Excuse me, why are my comments disappearing? Have I missed something? What’d I do?

121

Z 01.22.14 at 2:47 pm

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).

122

Belle Waring 01.22.14 at 2:51 pm

You’re banned Mao, I thought I had told you this earlier, but now I’ve extra double-dog told you with a giant post about it. GO. AWAY.

123

Belle Waring 01.22.14 at 2:52 pm

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

124

Z 01.22.14 at 2:53 pm

Also, in the interest of full disclosure, I append the post I had written before noticing Belle’s comment at 118.

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?

Spot on! Justifying this computation by appealing to String theory is exactly having it backwards. This is because mathematicians have found a way to make sense (and beautiful sense) of these computations that (many centuries later) physicists could envision such exotic (and in fact still largely controversial) model of space-time, not the other way round. And there are far, far fewer people who can give any reasonable argument about the existence of these extra-dimensions that people who can either give a rigorous proof of these identities or explain in lay terms why there are true.

125

Belle Waring 01.22.14 at 2:57 pm

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?

126

Z 01.22.14 at 3:00 pm

I promise I shut up after this one. Belle, please consider that the following question is absolutely completely sincere.

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

Is that insulting your mathematical ability or your intelligence (damn, I just checked the OED so fearful I was that “diminish” did not mean the same thing in English and in French)? I honestly intended it as an encouragement. Or is it like completely and totally the Barbie quote doing the bad work here?

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Belle Waring 01.22.14 at 3:01 pm

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?

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Belle Waring 01.22.14 at 3:03 pm

You might have gotten away with the latter (though it still smacks of ‘nice try, kiddo! someday you might even learn trigonometry!’) had you not included the former. I’m going to bed now too so I don’t wish to argue about it. Consider carefully though, is this how you treat the female students whom you wish to encourage in math?

129

Belle Waring 01.22.14 at 3:08 pm

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.

130

Z 01.22.14 at 3:14 pm

Belle 125. Ok! I understood. Here is, to be crystal clear, what I intended.

1) The video is a jumble of confused thing.

2) You, Belle, saw this and correctly remarked that if you were allowed the same power as the guy in the video, you could get different results.

3) You also expressed confusion about it all.

4) I meant to congratulate you on your step 2) and to reassure you that it is normal to have troubles with step 3). Contrary to many fallacies (like the divide by zero one), this one is not easily corrected and the right way to argue with these objects is not obvious at all. I don’t think any human being could devise it from scratch.

5) But what I managed was an insult on your step 2 (like *Poor Belle, she cannot even see why her reasoning is wrong at step 2*) which makes my comment later on the thread that your step 2) was correct all the more unfair.

My apologies. Please believe me that this what happened. The Barbie quote is quasi-legendary among people involved in math education, as is xkcd 385, and I honestly did not think that you could have thought I was quoting it in some literal way. But I guess we commenters are not as well-known that maybe we wished we were (I had perhaps the illusion that after 9 years or so of commenting on CT, I had somehow managed to show that I wasn’t a complete asshole).

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Belle Waring 01.22.14 at 3:29 pm

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.

132

Belle Waring 01.22.14 at 3:34 pm

Except now! Now, ironically, you can say the Barbie thing to me all day long and I won’t care. No, we do remember our long-time commenters, but you should consider that there are a ton of trolls who burn out like brief candles, often being remarkably awful on the way. I mean, I was joking in the above post but IRL we have about 450,000 comments on this blog…

133

Lee A. Arnold 01.22.14 at 4:33 pm

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.

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GiT 01.22.14 at 5:05 pm

For accuracy’s sake, I believe it is Malibu Stacey who says, “Math is hard. Let’s go bake cookies for the boys.” The (recalled) barbie doll satirized says, “Math class is tough.”

135

alkali 01.22.14 at 5:40 pm

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.

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mattski 01.22.14 at 5:44 pm

bourbaki @ (what is now) 88

For instance, one way to think of the real numbers is as an “action” on the set of points on the real line

Yes! I find this helpful. And indeed @ (what is now) 37 I offered a similar idea. Thinking of numbers as actions or processes is in a sense de-mystifying.

Better not to think of numbers as ‘things.’

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mattski 01.22.14 at 5:45 pm

Katherine @ 117

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

138

mattski 01.22.14 at 5:48 pm

*Can’t resist:

Alex K @ 102

That was wicked!

139

elm 01.22.14 at 6:14 pm

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:

Cancel the 6s: 16 / 64

Result: 1/4

140

Ragweed 01.22.14 at 8:13 pm

Belle @ 122-3 -I understand why you made that announcement sans comments. But let me say that I had a momentary flash of a standing ovation at the end of it.

Though, really, disemvowling is much more fun.

141

Bloix 01.22.14 at 8:13 pm

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.

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.

142

robotslave 01.22.14 at 8:34 pm

Lucie Rie Mann @108

If more budding string theorists were a bit more attentive in their pure mathematics courses, we might have more pure mathematicians, and fewer former string theorists.

Any attentive student of complex analysis will tell you that Riemann’s analytic continuation of the zeta function extends the domain to the entire complex plane except s=1, where it is undefined.

There are of course several methods of calculation, which happen to be convenient for various mathematical models in physics, that will produce a “value” of -1/12 for the zeta continuation as it approaches s=1, but this does not change the fact that the continuation is undefined at that point.

143

Walt 01.22.14 at 9:49 pm

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.

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Walt 01.22.14 at 9:56 pm

robotslave: 1 + 2 + 3 + … is zeta(-1), not zeta(1).

145

js. 01.22.14 at 10:08 pm

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.

146

Trader Joe 01.22.14 at 10:10 pm

Bloix @141′
Thank’s for the reminder about Gamow (I’d never have remembered the name)…I’m 100% sure that I have that book and have kept it all these years as something I know intuitively is too good to part with even though it surely must be >30 but < infinity years since I've last consulted it. What little I remember about topics such as these (which I find interesting in that brain teaser sort of way) were learned from that book….

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Larry Gonick 01.22.14 at 10:53 pm

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/.

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Bloix 01.22.14 at 11:37 pm

#143 – “The effect of the video is actually remove mathematical knowledge from the world.”

Yeah, I completely agree with this. Even worse – the whole attitude seems to be intended to make people conclude that math is for assholes. So why should we listen to experts about global warming? Evolution? GMO? Better to stick to the Bible.

#146 – My father, who was a generation or so younger than Gamow, met him several times and had tremendous respect for him. He gave me “1,23 …” (among other books like “Electronics for Boys and Girls” and “Flatland, by A. Square”) in hopes that I would be a scientist, too. Alas, I am a mere lawyer, but at least I don’t have the fear of science and math that many of my compatriots do.

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Ed Herdman 01.23.14 at 1:43 am

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.

150

TM 01.23.14 at 5:41 pm

“To frequently these days smart people are taken in by the “Malcolm Gladwell” effect. A desire to explain something outside of their field of interest with a simple counter-intuitive solution.”

I also tend to agree with 143, except I must note that worse things than stupid youtube videos *have happened*. What I agree with is that the video promotes the widespread view that Math is mostly arbitrary and useless. Now, if you are a math teacher, how are you going to convince your student that that result they came up with really makes no sense and here’s why? They might just say, duh, if 1+2+3+…=-1/12, then why shouldn’t my percentage be more than 100, or my probability negative, or this sum of squares come out negative?

Some of us try hard to teach students how to distinguish a meaningful question from a pointless one, and how to recognize when a result obviously makes no sense. The sum of all natural numbers is a meaningless question for normal purposes, and a negative number is a nonsense result for a sum of positives, unless you operate in a specific theoretical context that needs to be defined and explained before you can use it meaningfully. What is displayed in this video takes common sense out of math.

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JimV 01.23.14 at 6:50 pm

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.

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mattski 01.24.14 at 2:06 am

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.

I enjoyed it too, and not being expert in math, wasn’t offended by it’s sloppiness. Well, except for the ‘shift’, which did seem difficult to justify. That looked analogous to phase change, which–duh!–can either amplify or cancel out.

But I come back to the idea that numbers aren’t about the physical universe, despite the fact that they’re useful for describing it. Numbers, I’m thinking, are a human activity, more like baseball than gravity.

153

Belle Waring 01.24.14 at 2:22 am

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.

154

otpup 01.24.14 at 3:36 am

Belle, I think math people (especially that branch of applied mathematics called physics) are more platonists than you might think. There is that commonplace due to Wigner of the “unreasonable” success of heavily mathematical theories predicting things in the real world that were too wild to imagine. Now maybe that view point is wrong in some way, but it does effect the perspective of many, many people in the math and physical sciences. And many math people may eschew the platonist stance despite what they might really believe or suspect because they tend to be more humble than physics people and/or don’t want to take on the moral baggage that physics people inherited with nuclear weapons. “Math is just our little sandbox, not part of the real world and no effect on it.” Yeah, right.

155

john c. halasz 01.24.14 at 4:08 am

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.

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Belle Waring 01.24.14 at 9:39 am

The Mayans independently discovered zero. I actually considered writing about Sanskrit mathematics in grad school, not because I’m a math hot shot or anything, just because lots of things are both untranslated and not paid attention to by the non-mathematician Sanskrit readers in India itself. There are millions there, obviously, many with what’s pretty much native-level fluency, but they read it for religious reasons, not for historical mathematical research reasons. I would have had to learn Pali too, eh. My thinking was that I would translate it, ask mathematician friends what it was about if I really couldn’t determine it, and then see whether there were unknown results. Not unknown in the world, but more like, previously discovered by mathematicians writing in Sanskrit and then forgotten as the language user base switched so heavily to religious and ritual purposes. I discarded the idea as too practical.

157

P.M.Lawrence 01.24.14 at 10:02 am

158

Belle Waring 01.24.14 at 10:52 am

P.M. Lawrence, stop trolling me. Had you actually read the thread, rather than just pretended to have done so so that you could come down here to the bottom and insult me you would know that the commenter is a man (121); that he knows I am a woman (passim); that he works to educate young scientists math and strives to be non-sexist and thus was pained by the idea that he would be seen otherwise (121 ); that he has been commenting here for nine years (130); that, in his words, “[t]he Barbie quote is quasi-legendary among people involved in math education, as is xkcd 385, and I honestly did not think that you could have thought I was quoting it in some literal way” (130). So that far from failing to know the quote was offensive to a woman, Z rather thought that it was so obviously offensive that no one who thought him a person of any good will could ever think he was serious, and that he had hoped after nine years we would know he was a person of good will (130). So no, bitch, I don’t “have to ask him (?) to find out if he (?) was even giving it a moment’s thought, at the time.”

159

P.M.Lawrence 01.24.14 at 11:39 am

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.

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mattski 01.24.14 at 1:31 pm

No, man. Numbers are real and would exist even if humans never existed. Facts about squares are just true facts about squares.

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

161

Jim Buck 01.24.14 at 5:30 pm

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?

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TM 01.24.14 at 5:59 pm

Philosophy of Math on CT! That’s nice. I would so like to hear a really satisfactory account of what Math is. Most accounts are negative, dwelling on what math isn’t. “They are neither empirical objects, nor substances: when one counts such, one doesn’t find the numbers inside them.” (jch) While that sounds convincing, I would object that the whole concept of “empirical objects” is pretty screwed up. We think we know what empirical objects are but as soon as we look closer, the physics run into trouble. In that sense, maybe numbers aren’t even so different from “empirical objects”. I dimly remember somebody (I think I read it in Russell but he was quoting somebody else) asking what we mean when we refer to the “North Sea”. We think that’s an empirical physical entity right? But really it’s a concept or a classification invented by humans. There is a sense in which the North Sea, or the matter of which it is composed, exists objectively independently of humans perceiving it, and there’s a sense in which really there is no North Sea unless somebody draws a map and names it.

The other example I find highly instructive is that of color (that I think is from Russell). When you have a red flower with five petals, how is the property of redness fundamentally different from the property of fiveness? It is difficult to justify color but not number as an objective physical essence.

My interest now is mostly in education and there I prefer to stress that Math is indeed about the real world. At least that kind of Math that non-mathematicians should be familiar with.

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elm 01.24.14 at 6:20 pm

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.

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mattski 01.24.14 at 6:58 pm

@ 162

I love what you wrote. I love the way you put your finger on the problem of ‘scale’ or ‘resolution’ for lack of better terms. I think it is absolutely valid to say that “objects” are a function of our ‘degree of resolution.’ How far are we zooming in or zooming out in space… and in time? Because as we do so objects come and go.

So “objects” in the physical world (and yes, even “physical world” gets dubious the closer we look) are provisional. The world is in flux and formations come and go. ‘Identity’ is an appearance, but often a valid and useful way (a necessary way!) to think about the contents of reality. But if we want to be really rigorous in our observations wouldn’t we have to conclude that every object we encounter is drained of its identity by impermanence?

Mathematics though… beguilingly, has the appearance of being untouched by transience. Because it is nothing more than a collection of “rules.” And a rule can be thought of as an action. Indeed, math is a human game like language. We came up with it for the express purpose of describing the world. We ‘drape’ it over the world. But its relationship to reality is necessarily approximate. And a number, rather than resembling a “thing,” is more of an “instruction” to take a certain action.

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Nine 01.24.14 at 7:38 pm

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 …

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TM 01.24.14 at 7:44 pm

[158, 163, 165: Why oh why can’t people stop pursuing sidetracks?]

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Nine 01.24.14 at 7:47 pm

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

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TM 01.24.14 at 8:15 pm

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Belle Waring 01.25.14 at 8:21 am

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?

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John Holbo 01.26.14 at 6:52 am

Well, we still don’t know what math is, but I hope Lawrence learned something!

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Saurs 01.27.14 at 10:25 am

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.

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Z 01.27.14 at 1:51 pm

This thread is so painful to me: my favorite topic (the math sub-discipline called special values of L-function) discussed on my favorite blog; and all I managed was to provoke a clusterfuck of comments. I guess each man trolls the things he loves. My apologies again to John, Belle, Katherine and now P.M Lawrence for dragging him in this melee.

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Belle Waring 01.27.14 at 3:42 pm

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.

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P.M.Lawrence 01.28.14 at 1:23 am

Readers, I have been forbearing to follow these things up, because my earlier attempt seemed to make matters worse and I preferred to give matters a chance to settle. But I see that I keep being accused of deliberate trolling, of lying about what I did and did not read and understand and when, and so on – and in other posts, with ripples that may keep spreading.

So I just want readers to register that I have in fact noticed all this, and that I am being patient in the hope that, at some point, I can make my peace with anybody who might have been inadvertently offended, as well as clarifying matters. I regret if this reply is taken as fuel for flames; I realise the possibility, and I regret it if that puts off a settlement, but I also really want readers to notice that I have been holding off patiently rather than accepting the mistaken accusations of ill intent and so on. So now I will go back to waiting, unless someone wants to suggest some quicker yet safe and honourable way of making peace.

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Yama 01.28.14 at 12:02 pm

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

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Lucie Rie Mann 01.29.14 at 6:57 am

Well roboslave, at least I remember something, albeit vague, from 17 years and countless moles of kiln fumes ago. Thanks for the reminder and now I’ll go back to tweaking my firing schedule so that my turquoise matte stops pinholing.

I still don’t see why this discussion belongs here. Also it seems that someone insulted someone else which seems to happen in such pointless threads.

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