This should be news to nobody, and I can’t believe I have to say it out loud, but the sum:
is not well-defined. The series diverges. It does not make sense in any mathematically rigorous way to say that it “equals” anything.
Anyone who is trying to tell you otherwise in an attempt to show you “Simply the Most Astonishing Math You’ll Ever See” either doesn’t understand the first thing about what he’s talking about and lacks the good sense to thus remain silent, or else is lazily manipulating your desire to be surprised in order to attract web traffic to his blog.
The reason all of this matters to anyone but the community of those who already know better, who can easily write Plait off, is that it does real damage to the national discourse about math. It serves to tell people who don’t know better, who don’t know the zeta function from Catherine Zeta-Jones, that they are stupid. To tell them they are not smart enough to follow the technical mysteries of how:
To tell them that mathematics is a numinous, mysterious thing, beyond the ken of the man and woman on the street corner.
But Plait’s good news is false. This might seem obvious on its face, but let’s try and actually get at the subtleties that Plait was unwilling or unable to discuss.
Summing finite things is relatively straightforward:
You can even add many terms, and fractions:
Now this looks suspiciously close to 2.
If we keep going we get even closer:
Now I won’t actually prove this here, since this is a blog post and not a textbook, but what it means when we say that:
is that this series converges to 2. What this means is that the sequence of partial sums:
converges to 2. What this means is that if you challenge me with some small number, I can come up with a term in the sequence of partial sums such that the difference between this term and 2 is smaller than your number–and that the difference between all subsequent terms and 2 is also smaller than that number.
Still here? This is subtle stuff. Actually showing you the proof would take a bit more work. But I can give you the flavor of it. I can even tell you something I won’t prove.
Given any geometric series that converges, where the first term is “a” (1 in the case of our series above) and the ratio between successive terms is “r” (1/2 in our series above), the series sums to:
This is true only when:
-1 < r < 1.
However—and this is what mathematicians frequently do—we can generalize this result to numbers outside that range. Once we leave the range for which we have proven it to be true, the formula is now no longer a mathematical truth in and of itself, but rather a guide to deeper mathematics.
Take the example when r=2 and a=1. Then we have:
now this does not “equal” 1/(1-2)=1/(-1)=-1 in any sense other than by some sort of loose analogy, as this lucid, thoughtful, and mildly technical blog post by Matt Noonan explains.
One name for a version of this trick is the Abel sum, after the mathematician Niels Henrik Abel. (Who, by the way, is the subject of the book Abel’s Proof by Peter Pesic, which gives a wonderful flavor of mathematics to the non-mathematician.)
To say that the Abel sum of:
is -1 is intriguing. But it doesn’t drive web traffic.
also diverges. However, unlike:
it is something called Cesàro summable. Though the partial sums of the series do not converge, the average values of the partial sums do. This means that the Cesàro sum of the series is 1/2. It does not mean, as Plait wrote, that:
Again–Cesàro summation is a mathematically interesting process. But it isn’t the same as the mathematically precise idea of summing an infinite series. It is a generalization.
As Mark Chu-Carroll wrote in an excellent blog post quickly following on Plait’s, once you conflate Cesàro summation with taking the limit of an infinite series, any absurdity is possible:
Using the equality of an infinite series with its Cesaro sum, I can prove that 0=1, that the square root of 2 is a natural number, or that the moon is made of green cheese.
Just as the Cesàro sum is a generalization of what we normally mean by adding things, other generalizations exist. These are mathematically interesting and powerful. To belabor the point: they are not the same thing as addition. Summing an infinite series is also not the same thing as addition. It is related, but it is different.
Using one such technique, called zeta function regularization, Leonhard Euler, one of the most important mathematicians in history, and certainly the most prolific (or, if you prefer your prose purple, in Plait’s words, “a serious mathematical heavyweight”) showed that one could assign the value of -1/12 to the sum of all positive integers. This result was, years later, independently rediscovered by the Indian prodigy Srinivasa Ramanujan, who communicated his bewilderment in one of his early letters to G.H. Hardy, a prominent English mathematician.
And so, under this generalized meaning of addition, it is reasonable to state that the series:
is somehow intimately related to -1/12th, in a way that is subtle and mysterious. (Or, if you prefer Plait, “[r]eally bizarre, brain-melting”.) As Noonan sketches in more detail than I’ll go into here, the connection between -1/12 and the infinite sum has to do with some back-of-the-envelope complex analysis. Noonan recommends Hardy’s Divergent Series for more, and so too do I.
But it is simply wrong to state, as Plait does, that:
it’s simple math to show that
Plait ran a correction to his original post:
I originally wrote that the series 1+2+3+4+5… converges, but that’s not strictly true; it has a sum of -1/12 but by definition doesn’t converge to that.
Even his correction is nonsense. The series does not converge. Period. Saying it converges is not only not “strictly true”. It just isn’t true. Nor is it true to say it sums to -1/12.
Now all of this discussion is slightly technical. It requires some thought. But it does not require an advanced degree in mathematics to grasp the broad outlines. Anyone with a PhD in Astronomy who raced to publish a blog post on the subject after watching a poorly reasoned viral video should know better. You cannot credibly wring your hands about climate change denial–as Plait does–while yourself displaying such a cavalier and careless attitude towards mathematical truth.
To say say that there exists a generalized notion of a sum of infinite series under which the positive integers “sum” to -1 is interesting. It is educational. It conveys something of the way mathematics actually works–not by waving of hands and squinting of eyes, but by careful application of intellect–in a way that is accessible to the interested layperson. I’m flabbergasted just hitting control-c and control-v to convey the following sentence from Plait’s post in this space:
If you look at it from a distance, you lose the zigzag pattern and it’ll look more like a fuzzy flat line filling the vertical space between y=0 and y=1.
This is what Plait wrote to justify the claim that:
This is cheap sensationalism; Plait displays a willful disregard for rigor in pursuit of surprise and excitement.
But ultimately what is most galling about Plait’s piece is not his many errors of fact. It is his statement that the power of math stems from its counterintuitiveness, that what is remarkable about math is the fact that it will surprise you.
Many important mathematical results–I would venture to say most–are not in fact surprising. The fundamental theorem of algebra seems pretty reasonable. So too does the fundamental theorem of calculus.
The power of math stems not from the fact that these are reasonable conjectures, but rather that they are, upon admitting certain very basic axioms, certainties. Mathematics is one of the very few areas of human endeavor in which it is possible to know without doubt. (Though more can be said on that subject as well.)
To mention an example closer to the subject at hand, it seems perfectly reasonable that:
should equal 2. But until you have proved that fact, you aren’t doing math as mathematicians do it. Understanding something of the rigorous process involved in mathematical proof is possible for the non-mathematician, but it doesn’t figure in Plait’s overheated barking, which substitutes exaggerated ejaculations of wonder for actual mathematical exposition:
the result will carve out a piece of your soul and leave hollow space…the universe has lost its mind…my brain cells want to stop instantaneously and explode at the speed of light…We live our lives, we Earthbound apes, and we think we have it all figured out…something so truly twisted and bizarre that it makes you question just how well you truly see things.
It seems like piling on to point out that light does not travel instantaneously–a rhetorical error even closer to Plait’s astronomical field of expertise.
Plait’s supercilious attempt to stupefy widens the gulf between the knowledgable and the ignorant. It makes it that much harder to educate people about algebra, about compound interest, about evolution, about global warming. It makes it possible for people to say: “that stuff is beyond me; I don’t see how it can possibly be true, and so I won’t bother.”
His central thesis is that mathematics constitutes some “alternative reality” to everyday life. That is wrong, and it is pernicious.
So if you’re sitting there wondering how in the world:
scratch your head no longer.
first of all, sorry for my bad English. I find the way to summing divergent series and determines limit functions in their singular points. Even more, I was surprised when I discovered that the method can be applied to compute divergent integrals.
Konstantin, in what sense does i^2 = -1? That is an absurdity. How can the square root of a negative number exist? It’s nonsense! This should be news to nobody, and I can’t believe I have to say it out loud, but the value: sqrt(-1) is not well-defined. There is no solution in the real numbers. It does not make sense in any mathematically rigorous way to say that it “equals” anything. Anyone who is trying to tell you otherwise either doesn’t understand the first thing about what he’s talking about and lacks the good sense to thus remain silent, or else is lazily manipulating your desire to be surprised in order to attract web traffic to his blog.
See what I did there?
I could equally denounce the equation i^2 = -1 as nonsense because there is no number whose square is “actually equal” (whatever that means) to -1. To claim that 1 + 2 + 3 + … = -1/12 is unconditionally nonsense is, in itself, an expression of ignorance. Similar sums are used regularly throughout physics (i.e. in calculations involving the Casimir effect) and these procedures yield reasonable and well-defined answers.
Unless you also consider the Casimir effect and similar phenomena to be some “alternative reality” to everyday life.
The real question is not “Does this series converge?” or “Is the square root of a negative number equal to a real number?”, but rather “Is there a mathematically reasonable and consistent way to assign a value to a divergent series?” and similarly “Is there a mathematically reasonable and consistent way to assign a value to the square root of a negative number?”
To close off your mind to the possibility of deriving such a reasonable and consistent answers to problems that at first glance appear to be nonsense from the start is to close yourself off to some of the greatest wonders of mathematics.
I’m traveling at present with limited internet, so a fuller reply will have to wait. But you are throwing out the baby with the bathwater. One needn’t throw out imaginary numbers (or other constructions such as the square root of two) to object to the slap-happy way in which some math popularizers were recently manhandling divergent infinite series. More to come. Best regards,
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The significance is of what’s going on here should not be dismissed as easily as that. Whenever we talk about mathematical equalilty, the meaning is often subtle. Even when I say three apples are equal to three oranges, what I mean is not that the objects are literally identical, or even that the sets are identical. What numerical equality means is that the sets are idenified as members of an equivalence class under an abstract operation (namely counting.)
This is a sublty we forget about because we’ve all been proficient at counting since the age of four. But the operation of counting is actually quite hard to learn, and it’s implicaions are deeper than we realise.
Similarly, -1/12 and the divergent sum of the positive integers are identified under an abstract operation (in this case a set of rules for series summation designed to preserve the analytic continuity of complex functions.)
My point is that series summation and counting are both much deeper than they appear at first, and if I’d dismissed the implications of counting as offensive to common sense when I’d first encountered them, I very much doubt I’d ever have entered a mathematical world more complicated than the one my dog lives in.
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