Life is complicated. Why does my bank keep sending statements in Spanish, even though I've complained repeatedly that I don't speak Spanish? Where is my digital camera, and why do Canadians buy milk in plastic bags? There is nothing as nice as a solvable, well defined problem. I used to love finding bugs in fake proofs! For your amusement, here's a "proof" for 0=1 :

We start with an innocent looking natural logarithm

Use additivity of integration

And now we substitute y=x/2 in the first, and y=2x in the second integral, which yields

So, we now have ln(2) = 0, divide by ln(2) and voilà :-)

I believe somebody will explain the bug in the comments, so instead of spoiling your fun, I refer you there for debugging.

## 68 comments:

Infinity minus infinity is anything you want it to be. Also known as perturbative renormalisation, at least as (mis-)explained in many beginners' QFT textbooks.

wow, that was fast :-) indeed, abusing infinities is the most powerful device to produce nonsense.

Here's another one, I thought of in calculus class long ago (I don't know about originality) that works differently:

Start with x = 1

Then take d/dx of both sides:

1 = 0 !

I followed the standard rules for how to take D_x of whatever!

Or you can try x^2 = 1

Then 2x = 0, and those two contradict each other, and lots more similar fun can be had.

There is an explanation, but it does look scary.

I don't know what's scary about that? you've mixed up a function with the value of the function at a point. If f(x) = x, then Df/dx = 1, but D f(1)/dx = 0.

sorry, I don't understand it. Where is the mistake?

Thanks,

little stupid boy

Oh, "abusing infinities is the most powerful device to produce nonsense."

But that's what QED is all about? Maybe that caveat applies to the renormalization business etc., which I only know second hand as middle brow explanation, but sounds hokey as has been acknowledge by some (Feynman? Penrose?) Really, they talk of the polarization of the vacuum but I never see: an actual graph of the adjusted field around an electron, presumably much weaker near the infinitesimal center if had been extrapolated as 1/r^2 from far away. The sum of energy density/c^2 would still be more than the mass of the electron, even if not infinite, true? If so, the conservation and dynamics isn't really consistent, is it?

Hi Little Boy,

An equation like infinity minus infinity equals something doesn't make sense. It could be anything from 0 to infinity. That's what I've used in the second line - both integrals are infinite. Best,

B.

but isn't infinity = 1/0 ?

slb

...abusing infinities is the most powerful device to produce nonsense.Has the landscape grown beyond 10^1000 acceptable vacua? Glauber's salt (

sal mirabilis) is the appropriate pharma to treatunglaublich.Google

"glauber's salt" laxative 1230 hits

OK, maybe not scary, and there's an explanation there too. The point is, naive application of rules by rote without understanding the context can cause problems. That is maybe more "disturbing" to a person who thought they could always use legitimate rules, which are not like clearly-warned-against mistakes like using infinity or dividing by zero.

Hi slb (= lsb?),

whoever told you that clearly didn't know what he was doing. Infinity could as well be 1/0^2 or 1/0 + 5. The point is, one can't reverse dividing through zero, neither can one reverse adding infinity. You can have infinity + 5 = infinity, but you can't subtract infinity on both sides and get 0=5. You can have 5* infinity = infinity, but you can't multiply with zero to get 5 = 1. Best,

B.

Hi Neil:

naive application of rules by rote without understanding the context can cause problemsTrue. I wish those who decide on present school education would listen to you.

Best,

B.

My teaacher told us this. I think I understand now. Thanks. lb ;)

"abusing infinities is the most powerful device to produce nonsense."

Not quite so, think about the chiral anomaly :D

A harmless FREE theory with massless fermions and viola! damn triangles! ;)

uh, how do you get a triangle in a FREE theory?

We buy milk in bags because it's more enviromentally friendly! Except the bags are only one litre, so you use more of them...

Also, you can just cut the corner off, and because it's small it will stay fresh, so you don't have to fiddle with openning the carton or the jug.

I don't know anything about the history though. Just repeating opinions I've heard.

Alas the system isn't universal. It's available in Ontario, but not BC. Something about the activation energy of buying the pitcher to put the bags in.

Milk in bottles still tastes the best, though!

an innocent looking natural logarithmAlways nice to be reminded that logs are mappings of arithmetic and geometric sequences. Thanks.

Estas huyendo de la realidad...

[You´re running away from reality]

Wich could, by the way, explain why you cant find your camera. Need to concentrate more on the present less on past, future or platonic worlds.

Regarding the milk-bag thing... it was normal in europe before tetrabrick. If you buy it fresh you will still find this format for technical reasons (small old fashioned enterprises) or it migh just be marketing.

With spanish... er.. well this comment itself is some kind of meta-answer.

Note that any of this statements is a theorem. Luckily. Otherwise germans would rule the world by now.

Nice blog!.

You can do it without abusing infinities. Prove 1+1=3. Start with

4-10 = 9-15

Both is -6. Now, add 25/4 to both sides

4-10 + 25/4 = 9-15 + 25/4

Realize that both can be written as complete squares

(2-5/2)^2 = (3-5/2)^2

Now you just take the square root

2-5/2 = 3-5/2,

add 5/2 to both sides

2=3

and realize that 2 can be written as 1+1:

1+1=3

I hope it will be useful to find new discoveries about doubly, if not triply special relativity.

Dear Lubos,

nice :-)

sqrt(x^2)=|x|

will see if I can use it to renormalize gravity or so ;-)

Dear Rillian:

environmentally friendly? The better part of my family buys milk in glass bottles and recycles them. There is something that makes me feel awkward about these quabbly wobbly bags, don't know. I guess a waterbed is not for me ;-)

Best,

B.

Hi Ergodic,

Thanks for the kind and wise words. But I am not sure where reality is, so how can I run away from it? I'd hope if Germans ruled the worlds they'd be wise enough to step down.

Best,

B.

PS: That's not to say if Germans ruled the world they'd be wise enough to make sure you didn't notice ;-)

Dear Bee,

however funny it sounds, |x|=x is exactly one of the rules that underlies loop quantum gravity. ;-)

Even in 2+1 dimensions, the equivalence between gauge theory and gravity, even at the low-energy level, only works if this assumption is taken as true.

The Chern-Simons Lagrangian can have both signs and it is odd under parity while the "equivalent" Einstein-Hilbert Lagrangian is the absolute value and it is even under parity. :-)

Best

Lubos

lol Lubos, 2=3

now add twothirds 2/3 to each side

and multiply both sides by three

Some of mankind's greatest intellectual achievements to date:

1. The Lamb Shift

Experimental measurement:

1057.864 MHz

Theoretical calculation using QED:

Infinity minus infinity = 1058 MHz

2. The anomalous magnetic moment of the muon.

Experimental measurement:

g/2 = 1.0011659214(8)

Theoretical calculation using QED:

Infinity minus infinity = g/2 = 1.0011659208(6)

Well, the sneaky mistake stuff has been fun, but what about "real paradoxes" in math and logic? One of my favorites, which perhaps involves abusing infinities, is the supertask (an infinite number of subtasks carried out in a finite time using tricks like a summable infinite series: The steps are at t = (1 - 0.5^n) etc.) There are various examples, but I thought of this one, metaphysically scary IMHO:

What if the task is to magnify (or shrink) the number line? We can set up for 2x or 0.5x magnification each time. With any finite number of steps, we just get very tiny or very large numbers versus the original reference marks. But if supertasked, there is no way (?) to imagine the resulting view. I mean, either all infinity in a dot maybe, or all zero everywhere - or finally do we actually see the surreal numbers or their infinite inverses? (Surreal numbers look pretty fishy to me.)

You can say it doesn't matter, but if you believe in ideal possibilities and the reality of infinite steps in other areas, it could be problematical.

Another paradox about infinity: If I have a platonic dart to throw at the number line, it will hit somewhere, some number rational or irrational. But since that number has no finite size relative to the whole, we could argue that the chance of hitting any number, including that one, was "zero". Yet we did hit one, and have to ...

Physical issues of course set no limitation on the possible coherence and relevance of such paradoxes, but sometimes the subjects can be related.

many "paradoxes" disappear if one knows what one is talking about. see: measure zero/null set

Why does my bank keep sending statements in Spanish, even though I've complained repeatedly that I don't speak Spanish?Maybe they don't understand English?... ;)

I'm reminded of the story of how Einstein tried to divide by zero.

Too bad that he didn't have Lumo.

OK Bee, I looked at that link and the sublink about ideal in set theory. I have a partial grasp of the formalities, but I didn't see how it solved one or more of my paradoxes. The Wikipedia article on supertask implies there might be real logical problems deriving from such paradoxes.

Supertask

Hi Neil, sorry I don't have time right now, will have a look later. I was referring to your last 'paradox'. It's not zero, it has measure zero which isn't the same. In praxis, the dart will always pick a set with non-zero measure (it has a finite size - the real world is not 'ideal').

Best,

B.

OK thanks. For clarification and your curiosity meanwhile, the "dart" is a platonic-world entity with a literal point at the tip, not a physical dart, and we can make it strike along a limited range. Also, more relevant actually, REM to let me know if you ever start thinking about that multiple-measurement QM experiment again, tx.

OK Bee how about this one?:

1 = (-1*-1)^(1/2) = (-1)^(1/2)*(-1)^(1/2)= i^2 = -1

Hi Neil:

I meant to express that many apparent paradoxes vanish if one takes into account that we are dealing with reality. There is, in the physical world, no such paradox. Also on a mathematical level, everything is well defined, so there is no paradox either - it just seems to be unintuitive. E.g. there is an infinite amount of integers, yet their measure in R is zero. That is, you'd have to throw your dart uncountable infinitely many times before you'd hit 13. Yes, I am thinking about the experiment. It might take some more years before I come to a conclusion. Best,

B.

The sqare root of -1 can be either i or -i, as one sees most easily adding angles in the complex plane. You have to properly define the branch of the logarith before you deal with ln(-1) (rspt (-1)^1/2 = exp(1/2 ln(-1)).

The problem with QED is that it works. Anybody who criticizes must volunteer a better theory - perhaps one that subtracts smaller infinities. Lamb shift for U(91+):

459.8 (+/-)4.2 eV measured versus 463.95(+/-)0.50 eV calculated

PRL 95 233003 (2005)

PRL 94 223001 (2005)

PRL 97 253004 (2006)

PRL 91 073001 (2003)

Here's one that a 3rd grader can play with:

Consider the sum of this infinite series,

1, -1, 1, -1, 1, -1, ...

If we group it as

(1-1)+(1-1)+(1-1)+...

it appears to add up to 0. If on the other hand we re-group it as

1+(-1+1)+(-1+1)+...

it appears to add up to 1.

This is, of course, again due to an abuse of infinity, in this case the ambiguity in how the summation of an infinite series is defined. But once a clear definition is given for the sum of this type of infinite series, it will return a definite value. (This is related to the so-called Cesaro sum. Wish I know how to make a link here.)

In a sense, this example also belongs to the case of "infinity minus infinity", which can leads to multiple values depending on the rules of how to treat the infinities.

Yup--you can have all sorts of fun with things that have branch points.

(e.g. 1=(-1)^(2/3) = -1/2 + (Sqrt(3)/2)*i)

That is, you'd have to throw your dart uncountable infinitely many times before you'd hit 13.Right, if you specify

in advancewhat you are considering, that changes everything! This has further implications ...indeed, abusing infinities is the most powerful device to produce nonsense.If in step 2 instead of making the integral go from 0 we go to epsilon, with the intention of making epsilon go to zero eventually, and then do the change of variables, and make the lower limit of the two integrals equal (say to epsilon/2), we introduce a third term, an integral from epsilon/2 to 2*epsilon of 1/x. The first two terms are Bee's terms as epsilon -> zero. The third term is finite and independent of epsilon; contrary to expectations it does not vanish as epsilon -> zero.

So to me this example is somewhat subtler than most abuses of infinity.

Hi Arun, I am somewhat confused by your last comment. You say

The third term is finite and independent of epsilon; contrary to expectations it does not vanish as epsilon -> zero.- well, but that would be my expectation if one does the limits properly, after all we know the result better be ln(2)? Best,B.

Hi Chickenbreeder, your example is not an abuse of infinities - it's a series that simply doesn't converge and the limit is not defined. You say "in this case the ambiguity in how the summation of an infinite series is defined", is incorrect. Summation is a well defined procedure, there is no ambiguity how to define it, just that the partial sums keep jumping around and don't reach a limiting value.

Best,

B.

Bee, yes, we know what is wrong, but in general we expect the integral to depend on epsilon. It is interesting to try to try a similar mis-proof with 1/x^n. I haven't been able to get a good one yet.

Bee wrote:

"your example is not an abuse of infinities - it's a series that simply doesn't converge and the limit is not defined."

I can understand your reaction. In the ordinary sense the sum of the series does not converge as you said. But it's exactly my point that when infinity is involved it depends on the definition. The series I mentioned does converge in the sense of Cesaro. I should have put a link but here's the gist of it.

If the partial sum of the series up to the n-th term is S_n, the Cesaro sum for the first n terms is

C_n = (S_1 + S_2 + S_3 +...+ S_n)/n

The summation of a series is said to converge in the sense of Cesaro if C_n converges as n approaches infinity. For my example, the sum of the series does not converge in the ordinary sense but it converges to 1/2 in the sense of Cesaro.

(Note as well that if the sum of a series converges in the ordinary sense, it also converges in the Cesaro sense and the two produce identical value of the sum.)

My case is the situation when we have infinite (and seemingly equal) numbers of "1" and "-1" and we try to add them together. Based on the definition of Cesaro what we get is something akin to

infinity minus infinity = 1/2 ,

whereas the series cannot be evaluated at all based on the "ordinary" definition. This is what I meant when I said it all depends on the definition.

In physics, a lot of time we are able to evaluate "infinity minus infinity" and get a finite value out of it. But it's sometimes forgotten that the reason we can do that is due to a certain definition we use to treat infinity (or infinte sums). There is, thus, a hidden assumption that the definition we use reflects what's really happening in the real world.

My example of the Cesaro sum (which is a generalization of the ordinary sum) is by no mean exotic. The more I look at it, the more it appears that the "ordinary" mathematical analysis isn't enough to describe the physical world. Dirac's delta function is not continuous and not differentiable in the ordinary sense but it's very useful in physical theories (and it eventually finds a firm mathematical foundation in the theory of generalized function). Incidentally, all of the generalizations of ordinary analysis that are needed to extend our physical theories all have something to do with new ways of treating infinities.

Now Bee, you're being sacrilegious; only the gods can use and abuse infinities!

;-)

changcho

ChickenBreeder: I have heard anywhere that once a time Euler was convinced that the infinite sum you posted (1 - 1 + 1 -1 ...) resulted 1/2, just like you've shown in the "Cesaro sense".

Do you know if this historical remark is true? In this case, Euler probably invented the "Cesaro sum definition" before Cesaro, so why it has not Euler name? (By the way, people joke that half of mathematics was invented by Euler and we only call it by the name of the guy who discovered it after him...)

here's an old one that doesn't use calculus:

start from: a=b

multiply by a: a^2 = ab

subtract b^2: a^2-b^2 = ab-b^2

factorize: (a+b)(a-b) = b(a-b)

simplify: a+b = b

since a=b: 2b = b

simplify: 2 = 1

Uncle Al,

The problem with QED is not that it works. The problem is that it's not quantum electrodynamics. Quantizing classical electrodynamics leads to divergent integrals. The thing that "works" is "effective" field theory whose connection to the theory obtained by field quantization is broken by the invalid process of subtracting infinity from infinity.

As for volunteering a better theory, how can someone who is completely invisible do that?

Chris, I find your comment interesting since I always thought (albeit in middle-brow appreciation) that renormalization was trickery. So, how did those physical values given above (ostensibly as part of a sort of infinity - infinity, not pure math of it of course but per the boundaries etc.) turn out to be right? And why do so few scientists care about the phoniness of infinity - infinity, or is that really the point anyway?

clovis: I am not aware of the Euler story but it's certainly possible that a great mathematician like him might have thought about it.

If I remember correctly, Cesaro is a very modern (20th century?) man. It's also possible that in the 2 or 3 centuries between Eular and Cesaro someone else had come up with a similar idea.

My favorite piece of fallacious math is the proof that all positive integers are equal:

Let S(n)={1,2,3...n}. We want to prove that for all n, any two numbers x, y included in S(n) are equal to each other. Proof is by induction.

First, case n=1. Then only 1 is included in the set S(n), and 1 is equal to itself.

Next assume that any two numbers in S(n) are equal to each other, and consider S(n+1). Let x, y be two arbitrary numbers in S(n+1). Then x-1, y-1 belong to S(n), hence by assumption x-1 = y-1, hence x = y. Thus the proposition is proved by induction.

There is one sentence in the proof that is a glaring error, but it can easily pass over the head of an unprepared first-year undergrad.

Hi Alejandro:

Thanks, I like this one :-)

Let x, y be two arbitrary numbers in S(n+1). Then x-1, y-1 belong to S(n)Is exactly what was to be shown, so small wonder the proof follows. (It's evidently not the case. If one takes S(2) = {1,2} then 1-1 is not in S(1) ) Best,

B.

Chickenbreeder,

the series is 1/(1+z) with z -> 1. The series 1 - z + z^2 - z^3 + ...has a radius of convergence less than 1 because of the pole at z=-1.

there are reasons other than Cesaro summability to think of this as another representation of 1/2 :)

Neil,

The problem is that people are not honest about what is going on here. You can get any answer you want by subtracting infinity from infinity, so the constraints used to shape the theory after renormalization are all-important. Covariance, for example, is one thing that needs to be imposed (infinity minus infinity, being anything you want, is not necessarily covariant). Gauge invariance is another. The fact that a set of "reasonable" constraints here leads to numbers that agree well with experiment is not a vindication of quantum field theory, it is a vindication of the choice of constraints. We thus have something that was inspired by quantum field theory without actually deriving from it. I have no idea why particle physicists do not seem to recognise the obvious here, but it is certainly not because I don't point it out often enough.

Oh - and what's this "viola" stuff about. It looks like "voila" is what was meant. Is this String Theorists, or at least String Players demonstrating their omnipresence?

Hi Chris: Thanks for pointing out the typo, I've corrected that. It's more the omnipresence of Google spell check that was responsible in this case. Best, B.

If one can unambiguously deal with arising infinities then the theory is useful. One can like or not like any procedure that does that, one can even avoid it the point is, as Uncle said, that it works. However, the other point, as Chris said, is that one should keep in mind why it is that it works. This is definitely necessary if one wants to progress from applying a theory to understanding and going beyond it.

Chris, when you say 'Covariance' do you mean Lorentz invariance?

Hi Bee,

I used the word "covariance" because the expressions are not always Lorentz scalars.

BTW, I have a copy of the Scharf book ("Finite Quantum Electrodynamics") whose link you provided, but to my mind all it really proves is that some are capable of subtracting infinity from infinity in incredibly subtle and devious ways. This, I suppose, at least makes it relevant to your post.

Hi Carl,

Thanks for the clarification. I just meant to say you don't mean generally covariant there, but flat space etc. Yes, I admit I don't particularly like Scharf's book (it seems to me it's unnecessarily complicated) but if I recall that correctly his point is one can (after lots of mathematical fuss) formulate the theory such that it's finite from the beginning on. I am not so particularly thrilled by that because I don't think it really solves any problem that I personally am intrigued by, but it's interesting. Best,

B.

"one can (after lots of mathematical fuss) formulate the theory such that it's finite from the beginning on."

Being finite is not enough. It also needs to be axiomatic. All renormalization schemes involve introducing a spurious extra degree of freedom, whether it be a cutoff for the integrals or a complex number of spacetime dimensions, and then taking a spurious limit to get the finite theory, and Scharf is no different. "Finite quantum electrodynamics" to me means an axiomatic framework which gives finite answers without any of this nonsense. In fact, I would rather it just called "quantum electrodynamics" as its finiteness should go without saying (have you ever heard anyone talking about "finite thermodynamics"?)

arun: You are right :)

In fact, what you wrote serves to illustrate that there are situations when it makes more sense to consider the Cesaro sum than the "ordinary" sum. The function 1/(1+z) does have a well-defined value of 1/2 at z = 1!

This is my favorite example of a math puzzle because, as you said, it has so much more to it even though it looks so innocently simple.

Hi Chris: yes. I haven't said it's pretty but it works. I haven't said I like it, but one can do it. I think we basically agree. Best, B.

PS: Of course the point is that the result of renormalization eventually is independent of the scheme used. That's the problem with higher-dim. QFT where the result depends explicitly on the regulator and the way how it's introduced.

"Of course the point is that the result of renormalization eventually is independent of the scheme used."

This is tautological. If you permit only answers that you find acceptable, then these answers will be the only acceptable by definition.

Thanks for various comments about the suspect nature of subtracting infinities in renormalization. I wonder about the physical basis for it happening, the part about the polarization of the vacuum. As I said, I never see: an actual graph of the PV-adjusted field around an electron, presumably much weaker near the infinitesimal center than if had been extrapolated as 1/r^2 from far away. The sum of energy density/c^2 would still be more than the mass of the electron, even if not infinite, true? If so, the conservation and dynamics isn't really consistent, is it? In any case, it isn't just a decision to invoke a mathematical procedure, but something Nature does to avoid being a mess, true? Don't we have to explain that in satisfactory physical terms?

Chris: if one took it as a definition for 'allowed schemes', yes, it would be tautological to say the result has to be independent of the scheme. As a working prescription it's not because you have experimental reality constraints (data). My sentence above stated that the working schemes are those that agree with each other and with observation, and the others you better not use (i.e. various other abuses of infinities one could consider to get 'some' finite result). The requirement that results are indep. of the regulator is not an empty statement. Either way, if you want somebody to defend the merits of QFT I'm not the right person, I don't even like quantum mechanics to begin with. Best, B.

Hi Bee,

Not really something like 2=1 in my comment here, but just to mention, somewhat related to the present post, that unfortunately mathematics can be used in many ways to deceive people... of good heart...

See, e.g., the 2000 trick.

And as scientists we must be alert not to be deceived by our

ownmathematical tricks, right?... We are also people of good heart...Kind regards,

Christine

While you folks are on the subject of mathematical madness, I'd like to throw in one of my favourite crazy sounding math 'results', that 1+2+3+... = -1/12 (used to show that bosonic string theory is 26 dimensional)

here's a neat link on it: http://math.ucr.edu/home/baez/twf_ascii/week126

and the derivation of the casimir force given here: http://en.wikipedia.org/wiki/Casimir_effect uses the fact that the sum of all the cubes 1^3 + 2^3 + 3^3 +... = 1/120

Oh well, at least it's a positive number!

I still don't understand the idea of analytic continuation that makes this stuff work. Perhaps someone could point me to a good reference?

Thanks,

Aatish

Hi Aatish,

I currently don't have the reference at hand, but there is a paper from sometime in the middle 80ies by Stephen Wolfram - yes, the guy from Wolfram research - about Zeta function regularisation which explains all that stuff. Best,

B.

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