# Trust in Math

14 15 January 2008 04:25AM

Followup toExpecting Beauty

I was once reading a Robert Heinlein story - sadly I neglected to note down which story, but I do think it was a Heinlein - where one of the characters says something like, "Logic is a fine thing, but I have seen a perfectly logical proof that 2 = 1."  Authors are not to be confused with characters, but the line is voiced by one of Heinlein's trustworthy father figures.  I find myself worried that Heinlein may have meant it.

The classic proof that 2 = 1 runs thus.  First, let x = y = 1.  Then:

1. x  =  y
2. x2  =  xy
3. x2 - y=  xy - y2
4. (x + y)(x - y)  =  y(x - y)
5. x + y = y
6. 2 = 1

Now, you could look at that, and shrug, and say, "Well, logic doesn't always work."

Or, if you felt that math had rightfully earned just a bit more credibility than that, over the last thirty thousand years, then you might suspect the flaw lay in your use of math, rather than Math Itself.

You might suspect that the proof was not, in fact, "perfectly logical".

The novice goes astray and says:  "The Art failed me."
The master goes astray and says:  "I failed my Art."

Is this - gasp! - faith?  To believe that math is consistent, when you have seen with your own eyes a proof that it is not?  Are you supposed to just ignore the contrary evidence, my good Bayesian?

As I have remarked before, it seems worthwhile to distinguish "faith" that the sun will rise in the east just like the last hundred thousand times observed, from "faith" that tomorrow a green goblin will give you a bag of gold doubloons.  When first-order arithmetic has been observed to be internally consistent over the last ten million theorems proved in it, and you see a seeming proof of inconsistency, it is, perhaps, reasonable to double-check the proof.

You're not going to ignore the contrary evidence.  You're going to double-check it.  You're going to also take into account the last ten million times that first-order arithmetic has proven consistent, when you evaluate your new posterior confidence that 2 = 1 is not perfectly logical.  On that basis, you are going to evaluate a high probability that, if you check for a flaw, you are likely to find one.

But isn't this motivated skepticism?   The most fearful bane of students of bias?  You're applying a stronger standard of checking to incongruent evidence than congruent evidence?

Yes.  So it is necessary to be careful around this sort of reasoning, because it can induce belief hysteresis - a case where your final beliefs end up determined by the order in which you see the evidence.  When you add decision theory, unlike the case of pure probability theory, you have to decide whether to take costly actions to look for additional evidence, and you will do this based on the evidence you have seen so far.

Perhaps you should think to yourself, "Huh, if I didn't spot this flaw at first sight, then I may have accepted some flawed congruent evidence too.  What other mistaken proofs do I have in my head, whose absurdity is not at first apparent?"  Maybe you should apply stronger scrutiny to the next piece of congruent evidence you hear, just to balance things out.

Real faith, blind faith, would be if you looked at the proof and shrugged and said, "Seems like a valid proof to me, but I don't care, I believe in math."  That would be discarding the evidence.

You have a doubt.  Move to resolve it.  That is the purpose of a doubt.  After all, if the proof does hold up, you will have to discard first-order arithmetic.  It's not acceptable to be walking around with your mind containing both the belief that arithmetic is consistent, and what seems like a valid proof that 2 = 1.

Oh, and the flaw in the proof?  Simple technique for finding it:  Substitute 1 for both x and y, concretely evaluate the arithmetic on both sides of the equation, and find the first line where a true equation is followed by a false equation.  Whatever step was performed between those two equations, must have been illegal - illegal for some general reason, mind you; not illegal just because it led to a conclusion you don't like.

That's what Heinlein should have looked for - if, perhaps, he'd had a bit more faith in algebra.

AddedAndrew2 says the character was Jubal from Stranger in a Strange Land.

Charlie says that Heinlein did graduate work in math at UCLA and was a hardcore formalist.  I guess either Jubal wasn't expressing an authorial opinion, or Heinlein meant to convey "deceptively logical-seeming" by the phrase "perfectly logical".

If you don't already know the flaw in the algebra, there are spoilers in the comments ahead.

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Comment author: 15 January 2008 04:41:53AM 1 point [-]

Division by zero?

Genious.

Comment author: 15 January 2008 04:47:34AM 6 points [-]

Are you certain that 51 is prime?

Comment author: 15 January 2008 05:32:01AM 3 points [-]

Benquo: Even infinitesimal are not equal to zero. You don't even need infinitesimals in differential calculus. Instead, you can think dx and dy are just variables. You let them approach zero to see what would happen at the limit, but you don't set them equal to zero. I have always personally found infinitesimals a little disturbing, since one doesn't really need them anywhere.

I am a little puzzled by this; I don't know how they teach this stuff in the US, but in Finland, if my memory serves me correctly, they taught how this "proof" is wrong in elementary school. So only complete idiots would be fooled by this "logic".

Comment author: 22 June 2011 10:12:23PM *  2 points [-]

I think it was a bit later than elementary school, but showing why this "proof" is invalid was a test question at some point in my American academic career

Comment author: 15 January 2008 06:55:47AM 8 points [-]

In Heinlein's defense--if his character did say this, and if it was said by one of those fatherly "voice of Heinlein" characters, which I'm quite prepared to believe...

I would've read the quote as saying, in effect, "If a seemingly logical argument proves something I regard as absurd, I will be skeptical." That was, after all, the appropriate reaction for the person confronted with the proof that 2 = 1. The first reaction ought to be, "Well, I know 2 doesn't equal 1, so no matter how nice and logical the proof seems, there must be something wrong with it." Maybe then you look at the proof a bit harder and see the div-by-zero error in step 5. But if you don't find the error, you don't then run off and rebalance your checkbook based on your new knowledge that 2=1; rather, you say to yourself "Got to be a problem in there somewhere", and wait for someone to show you where it is.

So I would assume, in the book, that some sensible character offered a perfectly sensible argument why the heroes should believe something implausible, or do something they felt deep down that they oughtn't. And Fatherly Character responds by saying, essentially, "Just because you have an argument that seems solid right now, isn't enough reason to do something you really feel is wrong--it's more likely that there's a flaw in your argument that we haven't spotted yet."

Comment author: 12 June 2012 02:08:49AM -1 points [-]

And that, in turn, is very good advice for anyone who practices moral theorizing.

Comment author: 15 January 2008 07:41:34AM 0 points [-]

For me, the purpose of doubt is to motivate inquiry. When any particular doubt no longer serves inquiry, I retire it.

If the purpose of doubt were to eliminate doubt, it would be far more efficient simply never to doubt.

Therefore, I doubt your philosophy of doubt. Let the inquiry continue.

Comment author: 15 January 2008 08:09:19AM 0 points [-]

I assure you, if there is one thing that Robert Anson Heinlein considered holy, it was logic. Wait, maybe it was free love. But if there were TWO THINGS he considered holy...

If you read any of his future histories, you see tales of libertarian utopias set free by humans achieving, if not surpassing, the rationality of which evolved human minds are capable.

My favorite RAH excerpt, from Coventry:

First, they junked the concept of Justice. Examined semantically "justice" has no referent - there is no observable phenomenon in the space-time-matter continuum to which one can point, and say "This is justice". Science can deal only with that which can be observed and measured. Justice is not such a matter; therefore it can never have the same meaning to one as to another; any "noises" said about it will only add to confusion. [...] Since they had abandoned the concept of "justice", there could be no rational standards of punishment. Penology took its place with lycanthropy and other forgotten witchcrafts.

Comment author: 15 January 2008 09:37:35AM 1 point [-]

Saw this in the paper and thought They'll love this on Overcoming Bias

Did a pair of twins really get married by mistake?

Not sure if this story hit the US, but it was in every paper here. I must confess, my first thought was 'wow, weird.' Note to self - more proverbial salt with everything! (Less real salt, they tell me.)

Eliezer, could this post be linked as a follow-up to Your Strength As A Rationalist or I Defy The Data?

Comment author: 15 January 2008 09:44:48AM 0 points [-]

Ben, that sort of thing goes in the open thread.

Leo, Heinlein praised math to the very stars, but I'm not sure he was actually good at math. It's been a long time and I don't have the book in front of me, but I remember a scene in The Rolling Stones where the father is telling the kids they need to study advanced math, and using some mathobabble, and I don't think the father was making any sense...

Comment author: 15 January 2008 11:20:19AM 0 points [-]

I'm sorry you think so Eliezer - I thought it was relevant to this post, otherwise I wouldn't have linked to it.

This seems wrong is the rational response to 2=1 in the same way as it's the response to the headline 'Twins marry by mistake.' After all, if 'all is maths', shouldn't we trust in the internal consistency of the whole world, and check the evidence when it doesn't fit our best theories? I hope this doesn't hijack your intended train of discussion.

Comment author: 12 June 2012 02:10:49AM -1 points [-]

I think that conclusion is right, and the key lies in the word "check". Yes, we check the evidence when it doesn't fit what we thought. Sometimes the evidence is defective (you guys remember that FTL neutrino business?); sometimes, well... it isn't (Michelson-Morley experiment!).

Comment author: 15 January 2008 12:37:29PM 0 points [-]

Ben, most mathematical systems are not internally consistent. Things like "true" and "false" should be avoided when talking about theorems, "Provable in system X" is better. So only if the real world can be described with really simple math are it's laws internally consistent.

Comment author: 12 June 2012 02:11:43AM -1 points [-]

It seems perverse to call the Fundamental Theorem of Calculus anything less than true.

Comment author: 15 January 2008 02:03:47PM 0 points [-]

Another solid article.

One point of confusion for me: You talk about axiomatic *faith* in *logic* (which is necessary in some form to bootstrap your introspective thinking process), but then abruptly switch to talking about "the last ten million times that first-order arithmetic has proven consistent", a statement of *observed prior evidence* about *learned arithmetic*. Both points are valid, but it seemed a non sequiter to me to abruptly go from one to the other.

Oh well, off to cast half a vote in the Michigan Primary.

Comment author: 15 January 2008 02:33:01PM 0 points [-]

@Roko,

I'm familiar with the limit explanation. There are still plenty of cases (esp. in physics) where something like a dx is treated as exactly equal to zero in one place, and also cancels itself out elsewhere in the same equation, without causing any inaccuracy. You say that zero is just the limit, but it's at the limit that Calculus provides a mathematically precise description of, for instance, the area under a curve. Of course this doesn't mean that Calculus is not rigorous; only that in its typical symbolic expression, though you can usually manipulate things algebraically, when dealing with the magnitudes of the dx-type terms you have to pay attention to what you're actually saying. Algebra was invented so that you wouldn't have to pay attention to what you're saying, only follow the rules. But in actual fact you still have to look at the meaning of the equation from time to time. You can't be totally indifferent to the content of "x."

In other words, the limit rule by which you reconcile the Calculus may be internal to the Calculus, but it is external to the Algebra by which the Calculus is usually expressed.

The case is the same with the logic paradoxes, such as the Russell one; it's only by looking at what you're really saying that you can dismiss a logical problem as missing the point.

Comment author: 15 January 2008 03:44:51PM 0 points [-]

I think that was Jubal in Stranger in a Strange Land. I could be mistaken, however; all those father-figures seem to run together.

Comment author: 15 January 2008 06:38:29PM 0 points [-]

As I skimmed over the lines to see how 'smart' I was, figurin that the quicker I got 'there' the smarter I'd be able to feel, the first error I saw was in line 5. Bingo. How can that be! Ah yes, that line 4 div by zero thing. Bingo, again! feelin pretty smart! whew, gonna get more tea. Got back and looked at line 3. hmmm. Pesky zero. feelin a little uncertain. That dadgum line 2 looks pretty good. came from line one fair and square. Eventually over a number of cups of tea, i figured out why few of regulars had been posting on this post (i did notice). i appreciate their patience and EY's clue.

Gotta love this site. And tea :)

Comment author: 15 January 2008 06:56:27PM -1 points [-]

When one got past pre-adolescence, one realised that Heinlein's writing skills, such as they were, were in the service of a political philosophy somewhat to the right of Attila the Hun. Whatever floats your boat.

Comment author: 12 June 2012 02:14:14AM -2 points [-]

The right of Attila the Hun? Hyperbole if ever there was. After all, haven't we already mentioned Heinlein's position on free love? (Attila the Hun was not known for his tolerance of women or the LGBT community.)

Which is not to say that I'd want to live in the world of Starship Troopers.

Comment author: 15 January 2008 07:49:03PM 6 points [-]

When one got past pre-adolescence, one realised that Heinlein's writing skills, such as they were, were in the service of a political philosophy somewhat to the right of Attila the Hun. Whatever floats your boat.

Then one got past pre-adolescence by becoming an uninformed dolt. In fact, Heinlein's political views ranged from Upton Sinclair socialist and New Deal Democrat in the 30's, to hard-core libertarian later in life, but never corresponded to anything "right wing" except to those people who use "right wing" as a synonym for "I don't like it."

Leo, Heinlein praised math to the very stars, but I'm not sure he was actually good at math. It's been a long time and I don't have the book in front of me, but I remember a scene in The Rolling Stones where the father is telling the kids they need to study advanced math, and using some mathobabble, and I don't think the father was making any sense...

The kids, looking at some kind of map of mathematics, say "Dad, what's a hyper-ideal?" "Hyper-ideal" is a perfectly good term in algebraic topology. Heinlein did graduate work in maths at UCLA after his medical discharge. He did incline to being a hardcore formalist, as evidenced by the discussion of axiomatic systems and such in Rocket Ship Galileo.

And yes, it does worry me a little that I can quote a Heinlein juvenile by memory....

Comment author: 15 January 2008 08:11:51PM 2 points [-]

One of Heinlein's faults that he shared with certain individuals here is that he liked to treat all his conclusions as though they were as rigorously demonstrated - and easy to verify - as his mathematical arguments.

It's mostly a weakness in his earlier work - see Starship Troopers - and he had loosened up enough to acknowledge informalistic reasoning in later books.

I suspect we'd find the transition took place, as so many of his others did, in the middle of Stranger in a Strange Land.

Comment author: 15 January 2008 08:59:28PM 0 points [-]

Charlie (Colorado), I'd appreciate your thoughts on the difference between 'hard core libertarian' and 'right wing'. For me they map to pretty much the same territory, obviously not for you.

Comment author: 12 June 2012 02:15:36AM 0 points [-]

Libertarians are not socially rightist, for one thing. If you'd said Heinlein was a hardcore libertarian, I'd essentially agree.

Comment author: 15 January 2008 09:24:36PM 0 points [-]

Comment author: 15 January 2008 10:40:54PM 4 points [-]

tcpkac,

There's a theory of political alignment which separates economic views and social views. "Libertarian" maps to the right on economic issues, but to the left on social issues, what with the acceptance of same-sex partnerships and the push to legalize marijuana and all.

Comment author: 16 January 2008 02:11:42AM 0 points [-]

Hey, so, I figure this might be a good place to post a slightly on topic question. I'm currently reading "Scientific Reasoning: The Bayesian Approach" by Howson and Urbach. It seemed like a good place to start to learn Bayesian reasoning, although I don't know where the "normal" place to start would be. I'm working through the proofs by hand, making sure I understand each conclusion before moving to the next.

My question is "where do I go next?" What's a good book to follow up with?

Also, after reading this and "0 and 1 are not probabilities" I ran into the exact cognitive dissonance that Eliezer eluded to with his statement that "this would upset probability theorists is that we would need to rederive theorems previously obtained by assuming that we can marginalize over a joint probability by adding up all the pieces and having them sum to 1." The material is teaching that "P(t) == 1" is a fundamental theorem of the probability calculus and that "P(a) >= 0". These are then used in all succeeding derivations. After re-reading, I came to understand Eliezer's practical disagreement with a theoretical method.

So another question is: has anyone gone through the exercise of re-deriving the probability calculus, perhaps using "0 < P(a) < 1" or something similar, instead of the two previous rules?

Comment author: 16 January 2008 06:33:16AM 0 points [-]

I think dictatorship only implies a structure of government rather than its actual policies, which is where conservatives and libertarians would differ. This isn't to say that dictators have policies representative of polities as a whole, only that the two seem somewhat logically distinct.

Comment author: 16 January 2008 02:54:49PM 1 point [-]

Brandon Reinhardt,

I'd recommend Jaynes's magnum opus for Bayesian philosophy, keeping in mind that Jaynes isn't infallible in spite of the fact that he writes as if he is.

For practical Bayesian statistics, a good undergraduate intro course is in Bayesian Statistics: An Introduction. Material for upper level undergrad to low level grad can be found in Bayesian Data Analysis, 2nd ed.

Note that I haven't read Howson and Urbach, so I don't know what overlap there might be.

Comment author: 30 May 2012 02:43:15AM 0 points [-]

If you know some calculus, I recommend: Box and Tiao: Bayesian Inference in Statistical Analysis.

Comment author: 16 January 2008 03:05:31PM 0 points [-]

Whoops, misspelled your name, Brandon. Sorry!

Comment author: 16 January 2008 03:53:15PM 3 points [-]

"That is to say, the libertarian program and the right-wing program are identical in consequence, but libertarians pretend otherwise, through a process called "wishful thinking". An honest right-winger plans to hold a position of power in an ironclad dictatorship, where the libertarian hopes (or claims to hope) that the consequences of the policies he espouses won't actually be that ironclad dictatorship."

Since you don't like libertarians or right wingers, I assume you would describe yourself as left-wing. Given the stances on which libertarians and left-wingers agree, (which make up at least 50% of the libertarian platform) such as freedom of speech, drug de-criminalization, pro-gay marriage, pro-separation of church and state, etc... you'll have to explain to me how libertarians don't really believe those things, or how they actually do contribute to this ironclad dictatorship you think libertarians are secretly rooting for.

There are criticisms of libertarianism. But it seems rather silly to claim that a political philosophy that so emphatically supports reduction of the power of government is actually a front for dictatorship. Obviously they've just been really good at keeping that secret.

Comment author: 16 January 2008 07:01:37PM 0 points [-]

x-y=1-1=0

I don't trust in division by 0 to return rationals.

Comment author: 16 January 2008 07:39:50PM 1 point [-]

tcpkac, the first problem is coming up with a definition of "right wing" that has any operational value. If "hard core libertarian" meant "right wing", then you'd be including both the free-trade, free-love, cheap-drugs libertarian like me (and Heinlein, I think) with a Pat Buchanan nationalist, isolationist, Christian-privileging paleoconservative, a Ron Paul/Lew Rockwell racialist isolationist paleolibertarian, and, arguably, a Mussolini Italian Fascist into one category --- which seems to reduce "right wing" to vacuity. It happens that Philosoblog extends this discussion just today, while critiquing Jonah Goldberg's Liberal Fascism. It would seem that "right wing" primarily means "I think I'm left wing and I don't like it."

Elizer, I don't think that a skeptic like Heinlein meant any of his characters to be taken as completely authoritative. That said, I don't see any difficulty in reconciling "perfectly logical" in context --- he's using it as an example of using logic to arrive at an absurdity --- with "deceptively logical seeming."

Of course, to a formalist, the whole syllogism could be perfectly logical in the strong sense you're using, since we could construct an axiom system in which 1/0 is well defined.

Comment author: 17 January 2008 12:18:08PM -2 points [-]

Brian:

The mystery of libertarianism being identical, in practice, to fascism is easy to solve. Simply ignore the expressed goals (i.e. wishful thinking), and concentrate on the predictable consequences of the advocated program. Lo and behold, the program is the same as the fascists', and must therefore lead to the same results. QED.

And no, I'm not "left-wing". The wings are for loonies.

Comment author: 17 January 2008 12:53:53PM 0 points [-]

predictable consequences of the advocated program. The closest things to libertarianism that has ever existed are the classical liberal/capitalist states of the Anglosphere, which hardly seem fascist.

Lo and behold, the program is the same as the fascists' You just assumed for the hypothetical that the program was what the libertarians advocated contra the fascists and then moved on to your imagined results, which is a different thing from the program itself.

and must therefore lead to the same results. Predictable results lead to those same predictable results? Sounds like a circular argument to me.

Comment author: 17 January 2008 03:16:07PM 3 points [-]

Just to respond to the theme that 'right wing' is a meaningless label, not so. It originally arose from the seating arrangements in the French Assembly, where the right wing were the monarchists. Hence right wing became generally accepted as a label for the authoritarian defence of a monarchic, aristocratic, or oligarchic power structure. As these power structure tended to be the ones in place, you have the confusion with Conservatism (e.g. Torys). By a further semantic slide, it came, for some, to mean any authoritarian power structure with power concentrated in the hands of the few, hence the lumping together of the various 20thC dictatorships as right wing. For those who conceive the power of 'Big Business' to be oligarchic and oppressive, any political program favourising the large corporations is right wing. One source of confusion between 'right wing' and Libertarianism comes from the disingenuous protests that any politics which limit the power of the corporate world are 'attacking free enterprise' thus, attacking individual freedom. This is compounded by the myths attached to the notion of private property, where 'mine' as in 'my log cabin and my boots' is extended to 'my corporation over which I have Regalian powers' simply because I invested some bucks in it 30 years ago. Libertarianism as described here seems to be a peculiarly American movement, which would map somewhat but not completely to the European anarchists. Finally, of course individual politics are multi-dimensional. However, all countries which aren't dictatorships seem to end up with two party systems, so all those dimensions have to projected down, hopefully on a 'best-fit' basis, to the single axis most appropriate to the country in question.

Comment author: 17 January 2008 05:15:16PM 4 points [-]

Regarding calculus, it is possible to accept infinitesimals and thus view dx/dy as meaningful in an absolute sense, and not just as the outcome of a limit process. This is what is done in nonstandard analysis, in which infinitesimals are the reciprocals of superreal numbers that are infinite (although not equal to or equivalent to the infinite cardinals). This is in fact how both Newton and Leibniz thought of the matter. For Leibniz, a monad was a point surrounded by infinitesimals.

However, while infinitesimals are smaller than any positive real number, they are not equal to zero, strictly speaking. Therefore, they are irrelevant to the discussion of the steps in the Heinlein exercise, which is dealing with the actual zero. Indeed, this exercise is a reminder as to why dividing by zero is ruled out. Allowing it allows absurdities such as this exercise. 0/0 can be anything.

My late father was once asked by a young woman in the audience at one of his public lectures, "Is zero a real number?" He replied, "one of the finests, my dear, one of the finest."

Comment author: 18 January 2008 12:31:05AM -1 points [-]

Thank you, tcpkac, for your concise and clear exposition. All I have to add is that anarchy also leads directly to dictatorship.

Comment author: 18 January 2008 01:42:14AM 0 points [-]

Taking the principal determination for the log, find the error.

log[x^2] = log[ (-x)^2)]

2 log[x] = 2 log[-x]

log [x] = log [-x]

log[x] - log[-x] = 0

log[1] - log[-1] = 0

0 - iPi = 0

iPi = 0

Comment author: 18 January 2008 03:06:01AM 1 point [-]

However, all countries which aren't dictatorships seem to end up with two party systems

Um... no. This is simply wrong. You would have been better off claiming that all countries that aren't dictatorships end up with an n-party system - at least then any error would be concealed by the vast number of confirming examples. As it was, the opposite applies.

Comment author: 18 January 2008 07:01:29AM 2 points [-]

By a further semantic slide, it came, for some, to mean any authoritarian power structure with power concentrated in the hands of the few, hence the lumping together of the various 20thC dictatorships as right wing.

You've just argued that the Communist dictatorships of the 20th century, the USSR, China, Cuba. are "right wing", which seems to establish the vacuity of the term far better than anything I could have written.

Comment author: 18 January 2008 08:16:23AM 1 point [-]

Misanthropist, the error is in line two. a ln b = ln (a^b) holds only for positive a.

Comment author: 18 January 2008 09:35:04PM 0 points [-]

Oh, and regarding infinitesimals again, some have argued that the old medieval dispute about how many angels can dance on the head of a pin was really a debate about the existence or nonexistence of infinitesimals.

Comment author: 07 March 2008 09:02:31AM 0 points [-]

The discussion on this site seems to have many strands, but concerning Calculus - I was trying to figure it out for years and then came across smooth infinitesimal analysis. This is a version of analysis based on microstraightness and nilsquare infinitesimals. It reduces differential calculus to simple algebra, it does not lead to contradictions (such as the Banach-Tarski paradox), it facilitates physical derivations based on microadditivity, and it does not employ the dubious 'taking the standard part' trick of non-standard analysis (which is just Limit theory in disguise). The best book on the subject is A Primer of Infinitesimal Analysis by J L Bell.

Comment author: 20 August 2011 12:06:50AM 0 points [-]

"Huh, if I didn't spot this flaw at first sight, then I may have accepted some flawed congruent evidence too. What other mistaken proofs do I have in my head, whose absurdity is not at first apparent?"

Has this question ever been answered? It is one of those things I go around worrying about.

Comment author: 01 January 2012 04:43:16AM 0 points [-]

Why stop at 1=2? Once you get there, you're just a few simple steps away from proving that every number is actually equal to every other number.