1+1=2

[lederhosen]

ambitious_wench posted a link to a scan of part of Russell's proof that 1+1=2, and asked maths geeks to discuss why it was beautiful.

First off, I have to admit a couple of things: I'm not familiar enough with Bertrand's work to follow how the proof works, and I don't know that I'd call it beautiful. There are more than 300 pages of mathematical setup involved before he even gets to that point, and the proof isn't actually completed until eighty-something pages into the next volume of Principia Mathematica. Combined with aforementioned lack of familiarity on my part, that makes it rather too unwieldy to engage my sense of mathematical beauty. But I might be able to give at least some idea of why it's important.


Once upon a time there was a city in the middle of a swamp. It had been settled by mathematicians who arrived there a very long time ago.

The thing about the swamp was, you couldn't build there. If you tried, your buildings would just fall over.* So if you wanted to build something new - whether for practical purposes, to show off your skills, or just to admire the view - you had to build it in the city, and you had to build it on top of existing buildings, sort of like a circus acrobat standing on other people's shoulders. This was by no means a simple task; you had to attach it to those other buildings in just the right way, or you'd end up with something unstable.

Most of the time the unstable buildings would fall down pretty quickly, either of their own accord or because some other mathematician had wandered along, spotted a problem, and made a point of pushing them over.

But occasionally somebody would build something that looked pretty good, with only a very subtle flaw in it. And then it might stand up for years, long enough for other people to build their own structures on top of it... until at last some enterprising mathematician found the weak spot, and brought all of them crashing down.

Understandably, mathematicians don't like that sort of thing happening (not to their own buildings, at any rate) so as time went by and the city rose higher and higher, they gradually got more and more fussy about how things should be built - especially since some of the newer, fancier buildings required very elaborate techniques.

(Just as an example, one of the standard techniques for building a tower is a process called 'induction', which basically worked like this: the mathematician would place the first brick on a firm footing, call in some labourers, and show them how to put bricks securely on top of other bricks. Then the mathematician would wander off to fuss over something else, secure in the knowledge that the labourers would go on stacking bricks one at a time until they got to the top of the tower. But eventually a clever fellow - I think it might have been Cantor - showed that some building types couldn't be constructed that way; for certain really big types of tower, building one brick at a time - or two, or a hundred, or a thousand - wasn't 'fast enough'.)

So, as they built upwards, getting ever more fussy about architecture, they also started to look back downwards... because none of them wanted to find that they'd wasted all this effort building on top of something that wasn't sturdy. There was a certain attitude of "it's been around a long time and hasn't fallen over, so it must be sound", but this didn't entirely satisfy them, and gradually - around the end of the nineteenth century - they got more and more uncomfortable.

Because no matter how far down they went, they never found a completely secure foundation. Most mathematicians had believed that the first buildings in the city were built on solid bits of rock (or as they called them, 'axioms'). But as they tunneled further down, they found that some of the things that looked like axioms - for instance, "1+1=2" - were really very old buildings, resting on something deeper.**

Along the way, Bertrand Russell and others discovered something alarming: an entire wing of the Set Theory building - a building that served as foundation for many, many others, including in an odd sort of way the "1+1=2 building" - was structurally unsound. It turned out that one of its major pillars was resting on nothing but swamp, and if they kicked that out the entire wing would fall over. Through a great deal of lengthy and clever work, they managed to shore up the rest of the building and get it to the point where it would still support most of what had been built on it - not all of it, and to this day people occasionally come to grief by trying to build on top of the unsound parts of that building.

As part of that work, they had to verify that the remaining part of the building was okay, and this they did: they showed that well below the part of the Set Theory building that held up the "1+1=2" building were some deeper supports that looked pretty sturdy - whether axioms, or just more buildings resting on even deeper ones. In the course of Principia Mathematica they looked at which parts of the Set Theory building were still reliable (assuming those deeper supports held) and, to reassure some mathematicians who weren't sure what to believe any more, they made a point of showing that if those deeper supports were sound, which they look to be, then the "1+1=2" building was reliable.

(They also showed that this sort of safety analysis was the sort of thing that really did take hundreds of pages, even when condensed to shorthand that might as well be ancient Sumerian to most of us, which is probably why people don't do it very often.)

So, although it might not be exactly beautiful, it helps mathematicians sleep more soundly at night.

*No, they did not catch fire, and no, the fourth one did not stay up.
**They also had another problem with the genuine axioms - the axioms certainly looked solid, and after thousands of years of building on them everybody tended to assume that they were, but there was no absolute guarantee. In the end most of the mathematicians decided that if the central axioms themselves should ever fall over, there wasn't much they could do about it - that would be so bizarre a world that there was no way to approach it logically - so for the most part, they decided to treat them as sound.

Comments

[tenner.livejournal.com]

I love your math posts... you write about math very, very well.

The entire concept of proof is one reason that I never pursued pure mathematics as a course of study. (Okay, I actually have a minor in mathematics, but that was only because it was easy to pick that up en route to a physics degree.)

It seems to me that on some level you never really can prove anything, because every proof has some basic "given" that must be assumed in order to make the proof work. At best, this would lead to circular logic. (Given A, A proves B; given B, B proves A.) At worst, this would mean that nothing is provable, because 1+1=2, but we have to define addition and equality, the ideas of unity and cardinal numbers, and each of those definitions requires other assumptions. Your second footnote (**) pretty much sums up this idea, and I'm okay with it. It just doesn't appeal to me intellectually... it seems illogical to me, probably because I'm used to thinking of observation as the core principle of logic.

At least in the physical world, one can say 1 + 1 = 2 and know it's true because two measurements of 1 cm and 1 cm form a total length of 2 cm. Or two forces (1 N and 1 N) pointing in the same direction produce a net force of 2 N.

So, I appreciate mathematics for its entire concept of proof, but personally it seems to me that if you require proof based purely on logic instead of on logic and observation, you can't ever really prove anything.

My two cents.

[michiexile]

It is not the case that you cannot possibly prove anything. It is however the case that mathematics only ever deals in implications. All mathematics is true, because if the presuppositions hold, then the conclusions do hold, and if the presuppositions fail, then the statement is irrelevant.

However, the physical isn't quite as easy as you state it either. It is true, that up to subtle effects messing things up, 1cm + 1cm = 2cm. However, you do get things like dilation due to high sub-luminar speeds and other weird effects messing it up, so that it might not necessarily be the case that 1cm + 1cm = 2cm, especially not if you store your ruler apart from the objects you try to measure.

All that said, once we define 1, 2, + and = well enough, we would, as Russel and Whitehead, be able to prove beyond any doubt, that for these particular definitions, it really is the case that 1+1=2 and nothing else.

[mothwentbad.livejournal.com]

It's buildings, all the way down.

I made a Rudin exercise into a story once. Though I made a few choices that have no allegorical meaning.

You know...

[jazzmasterson.livejournal.com]

Posts like this one and the time you explained differentials and integration to me are why you really need to write a book.

[bunny-hugger.livejournal.com]

Good post. It made me start thinking about Descartes, and perhaps that is unsurprising since he was both a mathematician and a philosopher.

[tcpip.livejournal.com]

What is this with the sudden revival of interest in Bertie?!? He's popping up everywhere!