Mathematics is a bit like plumbing, don’t you think?
Not that I’ve ever done any proper plumbing. Electrical work I can manage, having paid careful attention when doing electricity in our high school physics classes. But plumbing is another matter altogether. Getting those joints water-tight requires just a bit more dexterity than I can manage.
That’s why it’s like maths. It’s those joints that are the problem. Let me explain.
Often in maths, when one is trying to prove some important result, one creates a series of theorems that build on one another to reach the final conclusion. If a proven result along the way is substantial enough – say if it takes more than half a page to prove – we call it a ‘theorem’, otherwise we call it a ‘lemma’. But a lemma is really just another theorem, only a bit smaller than usual.
Now a theorem is something that reads like this:
IF <bunch of premises> is true THEN <conclusion> is true.
Mathematicians try to make their premises as few and weak as possible, and their conclusion as strong as possible. For instance, we could prove a theorem that
IF x=2 THEN x2>0
This theorem is true, and easy to prove. But we can either make the conclusion stronger (ie more specific), turning it into say x2=4, or we can make the premises weaker, turning them into say ‘x is a real number’. We can’t do both however. We could, if we wanted, make the premises weaker and the conclusion stronger, but not by as much as what we just did. We could prove the theorem:
IF x>1 THEN x2>1
Now our premise is weaker (less specific) than before, and our conclusion is stronger (more specific) than before.
How this is like plumbing is as follows. Say we want to prove theorem D and, to do that we need to first prove theorems A, B and C. This might be because we need A to prove B, B to prove C and then C to prove D. So the conclusion of A becomes a premise for B, the conclusion of B becomes a premise for C and so on.
Now unless D is very easy to prove (eg like proving IF x=2 THEN x2>0), we won’t have much room for manouevre. If we make the premises of D too weak then we won’t be able to prove the conclusion, which is our ultimate goal. On the other hand, if we make the premises of D too strong we won’t be able to satisfy them with the conclusion of C.
This is like connecting a series of pipes (got you! And just in time. You thought I’d never get around to the plumbing didn’t you?). Each theorem is a pipe. Each pipe is a straight cylinder except that, at its inflow end, it flares out like a trumpet bell and, at its outflow end, it narrows. This enables pipes of the same diameter to snugly fit together. The end of each pipe has to fit inside the end of the pipe immediately downstream from it if the fit is to be tight enough to avoid a leak.
The inflow end represents the theorem’s premises. The stronger the premises, the narrower the inlet, and the stronger the conclusion the narrower the outlet. Now what happens if we make the premises of pipe D too strong? Then the downstream end (conclusions) of pipe C may not be able to fit inside the premise end of pipe D. If that happens then we need to narrow pipe C, which means making its premises stronger – hence a narrower inlet. That may then create a problem with fitting B into C, and so on. So if we are too ambitious in what we are trying to prove at D, that can create problems all the way back up the stream so that our premises at A are so narrow and restrictive as to make the whole combined theorem of no practical use (because the premises at A will hardly ever be satisfied). The theorem might only apply to left-handed, bearded Scottish taxidermists whose first and last names both begin with ‘X’.
Or, starting at the upstream end, if we start with premises that are too weak – non-specific – at A, our whole series of pipes will have to be so wide that the eventual conclusion at A will be as generalised as an astrologer’s forecast (‘Today will be a good day for working hard and believing in yourself!’). How can I relate this diffuse outcome to plumbing? Hmm, well if the pipe is too wide at the outlet then perhaps there won’t be enough pressure. Yes that’s it, no pressure. It would be like trying to water the garden with a big drainage pipe instead of a hose.
What one finds oneself doing then, as one tries to make the connection from A all the way to D, is going back and forth, loosening a premise here, tightening a conclusion there, then finding that with the premise we just loosened, the conclusion coming out of that pipe is too loose (wide) to fit into the next pipe so we tighten the premise again or alternatively look at the following pipe to see if we can get away with loosening the premises for that.
And now for one last bit that is completely unrelated, except that it also is like plumbing. Recently I’ve been working on something called Homology Theory, which is what you need to be able to tell the difference between a sphere and a donut. This discipline uses techniques of algebra to identify differences between different shapes. It often makes use of chains of things called modules, connected by functions, which are like arrows leading from one module to another. It’s a bit like how you can make a model of a molecule using plasticine for the atoms and straws for the chemical bonds (the modules are the plasticine and the functions are the straws, except it’s important to remember that the functions have a direction, as if water were flowing through the straw). An important part of a function going out of a module is something called the ‘Kernel’, which is a bit like the nucleus of the atom. Each function in a sense ‘projects’ the module from which it starts onto the module at which it ends, and the projection it makes is called the ‘Image’ of the function, which will be part or all of the target module. We call a chain of functions and modules ‘Exact’ if the image of the function coming in is identical to the Kernel of the function going out. Most chains are not Exact, but when a chain is Exact we can prove all sorts of useful things.
So this is more plumbing. If the pipe/straw (function) coming into a joint (module) is too wide (has too big an Image) it won’t fit the Kernel of the pipe/straw (function) going out. Then the liquid will spill out, we’ll have a big mess and we’ll never be able to tell our donuts from our spheres, let alone our circles and Klein Bottles. But if it all fits neatly, we’ll have an Exact Sequence, there’ll be no spillage, and we’ll be able to prove all sorts of useful things without needing a change of clothes.
I’m rather pleased with how many different metaphors I managed to mix together in that last couple of paragraphs. It leaves ‘to take arms against a sea of troubles’ far behind. I think the clarity may have suffered somewhat as a consequence though. Never mind.
Bondi Junction, December 2014