Wednesday, 27 June 2007

Lemma 1

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I'm suffering from some back pain and consequently it is pretty uncomfortable to sit at my desk for long periods. So I decided that I would write a little thing on one of my favourite things in maths, the lemma.

A lemma is a beautiful thing, although there is a bit of a grey area as to what constitutes a lemma and what constitutes a theorem. Both are true statements that are derived from some list of conditions; however, generally people distinguish a lemma from a theorem either by saying that a lemma is a stepping stone on the way to proving a much larger result (and the only reason for writing it down is that it is a useful construction along the way to getting the bigger result), or that a lemma is a true statement but that it is somehow "less important" than other results. Of course, some results are important and useful in the grand scheme of things by what they allow us to do, but I sometimes feel that classifying a result as a lemma because it is somehow worth less than a theorem is a bit sad really.

Recently I've proved my first theorem, i.e., I've taken some initial conditions and results and from that derived a new true statement (previously I've also shown something to be true by finding an example which satisfies certain conditions, but somehow this "proof" feels much less satisfying). A lemma is like a proverb (with the extra condition that a lemma is definitely true): it might not be completely neccessary, but very often knowing it makes life easier.

While it was very satisfying being able to write Theorem and put my name next to it, it was also pretty cool to get several results along the way and write Lemma next to them. In some ways it feels even more satisfying to divine a lemma, because it adds to the richness of the proof that one constructs. By creating a lemma as well it is possible that in future someone will be able to jump a couple of steps in some proof or construction that they are making, because they know that statement X is true due to your ingenuity.

That said, a lemma should have some usefulness. I proved a small result towards the end of last week, and extended it further over the weekend; the initial result and thought was this tiny little step in the big proof I did, but I realised very soon after that there was a much more general thing that one can say about these things. Now, my lemma is a true statement about some area of mathematics (about eigenvalues of eigenvectors of the meridian map in the space of patterns in the Kauffman skein of the annulus!) but at the same time it is unclear as to whether or not it us a useful piece of knowledge.

In one of the Ricky Gervais Show podcasts Ricky tells Karl that "knowledge without application is pointless," a statement that I don't exactly agree with. I take his point (or what I think his point is) but at the same time there is something about mathematics, especially pure mathematics, where you find yourself studying things, divining truths which really don't seem to have any effect on the real world whatsoever, whether the truth is "known" or otherwise. As it stands, I don't know if my lemma is something that someone else is ever going to use, I'm not even certain if it is a piece of maths that I will need in future.

It exists though, and even if I had never taken the time to work through and prove it it would still exist. And if that doesn't satisfy some condition of "importance" I don't know what does.

Here ends the first lemma. I'm going to see if there is anything else I can take for my back pain, because ibuprofen just isn't touching it at all.

4 comments:

mattiecore said...

I love Ricky Gervais, but I hate statements like the one you quoted...

hellmarx said...

I guess a lemma is considered to be a technical tool and people tend to thin that a lemma is a less imprtant result than a theorem. But if you just think of Zorn's Lemma - it's definitely one of the most amazing results in maths and equivalent to the axiom of choice!!! And it's still 'just a lemma'..

noisms said...

What about lemmings, though? That's what I want to know.

zero_zero_one said...

What about lemmings? They're some of the best videogame characters ever...

And they're the subject of rumours and myths when it comes to their apparent suicidal state of mind!