Monday, July 14, 2014

Definitions and Discoveries

At first glance it seems that it would be rather silly to mix up the ideas of a definition and a discovery, but at least in maths (and perhaps especially theoretical physics) it's surprisingly easy to do. Both processes produce rules that allow us to go on and derive new rules and descriptions -- the big difference is where they come from.

Might require a change of maths.
Suppose, for instance, that I want to construct a rule using the word "good". Then I would work out what the word "good" means for the rule by seeing how it works in the world -- I would discover it, rather than defining it. (There might be situations where you would choose to define it differently from how it works in the rest of the world, but you would have to be clear that you were doing something funny then.) This is rather like how we decide to write the description of gravity as pulling things together. Mathematically you could write about it as a force that pushes things apart, but that wouldn't describe the gravity we experience. Likewise, if you want to describe arithmetic as we know it, you can't decide whether 2+3 and 3+2 are the same thing. You discover that you don't get the expected results unless they are. (One might decide to define it differently anyway, but then one would no longer be talking about ordinary arithmetic.)

Things get more complicated if I have a procedure that involves putting "un" at the beginning of words. I can't go out and discover the meaning of "ungood", because it's not a word you can find in the world. Nothing about the way people talk changes if I interpret "ungood" as meaning "butterfly" or "convenient" or "pirouetting", because people don't say "ungood" in the first place. Situations like this tend to crop up in the intermediate mathematical steps of a physics problem. I've translated the situation into equations and now I want to solve them. Solving equations doesn't mean anything physically, although the solutions should be something we can translate back. That means there aren't clear physical requirements on what is or isn't allowed in trying to solve the equation. 

If I want to be able to use the "un" procedure, I have to come up with a meaning for "ungood". There's nothing stopping me from picking any definition I like -- but some will be more useful than others. I'd like to pick a definition that means the "un" procedure does the same thing it does to other rules. "Decided" becomes "undecided", "expected" becomes "unexpected" and "forgettable" becomes "unforgettable", all of which do mean things already. It makes sense that when "good" becomes "ungood", it means the opposite of "good". We might define "ungood" to mean "bad".

We don't have to define it that way, though. We could go by similar phonetics instead and define "ungood" to mean "unguent". The "un" rule wouldn't work as expected any more, of course. But if I were writing a speech-to-text program, say, that might be less important than the phonetics. Besides, it's not like the "un" rule is infallible anyway -- look at how "til" becomes "until". It's only if I want the definition to work in a system where the "un" rule always produces opposites that defining "ungood" to mean "unguent" would be inconsistent.

The same thing happens mathematically. If something doesn't exactly match up to some physical observable, we need to define what it means. Some definitions make more sense than others when it comes to translating back and those are the ones we choose. But unlike making a discovery, it is a choice. We decide that we want certain rules and procedures to do what they do elsewhere and choose to define things accordingly. The definition isn't forced upon us by the way the world works. 

Why do I care about such a subtle difference? If you just want to apply the rules, it doesn't really matter. But if you want to understand where they come from and what makes physics tick, you need to know what's required to describe the world and what's just a helpful way of thinking about intermediate maths. And that's why I'm quite sure I'm neither the first nor the last physics student to say "Oh is that just a definition? It all makes sense now..."

Savo 'lass a lalaith.