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.
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| 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.
