Re: OT: -1 (was Has anyone made a real conlang?)

On Tue, Apr 22, 2003 at 07:07:17AM -0700, Stone Gordonssen wrote:
> >And what, pray tell, is the purpose of these beasts? What problem can't be
> >solved with real numbers that it requires us to take the
> >sqrt of -1?
>
> Several physics and mechanics problems in quantum mechanics. Though, even as
> a math/comp.sci major, I've never felt at ease with imaginary numbers - I
> keep thinking that if we have to make up such numbers, there's something
> wrong in our reasoning or premes from the start.
[snip]

Then you should be equally uncomfortable with the idea of breaking up
vectors into vertical/horizontal components. The term "imaginary numbers"
is just a historical accident; they are no less imaginary than the "real"
numbers, which, for all we know, may be completely imaginary[1]. After
all, everything in the real world requires only fractions of sufficient
accuracy. The area of a circle is made up of finite atoms, and none of
them are fractional atoms. Irrationals (transcendentals, to be precise)
like pi are only necessary if we posit infinite accuracy---which in itself
is a stretch. We only accept it easily because we're so used to it in this
society. That does not make it any less imaginary than the so-called
'imaginary' numbers!

The same goes for negative numbers... how does it make sense (in terms of
analogies with the real world) to have actual entities that have the
opposite effect when you add them to other entities? One can argue that
(5 + (-4)) is a flawed way of writing (5-4). Negative numbers are just as
imaginary, in this sense.

The truth is, as someone so aptly put it, "God created the natural
numbers; everything else is the invention of man." The reason we have
negative numbers, fractional numbers, irrationals, etc., is because the
human mind works by abstractions, and abstractions like negatives,
fractions, etc., allow us to operate better in reasoning about quantities.
If 'imaginary' numbers ('complex numbers' is probably a better term) help
us understand the symmetry of fundamental particles, then more power to
it. Quaternions (numbers which have *3* distinct square roots of -1) have
been posited in representing 3-dimensional rotations. And you thought
'imaginary' numbers were a stretch... :-)


[1] If indeed the Axiom of Infinity is 'flawed', as some believe. But if
you're going to start questioning axioms, you might as well wonder about
whether mathematical sets actually exist, as posited by the first axiom of
set theory.


T

--
Once bitten, twice cry...