Re: Numbers and math

On Fri, 22 Sep 2000, H. S. Teoh wrote:

> Now, take this one step further -- combining these directional numbers
> (integral vectors, if you will) with each other, perhaps in simple
> combinations like Pythagorean triangles, and you have a culture that
> understands vectors but not negative numbers. Hmmm.... :-)
The Greeks...?

> > In your example, any vector space isomorphic to the complex plane, like
> > Euclidean two-space, would do just fine.  :-p
>
> Yep. That's right. I used complex numbers 'cos it sounds more impressive
> :-P
True.  :-p  But Euclidean two-space would probably be more indicative of
"understanding," because i, complex roots, phase shifts in waves (I
think...darnit, where's a physics major when you need one?) and so on are
*why* you use the complex plane as opposed to R^2.

> > And what does it mean to "understand" negative numbers?  Or to
> > "understand" complex numbers?  This isn't a trivial question!  <G>
>
> Perhaps claiming that they "understand" complex numbers is a bit
> far-fetched. But if a culture has different words for numbers depending on
> their direction -- e.g. 2-north is represented by a different word from
> 2-west, and so on for each integer number -- then one might ask, "so what
> is 1-north plus 4-east?" This could lead to the development of directional
> quantities (vectors) *before* the development of negative numbers and the
> like.
That would be neat.  :-)  Heck, I could still see them developing
fractions before negative numbers.

And I think Hindu mathematics may have had something darn close to
infinities, but don't quote me on that.  The only math history text I
have are Western-centric (because they're leading up to modern mathematics).

> > YHL the math major, alas
>
> Alas??? I find math to be very enlightening in learning different ways to
> think about things. I especially appreciate the courses I took on number
> theory and set theory.
It's great for that.  I really like algebra, myself.  But I find writing
and foreign language infinitely easier.  When you get right down to it,
I'd make a lousy mathematician.  I'm a math major because it's the
subject I need most help with, i.e. trying to learn it by myself, outside
of class, would be impossible.

(Yes, I'm a wimp.  Math geniuses now have permission to laugh.)

> Set theory is really neat because you get right down to the roots of it
> all, and have to deal with very fundamental issues such as vacuously true
> predicates, existence, incompleteness, etc.. In more "practical" math,
> such issues are often swept under the rug ("Assuming that X exists, the
> theorem says Y is true"). But in set theory, there is no rug to sweep
> things under -- you gotta deal with many apparently obvious statements and
> make sure there are no holes in your arguments.
Well, this goes into the entire issue of what constitutes proof, which is
less obvious than it seems at first (have you read any philosophy of
mathematics?).  Also, "more 'practical,'" do you mean applied math?
Because set theory can be applied in an awful lot of places.  When you
get into analysis and algebra (not applied, but the theoretical courses),
the TA's *kill* you for holes in arguments.  I suspect that in deeper set
theory you'd still find the rug-sweeping problem, but I'm not going into
that subfield.  :-p

ObConLang: what constitutes "proof" in a language?  It occurs to me that
writing mathematical proofs in Chevraqis (or any other language with
evidentiality) would be a lot of fun; I grant you there are only three
levels, but you could use the "probable" for reductio ad absurdum
hypotheses (things you're using to show they lead to a contradition), the
"plain" level of evidentiality while you're chasing implications, and the
"reportive" (something the speaker has witnessed or a "fundamental
truth") to make the final statement of your proof.  OTOH I wonder if it
would make math more difficult, in that a speaker would be
psychologically inclined to trust a statement in the reportive, and maybe
less likely to look for inconsistencies or contradictions?  I don't know.

YHL