--- In conlang@yahoogroups.com, Chris Bates
<chris.maths_student@N...> wrote:
> Apollo Hogan wrote:
>
> >On Fri, 13 Aug 2004, Chris Bates wrote:
> >
> >
> >
> >>I personally find a great deal in common between maths and
language. Its
> >>all the same... manipulating symbols (words, morphemes) according
to
> >>certain rules... and I wouldn't be surprised if the language
centers of
> >>the brain are active when someone is manipulating formulae etc.
For most
> >>maths problems I think visualization isn't actually that
useful.... you
> >>tell me what you visualize when solving problems in number
theory, group
> >>theory, or most of pure maths or statistics. The only part of
> >>mathematics where visualization is sometimes helpful is applied
maths,
> >>and even then not always. If you're working in a 4-dimensional
space how
> >>exactly do you visualize what's going on? I don't know about you
but my
> >>brain doesn't do pictures with more than 3 dimensions in them, so
if you
> >>ever want to do relativity you'll need to wean yourself off those
images
> >>in your brain a little. Pictures don't constitute proof and often
can be
> >>misleading.
> >>
> >>
> >
> >I'll throw my two kopeks in here. I do set-theoretic topology and
> >I must say that I can only do mathematics where I can have some
sort of
> >intuition of what is going on. This intuition is not necessarily
visual
> >(in that I can draw a picture) but it certainly doesn't seem
linguistic.
> >(My advisor does make fun of me for always drawing little pictures
when I
> >explain proofs to him :-) Purely formal/symbolic proofs do little
for me
> >until I can "unravel the symbols" and understand what's going on
underneath.
> >Thus I am terrible at things like algebra and number-theory which
can sometimes
> >be very formal and symbolic.
> >
> >However, there are many mathematicians I know who claim the
opposite. This
> >seems to be consistent with the idea that there are two approaches
to
> >mathematics: continuous and discrete or geometric and symbolic or
visual
> >and linguistic. (Granted both are necessary, but it seems many
people have
> >psychological leanings toward one or the other. I am more
geometrical/visual.)
> >
> >The point of this is that it seems that there is vitally a _non-
linguistic_
> >part of mathematical thinging/intuition.
> >
> >--Apollo Hogan
> >
> >
> >
> I'm not arguing against intuition, which can be a useful tool as
long as
> you use rigorous mathematics to try to back up what your intuition
tells
> you, but there is a difference between intuition and picturing
things
> visually. For me maths is mainly the occasional knowing intuition
which
> doesn't involve pictures or images much at all, and then symbolic
> manipulation that reminds me strongly of language. But I most
certainly
> do not think in pictures, and I truly do find that trying to draw
> pictures of most problems that are more than elementary in your
head or
> on paper simply confuses the issue and misleads you. For me maths is
> mainly symbolic manipulation with an occasional burst of intuition
to
> guide my steps.
Perhaps relevant to this discussion is that I heard that some
educators put people into three (or 4?) classes, based on their
primary method of learning -- visually (by sight), audibly (by
sound), and kineto-somethingly (by touching/doing). I'm weakest in
the audible learning area, though I did a fair amount of it in
college. I think I'm strongest in visual learning area (apparently
most people are), although I also have strong kineto-? learning --
I'm never really sure I've learned a new process until I've actually
done it. Often, I'll combine seeing+doing into drawing or graphing,
which helps me alot! Oddly enough, one of my hobbies (composing
classical music) makes use of my weak auditory area.
I'm a software engineer with minors in math and physics, and in order
to figure out new computer code, I've found that drawing class
diagrams and flowchart-like digrams with arrows all over the place
really helps. But its not the final picture that helps so much
(although that's useful as a reference later), as it is the process
of drawing it. And when I recently balanced my checkbook I ended up
plotting out my account over the last year and a half. That might be
going a bit too far, though.
In my Modern Physics class, I had to make myself graph the Lorenz
transformations before I could completely understnd them. Before
graphing them, I could solve the equations and get a correct result
without thinking about why the result works. After graphing it, I
gained insight and intuition about the whole thing, and I was able to
actually _see_ why relativistic paradoxes (like the barn and ladder
paradox) _had_ to occur (and how they had to be resolved).
Fortunately, the Lorenz transformations can be graphed in 2-D without
losing any essential information, however, I've also developed
several methods for visualising higher dimensions when the need
arises. Theoretically, at least one of these methods is arbitrarillay
nestable to any number of dimensions, although I don't think I would
want to go much above 8 or 9! But 4 and 5 dimensions are often not
that difficult for me. In a sense, we all visualize in 4-dimensions
anyway (space+time).
I find languages (or at least grammar) to be an extension of
mathematical principles as well. Not just predicate logic, but also
hierarchical trees (a familiar math/compsci concept). When I first
saw grammatical tree diagrams, I was amazed that there was so much in
common between linguistics and CS. For those who remember it, my
METAGRAM system was meant to be a way to textually diagram the
underlying tree structure of sentences (While also helping to
prototype Akathanu's grammar).
~Caleb