Re: How Mathematicians Think

[Russ Abbott]

Hi Nick,

Looks like a great book. Thanks for mentioning it.


-- Russ Abbott
______________________________________

 Professor, Computer Science
 California State University, Los Angeles

 cell:  310-621-3805
 blog: http://russabbott.blogspot.com/
 vita:  http://sites.google.com/site/russabbott/
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On Mon, Mar 22, 2010 at 12:35 PM, Nicholas Thompson <[hidden email]> wrote:

Glen,

you wrote

" Math is a language for disambiguation".

Forgive me if I have asked you this before:  Have you ever read Byers HOW
MATHEMATICIANS THINK?

If so, could you rub those two rocks together a little for me?

Thx,

N

Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
http://home.earthlink.net/~nickthompson/naturaldesigns/
http://www.cusf.org [City University of Santa Fe]




> [Original Message]
> From: glen e. p. ropella <[hidden email]>
> To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
> Date: 3/22/2010 12:20:16 PM
> Subject: Re: [FRIAM] (advice needed!)
>
> Thus spake Vladimyr Ivan Burachynsky circa 10-03-22 10:43 AM:
> > I opened a lid and are you the surprise in side?
> > Language issues are extremely complex and I am not sure which position
you
> > take, In fact I suspect you do not subscribe to either.
>
> I'm just an Eddington style typewriter.  I read a bunch of stuff.  It
> percolates around randomly.  Then I spew it back out without really
> knowing anything about what I'm saying. [grin]  And in the great
> tradition of psychological reflexion, I assume everyone else is the same.
>
> > I always suspected my father was slightly mad when he would begin
laughing
> > at something someone said, He explained that it sounded like something
> > obscene in Finnish or Yiddish.
>
> Exactly!  Those interested in language mismatch claim that lots of
> "interestingness" seems to come from language mismatches, including lots
> of humor.
>
> > I agree with your comments but unfortunately we often have to make
choices
> > between two bad options since there is nothing better. If we recognize
the
> > language trap how do we escape?
>
> I tend to keep reminding myself that my grasp of reality is very tenuous
> regardless of my (frequent) sporadic descents into the conviction that I
> have a very good understanding of it.  By continually reminding myself,
> I find that almost every time I remind myself while stuck in that
> conviction, the conviction is a direct result of being ensconced in a
> particular language.  As I age, however, I'm finding my own reminders
> more and more difficult to maintain.  So, I sporadically start arguments
> with people like those on this list and enlist them to help me remind
> myself.  (Yes, that's totally selfish ... But you'll rarely find me
> arguing that altruism is natural. ;-)
>
> I confess, though, that these constant reminders make me a jack of many
> trades, master of none.  And that can be a very bad thing.  Luckily, I'm
> a simulant and my job requires that I be that way.
>
> > Creating a new language such as mathematics did not solve our
difficulties
> > if anything it helped illuminate the issues.
>
> Yes!  I firmly agree with that!  Math is a language for disambiguation.
>
> --
> glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
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[lrudolph]

[CCd to the mathematical thinking list; followups
should be made mindfully.]

Nick, for some reason this one message in the thread
has never arrived here, though ones before and after
it have.  So I'm replying to Russ's reply to you, without
actually quoting Russ...well, actually I'm replying to
Glen, but I am only doing it because you picked up his
phrase which I'm about to tear into.

> > Glen,
> >
> > you wrote
> >
> > " Math is a language for disambiguation".
> >
> > Forgive me if I have asked you this before:  Have you ever read Byers HOW
> > MATHEMATICIANS THINK?

One thing I can say is that *this* mathematician thinks
that calling mathematics a "language" is neither helpful
nor accurate.  To put that unnegatively--I welcome
explanations as to why it is helpful and accurate to say it.

A natural human language (at least) has syntax, semantics,
and pragmatics: rules (more or less) determining how
to describe sayings in the language, rules (more or less)
for *interpreting* sayings in the language as *referring*
to Things in The World, and rules (more or less) for
*checking* these interpretations against The State
of The World (including in The World, of course, the
human social world).  Of these, mathematics *as such*
has--arguably--only syntax.  

Mathematical models, on the other hand, have all three;
and I think it is both accurate and helpful to say
"mathematical models are languages".  Whether they're
(all, or any) "languages for disambiguation", I'm
less sure: I'd rather say that, to they extent that
they are successful, mathematical models are languages
that help us control the amount of ambiguity in ways
that are useful for the purposes at hand.

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[glen e. p. ropella-2]

Thus spake [hidden email] circa 10-03-22 03:28 PM:
> One thing I can say is that *this* mathematician thinks
> that calling mathematics a "language" is neither helpful
> nor accurate.  To put that unnegatively--I welcome
> explanations as to why it is helpful and accurate to say it.

Excellent!  Thanks for knocking that chip off my shoulder. [grin]  I'll
toss out a defense of my assertion after I specifically address your
comments.

> A natural human language (at least) has syntax, semantics,
> and pragmatics: rules (more or less) determining how
> to describe sayings in the language, rules (more or less)
> for *interpreting* sayings in the language as *referring*
> to Things in The World, and rules (more or less) for
> *checking* these interpretations against The State
> of The World (including in The World, of course, the
> human social world).  Of these, mathematics *as such*
> has--arguably--only syntax.

You're making these assertions as if there is a _very_ clear separation
of syntax, semantics, and pragmatics in natural languages.  Now, I'm no
linguist; but my guess is that such distinctions are not as clear as
you're implying.

More importantly, I think it's fairly clear that math is NOT merely
syntax.  It may be true that pure deduction (or derivation) is pure
syntax.  But math, "real" math, whatever that may be, isn't wholly
deductive.  If math were purely syntactic, then Hilbert's programme
would not have failed.

> Mathematical models, on the other hand, have all three;
> and I think it is both accurate and helpful to say
> "mathematical models are languages".  Whether they're
> (all, or any) "languages for disambiguation", I'm
> less sure: I'd rather say that, to they extent that
> they are successful, mathematical models are languages
> that help us control the amount of ambiguity in ways
> that are useful for the purposes at hand.

Mathematical models are statements in the language of math.  So, I
definitely did NOT intend to say that a mathematical model is a language
for disambiguation.  Even in the case where one might call a particular
formal system a "mathematical model" (which is not the normal usage of
"mathematical model"), I would agree that they're languages; but I would
not assert that their purpose or best use is disambiguation.  Formal
systems, in my ignorant opinion, are for discovering the deductive
consequences of the axioms of that language... "playing it out" if you
will, what some of us call "simulation" or "numerical analysis".  (Sorry
for all the quotes... ;-)

No, I'm talking about math as a whole.  Math is basically a toolkit of
(sometimes incommensurate) methods for separating out and talking about
various different things and patterns.  Granted, math does focus quite a
bit more on syntax because its users tend to value quantity (metrics)
and clear distinction.  Natural language users tend to equally value
both metrics and smudging together disparate concepts (as in poetry).
Math is almost exclusively used to state things clearly and
unambiguously.  The best math is that which is used to bring clarity and
distinction to concepts that are otherwise conflated in natural languages.

Now, if you want to descend into a semantic argument about what
constitutes a "language" and what is merely a toolkit of methods, then
I'm willing to go that route; but there's really no need.  If you'd
like, I can change my words as follows:  Math is a meta-language, a
language of languages, most used for disambiguation.

But a language of languages is still a language.

p.s. I am NOT a mathematician.  So, my ability to "appeal to authority"
is lacking.  Caveat emptor. ;-)

--
glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com


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