Re: [cs-pdcdca] Re: [ProgSoc] A questions

[Bernard Doyle <bjd@nospam.it.uts.edu.au>]

On Tue, 17 Jun 2003, Vik wrote:

> > Encryption is mathematical warfare. Mathematics is the universal language.
> > Encryption is secrecy. Secrecy is a declaration of war.
>
> Is mathematics really the universal language? Or is it just another form
> of western cultural imperialism?
>
> (I don't have a hard opinion either way; it's just something that I've
> been pondering of late, and would like to see a bit of discussion on the
> issue)

The true nature of mathematics is an ongoing philosophical question.
Over the last hundred years there have been 4 main schools of thought:-

(1) Formalism. This basically says that mathematics is devoid of any
meaning and is just the set of all possible deductions from all possible
sets of consistent axioms using all possible rules of inference.
Unfortunately Godel's Theorem shoots this view to pieces because it proves
that there exist statements whose truth or falsity can never be demonstrated
from  the rules of deduction if they and the initial axioms are rich
enough to include our familiar arithjmetic of whole numbers.

(2) Inventionism. This says that mathematics is a purely human invention
like music or literature and is nothing more than the activities carried
out by mathematicians. If you hold the view that mathematics is a form
of western cultural imperialism, then I guess you are a holding a
variety of the inventionist view.

(3) Platonism. This view says that mathematics exists independently of
human beings. Mathematicians discover mathematics, but it exists "out
there" (wherever that may be). There is a lot to support this view.
For example, human cultures widely separated in space and time have all
discovered what we call "Pythagoras Theorem" (as well as many other
mathematical theorems). The fact that different cultures have independently
"discovered" certain mathematical facts also tends to discredit the
inventionist view. Most professional mathematicians operate as if
the Platonist view of mathematics is correct.

(4) Constructivism. This view of mathematics is a reaction to the
logical paradoxes of set theory and infinite sets that were first
postulated in the late 19th Century. Sensing that mathematics might be
lead into serious error by the manipulation of concepts like infinity
of which we have no concrete experience it rejects them altogether.
Basically, constructivists say mathematics should only contain
statements that that can be deduced in a finite sequence of step by
step constructions starting from the natural numbers. The confinement of
the allowable steps by constructivists removes a lot of mathematical tools
(such as the well-known "proof by contradiction").

A good introduction to the whole question of what is the nature of
mathematics can be found in "Pi in the Sky - Counting, thinking and
being" by John Barrow. (Penguin 1993). Just about every other book by John
Barrow contains a chapter on the different philosophies of mathematics as
well.

cheers,

Bernard

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>
> vik
>
> <http://www.progsoc.org/~vik>
> PGP: <http://www.progsoc.org/~vik/pgp.txt>
>
> "I think it would be a good idea." Ghandi, on Western Civilisation
>
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