On Aug 30, 1:37 am, Bruno Marchal <[EMAIL PROTECTED]> wrote:
> Le 29-août-07, à 12:48, [EMAIL PROTECTED] a écrit :
> > Any scientific theory (including Darwin's) *is* more accurate when
> > expressed in mathematical notation. You *can* draw a clear
> > distinction between the language used to express mathematical concepts
> > and the concept itself.
> > Pure math concepts themselves consist of:
> > Formal Systems, Relations and Differential Equations. They are
> > abstract concepts which are precisely defined
> Not necessarily ....
Well OK I take back the part about 'precisely defined'. But it seems
to me that all of mathematics can be classified after three different
categories - that is - there is natural 'three-fold' division of
mathematics. Threeness does seem to be fundamental to onotlogy at the
deepest level doesn't it? ;)
All of math is three things: At the most basic level - *Predicates*.
At a some what higher, more general level of abstractrion -
*Relations* (including categories and functions). Finally at the most
general level, differential equations. Relations could be thought of
as a special case of calculus, predicates in turn as a special case of
relations. BUt the most power (greatest level of generality) seems to
reside in analysis and calculus. Would you agree with this?
Predicates are the intrinsic aspect of math, relations
from....welll... they are...asbtracted relational properties of
predicates. Finally calulus seems the boundaries and limts
(literally! no pun intended) for the math-scape in which predicates
and relations reside.
> > and it is provable
> > matter to determine the equivalence (or not) of different symbolic
> > representations of them.
> Noooo..... I hope I will be able to prove this in due time to David,
> but even if you limit yourself to one prrograming language, it is
> provable that you have no general tools to see if two different
> programs compute the same function. At some point this is important to
> Mathematical reality kicks back! (This goes in your direction).
Sorry my mistake. But surely you can compare one specific instance of
a program with another specific instance.
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