pramANa (was Re: [Advaita-l] Re: Women and Vedas)
S Jayanarayanan
sjayana at yahoo.com
Tue Apr 11 23:38:33 CDT 2006
--- S Jayanarayanan <sjayana at yahoo.com> wrote:
> --- Vidyasankar Sundaresan <svidyasankar at hotmail.com> wrote:
[..]
> > Perhaps the
> > best English word to capture the spirit of the word pramANa in
> this
> > context
> > is "axiom," not theory nor corollary.
>
> PramANa is actually different from axiom, as I will explain below:
>
[..]
> Mathematics as it exists in the 21st century has gone beyond ideas
> like perception or comprehensibility.
>
> As an example, consider this axiomatic system:
>
There seem to be multiple meanings to the word "axiom", so there
could be shades of meanings where it can be taken as "self-evident
truth". From m-w.com:
axiom
1 : a maxim widely accepted on its intrinsic merit
2 : a statement accepted as true as the basis for argument or
inference : POSTULATE 1
3 : an established rule or principle or a self-evident truth
But it is definitely the case that modern mathematics, which makes
most use of the word, does not necessarily take it as referring to a
self-evident truth. For example,
http://planetmath.org/encyclopedia/Axiom.html
---
A great lesson learned by mathematics in the last 150 years is that
it is useful to strip the meaning away from the mathematical
assertions (axioms, postulates, propositions, theorems) and
definitions. This abstraction, one might even say formalization,
makes mathematical knowledge more general, capable of multiple
different meanings, and therefore useful in multiple contexts.
...
It is not correct to say that the axioms of field theory are
propositions that are regarded as true without proof. Rather, the
Field Axioms are a set of constraints. If any given system of
addition and multiplication tolerates these constraints, then one is
in a position to instantly know a great deal of extra information
about this system. There is a lot of bang for the formalist buck.
Modern mathematics formalizes its foundations to such an extent that
mathematical theories can be regarded as mathematical objects, and
logic itself can be regarded as a branch of mathematics. Frege,
Russell, Poincaré, Hilbert, and Gödel are some of the key figures in
this development.
In the modern understanding, a set of axioms is any collection of
formally stated assertions from which other formally stated
assertions follow by the application of certain well-defined rules.
In this view, logic becomes just another formal system. A set of
axioms should be consistent; it should be impossible to derive a
contradiction from the axiom. A set of axioms should also be
non-redundant; an assertion that can be deduced from other axioms
need not be regarded as an axiom.
---
-Kartik
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