Archive for November, 2010

Mathematics and beauty

Thursday, November 11th, 2010

Dear Neeraj or Amartya (or whoever you are),
 
You say
 
>One of the chief reasons why most
> of us are drawn to the notion of rigor today however, is not religious
> but simply because of the elegant, sublime, almost surreal nature of
> rigorous mathematical proofs.

First, I established in my previous mail/post, that the notion of “rigor” to which you refer is a culturally-specific notion of ”rigor” according to Christian metaphysics, and that it is non-rigorous, and contrary to the notion of “rigor” in various other systems of philosophy. Certainly, the church aims to dominate and become “universal”, by eliminating all others, and it uses various tricks to suggest that its notions are already “universal”, but that has not yet happened. So please don’t pretend like the church that that the notion of “rigor” is already universal, by using it without qualifying adjectives as your Western indoctrination taught you to do. Try to be honest and at least call it “rigor according to Christian theology”, or “rigor according to Christian mathematics”. The moment you apply those adjectives it becomes clear that this “rigor” is a matter of culturally-specific belief; that itself is enough to justify why I call it religious. (There are other reasons, but I won’t go into them here. My forthcoming book Euclid and Jesus explains this in more detail.)
 
I have no objection if you want to do math because you find it beautiful. (But why so many adjectives?  (”elegant, sublime, surreal”)  That suggests lack of confidence in what you are saying, for most people do not see that beauty; so maybe you are afraid that you are just deluding yourself, as so many people do about so many things.)
 
My first objection to formal math is this: why impose it on school kids?  (more…)

Calculus without Limits: New Foundations for Mathematics

Wednesday, November 10th, 2010

Dear Mr Jain,
 
For a masters level math student you seem quite enterprising.
 
You ask:
 
> Going through the material on your website, I was not able to decide
> whether you envision a paradigm shift only in terms of the teaching of
> calculus or a shift in its foundations too.

A serious discussion will require that you read up my book Cultural Foundations of Mathematics (Pearson Longman, 2007), or at least some of the related papers available from my website.
 
However, let me explain briefly. What is the point of a mathematical proof? According to me it gives neither “eternal truths” (as Plato thought), nor “universal truths” (as the medieval church propagated), nor “necessary truths” (as formalism, or the theoreis of Tarski and Wittgenstein make out). Why? Because formal proof is entirely metaphysical, and depends upon logic. But why use 2-valued logic, why not Buddhist logic of catuskoti or the Jain logic of syadavada, or quantum logic, or the logic of natural language? Indeed, why not use one of the infinity of different logics one can conceive of? Clearly, the theorems of mathematics will change with logic. All proofs by contradiction, for example, would fail with Buddhist logic or syadavada. The difficulties are easily understood through Indian tradition: Naiyayikas (who used 2-valued logic, later incorrectly attributed to Aristotle) and Buddhists who did not, could not agree on the most elementary propositions in a thousand years of debate. (You can learn more about these logics by looking at my article on non-Western logic at http://ckraju.net/papers/Nonwestern-logic.pdf.)
  (more…)