Calculus without Limits: New Foundations for Mathematics

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.)
 
Formalism started off with a rejection of emirical proofs. Hilbert and Russell rejected the proof of the side-angle-side theorem in the Elements  which involved picking up a triangle, moving it in space, and putting it on top of another triangle to show that the two coincided. They replaced it with a postulate, and that is how you must have learnt geometry in school, using the side-angle-side postulate. (Check out the section on the SAS postulate in http://ckraju.net/papers/Euclid.pdf. Having eliminate the empirical, formalism does not allow us to say, the real world is like this, therefore the nature of logic must be like that. If, however, we “adjust” formalism, and do allow empirical techniques to be used to determine logic, why should they not be used directly to prove theorems? Moreover, how do we know that the logic of the real world (at any scale) is 2-valued? To begin with, take a look at my paper on “Quantum mechanical time” (on the arxiv) which argues that time is “structured”, and that hence the logic of the real world is quasi truth-functional.  (This is called the structured-time interpretation of quantum mechanics.)
 
Since present-day math has no way to decide logic, the conclusion is that present-day mathematical proof leads only to “cultural truth”. Which culture? That is obvious. The possible-world semantics of Tarski is exactly like Christian rational theology, except for the reference to God. That is, where Christian theologians spoke of “all possible worlds that God could create”, possible-world semantics speaks of “all possible worlds”, where world is understood in the sense of Wittgenstein. 
 
There are other ways to see the strong cultural bias in present-day formal mathematics. All Indian systems of philosophy (from the materialist Lokayata to  Advaita Vedanta, and, of course, Buddhists and Jains) accepted the pratyaksa (or the empirically manifest) as the first means of proof. They did not have one notion of proof for law (calling in witnesses), another for physics (performing experiments) and a third notion for mathematics; they had only one notion of proof which applied to everything. So, regarding a deductive mathematical proof, as more certain than an empirical proof, involves a denial of ALL Indian philosophies at one stroke. Likewise the postulates of present-day set theory which lead to the continuum are biased against Islamic thinkers who advocated atomicity. I hope you get the point: the claim that metaphysical mathematical proof is something special is redible ONLY from the perspective of Christian metaphyiscs (which perspective is what people are taught through Western education).  
 
Note also that this is NOT Christianity as it originally started, but is a highly politicised form of Christianity which teaches how everything from Crusades to genocide (in Americas and Australia) to slavery and racism is the highest form of morality. Note also, how, today, the value of a mathematical theorem can only be credibly decided only by Westerners, so that this metaphysics automatically entails subordination to Western authority, along the lines of the dirty politics of soft power.
 
So, I reject the present-day notion of mathematical proof as unconvincing and non-secular (it goes against your own Jain traditions, for example, assuming you still hold on to them), and, indeed, culturally and politically poisionous. Accordingly, I reject all present-day mathematical theorems, unless they can be proved by other means than formal proof.  You could throw all those theorems into the sea (but take care not to poison it)!
 
So, we need to start mathematics afresh. What, then, is the value of calculus with limits? What is the value of being able to (formally) prove the “existence” of limits starting from the postulates of set theory? Nothing. Not even zero. So, ”calculus without limits” is not merely a new pedagogy, it is a new pedagogy based on an alternative foundation of mathematics which is more universally acceptable.
 
I will elaborate on that and answer your other questions in my next mail.
 
All best,
 
 
C. K. Raju
 

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