Dear Mr Jain,
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\nFor a masters level math student you seem quite enterprising.
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\nYou ask:
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\n> Going through the material on your website, I was not able to decide
\n> whether you envision a paradigm shift only in terms of the teaching of
\n> calculus or a shift in its foundations too.<\/p>\n
A\u00a0serious discussion will require that you read up my book Cultural Foundations of Mathematics <\/em>(Pearson Longman, 2007), or at least some of the related papers available from my website. Dear Mr Jain, \u00a0 For a masters level math student you seem quite enterprising. \u00a0 You ask: \u00a0 > 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\u00a0serious 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. \u00a0 However, let me explain briefly.\u00a0What is the point of a mathematical proof?\u00a0According to me\u00a0it 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\u00a0or syadavada. The difficulties are easily understood through\u00a0Indian\u00a0tradition: 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\u00a0article on non-Western logic at http:\/\/ckraju.net\/papers\/Nonwestern-logic.pdf.) \u00a0<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2,3],"tags":[],"class_list":["post-47","post","type-post","status-publish","format-standard","hentry","category-education","category-history-and-philosophy-of-mathematics"],"_links":{"self":[{"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=\/wp\/v2\/posts\/47","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=47"}],"version-history":[{"count":0,"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=\/wp\/v2\/posts\/47\/revisions"}],"wp:attachment":[{"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=47"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=47"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ckraju.net\/wordpress_F\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=47"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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\nHowever, let me explain briefly.\u00a0What is the point of a mathematical proof?\u00a0According to me\u00a0it 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\u00a0or syadavada. The difficulties are easily understood through\u00a0Indian\u00a0tradition: 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\u00a0article on non-Western logic at http:\/\/ckraju.net\/papers\/Nonwestern-logic.pdf<\/span><\/a>.)
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\nFormalism started off with a rejection of emirical proofs. Hilbert and Russell rejected the proof of the side-angle-side theorem in the Elements<\/em>\u00a0\u00a0which 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<\/em>. (Check out the section on the SAS postulate in http:\/\/ckraju.net\/papers\/Euclid.pdf<\/span><\/a>. 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\u00a0to be used to determine logic, why should they not be used directly\u00a0to prove theorems? Moreover, how do we know that the logic of the real world (at any scale) is\u00a02-valued?\u00a0To begin with, take a look at my paper on “Quantum mechanical time”\u00a0(on the arxiv) which argues that time is “structured”, and that\u00a0hence the logic of the real world is quasi truth-functional.\u00a0 (This is\u00a0called the structured-time interpretation of quantum mechanics.)
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\nSince present-day math has no way to decide logic, the conclusion is that present-day mathematical\u00a0proof leads only to “cultural truth”. Which culture? That is obvious. The possible-world semantics of Tarski is\u00a0exactly like Christian rational\u00a0theology, except for the reference to God.\u00a0That is, where Christian theologians spoke of “all possible worlds that\u00a0God could\u00a0create”,\u00a0possible-world semantics speaks of “all possible worlds”,\u00a0where world is understood in the sense of Wittgenstein.\u00a0
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\nThere are other ways to see the strong\u00a0cultural bias in present-day formal mathematics. All<\/em> Indian systems of philosophy (from the materialist Lokayata to \u00a0Advaita Vedanta, and, of course, Buddhists and Jains) accepted the pratyaksa<\/em> (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,\u00a0as more<\/em> certain than an empirical proof, involves a denial of ALL Indian philosophies at one stroke. Likewise the postulates of present-day\u00a0set theory\u00a0which 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\u00a0ONLY from the perspective of Christian metaphyiscs (which perspective is what people are taught through Western education). \u00a0
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\nNote also that this is\u00a0NOT 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)\u00a0to slavery and racism is the highest form of morality. Note also, how, today, the value<\/em> of a mathematical theorem can only be credibly decided only\u00a0by\u00a0Westerners, so that\u00a0this metaphysics automatically entails\u00a0subordination to Western authority,\u00a0along the lines of the\u00a0dirty politics of soft power.
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\nSo, I reject the present-day notion of mathematical proof as unconvincing and\u00a0non-secular (it goes against your own Jain traditions, for example,\u00a0assuming you still hold on to them), and, indeed, culturally and politically poisionous. Accordingly,\u00a0I\u00a0reject all<\/em>\u00a0present-day mathematical theorems, unless they can be\u00a0proved by other means than formal proof.\u00a0\u00a0You could throw all those theorems into the sea (but take care not to poison it)!
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\nSo, we need to start mathematics afresh.\u00a0What, then,\u00a0is the value of calculus with limits?\u00a0What is the value\u00a0of being able to (formally) prove<\/em> the “existence” of limits starting from the postulates of\u00a0set theory?\u00a0Nothing. Not even zero. So,\u00a0“calculus without limits” is not\u00a0merely a new pedagogy, it is a new pedagogy\u00a0based on an alternative foundation of mathematics\u00a0which is more\u00a0universally acceptable.
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\nI\u00a0will elaborate on that and answer your other questions in my next mail.
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\nAll best,
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\nC. K. Raju
\n\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"