Calculus without Limits C. K. Raju Recap Calculus without Limits: Introduction Why do formal the Theory mathematics? How proof varies A Critique of Formal Mathematics with logic Part 2: The Choice of Logic Underlying Proof How logic varies with physics How logic varies with culture C. K. Raju Summary and Conclusions Inmantec, Ghaziabad and Centre for Studies in Civilizations, New Delhi
Calculus without Limits C. K. Raju Recap Calculus without Limits: Introduction Why do formal the Theory mathematics? How proof varies A Critique of Formal Mathematics with logic Part 2: The Choice of Logic Underlying Proof How logic varies with physics How logic varies with culture C. K. Raju Summary and Conclusions Inmantec, Ghaziabad and Centre for Studies in Civilizations, New Delhi
Calculus without Outline Limits C. K. Raju Recap Recap Introduction Why do formal Introduction mathematics? How proof varies with logic Why do formal mathematics? How logic varies with physics How logic varies How proof varies with logic with culture Summary and Conclusions How logic varies with physics How logic varies with culture Summary and Conclusions
Calculus without Recap Limits C. K. Raju Recap Introduction ◮ Calculus with limits enormously difficult to teach. Why do formal mathematics? How proof varies with logic How logic varies with physics How logic varies with culture Summary and Conclusions
Calculus without Recap Limits C. K. Raju Recap Introduction ◮ Calculus with limits enormously difficult to teach. Why do formal mathematics? ◮ Standard calculus books do not teach the precise How proof varies dx , e x etc. definition of limits, d with logic How logic varies with physics How logic varies with culture Summary and Conclusions
Calculus without Recap Limits C. K. Raju Recap Introduction ◮ Calculus with limits enormously difficult to teach. Why do formal mathematics? ◮ Standard calculus books do not teach the precise How proof varies dx , e x etc. definition of limits, d with logic How logic varies ◮ Teach calculus as a bunch of rules to be used without with physics question—a useless skill for symbolic manipulation How logic varies with culture can be better done by low-cost machines. Summary and Conclusions
Calculus without Recap Limits C. K. Raju Recap Introduction ◮ Calculus with limits enormously difficult to teach. Why do formal mathematics? ◮ Standard calculus books do not teach the precise How proof varies dx , e x etc. definition of limits, d with logic How logic varies ◮ Teach calculus as a bunch of rules to be used without with physics question—a useless skill for symbolic manipulation How logic varies with culture can be better done by low-cost machines. Summary and ◮ Why teach humans to behave like low-cost Conclusions machines?
Calculus without Recap Limits C. K. Raju Recap Introduction ◮ Calculus with limits enormously difficult to teach. Why do formal mathematics? ◮ Standard calculus books do not teach the precise How proof varies dx , e x etc. definition of limits, d with logic How logic varies ◮ Teach calculus as a bunch of rules to be used without with physics question—a useless skill for symbolic manipulation How logic varies with culture can be better done by low-cost machines. Summary and ◮ Why teach humans to behave like low-cost Conclusions machines? ◮ Limits required since visual intuition denied. But calculus texts rely heavily on visual intuition.
Calculus without Recap 2 Limits Axioms and definitions arbitrary C. K. Raju Recap ◮ Limits depend upon R . But why R ? Why not Introduction something larger? Why do formal mathematics? How proof varies with logic How logic varies with physics How logic varies with culture Summary and Conclusions
Calculus without Recap 2 Limits Axioms and definitions arbitrary C. K. Raju Recap ◮ Limits depend upon R . But why R ? Why not Introduction something larger? Why do formal mathematics? ◮ R constructed using set theory. But set theory How proof varies probably inconsistent since it requires 2 standards of with logic proof, one for mathematics, and one for How logic varies with physics metamathematics. How logic varies with culture Summary and Conclusions
Calculus without Recap 2 Limits Axioms and definitions arbitrary C. K. Raju Recap ◮ Limits depend upon R . But why R ? Why not Introduction something larger? Why do formal mathematics? ◮ R constructed using set theory. But set theory How proof varies probably inconsistent since it requires 2 standards of with logic proof, one for mathematics, and one for How logic varies with physics metamathematics. How logic varies with culture ◮ ǫ – δ definition of derivative and Reimann integral had Summary and to be abandoned in favour of Schwartz theory and Conclusions Lebesgue for applications of calculus to physics.
Calculus without Recap 2 Limits Axioms and definitions arbitrary C. K. Raju Recap ◮ Limits depend upon R . But why R ? Why not Introduction something larger? Why do formal mathematics? ◮ R constructed using set theory. But set theory How proof varies probably inconsistent since it requires 2 standards of with logic proof, one for mathematics, and one for How logic varies with physics metamathematics. How logic varies with culture ◮ ǫ – δ definition of derivative and Reimann integral had Summary and to be abandoned in favour of Schwartz theory and Conclusions Lebesgue for applications of calculus to physics. ◮ However, Schwartz theory incomplete: lacks notion of product of distributions (required for physics).
Calculus without Recap 2 Limits Axioms and definitions arbitrary C. K. Raju Recap ◮ Limits depend upon R . But why R ? Why not Introduction something larger? Why do formal mathematics? ◮ R constructed using set theory. But set theory How proof varies probably inconsistent since it requires 2 standards of with logic proof, one for mathematics, and one for How logic varies with physics metamathematics. How logic varies with culture ◮ ǫ – δ definition of derivative and Reimann integral had Summary and to be abandoned in favour of Schwartz theory and Conclusions Lebesgue for applications of calculus to physics. ◮ However, Schwartz theory incomplete: lacks notion of product of distributions (required for physics). ◮ No general rule available for choosing between different definitions and axiom sets: only way is to rely on Western mathematical authority.
Calculus without Universal math Limits C. K. Raju Recap ◮ Mathematics has long been regarded in the West as Introduction Why do formal universal; not merely global, but universal. mathematics? How proof varies with logic How logic varies with physics How logic varies with culture Summary and Conclusions 1 Christiaan Huygens, The Celestial Worlds Discover’d: or Conjectures Concerning the Inhabitants, Plants, and Productions of the Worlds in the Planets , London, 1698, p. 86.
Calculus without Universal math Limits C. K. Raju Recap ◮ Mathematics has long been regarded in the West as Introduction Why do formal universal; not merely global, but universal. mathematics? ◮ This thought is captured in the following remark from How proof varies with logic Huygens 1 How logic varies with physics no matter how inhabitants of other planets How logic varies might differ from man in other ways, they with culture must agree in music and geometry, since Summary and Conclusions [music and geometry] are everywhere immutably the same, and always will be so. 1 Christiaan Huygens, The Celestial Worlds Discover’d: or Conjectures Concerning the Inhabitants, Plants, and Productions of the Worlds in the Planets , London, 1698, p. 86.
Calculus without Modified claim Limits C. K. Raju Recap Introduction ◮ This claim modified with advent of formal Why do formal mathematics? mathematics. How proof varies with logic How logic varies with physics How logic varies with culture Summary and Conclusions
Calculus without Modified claim Limits C. K. Raju Recap Introduction ◮ This claim modified with advent of formal Why do formal mathematics? mathematics. How proof varies with logic ◮ Arbitrariness in axioms (for R ) or definitions (of How logic varies derivative) are easy to understand, and formal math with physics How logic varies accepts this arbitrariness. with culture Summary and Conclusions
Calculus without Modified claim Limits C. K. Raju Recap Introduction ◮ This claim modified with advent of formal Why do formal mathematics? mathematics. How proof varies with logic ◮ Arbitrariness in axioms (for R ) or definitions (of How logic varies derivative) are easy to understand, and formal math with physics How logic varies accepts this arbitrariness. with culture ◮ Not permitted in practice (paper with different notion Summary and Conclusions of derivative not likely to be published).
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