## Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th C. CEThe Volume Examines, In Depth, The Implications Of Indian History And Philosophy For Contemporary Mathematics And Science. The Conclusions Challenge Current Formal Mathematics And Its Basis In The Western Dogma That Deduction Is Infallible (Or That It Is Less Fallible Than Induction). The Development Of The Calculus In India, Over A Thousand Years, Is Exhaustively Documented In This Volume, Along With Novel Insights, And Is Related To The Key Sources Of Wealth-Monsoon-Dependent Agriculture And Navigation Required For Overseas Trade - And The Corresponding Requirement Of Timekeeping. Refecting The Usual Double Standard Of Evidence Used To Construct Eurocentric History, A Single, New Standard Of Evidence For Transmissions Is Proposed. Using This, It Is Pointed Out That Jesuits In Cochin, Following The Toledo Model Of Translation, Had Long-Term Opportunity To Transmit Indian Calculus Texts To Europe. The European Navigational Problem Of Determining Latitude, Longitude, And Loxodromes, And The 1582 Gregorian Calendar-Reform, Provided Ample Motivation. The Mathematics In These Earlier Indian Texts Suddenly Starts Appearing In European Works From The Mid-16Th Century Onwards, Providing Compelling Circumstantial Evidence. While The Calculus In India Had Valid Pramana, This Differed From Western Notions Of Proof, And The Indian (Algorismus) Notion Of Number Differed From The European (Abacus) Notion. Hence, Like Their Earlier Difficulties With The Algorismus, Europeans Had Difficulties In Understanding The Calculus, Which, Like Computer Technology, Enhanced The Ability To Calculate, Albeit In A Way Regarded As Epistemologically Insecure. Present-Day Difficulties In Learning Mathematics Are Related, Via Phylogeny Is Ontogeny , To These Historical Difficulties In Assimilating Imported Mathematics. An Appendix Takes Up Further Contemporary Implications Of The New Philosophy Of Mathematics For The Extension Of The Calculus, Which Is Needed To Handle The Infinities Arising In The Study Of Shock Waves And The Renormalization Problem Of Quantum Field Theory. |

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### Contents

List of Boxes Tables and Figures | xv |

Preface | xxvii |

Introduction | xxxv |

Euclid and Hilbert | 3 |

Proof vs Pramdna | 59 |

Boxes | 64 |

Infinite Series and 7r | 109 |

pramdna for the Indian infinite series in contrast to Newton etc who could | 185 |

Models of Information Transmission | 267 |

How and Why the Calculus Was Imported into Europe | 321 |

Figures | 372 |

Numbers in Calculus Algol ismus and Computers | 375 |

Math Wars and the Epistemic Divide in Mathematics | 411 |

difficulties in understanding the algorismus and the calculus are here | 420 |

A Distributions Renormalization and Shocks | 425 |

The Feynman diagrams for electron and photon selfenergy | 441 |

### Common terms and phrases

accurate Alexandria algorismus Almagest Arabic Archimedes Aristotle Aryabhata Aryabhatiya astronomy attributed authority belief Bhaskara Bhaskara II Brahmagupta Buddhist C. K. Raju calculation calculus calendar centuries Christian circumference cited earlier Clavius Cochin corresponding cultural Delhi derived difference difficulties earth Elements empirical epistemological equations Euclid Europe European navigators example finite formal mathematics formula function geometry Greek Hence India Indian mathematics Indian tradition infinite series infinitesimals interpolation Jesuits kamdl knowledge known latitude Leibniz logic longitude math mathematical proof mathematicians measure method Newton non-representable notion of mathematical obtained philosophy of mathematics physical Plato practical pratyaksa precise trigonometric values present-day problem Proclus Ptolemy quantum quantum field theory racist radius rational theology regarded religious Roman sexagesimal Siddhdnta sine values social sources square standard of evidence sulba sutra supertasks technique term texts theorem theory trans translated transmission triangle truth understanding universal Vasco da Gama Western historians Yuktibhdsd zero