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Calculus without Limits C. K. Raju Calculus without Limits: the Theory A Critique of the History of Mathematics Part 1: Euclid and all that C. K. Raju Inmantec, Ghaziabad and Centre for Studies in Civilizations, New Delhi Calculus without


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SLIDE 1

Calculus without Limits

  • C. K. Raju

Calculus without Limits: the Theory

A Critique of the History of Mathematics Part 1: Euclid and all that

  • C. K. Raju

Inmantec, Ghaziabad and Centre for Studies in Civilizations, New Delhi

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SLIDE 2

Calculus without Limits

  • C. K. Raju

Calculus without Limits: the Theory

A Critique of the History of Mathematics Part 1: Euclid and all that

  • C. K. Raju

Inmantec, Ghaziabad and Centre for Studies in Civilizations, New Delhi

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SLIDE 3

Calculus without Limits

  • C. K. Raju

Outline

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SLIDE 4

Calculus without Limits

  • C. K. Raju

Introduction

◮ We saw that teaching calculus with limits involves

practical difficulties.

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SLIDE 5

Calculus without Limits

  • C. K. Raju

Introduction

◮ We saw that teaching calculus with limits involves

practical difficulties.

◮ It offers no particular practical advantage.

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SLIDE 6

Calculus without Limits

  • C. K. Raju

Introduction

◮ We saw that teaching calculus with limits involves

practical difficulties.

◮ It offers no particular practical advantage. ◮ And is maintained through an incorrect claim (“rigor”)

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SLIDE 7

Calculus without Limits

  • C. K. Raju

Introduction

◮ We saw that teaching calculus with limits involves

practical difficulties.

◮ It offers no particular practical advantage. ◮ And is maintained through an incorrect claim (“rigor”) ◮ Which involves culturally specific beliefs.

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SLIDE 8

Calculus without Limits

  • C. K. Raju

Introduction

◮ We saw that teaching calculus with limits involves

practical difficulties.

◮ It offers no particular practical advantage. ◮ And is maintained through an incorrect claim (“rigor”) ◮ Which involves culturally specific beliefs. ◮ This intrusion of culture into a secular science is

maintained by an appeal to history.

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SLIDE 9

Calculus without Limits

  • C. K. Raju

Euclid

the stock claim

◮ It is claimed that Greeks, and particularly Euclid

invented the notion of mathematical proof.

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SLIDE 10

Calculus without Limits

  • C. K. Raju

Euclid

the stock claim

◮ It is claimed that Greeks, and particularly Euclid

invented the notion of mathematical proof.

◮ That other cultures which lacked this notion did not

really do mathematics,

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SLIDE 11

Calculus without Limits

  • C. K. Raju

Euclid

the stock claim

◮ It is claimed that Greeks, and particularly Euclid

invented the notion of mathematical proof.

◮ That other cultures which lacked this notion did not

really do mathematics,

◮ and the knowledge they had was inferior.

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SLIDE 12

Calculus without Limits

  • C. K. Raju

The stock claim

contd.

◮ This is stated quite explicitly, for example, by the

historian Rouse Ball The history of mathematics cannot with certainty be traced back to any school or period before that of the. . . Greeks. . . . Though all early races. . . knew something of numeration yet the rules. . . were neither deduced from nor did they form part of any science.1

  • 1W. W. Rouse Ball, A Short Account of the History of Mathematics,

Dover, New York, 1960, pp. 1–2, emphasis mine.

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SLIDE 13

Calculus without Limits

  • C. K. Raju

Another example

Pythagoras theorem

◮ Everyone has heard of the Pythagoras theorem

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SLIDE 14

Calculus without Limits

  • C. K. Raju

Another example

Pythagoras theorem

◮ Everyone has heard of the Pythagoras theorem ◮ Does anyone know any evidence which connects this

theorem to Pythagoras?

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SLIDE 15

Calculus without Limits

  • C. K. Raju

Another example

Pythagoras theorem

◮ Everyone has heard of the Pythagoras theorem ◮ Does anyone know any evidence which connects this

theorem to Pythagoras?

◮ There is none. Proclus, who comes a 1000 years

after Pythagoras, says there is a “rumor” that Pythagoras sacrificed an ox when he found a proof

  • f the theorem.
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SLIDE 16

Calculus without Limits

  • C. K. Raju

Another example

Pythagoras theorem

◮ Everyone has heard of the Pythagoras theorem ◮ Does anyone know any evidence which connects this

theorem to Pythagoras?

◮ There is none. Proclus, who comes a 1000 years

after Pythagoras, says there is a “rumor” that Pythagoras sacrificed an ox when he found a proof

  • f the theorem.

◮ Does anyone know what that proof was which was

supposedly found by Pythagoras?

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SLIDE 17

Calculus without Limits

  • C. K. Raju

Another example

Pythagoras theorem

◮ Everyone has heard of the Pythagoras theorem ◮ Does anyone know any evidence which connects this

theorem to Pythagoras?

◮ There is none. Proclus, who comes a 1000 years

after Pythagoras, says there is a “rumor” that Pythagoras sacrificed an ox when he found a proof

  • f the theorem.

◮ Does anyone know what that proof was which was

supposedly found by Pythagoras?

◮ Was it a “deductive” proof or did it involve the

empirical?

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SLIDE 18

Calculus without Limits

  • C. K. Raju

Did Pythagoras have a deductive proof?

◮ Greek proofs of the Pythagorean theorem should

presumably be found in the Elements.

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SLIDE 19

Calculus without Limits

  • C. K. Raju

Did Pythagoras have a deductive proof?

◮ Greek proofs of the Pythagorean theorem should

presumably be found in the Elements.

◮ But all known manuscripts of the Elements use

empirical means of proof.

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SLIDE 20

Calculus without Limits

  • C. K. Raju

Did Pythagoras have a deductive proof?

◮ Greek proofs of the Pythagorean theorem should

presumably be found in the Elements.

◮ But all known manuscripts of the Elements use

empirical means of proof.

◮ We saw an example in Elements 1.1

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SLIDE 21

Calculus without Limits

  • C. K. Raju

Did Pythagoras have a deductive proof?

◮ Greek proofs of the Pythagorean theorem should

presumably be found in the Elements.

◮ But all known manuscripts of the Elements use

empirical means of proof.

◮ We saw an example in Elements 1.1 ◮ Another example is Elements 1.4 (Side-angle-side)

theorem.

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SLIDE 22

Calculus without Limits

  • C. K. Raju

Did Pythagoras have a deductive proof?

◮ Greek proofs of the Pythagorean theorem should

presumably be found in the Elements.

◮ But all known manuscripts of the Elements use

empirical means of proof.

◮ We saw an example in Elements 1.1 ◮ Another example is Elements 1.4 (Side-angle-side)

theorem.

◮ Which proved the equality of two triangle by

“applying” one triangle to another.

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SLIDE 23

Calculus without Limits

  • C. K. Raju

Did Pythagoras have a deductive proof?

◮ Greek proofs of the Pythagorean theorem should

presumably be found in the Elements.

◮ But all known manuscripts of the Elements use

empirical means of proof.

◮ We saw an example in Elements 1.1 ◮ Another example is Elements 1.4 (Side-angle-side)

theorem.

◮ Which proved the equality of two triangle by

“applying” one triangle to another.

◮ Empirical proofs of the Pythagorean theorem are

very easy, and were known to other cultures, such as Egyptian and India.

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SLIDE 24

Calculus without Limits

  • C. K. Raju

“Euclid”

◮ Similarly, the Elements is attributed to “Euclid”

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SLIDE 25

Calculus without Limits

  • C. K. Raju

“Euclid”

◮ Similarly, the Elements is attributed to “Euclid” ◮ Supposedly a giant of mathematics.

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SLIDE 26

Calculus without Limits

  • C. K. Raju

“Euclid”

◮ Similarly, the Elements is attributed to “Euclid” ◮ Supposedly a giant of mathematics. ◮ Does anyone know the evidence that “Euclid” was

the author of the Elements?

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SLIDE 27

Calculus without Limits

  • C. K. Raju

Theon, not Euclid

◮ Euclid’s name is not mentioned in any Greek

manuscripts of the Elements

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SLIDE 28

Calculus without Limits

  • C. K. Raju

Theon, not Euclid

◮ Euclid’s name is not mentioned in any Greek

manuscripts of the Elements

◮ They all claim to be based on the lectures of Theon.

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SLIDE 29

Calculus without Limits

  • C. K. Raju

Theon, not Euclid

◮ Euclid’s name is not mentioned in any Greek

manuscripts of the Elements

◮ They all claim to be based on the lectures of Theon. ◮ Euclid’s name is also not mentioned in any

commentaries on the Elements,

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SLIDE 30

Calculus without Limits

  • C. K. Raju

Theon, not Euclid

◮ Euclid’s name is not mentioned in any Greek

manuscripts of the Elements

◮ They all claim to be based on the lectures of Theon. ◮ Euclid’s name is also not mentioned in any

commentaries on the Elements,

◮ They all speak anonymously of the “author of the

Elements”.

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Calculus without Limits

  • C. K. Raju

◮ This is confirmed by Sir Thomas Heath, a leading

authority on “Euclid”. Elements.2 All our Greek texts of the Elements up to a century ago. . . purport in their titles to be either ‘from the edition of Theon’. . . or ‘from the lectures of Theon’.

2Sir Thomas Heath, A History of Greek Mathematics, Dover, New

York, 1981, p. 360.

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SLIDE 32

Calculus without Limits

  • C. K. Raju

◮ This is confirmed by Sir Thomas Heath, a leading

authority on “Euclid”. Elements.2 All our Greek texts of the Elements up to a century ago. . . purport in their titles to be either ‘from the edition of Theon’. . . or ‘from the lectures of Theon’.

◮ Euclid’s name does not appear even in Greek

commentaries on the Elements because the Greek commentaries “commonly speak of the writer of the Elements instead of using his name.”

2Sir Thomas Heath, A History of Greek Mathematics, Dover, New

York, 1981, p. 360.

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SLIDE 33

Calculus without Limits

  • C. K. Raju

The evidence for Euclid

A single remark by Proclus

Not much younger than these [pupils of Plato] is Euclid, who put together the Elements,. . . bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man [must have] lived in the time of the first Ptolemy; for Archimedes, who followed closely the first [Ptolemy? book?] makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorter way to study geometry. . . to which he replied that there was no royal road to

  • geometry. He is therefore younger than Plato’s

circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says.”

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SLIDE 34

Calculus without Limits

  • C. K. Raju

“Proclus’ remark

◮ The vague remark is attributed to Proclus

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SLIDE 35

Calculus without Limits

  • C. K. Raju

“Proclus’ remark

◮ The vague remark is attributed to Proclus ◮ who came 800 years after “Euclid’s” supposed date.

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SLIDE 36

Calculus without Limits

  • C. K. Raju

“Proclus’ remark

◮ The vague remark is attributed to Proclus ◮ who came 800 years after “Euclid’s” supposed date. ◮ and says that no one else mentioned “Euclid” earlier.

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SLIDE 37

Calculus without Limits

  • C. K. Raju

“Proclus’ remark

◮ The vague remark is attributed to Proclus ◮ who came 800 years after “Euclid’s” supposed date. ◮ and says that no one else mentioned “Euclid” earlier. ◮ And even that remark itself comes from a manuscript

which comes from another 800 years after Proclus.

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SLIDE 38

Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ The remarks mentions a citation of “Euclid” by

Archimedes.

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SLIDE 39

Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ The remarks mentions a citation of “Euclid” by

Archimedes.

◮ A citation of the Elements (not “Euclid”) is indeed

found in a manuscript

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SLIDE 40

Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ The remarks mentions a citation of “Euclid” by

Archimedes.

◮ A citation of the Elements (not “Euclid”) is indeed

found in a manuscript

◮ attributed to Archimedes but coming from 1800 years

after him.

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SLIDE 41

Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ The remarks mentions a citation of “Euclid” by

Archimedes.

◮ A citation of the Elements (not “Euclid”) is indeed

found in a manuscript

◮ attributed to Archimedes but coming from 1800 years

after him.

◮ The citation is regarded as spurious since it is

isolated

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SLIDE 42

Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ The remarks mentions a citation of “Euclid” by

Archimedes.

◮ A citation of the Elements (not “Euclid”) is indeed

found in a manuscript

◮ attributed to Archimedes but coming from 1800 years

after him.

◮ The citation is regarded as spurious since it is

isolated

◮ and it was not the custom in Archimedes’ time to

make such citations.

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Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ Since the author of the “Proclus’ remark” knew of the

spurious Arhicmedes citation

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SLIDE 44

Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ Since the author of the “Proclus’ remark” knew of the

spurious Arhicmedes citation

◮ which comes from a thousand years after Proclus

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SLIDE 45

Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ Since the author of the “Proclus’ remark” knew of the

spurious Arhicmedes citation

◮ which comes from a thousand years after Proclus ◮ the “Proclus’ remark” is itself spurious.

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Calculus without Limits

  • C. K. Raju

“Proclus’ remark”

contd.

◮ Since the author of the “Proclus’ remark” knew of the

spurious Arhicmedes citation

◮ which comes from a thousand years after Proclus ◮ the “Proclus’ remark” is itself spurious. ◮ So, there is no evidence that “Euclid” even existed.

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SLIDE 47

Calculus without Limits

  • C. K. Raju

Does it matter?

◮ Does it matter whether or not “Euclid” existed?

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SLIDE 48

Calculus without Limits

  • C. K. Raju

Does it matter?

◮ Does it matter whether or not “Euclid” existed? ◮ After all, there is the book, Elements.

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SLIDE 49

Calculus without Limits

  • C. K. Raju

Does it matter?

◮ Does it matter whether or not “Euclid” existed? ◮ After all, there is the book, Elements. ◮ It does, because if the book was written by Theon

(4th. c.) or after him, its purpose and interpretation would change completely.

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SLIDE 50

Calculus without Limits

  • C. K. Raju

Does it matter?

◮ Does it matter whether or not “Euclid” existed? ◮ After all, there is the book, Elements. ◮ It does, because if the book was written by Theon

(4th. c.) or after him, its purpose and interpretation would change completely.

◮ This was a time when a religious war was going on.

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SLIDE 51

Calculus without Limits

  • C. K. Raju

Does it matter?

◮ Does it matter whether or not “Euclid” existed? ◮ After all, there is the book, Elements. ◮ It does, because if the book was written by Theon

(4th. c.) or after him, its purpose and interpretation would change completely.

◮ This was a time when a religious war was going on. ◮ How does a religious war concern mathematics?

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SLIDE 52

Calculus without Limits

  • C. K. Raju

Mathematics and religion

◮ What does the word “mathematics” mean? What is

its derivation.

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SLIDE 53

Calculus without Limits

  • C. K. Raju

Mathematics and religion

◮ What does the word “mathematics” mean? What is

its derivation.

◮ Mathematics derives from “mathesis” meaning

learning.

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SLIDE 54

Calculus without Limits

  • C. K. Raju

Mathematics and religion

◮ What does the word “mathematics” mean? What is

its derivation.

◮ Mathematics derives from “mathesis” meaning

learning.

◮ Plato (Meno) explained that “all learning is

recollection of eternal ideas in the soul”.

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SLIDE 55

Calculus without Limits

  • C. K. Raju

Mathematics and religion

◮ What does the word “mathematics” mean? What is

its derivation.

◮ Mathematics derives from “mathesis” meaning

learning.

◮ Plato (Meno) explained that “all learning is

recollection of eternal ideas in the soul”.

◮ Plato (Republic) prescribed the teaching of

mathematics to make people virtuous, since it improves their soul.

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Calculus without Limits

  • C. K. Raju

Proclus on the Elements

◮ This is repeated by Proclus in his Com-

mentary on the Elements from a thousand years later. learning (µα′θησιζ [mathesiz]) is recollection of the eternal ideas in the soul; and this is why the study that especially brings us the recollection of these ideas is called the science concerned with learning (µα′θηµαθικη′ [mathematike])3

3Proclus, Commentary, p. 38.

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Calculus without Limits

  • C. K. Raju

Proclus on the Elements

contd.

◮ Proclus’ Neoplatonist beliefs were under attack by

the Christian church.

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SLIDE 58

Calculus without Limits

  • C. K. Raju

Proclus on the Elements

contd.

◮ Proclus’ Neoplatonist beliefs were under attack by

the Christian church.

◮ And he defended them using mathematics.

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SLIDE 59

Calculus without Limits

  • C. K. Raju

Proclus on the Elements

contd.

◮ Proclus’ Neoplatonist beliefs were under attack by

the Christian church.

◮ And he defended them using mathematics. ◮ But present-day historians maintain that he did not

know what he was writing about!

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SLIDE 60

Calculus without Limits

  • C. K. Raju

Origin of “Euclid”

◮ Stories about “Euclid” originated with Latin texts.

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SLIDE 61

Calculus without Limits

  • C. K. Raju

Origin of “Euclid”

◮ Stories about “Euclid” originated with Latin texts. ◮ These texts came from the mass translation of an

Arabic library captured during the Crusades.

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SLIDE 62

Calculus without Limits

  • C. K. Raju

Origin of “Euclid”

◮ Stories about “Euclid” originated with Latin texts. ◮ These texts came from the mass translation of an

Arabic library captured during the Crusades.

◮ At time the Christian church was fighting a religious

war (Crusades) against the Muslim Arabs?

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SLIDE 63

Calculus without Limits

  • C. K. Raju

Origin of “Euclid”

◮ Stories about “Euclid” originated with Latin texts. ◮ These texts came from the mass translation of an

Arabic library captured during the Crusades.

◮ At time the Christian church was fighting a religious

war (Crusades) against the Muslim Arabs?

◮ It had a 800 year old tradition of burning books.

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SLIDE 64

Calculus without Limits

  • C. K. Raju

Origin of “Euclid”

◮ Stories about “Euclid” originated with Latin texts. ◮ These texts came from the mass translation of an

Arabic library captured during the Crusades.

◮ At time the Christian church was fighting a religious

war (Crusades) against the Muslim Arabs?

◮ It had a 800 year old tradition of burning books. ◮ How was it possible to learn from the books of the

enemy?

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SLIDE 65

Calculus without Limits

  • C. K. Raju

The key to geometry

◮ The captured library at Toledo was a vast library.

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SLIDE 66

Calculus without Limits

  • C. K. Raju

The key to geometry

◮ The captured library at Toledo was a vast library. ◮ Containing accumulated knowledge from around the

world.

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SLIDE 67

Calculus without Limits

  • C. K. Raju

The key to geometry

◮ The captured library at Toledo was a vast library. ◮ Containing accumulated knowledge from around the

world.

◮ All useful knowledge in it was attributed to “Greeks”

using the flimsiest of evidence.

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SLIDE 68

Calculus without Limits

  • C. K. Raju

The key to geometry

◮ The captured library at Toledo was a vast library. ◮ Containing accumulated knowledge from around the

world.

◮ All useful knowledge in it was attributed to “Greeks”

using the flimsiest of evidence.

◮ The Arabic “uclides” meaning “ucli” (key) + “des”

(geometry) hence “Key to geometry” was interpreted as the name of a Greek author “Uclides”

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SLIDE 69

Calculus without Limits

  • C. K. Raju

The key to geometry

◮ The captured library at Toledo was a vast library. ◮ Containing accumulated knowledge from around the

world.

◮ All useful knowledge in it was attributed to “Greeks”

using the flimsiest of evidence.

◮ The Arabic “uclides” meaning “ucli” (key) + “des”

(geometry) hence “Key to geometry” was interpreted as the name of a Greek author “Uclides”

◮ The text was also reinterpreted in line with Christian

theology.

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SLIDE 70

Calculus without Limits

  • C. K. Raju

Conclusions

◮ Mathematics in the West has related to religion since

Plato.

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SLIDE 71

Calculus without Limits

  • C. K. Raju

Conclusions

◮ Mathematics in the West has related to religion since

Plato.

◮ During the Crusades history was distorted by

attributing all world knowledge indiscriminately to early Greeks real or imagined (like “Euclid”).

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SLIDE 72

Calculus without Limits

  • C. K. Raju

Conclusions

◮ Mathematics in the West has related to religion since

Plato.

◮ During the Crusades history was distorted by

attributing all world knowledge indiscriminately to early Greeks real or imagined (like “Euclid”).

◮ The difficulties of mathematics thus arise because

theology has crept into it.

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SLIDE 73

Calculus without Limits

  • C. K. Raju

Conclusions

◮ Mathematics in the West has related to religion since

Plato.

◮ During the Crusades history was distorted by

attributing all world knowledge indiscriminately to early Greeks real or imagined (like “Euclid”).

◮ The difficulties of mathematics thus arise because

theology has crept into it.

◮ Wy not make mathematics easy by eliminating the

theology

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SLIDE 74

Calculus without Limits

  • C. K. Raju

Conclusions

◮ Mathematics in the West has related to religion since

Plato.

◮ During the Crusades history was distorted by

attributing all world knowledge indiscriminately to early Greeks real or imagined (like “Euclid”).

◮ The difficulties of mathematics thus arise because

theology has crept into it.

◮ Wy not make mathematics easy by eliminating the

theology

◮ and focussing on its practical applications?

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SLIDE 75

Calculus without Limits

  • C. K. Raju

Further reading

◮ Is Science Western in Origin? (Multiversity, Penang,

2009)

◮ ◮ “Towards Equity in Math Education 1. Good-Bye

Euclid!”, Bhartiya Samajik Chintan 7 (4) (New Series) (2009), pp. 255–264.

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SLIDE 76

Calculus without Limits

  • C. K. Raju

Further reading

◮ Is Science Western in Origin? (Multiversity, Penang,

2009)

◮ ◮ “Towards Equity in Math Education 1. Good-Bye

Euclid!”, Bhartiya Samajik Chintan 7 (4) (New Series) (2009), pp. 255–264.

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SLIDE 77

Calculus without Limits

  • C. K. Raju

Further reading

◮ Is Science Western in Origin? (Multiversity, Penang,

2009)

◮ ◮ “Towards Equity in Math Education 1. Good-Bye

Euclid!”, Bhartiya Samajik Chintan 7 (4) (New Series) (2009), pp. 255–264.

◮ “Teaching racist history”, Indian Journal of

Secularism 11(4) (2008) pp. 25-28.