The Rise of Modern Mathematics

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The Rise of Modern Mathematics

• The Renaissance Continued

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How did he do that?

• To disprove Aristotelian scholars, Galileo climbs
  to top of tower and drops a canon ball and a
  wooden ball.
• Both hit ground at same time.

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Europe Divided

• The Reformation 1517-1588
• Grievances against Roman Cat. Church
• Europe splits Protestants (N) and Catholics (S)
• Under Pope, Spain Invades England
• 1588 Defeat of Armada
• King James Bible - 1611

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Tension Builds

• Thirty Years War
• Lutherans (Protestants) vs. Calvinists
  (Catholics)
• Religious, military, and political war
• English Civil War
• King James dies in 1625, rep. By Charles I
• Conflicts between Charles and Parliament over
  religion
• 1640 Parliament tries to reform religion

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Implications for Mathematics

• Mathematics is confined
• War between mathematicians and church
• Some ideas withheld, lost, destroyed
• Regardless.
• Mathematics thrives more than ever
• Effects of printing continues, ideas easily
  shared
• Battle of Scholars continues

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Rise of modern mathematics

• Mysticism to theory
• Concrete to Abstract
• Push for logical thinking
• Birth of calculus
• Calculations become easier
• The familiar look of mathematics emerges

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Galileo The Rebel

• (1564-1642) Born in Pisa
• U. of Pisa as medical student, left in 1585 to
  study mathematics
• 1589 Lectureship at Pisa
• Sorcery at the Tower
• Two gods at time God and Aristotle
• He challenged both
• Eventually is run out of Pisa (1592)
• Becomes Math Prof. at U. of Padua

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Galileo and the Telescope

• Sidereus Nuncius (The Starry Messenger)
• Discovers four satellites around Jupiter
• Challenges Mathematics of Aristotelian Univ.
• Does not sit well with Aristotelian Padua
• 1610 Return to Pisa
• Galileo takes a stand
• If church has a problem, they should disprove him.

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Mathematics Devil

• Church sees this as assailing the authority of
  scripture
• Church denounces all mathematics
• Mathematicians are enemies of true religion
• Galileo meets Cardinal Bellarmine
• No teaching, defending or discussing!!

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Second Chance

• 1623 New Pope, Urban VIII allows Galileo to
  write about theory of Copernicus
• Must be presented as hypothesis
• Other arguments must be equally presented
• Writes Dialogue
• Conversations between Salviati (Galileo), and
  Simplico (Aristotelian philosopher/bumbling
  idiot)
• Church redoubles persecution

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Galileo Meets Inquisition

• At age 70, he is called before Inquisition
• Had to confess errors and sign recognition of
  authority of Church
• Dialogue joins other works
• Galileo sentence to permanent house arrest
• Writes Discourses and Mathematical Demonstrations
  Concerning Two New Sciences
• Projectile motion and gravitational acceleration

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But what does it all mean?

• Somebody had to do it.
• Maker of Mathematicians
• Known for idea that mathematics is the vehicle of
  scientific exploration
• Universe could be understood through direct
  observation and mathematical reasoning

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John Napier

• 1550 in Merchiston Castle, Edinburgh, Scotland
• Math was a hobby Focused on theology
• Fierce Calvinist Proves that Catholic church
  is beast through interpretation of Revelation
• Known for ingenuity and imagination
• Greatest accomplishments involved easing
  calculations
• Napiers Bones
• Mechanically multiplying, dividing, taking square
  and cube roots

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The Logarithms

• Printed in Descriptio
• Different from modern logarithms
• If N 107 (1 10-7)n ,
  then Nap.log N n.
• Henry Briggs suggested using base 10
• Birth of modern logarithms
• Less time for calculations, fewer errors in
  calculations
• Huge advancement for mathematics

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Johannes Kepler

• Believed in math simplicity of nature
• Principle of Continuity
• Enrolls at U. of Tubingen
• Wanted to become Lutheran minister
• Also wanted to fine tune and strengthen
  Copernican ideas
• 1593 Prepares dissertation supporting
  sun-centered Universe, but not allowed to present.

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Concerning Motion of Planets

• 1609 Published Astronomia Nova which contains
  three laws
• p. 335
• Ellipse was not perfect like circle
• Could not explain forces
• Paves way for Newton
• Excommunicated in 1611

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Descartes

• Born in France in 1596
• Health problems from birth to death
• Unlike others, pure focus was mathematics
• Considered himself devoted Roman Catholic
• Hindered publication of findings and possibly
  even the revelation of some of his ideas
• Settles in Holland in 1637

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While in Holland.

• Completes Le Monde (The World)
• Affirms heliocentric theory
• On eve of completion, he learns of fate of
  Galileos Dialogue
• But on no account will I publish anything that
  contains a word that might displease the Church
• Instead, his first principle published work is

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Discourse de la Methode - 1637

• Discourse on the Method of Rightly Conducting the
  Reason in the Search of the Truth
• Includes summary of Le Monde because
• I resolved to leave all this world to them and
  to speak solely of what would happen in a new
  world, if God were now to create

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Birth of Logical Thinking

• Systematic Doubt
• If you would be a real seeker after truth, you
  must at least once in your life doubt, as far as
  possible, all things.
• Descartes begins with own existence
• I think, therefore I am.
• This directly models our modern proof techniques
• 3 appendices that model this thinking
• La Dioptrique, Les Meteores, and La Geometrie

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La Geometrie

• Formation of Analytic Geometry
• Coordinate system
• Modern algebraic notation
• Purely algebraic method of finding normal to
  curve
• Law of Signs
• Uses negative and imaginary roots

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And FinallyThe Principia

• Principia Philosophaie 1644
• All physical explanations could be sought and
  nature could be framed in terms of mathematical
  and mechanical laws alone
• Provides theory of universe contained in
  vorticies
• Places universe on mathematical foundation

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In the End

• Despite efforts, works are placed on the Index of
  Prohibited Books by the Holy Office.
• Dies in 1650 of pneumonia
• Funeral oration is prohibited
• Even though he did not want to displease Church,
  he broke religious barriers for future scholars

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The Calculus Controversy
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• Isaac Newton and Gottfried Leibniz had very
  different views of calculus. Newtons calculus
  was based on limits and concrete reality, while
  Leibniz focused more on the infinite and the
  abstract. However, regardless of the divergent
  paths these two scholars chose to venture down,
  the question of who took the first step remained
  the primary issue of debate.

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• Unaware that Newton was reported to have
  discovered similar methods, Leibniz discovered
  his calculus in Paris between 1673 and 1676. By
  1676, Leibniz realized that he was onto something
  big he just didnt realize that Newton was on to
  the same big discovery because Newton was staying
  quiet about his breakthroughs. In fact, it was
  actually the delayed publication of Newtons
  findings that caused the entire controversy.
  Leibniz published the first account of
  differential calculus in 1684 and then published
  the explanation of integral calculus in 1686.

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• There was speculation that Leibniz may have
  gleaned some insights from two of Newtons
  manuscripts on fluxions, and that is what sparked
  his understanding of calculus. Many believed
  that Leibniz used Newtons unpublished ideas,
  created a new notation and then published it as
  his own, which would obviously constitute
  plagiarism.

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Statement made by Newton

• In letters that passed between me and that most
  excellent geometer, G.W. Leibniz, ten years ago,
  when I signified that I knew a method of
  determining maxima and minima, of drawing
  tangents and the like, and when I concealed it in
  transposed letters the most distinguished man
  wrote back that he had fallen upon a method of
  the same kind, and communicated his method, which
  hardly differed from mine, except in his form of
  words and symbols.

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• Leibniz demanded that the Royal Society intervene
  with the calculus controversy. In the early
  spring of 1712, the Royal Society responded to
  Leibnizs appeal by appointing a committee of
  eleven to examine all documents in the societys
  possession that bore on the matter. The
  committee consisted of two foreigners, one a
  mathematician. Six members were mathematicians,
  and at least seven were intimate personal friends
  of Newton, who had been president of the Royal
  Society since 1703.

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Hypothesis Formation

• In 1712 who received credit for the discovery of
  calculus? Why?
• Who is credited now with the discovery of
  calculus? Why?

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Collect Data

• collect data to determine whether each hypothesis
  should be accepted, rejected, or characterized as
  more info needed.
• Come back to the classroom and report your
  findings.