MATH 533 HU Heliocentric vs G

  • Heliocentric vs. geocentric solar system, Kepler’s laws, Inductive reasoning and Galileo
  • Analytic geometry and deductive reasoning: Descartes and Fermat—What purpose does coordinate
    geometry have for Descartes, and what relationship does it have with geometric construction and
    problem-solving? How does it relate to his Method?
  • Indivisibles: Galileo’s paradoxes of the infinite, Cavalieri’s principle
  • Was there any work before Newton and Leibniz about finding equations of tangent and normal
    lines to curves? or about quadrature (areas under curves)? If so, why do we credit Newton and
    Leibniz with the discovery of calculus instead of Isaac Barrow and some of these other folks?
  • Isaac Newton’s method of fluxions, fluents, moments
  • Newton’s Principia Mathematica—What is it about, why is it important, how does it differ from his
    earlier version of calculus with fluxions?
  • G.W. Leibniz’s infinitesimals—How are these different from indivisibles? How are they different
    from Newton’s infinitesimals? What complaints do some contemporaries have about them?
  • Who are some folks who applied calculus ideas in the early 1700s and what is an example of how
    they did?
  • Why are power series important in the development of calc? Why was it important that Euler’s
    work with infinite series and analysis was so formal and symbolic? How does this influence
    Lagrange’s development of calculus?
  • How does the idea of function change between 1700 and 1850?
  • Why does each of the following examples of 19th-century mathematics reflect a crisis of the
    Enlightenment perspective of the previous century? How did things not work quite so smoothly as
    previously thought? Examples:

o Fourierseries
o Insolvabilityof5th-degreepolynomials,andofthe3toughGreekconstructionproblems
o Non-Euclideangeometry(whatisthis,whodowecreditwithitsdiscovery,andwhydoes

it reflect a change of perspective?)
o Infinitesimalsgetreplacedbylimits(when,bywho?)

  • What is set theory and how does it relate to the axiomatization of mathematics? Who are the
    characters in that story?
  • What are formalism, platonism, intuitionism, and logicism? More specifically:

o Whywereintuitionists(constructivists)opposedtoCantor’sideasaboutinfinitesets?
o Whatareacoupleimportantmathematicalideaswemightnothavehadifwewereall

platonists, and why?
o HowdoGødel’sincompletenesstheoremschallengetheformalistprogram?
o Asaformalist,whydoesitmakesensethatHilbertthoughttheContinuumHypothesis

was an important question?

  • Chinese math examples: counting boards, number representations and calculation methods, systems
    of equations, magic squares, Pythagorean theorem, Chinese remainder theorem
  • Mayan math: calendars, pure base-20 system vs. positional almost-base-20 solar calendar system,
    zero
  • Inca math: what were quipus for, how were numbers represented, yupanas (weird abacuses) and
    calculation
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