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
problemsolving? 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 19thcentury 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 Insolvabilityof5thdegreepolynomials,andofthe3toughGreekconstructionproblems
o NonEuclideangeometry(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 base20 system vs. positional almostbase20 solar calendar system,
zero 
Inca math: what were quipus for, how were numbers represented, yupanas (weird abacuses) and
calculation