E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 1: Mathematical rootsIn the same way as one has distinguished the canons of rhetorics: memory, invention, deliv-ery, style, and arrangement, or combined the trivium: grammar, logic and rhetorics , with thequadrivium arithmetic, geometry, music, and astronomy, to get the seven liberal arts andsciences, one has also tried to organize all mathematical activities.Historically, one has dis-tinguished eight ancientroots of mathematics.These are 8 activities, eachof which suggest a key areain mathematics:counting and sorting arithmeticspacing and distancinggeometrypositioning and locatingtopologysurveying and angulatingtrigonometrybalancing and weighingstaticsmoving and hittingdynamicsguessing and judgingprobabilitycollecting and orderingalgorithmsTo morph these 8 roots to the 12 mathematical areas we cover in this class, we complementedthe ancient roots by calculus, numerics and computer science, merge trigonometry with geometry,separate arithmetic into number theory, algebra and arithmetic and change statics to analysis.Lets call this more modernadaptation the12 modern roots ofMathematics:counting and sorting arithmeticspacing and distancinggeometrypositioning and locatingtopologydividing and comparingnumber theorybalancing and weighinganalysismoving and hittingdynamicsguessing and judgingprobabilitycollecting and orderingalgorithmsslicing and stackingcalculusoperating and memorizingcomputer scienceoptimizing and planningnumericsmanipulating and solvingalgebraWhile relating math-ematical areas withhuman activities isuseful, it can makemore sense to pairthese 12 major areaswith one or two ex-amples of topics whichappear in this area.These 12 topics will bethe 12 lectures of thiscourse.Arithmetics numbers and number systemsGeometryinvariance, symmetries, measurement, mapsNumber theoryDiophantine equations, factorizationsAlgebraalgebraic and discrete structuresCalculuslimits, derivatives, integralsSet Theoryset theory, foundations and forma lismsProbability combinatorics, measure theory and statisticsTopologypolyhedra, topological spaces, ma nif oldsAnalysis extrema, estimates, variation, measureNumerics numerical schemes, codes, cryptologyDynamics differential equations, mapsAlgorithmscomputer science, artificial intelligenceOf course, like any classification, this division is rather arbitrary and also a matter of personalpreferences. The 2010 AMS classification for example distinguishes 63 areas of mathematics.In MSC 2010, many of the main areas are broken off into even finer pieces. Additionally, thereare fields which relate with other areas of science, like economics, biology or physics:00 General01 History and biography03 Mathematical logic and foundations05 Combinatorics06 Lattices, ordered algebraic structures08 General algebraic systems11 Number theory12 Field theory and pol ynomials13 Commutative rings and algebras14 Algebraic geometry15 Linear/multi-linear algebra; matrix theory16 Associative rings and algebras17 Non-associative rings and algebras18 Category theory, homological algebra19 K-theory20 Group theory and generalizations45 Integral equations46 Functional analysis47 Operator theory49 Calculus of variations, optimization51 Geometry52 Convex and discrete geometry53 Differential geometry54 General topology55 Algebraic topology57 Manifolds and cell complexes58 Global analysis, analysis on manifolds60 P r obability theory and stochastic processes62 Statistics65 Numerical analysis68 Computer science70 Mechanics of particles and systems22 Topological groups, Lie groups26 Real functions28 Measure and integration30 Functions of a complex variable31 Potential theory32 Several complex variables, analytic spaces33 Special functions34 Ordinary differential equations35 Partial differential equations37 Dynamical systems and ergodic theory39 Diff eren ce and functional equations40 Sequences, series, summability41 Approximations and expansions42 Fourier analysis43 Abstract harmonic analysi s44 Integral transforms, operational calculus74 Mechanics of deformable solids76 Fluid mechanics78 Optics, electromagnetic theory80 Classical thermodynamics, heat transfer81 Quantum theory82 Statistical mechanics, structure of matter83 Relativity and gravitational theory85 Astronomy and astrophysics86 Geophysics90 Operations research, math. programming91 Game theory, Economics Social and Behavioral Sciences92 Biol ogy and other natural sciences93 Systems theory and control94 Information and communication, circuits97 Mathematics educationWhat are hot spots in mathe-matics today? Michael Athiyahidentified in the year 2000 thefollowing 6 hot spots in thedevelopment of mathematics:local and globallow and high dimensioncommutative and non-commutativelinear and nonlineargeometry and algebraphysics and mathematicsAlso this choice is of course highly personal. One can easily add an other 12 of such polarizingquantities which help to distinguish or parametrize different parts of mat hematical areas, especiallythe ambivalent pairs which are ”hot”:regularity and randomnessintegrable and non-integrableinvariants and perturbationsexperimental and deductivepolynomial and exponentialapplied and abstractdiscrete and continuousexistence and constructionfinite dim and infinite dimensionaltopological and differential geometricpractical and theoreticalaxiomatic and case basedAn other p ossibility to refine the fields of mathematics is to combine different of the 12 areas.Examples are probabilistic number theory, algebraic geometry, numerical analysis, ge-ometric number theory, numerical algebra, algebraic topology, geometric probability,algebraic number theory, dynamical probability = stochastic processes. Almost everypair is an actual field. Finally, lets give a short answer to the question: What is Mathematics?Mathematics is the science of structure.The goal is to illustrate some of these structures fro m a historical point of view.E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 2: ArithmeticThe first mathematical steps were done by Babylonian, Egyptian, Chinese, Indian and Greekthinkers. The oldest mathematical discipline is Arit hmetic, the theory of manipulating numbers.Everything starts with the class of natural numbers 1, 2, 3, 4... where one can add and multiply.While addition is natural like when adding 3 sticks to 5 sticks to get 8 sticks, the multiplicativeoperation ∗ is more subtle: 3 ∗4 can be