Course Overview

Math 390

Dr. Janssen

About this course

“God made the integers. All the rest is the work of man.”

—Leopold Kronecker

Introduction

Not a typical math course
Emphasis on reading, writing, discussion (and some historical mathematics)
Informal Goal: Ask more questions than we answer

Course Liturgies

(Most) classes: a mix of lecture and hands-on historical problems
Occasional Friday philosophy discussions
Homework due every other Wednesday

Required Resources

Katz
Hersch
MTEF

Major Assessments

Historical Term Paper
Personal philosophy of mathematics

Questions?

History of Math in a Very Large Nutshell

Beginnings

Ancients

(Pre-)Mathematical writings exist as far back as we have records, across all cultural barriers
Example: Genesis
First writings done by scribes, practical mathematics
Most of the oldest surviving writing comes from Mesopotamia (Iraq) and Egypt

Greek Mathematics

The “Greeks”

First to put logic and proof at the center of mathematics
Most growth 600BC—400AD
Estimate: 1000 total working mathematicians over the thousand years; only 150 mentioned in surviving texts
Dominant form of Greek mathematics: geometry

Famous Greek Mathematicians

Thales
Pythagoras
Euclid
Archimedes
Hypatia

Non-Western Mathematics

Indian Mathematics

Brahmagupta and Bhāskara II: first to work with negative quantities
Decimal numeration system (prior to year 600)
Introduced the sine of an angle
Methods for cube/square roots
Arithmetic progressions
Quadratic formula

The Islamic Empire (750 AD)

The Islamic Empire

Muhammad ibn Mūsa al-Khwārizmī

Mid-9th century
Book: al-jabr w’al muqābala
Robert of Chester (1145): Latinized al-jabr to algebra
“dixit algorismi”
- \(\to\) algorism
- \(\to\) algorithm

Medievals

Eleventh and Twelfth Centuries

First universities established in Bologna, Oxford, Paris
Interest in kinematics and notions of instantaneous velocity
Nicole Oresme (U. Paris) — represent changing quantities
Trade with Islamic Empire
Liber Abacci (1202) by Leonardo of Pisa (Fibonacci)

Fifteenth and Sixteenth Centuries

Maya developed base 20 numeration, calendars
Sophisticated mathematics done in China, India, Arabic cultures
Insulated from one another and Europe
Europeans develop navigation in 15th century; six trig functions standardized

Algebra and Calculus

Viète and Descartes

Three Innovations in Algebra

Search for a general solution to quintic
Algebra linked to coordinate geometry
Fermat’s new problems/Fermat’s Last “Theorem”: \(x^n +y^n \ne z^n\) for all \(n\ge 3, x,y,z\ge 1\)

Calculus and Applied Mathematics

Until recently, “applied math” generally meant “physics”
Galileo and Kepler used Greek geometry of conic sections to describe the solar system
Study of motion raised difficult questions

Newton and Leibniz

Euler

The most prolific mathematician in history
Introduced many conventions and notations still in use, e.g., function notation
Popularized the use of \(\pi\), \(e\)

Wrote on math and physics, as well as astronomy, engineering, and philosophy–over 850 publications in total

Euler’s eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye. Euler blamed the cartography he performed for the St. Petersburg Academy for his condition, but the cause of his blindness remains the subject of speculation. Euler’s vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as “Cyclops”. Euler remarked on his loss of vision, stating “Now I will have fewer distractions.” In 1766 a cataract in his left eye was discovered, and a few weeks later a failed surgical restoration rendered him almost totally blind. However, his condition appeared to have little effect on his productivity. With the aid of his scribes, Euler’s productivity in many areas of study increased and in 1775 he produced, on average, one mathematical paper every week.

Rigor

Professionalization

Aftermath of French Revolution — new emphasis on education
Mathematicians were expected to teach, and students to learn
New emphasis on clarity, precision and rigor; how can a student understand what a teacher does not?

C.F. Gauss

Could do arithmetic at age 3
Disquisitiones Arithmeticae (1801)
Number theory is the “queen of mathematics”
Work spanned pure and applied mathematics, and physics

19th Century

Rigor and Foundations

Weierstrauss formally defined continuity
Cantor introduced the notion of a set
Foundations of calculus (analysis) set
Dedekind and Peano investigated the foundations of arithmetic
Galois/Abel: insolubility of the quintic

Geometry

Centuries-long exploration of the parallel postulate concludes
Gauss, Bolyai, Lobachevsky, Riemann discovered non-Euclidean geometries
Einstein needed Riemannian geometry to describe his theory of gravitation

Twentieth-Century Developments

Hilbert’s 23 Problems
Abstraction
Proliferation
Specialization
Computation
Data
Clay Math Institute’s Millennium Problems
Wiles’ Theorem and progress on Twin Primes

David Hilbert (ICM 1900) — 23 problems to guide mathematical work in the 20th century Mathematical knowledge continues to increase; more new mathematics has been produced after 1969 than in all of recorded human history before it

Trend toward unity in abstraction
Many old questions answered with new techniques
Sheer amount of new mathematics caused a growth in specialization; I am an algebraist by training, please don’t ask me anything about differential equations
This is not unique to math, though

At the dawn of the 21st century, the Clay Mathematics Institute produce the Millennium problems, 7 problems with million dollar bounties - One dates back to Poincaré’s work, which was generalized in the 20th century to higher dimensions. It was solved in dimension 5 and higher in the 1960s, and in dimension 4 in the 1980s. But it wasn’t until 2002 that the three-dimensional problem was solved by Grigory Perelman. He refused the prize

Four Color Theorem
Sphere packing
- Given a collection of equal-sized marbles, how can you arrange them to pack them as tight as possible? Kepler had a guess, and a proposed proof was given in 1998, but it required a large number of cases to be checked by computer
- The proof was checked by 12 experts who said they were 99% sure it was right. Is that enough? Cf. the four-color theorem

Number Theory - Yitang Zhang’s progress on the twin prime conjecture - Ron Rivest, Adi Shamir, and Leonard Adleman (1970s) discovered that primes could be use to construct public key cryptosystems - This is how we encrypt information we send around the internet - This also incentivizes people to look for better ways to factor large numbers—it’s useful for breaking RSA!