The Arithmetic of Reasoning: Logic and Boolean Algebra -Jeff Johnson -Mike Grassel

OVERVIEW  Development of Logic and Boolean Algebra was important to the development of computers.  Computers keep track of finances, correct our grammar, calculate taxes, etc.  Computers reduce human reasoning to mechanical processes by simple logic

The Mathematicians  Four mathematicians collectively transformed reasoning from words to symbols to numbers.  The transformation has led to the modern computer.  The four mathematicians were Gottfried Wilhelm Leibniz, Augustus De Morgan, George Boole, and Charles Sanders Pierce

Gottfried Wilhelm Leibniz

Leibniz  1694: created a mechanical calculating device called the “Stepped Reckoner”.  The “Stepped Reckoner” knew how to add, multiply, subtract, and divide.  Became interested by vision of “Calculus of Logic”.  Starting with a few basic logical assumptions, Leibniz wanted a system to work mechanically by a simple set of rules with this system in which new statements are derived from ones already known.

AUGUSTUS DE MORGAN

De Morgan  Born in Madras, India and was blind in one eye.  Graduated with honors from Trinity College in Cambridge.  By the age of 22, he was a mathematics professor at London University  De Morgan thought the 19 th century separation between math and logic was harmful  Wrote textbooks and popular articles on logic.  Put many mathematical concepts on logical basis.  Worked to make logic more mathematical.  Emphasized logical relations as objects worth of detached study.

George Boole

Boole  Son of an English tradesman with no money or privileges  Taught himself Greek and Latin  Acquired enough education to become elementary school teacher  At 20, he began studying mathematics “seriously”.  1849: Boole became a professor of Mathematics at Queens College in Dublin

Boole (cont.)  Wrote two books that enhanced the field of logic  1847:published “The Mathematical Analysis of Logic”, which helped lay the foundation for the numerical and algebraic treatment of logical reasoning.  1854: published “An Investigation of the Laws of Thought” which elaborated and codified ideas which he explored in his previous writing, “The Mathematical Analysis of Logic”.

Boole (cont…)  Boole’s symbolic approach to logic led to the development of Boolean algebra  Boolean algebra is the basis for modern computer logic system  Key element of Boole’s work: systematic treatment of statements as objects whose truth values can be combined by logical operations.  These operations are calculated the same way as numbers are added and subtracted

CS Pierce   Son of Harvard mathematician and astronomer  Resurrected and extended De Morgan’s contributions to the mathematical theory of logic  Interests in philosophy and logic led him to an “algebra of logic”.

CS Pierce’s “Algebra of Logic”  Claimed other mathematicians want to get to conclusions as quickly as possible  Will to skip steps when they know where argument is heading  Logicians want to analyze deductions as carefully as possible  Break deductions down into small simple steps

Timeline  4 th century BC: Aristotle’s logical syllogisms  1642: Blaise Pascal invents Pascaline (can only add and subtract)  1694: GW Leibniz invents Stepped Reckoner (add, subtract, multiply, divide)  : De Morgan beings to piece together mathematics of logic  1847: George Boole publishes “The Mathematical Analysis of Logic”  1854: Boole publishes “An Investigation of the Laws of Thought”  : CS Pierce reduce reduction of mathematical reasoning to long strings (critical prerequisite for the computer age)  2005: Jeff Johnson / Mike Grassel undergraduate presentation on History of Logic

References  Berlinghoff, William P and Fernando Q Gouvea. Math through the Ages: A Gentle History for Teachers and Others. Oxton House Publishers,  “The Mathematics of Boolean Algebra”, Monk, Donald.  “Boolean Algebra”.  “Hands on Puzzles for Thinking Fun”,