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
 Wrote textbooks 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
19th century separation
between math and logic
was harmful
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
 1839-1914
 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
 4th 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)
1806-1871: 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”
1839-1914: 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, 2002.
 “The Mathematics of Boolean Algebra”, Monk, Donald.
http://plato.stanford.edu/entries/boolalg-math/
 “Boolean Algebra”.
http://www.eelab.usyd.edu.au/digital_tutorial/chapter4/4_0.html
 “Hands on Puzzles for Thinking Fun”, http://www.logicpuzzlemuseum.org/