The Arithmetic of Reasoning: The Arithmetic of Reasoning: Logic and Boolean AlgebraLogic and Boolean Algebra

-Jeff Johnson-Jeff Johnson

-Mike Grassel-Mike Grassel

OVERVIEWOVERVIEW Development of Logic Development of Logic

and Boolean Algebra was and Boolean Algebra was important to the important to the development of development of computers.computers.

Computers keep track of Computers keep track of finances, correct our finances, correct our grammar, calculate taxes, grammar, calculate taxes, etc.etc.

Computers reduce Computers reduce human reasoning to human reasoning to mechanical processes by mechanical processes by simple logicsimple logic

The MathematiciansThe Mathematicians Four mathematicians Four mathematicians

collectively transformed collectively transformed reasoning from words to reasoning from words to symbols to numbers.symbols to numbers.

The transformation has The transformation has led to the modern led to the modern computer.computer.

The four mathematicians The four mathematicians were Gottfried Wilhelm were Gottfried Wilhelm Leibniz, Augustus De Leibniz, Augustus De Morgan, George Boole, Morgan, George Boole, and Charles Sanders and Charles Sanders PiercePierce

Gottfried Wilhelm LeibnizGottfried Wilhelm Leibniz

LeibnizLeibniz

1694: created a mechanical calculating device 1694: created a mechanical calculating device called the “Stepped Reckoner”.called the “Stepped Reckoner”.

The “Stepped Reckoner” knew how to add, The “Stepped Reckoner” knew how to add, multiply, subtract, and divide.multiply, subtract, and divide.

Became interested by vision of “Calculus of Became interested by vision of “Calculus of Logic”.Logic”.

Starting with a few basic logical assumptions, Starting with a few basic logical assumptions, Leibniz wanted a system to work mechanically Leibniz wanted a system to work mechanically by a simple set of rules with this system in which by a simple set of rules with this system in which new statements are derived from ones already new statements are derived from ones already known.known.

AUGUSTUS DE MORGANAUGUSTUS DE MORGAN

De MorganDe Morgan

Born in Madras, India and Born in Madras, India and was blind in one eye.was blind in one eye.

Graduated with honors Graduated with honors from Trinity College in from Trinity College in Cambridge.Cambridge.

By the age of 22, he was By the age of 22, he was a mathematics professor a mathematics professor at London Universityat London University

De Morgan thought the De Morgan thought the 1919thth century separation century separation between math and logic between math and logic was harmfulwas harmful

Wrote textbooks and Wrote textbooks and popular articles on logic.popular articles on logic.

Put many mathematical Put many mathematical concepts on logical basis.concepts on logical basis.

Worked to make logic Worked to make logic more mathematical.more mathematical.

Emphasized logical Emphasized logical relations as objects worth relations as objects worth of detached study.of detached study.

George BooleGeorge Boole

BooleBoole

Son of an English tradesman with no Son of an English tradesman with no money or privilegesmoney or privileges

Taught himself Greek and LatinTaught himself Greek and Latin Acquired enough education to become Acquired enough education to become

elementary school teacherelementary school teacher At 20, he began studying mathematics At 20, he began studying mathematics

“seriously”.“seriously”. 1849: Boole became a professor of 1849: Boole became a professor of

Mathematics at Queens College in DublinMathematics at Queens College in Dublin

Boole (cont.)Boole (cont.)

Wrote two books that enhanced the field of logicWrote two books that enhanced the field of logic 1847:published “The Mathematical Analysis of 1847:published “The Mathematical Analysis of

Logic”, which helped lay the foundation for the Logic”, which helped lay the foundation for the numerical and algebraic treatment of logical numerical and algebraic treatment of logical reasoning.reasoning.

1854: published “An Investigation of the Laws of 1854: published “An Investigation of the Laws of Thought” which elaborated and codified ideas Thought” which elaborated and codified ideas which he explored in his previous writing, “The which he explored in his previous writing, “The Mathematical Analysis of Logic”.Mathematical Analysis of Logic”.

Boole (cont…)Boole (cont…)

Boole’s symbolic approach to logic led to the Boole’s symbolic approach to logic led to the development of Boolean algebradevelopment of Boolean algebra

Boolean algebra is the basis for modern Boolean algebra is the basis for modern computer logic systemcomputer logic system

Key element of Boole’s work: systematic Key element of Boole’s work: systematic treatment of statements as objects whose truth treatment of statements as objects whose truth values can be combined by logical operations.values can be combined by logical operations.

These operations are calculated the same way These operations are calculated the same way as numbers are added and subtractedas numbers are added and subtracted

CS PierceCS Pierce

1839-19141839-1914 Son of Harvard mathematician and Son of Harvard mathematician and

astronomerastronomer Resurrected and extended De Morgan’s Resurrected and extended De Morgan’s

contributions to the mathematical theory of contributions to the mathematical theory of logiclogic

Interests in philosophy and logic led him to Interests in philosophy and logic led him to an “algebra of logic”.an “algebra of logic”.

CS Pierce’s “Algebra of Logic”CS Pierce’s “Algebra of Logic”

Claimed other mathematicians want to get Claimed other mathematicians want to get to conclusions as quickly as possibleto conclusions as quickly as possible

Will to skip steps when they know where Will to skip steps when they know where argument is headingargument is heading

Logicians want to analyze deductions as Logicians want to analyze deductions as carefully as possiblecarefully as possible

Break deductions down into small simple Break deductions down into small simple stepssteps

TimelineTimeline 44thth century BC: Aristotle’s logical syllogisms century BC: Aristotle’s logical syllogisms 1642: Blaise Pascal invents 1642: Blaise Pascal invents PascalinePascaline (can only add and subtract) (can only add and subtract) 1694: GW Leibniz invents 1694: GW Leibniz invents Stepped ReckonerStepped Reckoner (add, subtract, multiply, (add, subtract, multiply,

divide)divide) 1806-1871: De Morgan beings to piece together mathematics of logic1806-1871: De Morgan beings to piece together mathematics of logic 1847: George Boole publishes “The Mathematical Analysis of Logic” 1847: George Boole publishes “The Mathematical Analysis of Logic” 1854: Boole publishes “An Investigation of the Laws of Thought”1854: Boole publishes “An Investigation of the Laws of Thought” 1839-1914: CS Pierce reduce reduction of mathematical reasoning to long 1839-1914: CS Pierce reduce reduction of mathematical reasoning to long

strings (critical prerequisite for the computer age) strings (critical prerequisite for the computer age) 2005: Jeff Johnson / Mike Grassel undergraduate presentation on History 2005: Jeff Johnson / Mike Grassel undergraduate presentation on History

of Logic of Logic

References References Berlinghoff, William P and Fernando Q Gouvea. Berlinghoff, William P and Fernando Q Gouvea. Math through the Ages: A Math through the Ages: A

Gentle History for Teachers and OthersGentle History for Teachers and Others. Oxton House Publishers, 2002. . Oxton House Publishers, 2002. ““The Mathematics of Boolean Algebra”, Monk, Donald. The Mathematics of Boolean Algebra”, Monk, Donald.

http://plato.stanford.edu/entries/boolalg-math/http://plato.stanford.edu/entries/boolalg-math/ ““Boolean Algebra”. Boolean Algebra”.

http://www.eelab.usyd.edu.au/digital_tutorial/chapter4/4_0.htmlhttp://www.eelab.usyd.edu.au/digital_tutorial/chapter4/4_0.html ““Hands on Puzzles for Thinking Fun”, Hands on Puzzles for Thinking Fun”, http://www.logicpuzzlemuseum.org/http://www.logicpuzzlemuseum.org/