History 1931 A.D.

1931 The CRC by Paul Koenig

Godel's Incompleteness Theorem by Maggie Cooper.

First edition of “Standard Mathematical Tables” is published by the Chemical Rubber Corporation in 1931. Affectionately referred to as “the CRC,” later editions of the book would become a standard reference in assisting with pencil and paper calculations.

Though containing a wide variety of mathematical tables, the most important were the tables of logarithmic and trigonometric values and tables of formulae for integration and mensuration. All tables were meant to simplify ordinary calculation to a reasonable precision.

Logarithm tables simplified complicated multiplication and division problems to simple addition and subtraction giving a result generally accurate to four to six significant digits. Other tables in the CRC would give the most used formulas and numerical values, thus preventing errors from faulty memorization and derivation. The time saved from the readiness of the information was another key to the books success.

One calculating device was both inexpensive and prevalent during the heyday of the CRC. For those wishing quicker but less accurate computations, the slide rule was a popular choice. By adding logarithms graphically, rapid multiplications could be performed. However, limited space on most slide rules limited the precision to three significant digits.

By the 1960’s and 70’s, inexpensive electronic calculators rapidly began replacing both the slide rule and books of mathematical tables such as the CRC for simple operations such as multiplication and division. In the 70’s and 80’s, scientific and programmable calculators took on the tasks of performing trigonometric computations and more complicated formulae. By 1988, when Hewlett-Packard released the HP 28S, the first inexpensive pocket calculator capable of symbolic operations, all of the functions of slide rules and the CRC could be performed on electronic devices with much greater precision.

Today, teachers, especially in the physical sciences and statistics, must warn students about the perils of TOO MUCH precision in an answer. An answer more precise than the data that began the computation is both misleading and wrong. Also, teachers must instruct students on the limits of the computers’ method of solution for certain problems. When calculations were accurate to three to six digits rather than eight to twelve, these concerns were less pressing.

Author: Paul Koenig

References:
CRC Press. Standard Mathematical Tables.

Hewlett-Packard. 1997. HP 35 Pocket Calculator. <http://www.hp.com/abouthp/features/hp35calculator/sliderule/>

Hicks, David. 1998. Slide Rules. <http://www.hpmuseum.org/sliderul.htm>

Meeus, Jean. 1991. Astronomical Algorithms. Richmond: Willman-Bell. 15-22

Triola, Mario. 1995. Elementary Statistics. Reading: Addison-Wesley. 12-4

Stewart, James. 1994. Calculus. Pacific Grove: Brooks/Cole. A42-5

In 1931 something amazing happened that effected most directly the field of math logic, but had implications in many other fields. An article appeared in a German technical journal that disproved the belief that there would someday be a discovery that would unite the branches of mathematics together and that there was no longer any need to try to find axioms that would be a basis of truths for all mathematics. These "unforseen limitations on the methods of formal mathematics," came from the publications of Kurt Godel, a mathematician and logician born in 1906 in present day Czechoslovakia who died in 1978 in the United States.

One of Godel's publications known as the "undecidablity theorem" which proved that with in a formal system, there are questions that exist that are neither provable nor disprovable on the basis of axioms that define that system. The "incompleteness theorem" showed there will be a contradiction of statements when the system has "decidability of all questions."

Godel's work contradicted the work that other mathematicians had been working on. One accepted view that Godel contradicted was Hilberts' system for all classical math making, as well as the work of those who were trying to prove consistency and completeness of Zermelo-Frankel's formalism for set theory and logic by using Peano's axioms for natural numbers. Godel said that if one could support Peano's axioms (to provide a formal description of the process of counting) the support could not be consistent (defined as a desirable property of a logical system which says there are no statements which the system regards as both true and false), and at the same time complete (defined as a desirable property of a logical system that says it can prove, one way or the other, any statement that it knows how to address). Also, this support could not prove itself consistent with out proving itself inconsistent. Godel's theorems were a shock to many when they showed that math was forever destined to remain incomplete; math is not a finished object.

His school of philosophical thought, logical postivism, that if a thing is not emperically verifiable, it is meaningless, influenced these theorems and show up in the implications that these theorems had in applications to art through Escher, philosophy through Zen Buddism, language, and computers. Godel's contributions in 1931 was a major event in the math history time line, but other time line's as well.

Author: Maggie Cooper

References:
Eves, Howard, An Introduction to the History of Mathematics, 6th ed, Saunders College Publishing, San Diego, 1990, pgs634-635, 685.

Greenberg, Marvin Jay, Euclidean and Non Eulclidean Geometries, 3rd ed, WH Freeman and Company, New York, 1993, pgs 297, 300, 305.

http://204.249.215.251/clubs/physics/godel.html
http://acnet.pratt.edu/~arch543p/help/Godel.html
http://www.chaos.org.uk/~eddy/math/Godel.html
http://www.exploratorium.edu/complexity/Complexicon/godel.html

Last updated September 1998