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2011 | OriginalPaper | Buchkapitel

1. Mathematics, Computation, and Economics

verfasst von : Kamran Dadkhah

Erschienen in: Foundations of Mathematical and Computational Economics

Verlag: Springer Berlin Heidelberg

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Abstract

Many believe that mathematics is one of the most beautiful creations of humankind, second only to music. The word creation, however, may be disputed. Have humans created something called mathematics? Then how is it that we increasingly discover that the world, and indeed the universe around us, obey one or another mathematical law? Perhaps to the faithful, the answer is clear: A higher power is the greatest mathematician of all. A worldly answer may be that humans discovered, rather than created, mathematics. Thus, wherever we look, we see mathematical laws at work. Whether humans created mathematics and it just so happened that the world seems to be mathematical, or the universe is a giant math problem and human beings are discovering it, we cannot deny that mathematics is extremely useful in every branch of science and technology, and even in everyday life. Without math, most likely we would be still living in caves. But what is mathematics? And why are so many afraid of it?

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Nächstes Kapitel Basic Mathematical Concepts and Methods

Omar Khayyam (1048–1131), the great Iranian mathematician, is better known in the West for his Rubaiyat, a book of poetry freely translated by Edward J. Fitzgerald. Khayyam has several treatises on mathematics including Treatise on Demonstration of Problems of Algebra, from which the quotation in the text is taken. He worked with a group of learned men on the compilation of astronomical tables and the reform of the calendar, which resulted in the Jalali calendar (after Jalaleddin Malik Shah Seljuq), a quite accurate calendar for his time and for many centuries to come. Khayyam measured the length of a year with amazing accuracy. He also knew the Pascal triangle and the coefficients of the binomial expansion, but because of the limitation of mathematics in his time, he could not express them in the general form of today. Khayyam was also a philosopher, and his poetry is mostly musings on the meaning of life.

The Algebra of Omar Khayyam, translated by Daoud Kasir, New York, Teachers College, Columbia University, 1931, p. 61.

Gioseppe Peano (1858–1932), an Italian mathematician, contributed to the study of differential equations and was a founder of mathematical logic. Bertrand Russell wrote in his autobiography that the 1900 International Congress of Philosophy “was the turning point of my intellectual life, because there I met Peano.”

Euclid of Alexandria (~325BC–~265BC) was the most famous mathematician of antiquity, whose book The Elements brought together the mathematical knowledge of his day with rigor and clarity. Indeed, the geometry we learn and use in high schools today is based on Euclid’s axioms and, therefore, referred to as Euclidean geometry. The Elements is still available in bookstores and from online bookshops.

David Hilbert (1862–1943), one of the greatest mathematicians of the twentieth century, is best known for his work for infinite dimensional space referred to as Hilbert space. In a speech he challenged mathematicians to solve 23 fundamental problems, a challenge that is still partly open. Hilbert also contributed to mathematical physics. For more on Hilbert’s problems, which have been solved, and who has solved them, see The Honors Class, Hilbert’s Problems and Their Solvers by Ben H. Yendell (2002).

Nikolai Ivanovich Lobachevsky (1792–1856), the great Russian mathematician, presented his results on non-Euclidean geometry in 1826, although not many of his contemporaries understood it. János Bolyai (1802–1860), a Hungarian mathematician, independently discovered hyperbolic geometry.

German mathematician Georg Friedrich Berhard Riemann (1826–1866), who despite his short life made brilliant contributions to mathematics. Of him it is said that “he touched nothing that he did not in some measure revolutionize.” [Men of Mathematics, by E. T. Bell (1937)].

Set theory was founded in the late nineteenth century by Georg Cantor (1845–1918). He based his analysis on three axioms. But soon these axioms ran into paradoxes, the discovery of the the most famous of them being due to Bertrand Russell. The interested reader is referred to Mathematics, The Loss of Certainty, by Morris Kline (1980). To avoid paradoxes and to assure consistent deductions, set theory is based on Zermelo-Frankel axioms plus the axiom of choice. A discussion of these axioms is well beyond this book.

Usually, a lemma is a mini-theorem.

Kurt Gödel (1906–1978) was born in Brno, Czech Republic, but spent more than half of his life in the United States doing research at the Institute for Advanced Study and at Princeton University, although he did not have lecturing duties. Gödel’s fame rests on his three theorems in mathematical logic: the completeness theorem for predicate calculus (1930), the incompleteness theorem for arithmetic (1930), and the theorem on consistency of the axiom of choice with continuum hypothesis (1938).

Paul Joseph Cohen (1934-2007) was a professor in the mathematics department at Stanford University. He received a Fields Medal (the equivalent of a Nobel Prize in mathematics) in 1966.

In particular, I recommend the following books: Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures (2008) by James Robert Brown; The Mathematical Experience (1981) by Philip Davis and Reuben Hersh; and The Road to Reality, A Complete Guide to the Laws of the Universe (2005) by Roger Penrose. The last book is concerned with the applications of mathematics to physics.

Alfred North Whitehead (1861–1947) was a British mathematician and a collaborator of Russell.

British mathematician and philosopher, Bertrand Arthur William Russell (1872–1970) is considered one of the most important logicians of the last century. In 1950, he won the Nobel Prize in literature “in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought.”

A prominent intuitionist is Luitzen Egbertus Jan Brouwer (1881–1966) of the Netherlands. Economists will become familiar with his name and work through his fixed point theorem that is crucial for proving the existence of equilibrium.

It took 362 pages in Russell’s Principia Mathematica to show that 1 + 1 = 2.

A glimpse of the history of the evolution of mathematics in the twentieth century and its relation to economics is captured in E. Roy Weintraub, How Economics Became a Mathematical Science (2002).

For a more comprehensive list, the reader may want to consult Notable Women in Mathematics, A Biographical Dictionary (1998), edited by Charlene Morrow and Teri Perl.

Joseph A. Schumpeter, “The Common Sense of Econometrics,” Econometrica, 1, 1933, pp. 5–6.

Quoted in Leon Smolinski, “Karl Marx and Mathematical Economics,” Journal of Political Economy, 81, 1973, p. 1200.

Titel: Mathematics, Computation, and Economics
verfasst von: Kamran Dadkhah
Verlag: Springer Berlin Heidelberg
Buch: Foundations of Mathematical and Computational Economics
Print ISBN: 978-3-642-13747-1

Electronic ISBN: 978-3-642-13748-8

Copyright-Jahr: 2011
DOI: https://doi.org/10.1007/978-3-642-13748-8_1