Bruce J. Petrie
Dissertation & Research
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What follows below is composed of selections and minor modifications from my official thesis proposal submitted to the IHPST on April 16 2010. Selections have been removed from the original and will not be made available online here. Content from my dissertation chapters have been, are being, or will be revised for journal publication. Please see the CV section for those which have been made available as of May 2011.
The Roots of Transcendental Numbers: 1700 – 1900
"Each new major achievement in the theory of transcendental numbers is linked with the emergence of a new method." [1]
When the prominent mathematician David Hilbert gave his famous 1900 Paris Address he claimed “We know that every age has its own problems, which the following age either solves or casts away as profitless to be replaced by new ones…The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied."[2] During the address, Hilbert listed ten problems (although twenty-three later appeared in print) that would direct mathematical research for the new century. One of his ten problems concerned the transcendence of two numbers 2^√2 and e^π. In 1900 transcendental number theory was the frontier of mathematical research.
My doctoral dissertation will be an investigation of an important topic in the history of modern mathematics, the origin and development of transcendental number theory. The term transcendence has been used to describe various mathematical objects. For instance, Gottfried Leibniz (1646 – 1716) used the term to describe progressions and Leonhard Euler (1707 – 1783) used the term to describe functions. My thesis concentrates instead on how the term transcendence is applied to numbers during the eighteenth and nineteenth centuries particularly in France, Germany, and Russia by uncovering the specific catalysts, intuitions, and motivations of mathematicians making notable contributions to transcendental number theory especially Euler, Johann Lambert (1728 – 1777), Joseph Liouville (1809 – 1882), Charles Hermite (1822 – 1901), and Ferdinand Von Lindemann (1852 – 1939). The modern definition of a transcendental number is a number which is not the root of any single-variable non-zero polynomial with rational coefficients. In 1844, Liouville used this definition and became the first person to demonstrate a number was transcendental (actually a whole family of them) and thus proved their mathematical existence. The number e was proven to be transcendental by Hermite in 1873 and the number π was proven to be transcendental by Lindemann in 1882. Naum Fel’dman and Andrei Shidlovskii (1967) claimed that advances in transcendental number theory were linked to the emergence of new mathematical methods. My research aims to evaluate the claims by Fel’dman and Shidlovskii by thoroughly examining the methodology of Liouville, Hermite, and Lindemann in their respective achievements within transcendental number theory.
Little work has been done concerning the history of transcendental numbers. Historical treatments are often mathematical papers or texts which involve a brief and insufficient history of the subject to provide some heritage relevant to the mathematics contained in the paper or text. As it stands, we are lacking a comprehensive history of transcendental numbers. For instance, in remarking how problems in modern number theory can be related to some works of Euler, Paul Erdös and Underwood Dudley (1983) stated “As far as we know, Euler was the first to define transcendental numbers as numbers which are not roots of algebraic equations”[3]. My research suggests otherwise (Petrie, 2010)[4]. It is interesting that historians of mathematics have had difficulty pinpointing the origin of this modern definition and is likely a result of the scarcity of ample research on the topic. Although this project does find some justification in understanding the development of modern mathematics, my research is more significant to properly understanding the mathematics of the past on its own terms. A proper history of transcendental number theory will not only fill a void in the literature but also exemplify differences in eighteenth century and nineteenth century paradigms of mathematics.
The heritage approach to studying the history of mathematics described by Ivor Grattan-Guinness (2004) can be observed in the history of mathematical transcendence because research often takes the form of cursory anecdotes relating to an ancient geometric problem. This “squaring of the circle” problem started in another paradigm emphasizing mathematical construction when Greek philosopher Anaxagoras (499BC – 428BC) tried to construct, using only a straight edge and compass, a square with area equal to that of the unit circle, which has area equal to π. Thus, he needed to create a length of √ π. Lindemann proved this impossible when he proved that π is transcendental and therefore not constructible because all numbers which can be constructed using a straight edge and compass are algebraic. Although many historians of mathematics attribute the quadrature of the circle as the mechanism driving the development of the whole of mathematical transcendence, Jesper Lützen (1990) claims that Liouville was inspired by the correspondence between Christian Goldbach (1690 – 1764) and Daniel Bernoulli (1700 – 1782) concerning Joseph-Louis Lagrange’s (1736 – 1813) formula for continued fractions. I want to discern how the concept of a transcendental number materialized in mathematics and reveal the conditions necessary for Liouville to construct his approximation theorem and the first family of transcendental numbers.
The history of transcendental number theory is largely untreated, especially by professional historians. This project will fill a void in the current academic literature by offering a historically informed alternative to the common mathematical accounts of the origin and development of transcendental number theory and highlight paradigmatic differences between eighteenth and nineteenth century mathematics. My research will not only satisfy historical curiosities of a mathematical audience by detailing episodes relevant to modern mathematicians but also to historians wanting a comprehensive treatment of the birth of a fascinating branch of number theory during one of the most turbulent periods in the history of mathematics.
[1] Fel’dman, N. I. & Shidlovski, A. B.. "The Development and Present State of the Theory of Transcendental Numbers.” Russian Math. Surv. 22 (1967): 1-79.
[2] Hilbert, D.. “Mathematical Problems.” Bull. Amer. Math. Soc. 8 (1902): 437-739.
[3] Erdös, P. & Dudley, U.. “Some Remarks and Problems in Number Theory Related to the Work of Euler.” Mathematics Magazine 56 (1983): 292-8.
[4] Petrie, B. J.. “Leonhard Euler’s Use and Understanding of Mathematical Transcedence.” Proceedings of the Canadian Society for History and Philosophy of Mathematics, 23 (2010): 241-250.