Amateur mathematicians

SUMMARY: Mathematics has appealed to amateurs as a recreation and even without the rigor of the academy and peer review they have made strong contributions.

Historically, amateurs around the world have made significant contributions to mathematics in amazing and diverse ways. Can anyone now contribute to the development of contemporary mathematics, or can only professionally trained individuals do so? Answering this question requires reflection on both the ways in which mathematical research develops and the nature of the community that defines who is accepted as a mathematician.

The Nature of Mathematics

There are many examples of self-taught mathematicians or part-time mathematicians whose main professions or training was in another field. Some of these are well known in the history of mathematics, like Albert Einstein (1879–1955), who showed an early interest in mathematics by teaching himself geometric concepts at the age of 12. Gottfried Leibniz’s (1646–1716) and Pierre de Fermat’s (1601–1665) initial formal training was in law, not mathematics. Srinivasa Ramanujan (1887–1920) is cited as a mathematical genius who was self-taught.

People such as these raise the question about the nature of mathematical ability. Experiments on very young children have indicated that all individuals have the innate ability to recognize quantitative differences when the quantities are small. Further, lesions in the angular gyrus within the inferior parietal cortex of the brain can significantly impair mathematical ability, while the inferior parietal lobe region of Einstein’s brain was 15 times larger than normal. In approaching the physical world, humans utilize number sense, pattern identification, and spatial awareness. These concepts contribute to mathematical reasoning. Contemporary mathematics as an academic discipline often requires high-level abstraction and complex symbolization.

Mathematicians like Reuben Hersh and Ian Stewart hold that abstract mathematical objects are cultural creations. Although it is true that any contribution to mathematics must take an account of the cultural context, this still leaves open the question of whether only individuals professionally trained in these traditions can make contributions to the development of mathematics. For example, Leibniz sought a tutor, and Einstein immersed himself in these cultural traditions. Furthermore, over the course of the twentieth century, mathematics became increasingly professionalized. Professions consisted of individuals with specialized training who, as a result, were granted a large degree of autonomy and self-policing oversight in determining what does and does not constitute acceptable mathematical thought by setting the appropriate standards, methods, and problems of the discipline. These functions have been embodied in institutions, such as mathematics departments in universities and mathematical periodicals, societies, and conventions.

Some conferences and lectures are by invitation only, and, in other cases, a conference or session organizer selects from submissions. Journal editors and reviewers decide what is appropriate for publication. In this way, the people in the mathematical community determine standards and recognition or rejection of ideas and results. Over the twentieth century, mathematics developed through higher levels of generalization and abstraction using the axiomatic method, by cross-fertilization among different mathematical fields, by developing new mathematical theories in an attempt to solve a given mathematical problem, and by examining the foundations of mathematics as a mathematical problem. All of these processes require immersion within the discipline.

Amateur Contributions

It seemed that, by the mid-twentieth century, the mathematical universalist was a thing of the past, which led to the question of whether there was any space for the amateur mathematician. The mathematical profession generally holds that an individual without formal credentials in mathematics could not engage in significant mathematical research or make any meaningful contributions to the discipline. However, there are several areas that are, in theory, still open to amateurs. The development of new forms of mathematics from nonmathematical considerations, the applications of abstract mathematics to real world problems, and discoveries of solutions to specific mathematical problems are three possible ways in which amateurs can make contributions. Further, amateurs can identify mathematical problems, topics, and subject matters that professionals do not recognize. They can conceptualize mathematical problems in ways that the professionals cannot with new definitions or proofs. They can develop new symbolic notations that assist in solving existing mathematical problems. Finally, they can develop new methods for solving mathematical problems.

High school students have even published their discoveries, such as Ryan Morgan in 1994. Someone outside the profession can have fresh, fruitful insights. Indeed, it has been argued that disciplines go through periods of normal change in which there is development of existing paradigms and revolutionary periods in which basic paradigms change. Often, the revolutionary stage is initiated by individuals outside or at the margin of the discipline. Stock market analyst Robert Prechter Jr.’s (1949–) love of mathematics, deep belief in the mathematical structure of the universe, and search for innovative ways to apply mathematics to develop an understanding of the real world make him an interesting amateur mathematician. One of Prechter’s goals has been to identify Fibonacci growth patterns in the stock markets. Prechter stated, “We would love to see Leonardo Fibonacci (c. 1175–1240) at least make the list of contenders for the real Man of the Millennium.”

Obstacles

However, an individual may face an uphill battle to have his or her work understood and accepted. This battle is made even more difficult by the existence of a plethora of what are often called “mathematical cranks” (individuals who claim to be able to solve all sorts of mathematical problems, but often just produce aimless ramblings). English philosopher Thomas Hobbes (1588–1679), who claimed to be able to square the circle, was such a person. Others have submitted proofs of Fermat’s Last Theorem. Notable mathematicians are bombarded by such claims, making them less receptive to genuine amateur innovators. For instance, Ramanujan wrote letters about his work to mathematicians outside India. However, his mathematical writing was not the same as the standard communication at the time, and he was ignored until Godfrey “G. H.” Hardy (1877–1947) looked beyond the stylistic and notation issues and recognized his genius. This recognition was the beginning of a fruitful and well-known collaboration between them. Hardy noted:

What was to be done in the way of teaching him modern mathematics? The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems of complex multiplication, to orders unheard of, whose mastery of continued fractions was, on the formal side at any rate, beyond that of any mathematician in the world, who had found for himself the functional equation of the Zeta-function, and the dominant terms of many of the most famous problems in the analytic theory of numbers; and he had never heard of a doubly periodic function or of Cauchy’s theorem, and had indeed but the vaguest idea of what a function of a complex variable was. His ideas as to what constituted a mathematical proof were of the most shadowy description. All his results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account.

Oliver Heaviside’s (1850–1925) operator calculus work was not well received until Thomas Bromwich (1875–1929) justified the theory. Radio engineer and high school teacher Kurt Heegner’s (1893–1965) algebraic number theory result was initially dismissed, but number theorist Harold Stark (1939–) filled in the gaps and the Stark–Heegner theorem is named for them. Another notable example is Thomas Fuller (1710–1790), a slave who could perform remarkable mental calculations. In 1788, abolitionists interviewed Fuller in order to demonstrate the superior intellectual abilities of African Americans. Historians do not know exactly how Thomas Fuller performed his calculations. However, they theorize that the algorithms he used were probably based on traditional African counting systems.

Dutch graphic artist M.C. Escher (1898–1972) and San Diego homemaker Marjorie Rice (1923–2017) were cited as amateur mathematicians. They developed innovative approaches to geometric tiling and tessellations, which were introduced to the mathematical community by mathematicians like Doris Schattschneider (1939–). Some self-taught mathematicians are noted both for their work and for their other contributions to the mathematical community, such as Artemas Martin (1835–1918), who not only published articles but also was cited as having founded journals like the American Mathematical Monthly that paved the way for others who followed.

Some mathematicians have given stylistic advice to those who want to be taken seriously. Others identify mathematical puzzles or problems that could be solved by the amateur. For instance, some have noted that the question of whether P=NP in theoretical computer science might be solved by an amateur, and others have noted the Beal Conjecture, named for Andrew Beal (1952–), a self-made billionaire and banker. Mathematicians and historians continue to publicize results from amateurs who might not otherwise be as known to the community, such as Mehmet Nadir (1856–1927), who is noted as an amateur mathematician in Ottoman Turkey, and geometric theorems on Japanese wooden tablets in temples that predate the work of Western mathematicians.

Aubrey de Grey, a master in the field of biology, was considered an amateur mathematician because of his lack of formal, specific training. However, like in most other fields that de Grey studied, he excelled and stunned the mathematical world with his discovery regarding the Hadwiger-Nelson problem. The problem involved finding the chromatic number of a plane, which mathematicians previously believed was no fewer than four but less than seven. De Grey, however, published an article in which he claimed that the number could be no less than five. Many mathematicians were able to weigh in on de Grey's findings thanks to the invention of Polymath by Timothy Gowers, a mathematician at the University of Cambridge. Polymath allows for mathematical problems to be solved publically online. Though de Grey did not solve the problem, his findings pushed it forward in a significant way.

Bibliography

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Allaire, Patricia, and Antonella Cupillari. “Artemas Martin: An Amateur Mathematician of the Nineteenth Century and His Contribution to Mathematics.” College Mathematics Journal, vol. 31, no. 1 (2000).

Andrews, George. “Srinivasa Ramanujan (1887–1920): The Centenary of a Remarkable Mathematician.” The Institute of Mathematics and Its Applications, 14 Apr. 2020, ima.org.uk/13780/srinivasa-ramanujan-1887-1920-the-centenary-of-a-remarkable-mathematician/. Accessed 22 Oct. 2024. De Grey, Aubrey D. "The Chromatic Number of the Plane Is at Least 5." ArXiv, 2018, arxiv.org/abs/1804.02385. Accessed 23 Oct. 2024.

Johnson, George. “Genius or Gibberish? The Strange World of the Math Crank.” The New York Times, 9 Feb. 1999, www.nytimes.com/1999/02/09/science/genius-or-gibberish-the-strange-world-of-the-math-crank.html. Accessed 22 Oct. 2024.

Lamb, Evelyn. “Decades-Old Graph Problem Yields to Amateur Mathematician.” Quanta Magazine, 17 Apr. 2018, www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/. Accessed 22 Oct. 2024.

Normile, Dennis. “‘Amateur’ Proofs Blend Religion and Scholarship in Ancient Japan: A 300-Year-Old Japanese Art Form Presents Some Surprising Mathematical Discoveries on Elegant Wooden Tablets.” Science, vol. 307, no. 5716. 18 Mar. 2005.

Prechter, Robert, Jr. Beautiful Pictures From the Gallery of Phinance, New Classics Library, 2010.

Schattschneider, Doris. “In Praise of Amateurs.” The Mathematical Gardner. Edited by David Klamer, Wadsworth, 1981.