**A history of mathematical rigor**

**By Jon Alter ’21**

Professor Joan Richards from Brown University presented at the Dartmouth College Mathematics Colloquium on September 15. Her lecture recounted historical changes to the idea of mathematical rigor both pre- and post-Enlightenment.

Contemporarily, mathematical proofs require thoroughness, completeness, and exactness in every detail from start to finish. However, this has not always been the prevailing standard for writing proofs. Richards argues that the European Enlightenment marked a shift in mathematical proof-writing toward widespread accessibility.

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Rigor, as defined by the Merriam-Webster dictionary, is “harsh inflexibility in opinion, temper, or judgment.”^{2} Our modern definition of rigor stems from the historical changes in mathematical rigor, which, today, describes the thorough and complete nature of constructing a mathematical idea through a logical sequence. Rigor, inherent in its definition, is denoted by rigidness. Yet Richards indicates that this rigidness was not always a requirement of mathematical proofs, and perhaps for good reason.

First, we must consider Isaac Newton’s role in the early stages of devising modern mathematics. Newton was a core Enlightenment thinker in late 17^{th} century England, responsible for instilling mathematical and logical appreciation in his fellow Europeans. Newton expressed his belief that the world could best be understood through the study of mathematics. This idea pervaded the ideologies of later Enlightenment thinkers, including Voltaire. Mathematics became a fundamental academic study in Europe – until the 1730s.

Georges-Louis Leclerc, Comte de Buffon, a French mathematician, translated Newton’s works into French, though ultimately chose to abandon the study of math to instead focus on natural history. Buffon desired to understand the realness of the world, stipulating that mathematics was a contrived study that merely allowed humans to study themselves. Buffon was the first to sense issue with mathematical rigor, under the context that, without proving mathematical ideas in full, the entire study is improper, and, to a degree, false.

Buffon’s view largely prevailed until the mid-to-late-1700s, when the *Encyclopedié* was written as a reference book comprised of Enlightenment thoughts. One of the book’s lead editors was Jean le Rond D’Alembert, a French mathematician who disagreed vehemently with Buffon’s skepticism. D’Alembert fought against Buffon’s assertions by including a math reference book in the *Encyclopedié* for French citizens to study. D’Alembert claimed that, contrary to Buffon, math allowed for a better conception of the world, and should be a necessary part of academic study. Similarly, D’Alembert stated that mathematical rigor “consists in reducing everything to the simplest principles. From which it follows that rigor… necessarily entails the most natural and direct method.” Richards explains that, unlike Buffon, D’Alembert saw rigor as the ability to reduce mathematical complexity to core concepts that are both true and accessible. . Without strict rigidity of defining minor details while writing a proof, math can be more easily understood by a larger populace, in turn enabling more to understand the world.

D’Alembert’s description of rigor continued to define mathematics and the Enlightenment in Europe. Math became a fundamental aspect of European education, as D’Alembert and his disciples made it clear that math would enable the success of future visionaries. This was until the Enlightenment era ended with the French Revolution when Napoleon, a proponent of Enlightenment ideology, was overthrown. With Napoleon, some ideas of the Enlightenment era were similarly scraped.

At the *École Polytechnique*, a French engineering institution founded during the Enlightenment, Augustin-Louis Cauchy became the new head of the mathematics department. He immediately changed the teaching of mathematics to that which used classical rigor, using a definition more in line with that of Buffon. Cauchy explained that “by determining these conditions and these values, and by fixing precisely the sense of all the notations I use, I make all uncertainty disappear.” Cauchy’s stipulation returned to a requisite defining of all steps and logic while constructing a proof. Richards concludes that the Enlightenment idea of accessibility was revoked in place of inductive and deductive reason. Shortcuts that allowed for truthful math with greater understanding were no longer warranted in its instruction, instead giving way to the rigidly rigorous mathematics that is taught and expected today.

**References:**

- Richards, J. (2017, September 15). Mathematics and its Rigors. Lecture presented at Mathematics Colloquium in Dartmouth College, Hanover.

- Retrieved September 18, 2017, from https://www.merriam-webster.com/dictionary/rigor