The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense.

Note that some phrases, like "in general", appear below in more than one section.

These terms discuss mathematics as mathematicians think of it; they connote common intellectual strategies or notions the investigation of which somehow underlies much of mathematics.

—The proof that there are infinitely many prime numbers.

—The proof of the irrationality of the square root of two.

Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context.

Mihalis Constantine Dafermos is a Professor in Mathematical Physics at the University of Cambridge.

He studied mathematics at Harvard University and was awarded a BA in 1997. His PhD thesis titled Stability and Instability of the Cauchy Horizon for the Spherically Symmetric Einstein-Maxwell-Scalar Field Equations was written under the supervision of Demetrios Christodoulou at Princeton University.

He has won the Adams Prize writing on the subject Differential Equations in 2004 and the Whitehead Prize in 2009 for "his work on the rigorous analysis of hyperbolic partial differential equations in general relativity."

Thomas Paine (January 29, 1737 (NS February 9, 1737) – June 8, 1809) was an English-American author, pamphleteer, radical, inventor, intellectual, revolutionary, and one of the Founding Fathers of the United States. He has been called "a corsetmaker by trade, a journalist by profession, and a propagandist by inclination."

Born in Thetford, England, in the county of Norfolk, Paine immigrated to the British American colonies in 1774 in time to participate in the American Revolution. His principal contributions were the powerful, widely read pamphlet Common Sense (1776), the all-time best-selling American book that advocated colonial America's independence from the Kingdom of Great Britain, and The American Crisis (1776–83), a pro-revolutionary pamphlet series. Common Sense was so influential that John Adams said, "Without the pen of the author of Common Sense, the sword of Washington would have been raised in vain.”

Paine lived in France for most of the 1790s, becoming deeply involved in the French Revolution. He wrote the Rights of Man (1791), in part a defense of the French Revolution against its critics. His attacks on British writer Edmund Burke led to a trial and conviction in absentia in 1792 for the crime of seditious libel. In 1792, despite not speaking French, he was elected to the French National Convention. The Girondists regarded him as an ally. Consequently, the Montagnards, especially Robespierre, regarded him as an enemy. In December of 1793, he was arrested and imprisoned in Paris, then released in 1794. He became notorious because of The Age of Reason (1793–94), his book that advocates deism, promotes reason and freethinking, and argues against institutionalized religion in general and Christian doctrine in particular. He also wrote the pamphlet Agrarian Justice (1795), discussing the origins of property, and introduced the concept of a guaranteed minimum income.

Penn Fraser Jillette (born March 5, 1955) is an American illusionist, comedian, musician, and best-selling author known for his work with fellow magician Teller in the team Penn & Teller, and advocacy of atheism, scientific skepticism, libertarianism and free market capitalism.

Jillette was born in Greenfield, Massachusetts. His mother, Valda R. Jillette (née Parks) (November 8, 1909 – January 1, 2000), was a secretary, and his father, Samuel H. Jillette (March 14, 1912 – February 14, 1999), worked at Greenfield's Franklin County Jail. Jillette became disenchanted with traditional illusionist acts that presented the craft as authentic magic, such as The Amazing Kreskin on The Tonight Show Starring Johnny Carson. At age eighteen, he saw a show by illusionist James Randi, and became enamored of his approach to magic that openly acknowledged deception as entertainment rather than a mysterious supernatural power. Jillette regularly acknowledges Randi as the one person on the planet he loves the most besides members of his family.