Gian-Carlo Rota - Collected Quotes

"We often hear that mathematics consists mainly in ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences’? A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks." (Gian-Carlo Rota, "Discrete Thoughts", 1981) 

"The progress of mathematics can be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example, the theory of enumeration appear to be boundless. Rules for the enumeration of sets subject to various conditions, or combinatorial objects as they are often called, appear to obey an indefinite variety of and seem to lead to a welter of generating functions. We are at first led to suspect that the class of objects with a common property that may be enumerated is indeed infinite and unclassifiable." (Gian-Carlo Rota [Preface to (Ian P Goulden and David M Jackson, "Combinatorial Enumeration", 1983)])

"There are two kinds of mistakes. There are fatal mistakes that destroy a theory; but there are also contingent ones, which are useful in testing the stability of a theory." (Gian-Carlo Rota, [lecture] 1996)

"Mathematical logic deals not with the truth but only with the game of truth." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Nature imitates mathematics." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"Mathematical beauty and mathematical truth share the fundamental property of objectivity, that of being inescapably context-dependent. Mathematical beauty and mathematical truth, like any other objective characteristics of mathematics, are subject to the laws of the real world, on a par with the laws of physics." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", Synthese, 111(2), 1997)

"[...] the beauty of a piece of mathematics is strongly dependent upon schools and periods of history. A theorem that is in one context thought to be beautiful may in a different context appear trivial. [...] Undoubtedly, many occurrences of mathematical beauty eventually fade or fall into triviality as mathematics progresses." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 1997)

"The lack of beauty in a piece of mathematics is of frequent occurrence, and it is a strong motivation for further mathematical research. Lack of beauty is associated with lack of definitiveness. A beautiful proof is more often than not the definitive proof (though a definitive proof need not be beautiful); a beautiful theorem is not likely to be improved upon or generalized." (Gian-Carlo Rota, "The phenomenology of mathematical proof", Synthese, 111(2), 1997)

"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. […] A beautiful theorem may not be blessed with an equally beautiful proof; beautiful theorems with ugly proofs frequently occur. When a beautiful theorem is missing a beautiful proof, attempts are made by mathematicians to provide new proofs that will match the beauty of the theorem, with varying success. It is, however, impossible to find beautiful proofs of theorems that are not beautiful." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", Synthese, 111(2), 1997)