On Rigor (1990-1999)

"Regarding stability, the state trajectories of a system tend to equilibrium. In the simplest case they converge to one point (or different points from different initial states), more commonly to one (or several, according to initial state) fixed point or limit cycle(s) or even torus(es) of characteristic equilibrial behaviour. All this is, in a rigorous sense, contingent upon describing a potential, as a special summation of the multitude of forces acting upon the state in question, and finding the fixed points, cycles, etc., to be minima of the potential function. It is often more convenient to use the equivalent jargon of 'attractors' so that the state of a system is 'attracted' to an equilibrial behaviour. In any case, once in equilibrial conditions, the system returns to its limit, equilibrial behaviour after small, arbitrary, and random perturbations." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof – which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane, "Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"Many pages have been expended on polemics in favor of rigor over intuition, or of intuition over rigor. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigor." (Ian Stewart,"Concepts of Modern Mathematics", 1995)

"Scientists reach their conclusions for the damnedest of reasons: intuition, guesses, redirections after wild-goose chases, all combing with a dollop of rigorous observation and logical reasoning to be sure […] This messy and personal side of science should not be disparaged, or covered up, by scientists for two major reasons. First, scientists should proudly show this human face to display their kinship with all other modes of creative human thought […] Second, while biases and references often impede understanding, these mental idiosyncrasies may also serve as powerful, if quirky and personal, guides to solutions. (Stephen J Gould, "Dinosaur in a Haystack: Reflections in natural history", 1995)

"Factoring big numbers is a strange kind of mathematics that closely resembles the experimental sciences, where nature has the last and definitive word. […] as with the experimental sciences, both rigorous and heuristic analyses can be valuable in understanding the subject and moving it forward. And, as with the experimental sciences, there is sometimes a tension between pure and applied practitioners." (Carl B Pomerance, "A Tale of Two Sieves", The Notices of the American Mathematical Society 43, 1996)

"Empirical evidence can never establish mathematical existence - nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful." (Richard Courant, "The Parsimonious Universe, Stefan Hildebrandt & Anthony Tromba", 1996)

"Factoring big numbers is a strange kind of mathematics that closely resembles the experimental sciences, where nature has the last and definitive word. […] as with the experimental sciences, both rigorous and heuristic analyses can be valuable in understanding the subject and moving it forward. And, as with the experimental sciences, there is sometimes a tension between pure and applied practitioners. (Carl B Pomerance, "A Tale of Two Sieves", The Notices of the American Mathematical Society 43, 1996)

"The reason why a 'crude', experimental approach is not adequate for determining mathematical truth lies in the nature of what mathematics is and is intended to be. Though its roots lie in the physical world, mathematics is a precise and idealized discipline. The 'points', 'lines', 'planes', and other ideal constructs of mathematics have no exact counterpart in reality. What the mathematician does is to take a totally abstract, idealized view of the world, and reason with his abstractions in an entirely precise and rigorous fashion. (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"But as often happens with rigorous theorems in physics, the more serious the conclusions which follow from proven assertions, the more carefully one must examine the initial premises." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it’s not that one approach is right and the other wrong – the methods one chooses to use are largely a matter of taste and training." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)