Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

420 alasdair urquhartand Paul Lévy, whom they persecuted because of their lack of rigour. In reply,Saunders Mac Lane dismisses Mandelbrot as ‘a notorious non-mathematician’(Mac Lane, 1997, p.149).I shall not discuss the problem of demarcation of mathematics from otherareas. It is clear that there is a great deal of activity in areas of science thatdo not fall within the restricted boundaries of rigorous mathematics, but arein some sense mathematical; the work of Mandelbrot certainly falls into thiscategory, as does the seminal work of Mitchell Feigenbaum in the area ofdynamical systems. Rather, I shall take for granted that there is a more or lessclear distinction to be made between fully rigorous work, and work that islacking rigour, even though it may be broadly mathematical in some sense.Instead, the topic that I shall discuss in the remainder of this essay is the processwhereby the more speculative parts of mathematical work are incorporatedinto the body of rigorous research.16.2 Proof in physics and mathematicsIf we define a proof as something that conveys conviction that a given assertionis true, then it is obvious that proofs do not have to obey the laws of logic orclassical mathematics. In this section, we address the question of what is meantby a proof, as it appears to mathematicians and physicists. The combinatorialistX. G. Viennot sets the scene in the following quotation:Usually, in the physics literature, the distinction between establishing a formulafrom computer experiments and giving a mathematical proof is not clearlystated ... When the number of known elements of the sequence is much biggerthan the number of elements needed to guess the exact formula, the probabilitythat the formula is wrong is infinitesimal, and thus the formula can be consideredas an experimentally true statement. But mathematically, it is just a conjecturewaiting for a proof, and maybe more: a crystal-clear [bijective] understanding.(Viennot, 1992, p.412)As a result of the evolution of mathematics in the 20th century, the notionof mathematical proof is quite standardized. In a rigorous mathematical argument,all objects are to be defined as set-theoretical objects on the basis ofZermelo–Fraenkel set theory, and their fundamental properties are to beestablished on the basis of the usual set-theoretical axioms. The resultingconsensus on the basis of mathematics gives contemporary mathematics considerableunity and cohesion. The fact that from time to time, mathematicianshave advocated alternative foundations such as category theory does not really