The Proof Is in the Process: A Preamble for a Philosophy of Computer-Assisted Mathematics

Abstract

Mechanization tends to emphasize practice rather than theory, deeds rather than words, explicit answers rather than existence statements, definitions that are formalized rather than behavioristic, local rather than global phenomena, the limited rather than the infinite, the concrete rather than the abstract, and one could almost say, the scientific rather than the artistic.

Impact of Informatics on Mathematics and Its Teaching

“…An algorithm is a problem solving procedure, which can be applied on a set of instances of the problem and that produce, in a finite number of constructive, unambiguous and organised steps, the answer to the problem for any of those instances. 5 The Theory of Complexity (see [9] for instance) defines a problem as a couple (I, Q) where I is a set of instances and Q a question that can concern any instance of I 6 . An algorithm A solves a problem P = (I, Q) if for any instance of I, A give an right answer to Q after a finite number of steps.…”

“…Experimental mathematics and role of the computer Informatics allowed to develop or renew experimental aspects of mathematics [2,22]. From an epistemological point of view, to a certain extend, the nature of mathematics can be considered as unchanged with computer-assisted mathematics, but works like [5,6] show that there are changes, due to the use of computer and influence of informatics, that must be considered in the practice of the mathematics. Hamming [10] illustrate well this point:…”

Some Reflections on Mathematics and Its Relation to Computer Science

“…What particular characteristics are there to the computer that help to understand or clarify its (potential) effect on mathematics? In [13] I discuss three features which I consider as characteristic to computer-assisted explorative mathematics: human-computer interaction; the significance of time and processes and the internalization of mathematics into the machine.…”

“…I discuss two of them in more detail here. 13 First of all, with current programming environments, much of the algorithms being used are no longer explicitly programmed by the mathematician who uses them but are instead called by their name. This means that part of the knowledge that is used within computer-assisted mathematics is unknown to the mathematician unless he/she has bothered him/herself with looking at the details of the subroutines called by the main program.…”