Pi on Earth, or Mathematics in the Real World

Abstract

We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one's goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one's means/ends ratio. Our story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like.

Identifying logical evidence

“…One cannot be expected to make sound conclusions about how we come to be justified in believing scientific theories without taking notice of how scientists go about justifying their own theories (and indeed go about experimenting generally). The same can be said for mathematics, and the last 20 years has seen a far greater appreciation of mathematical epistemology by looking in detail at the practice of working mathematicians [see, for example, Giaquinto (2007), Van Bendegem (2003), and Van Kerkhove and Van Bendegem (2008]. The assumption underlying all of these uses of a practice-based approach to the epistemology of a particular field is that scientists and mathematicians are generally very good at recognising what constitutes suitable evidence for a theory within their field, and what the plausible means are to justify a proposition or theory.…”

Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups

“…rivalrous and excludable) good that can form the basis for an incentive structure that motivates scientists, leading to them working harder and dividing up their efforts across different problems and methodologies (Merton 1957, Zollman 2018. The priority rule for publications is written into scientific history by discoveries which are named 1 Fallis (1997) and Van Kerkhove and Van Bendegem (2008) have footnotes on the Classification Theorem, but don't discuss it in depth. Fallis (2003) and Johansen and Misfeldt (2016) mention it as a potential parallel to the problems raised by the Four Colour Theorem.…”

“…We believe that, although they have been overlooked, massively collaborative efforts such as the Classification of Finite Simple Groups pose similar challenges to the core properties of proof, particularly to the way these properties have been framed in individualistic terms. 1 1 Fallis (1997) and Van Kerkhove and Van Bendegem (2008) have footnotes on the Classification Theorem, but don't discuss it in depth. Fallis (2003) and Johansen and Misfeldt (2016) mention it as a potential parallel to the problems raised by the Four Colour Theorem.…”

“… 1 Fallis (1997) and Van Kerkhove and Van Bendegem (2008) have footnotes on the Classification Theorem, but don't discuss it in depth. Fallis (2003) and Johansen and Misfeldt (2016) mention it as a potential parallel to the problems raised by the Four Colour Theorem.…”

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

“…A complementary approach which does take the practice of computer-assisted mathematics more seriously seems necessary in order to get a more balanced account of the impact of the computer on (the philosophy of) mathematics. This has already been argued to some extent by (van Kerkhove and van Bendegem 2008) where it is stated that we should account for the practices underpinning formal proofs, including the use of experimental methods (Idem, p. 434):…”