Observations, problems and conjectures in number theory – the history of the prime number theorem. (English) Zbl 0848.00008

Echeverria, Javier

Echeverria, Javier (ed.) et al., The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. Grundlagen der Kommunikation und Kognition. 230-252 (1992).

Rudolf Carnap’s distinction between formal sciences and real sciences separated mathematics (and logic) strictly from the rest of sciences. This distinction and the doctrine of the a priori character of mathematical knowledge was criticized by Quine, Putnam, Lakatos and others. Against mathematical apriorism they maintained the connection of mathematics with empirical, or quasi-empirical, methods. Even if this connection is conceded, it deserves clarification, that conclusions from the mathematicians’ heuristics, i.e., from the way they choose to search for results, will hardly refute the epistemological thesis of mathematical apriorism. The author does not fall into the traps of overestimating the epistemological significance of mathematical practice, as becomes clear, however, only from the last paragraph where he writes that he does not “argue in favour of the existence of an empirical basis in mathematics” (p. 252).
Concerning “experimental methods” in mathematics and the quasi-empirical character of mathematical procedures, the author provides an instructive example from number theory making it plausible that one who wants to know how mathematics goes on, should not restrict his interest to the analysis of mathematical ideas and results, but should also take into account the ways of doing mathematics.
The author discusses in some detail the history of the prime number theorem from Euler up to its elementary proofs in the mid 20th century. He especially presents the use of prime number tables. He maintains that these tables “play the same epistemological rôle in NT [number theory] as empirical measures and data in experimental sciences” (p. 250). [One should qualify: not really the same rôle, because empirical measures and data constitute the empirical base of experimental sciences, a base which is not maintained for mathematics].
For the entire collection see [Zbl 0839.00019].

Reviewer: V.Peckhaus (Erlangen)

MSC:

00A30	Philosophy of mathematics
11-03	History of number theory
01A45	History of mathematics in the 17th century
01A50	History of mathematics in the 18th century
01A55	History of mathematics in the 19th century
01A60	History of mathematics in the 20th century
11A41	Primes

Keywords:

empirical methods; experimental methods; number theory; prime number theorem; prime number tables