En gren av vetenskapsfilosofin kallas matematikens filosofi och berör matematikens karaktär och grunder. Jag har just läst ett bokkapitel* som sammanfattar hur huvudfåran i matematikens filosofi tänker och som också ger en kritik och förslag på ett annat synsätt. Jag misstänker att fler än jag kan finna de 13 kontrasteringarna upplysande och stimulerande:
|Huvudfårans synsätt||Celluccis alternativa synsätt|
|1. [T]he reflection on mathematics is the task of a specialized discipline, the philosophy of mathematics, starting with Frege, characterized by its own problems and methods, and in a sense “the easiest part of philosophy”.||1′. [E]ntrusting reflection on mathematics to a specialized discipline poses serious limitations, because one cannot assume that philosophical problems occur in mathematics in an especially pure, or especially simplified, form.|
|2. [T]he main problem in the philosophy of mathematics is the justification of mathematics.||2′. [T]he main problem in the reflection on mathematics is discovery.|
|3. [A]nother important problem in the philosophy of mathematics is the existence of mathematical objects.||3′. [T]he problem of the existence of mathematical objects is irrelevant to mathematics because, as Locke pointed out, “all the discourses of the mathematicians about the squaring of a circle” – or any other geometrical figure – “concern not the existence of any of those figures”, and their proofs “are the same whether there be any square or circle existing the world, or no”.|
|4. [T]he philosophy of mathematics does not add to mathematics.||4′. [T]he reflection on mathematics is relevant to the progress of mathematics.|
|5. [T]he philosophy of mathematics does not require any detailed knowledge of mathematics, because its main aim – the justification of mathematics through a clarification of its foundations – does not require any detailed knowledge of the edifice built up on such foundations.||5′. [T]he reflection on mathematics does in fact require detailed knowledge of mathematics.|
|6. [M]athematics is theorem proving because it “is a collection of proofs”.||6′. [M]athematics is problem solving.|
|7. [T]he method of mathematics is the axiomatic method.||7′. [T]he method of mathematics is the analytic method, a method which, unlike the axiomatic method, does not start from axioms which are given once and for all and are used to prove any theorem, nor does it proceed forwards from axioms to theorems, but proceeds backwards from problems to hypotheses.|
|8. [T]he logic of mathematics is deductive logic.||8′. [T]he logic of mathematics is not deductive logic but a broader logic, dealing with non-deductive (inductive, analogical, metaphorical, metonymical, etc.) inferences in addition to deductive inferences.|
|9. [M]athematical discovery is an irrational process based on intuition, not on logic.||9′. [M]athematics is a rational activity at any stage, including the most important one: discovery.|
|10. [I]n addition to mathematical discovery, mathematical justification too is based on intuition.||10′. [J]ustification is not based on intuition but on the fact that the hypotheses used in mathematics are plausible, i.e., compatible with the existing knowledge, in the sense that, if one compares the reasons for and against the hypotheses, the reasons for prevail.|
|11. [M]athematics is a body of truths – indeed a body of absolutely certain and hence irrefutable truths.||11′. [M]athematics is a body of knowledge but contains no truths.|
|12. [T]he question of the applicability of mathematics to the physical sciences is inessential for the philosophy of mathematics.||12′. [T]he question of the applicability of mathematics to the physical sciences is important for the reflection on mathematics.|
|13. [M]athematics is based only on conceptual thought.||13′. [M]athematics is based not only on conceptual thought but also on perception, which plays an important role in it, for example, in diagrams.|
Alla punkterna utvecklas i bokkapitlet. Slutsats:
The arguments sketched above provide reasons for rejecting the dominant view. In short, the rejection is motivated by the fact that the dominant view does not explain how mathematical problems arise and are solved. Rather, it presents mathematics as an artificial construction, which does not reflect its important aspects, and omits those features which make mathematics a vital discipline. Thus the dominant view does not account for the richness, multifariousness, dynamism and flexibility of mathematical experience.
Övertygas du av Celluccis argumentation? Själv är jag osäker men fann den i vilket fall riktigt underhållande.
*Cellucci, Carlo (2006). ”‘Introduction’ to Filosofia e matematica.” I Hirsch, Reuben (red.), 18 Unconventional Essays on the Nature of Mathematics. Berlin: Springer.