Tijdschrift Voor Filosofie 46 (4):611 - 642 (1984)

Philosophers of science don't very often discuss the place of mathematics between other sciences or the meaning of mathematics for other sciences. They consider mathematics as a formal language with mainly analytical statements about the use of symbols (Carnap, Russell, Ayer ). Originally Wittgenstein defended this formalistic interpretation of mathematics in his TLP. Gradually, however, he develops himself towards an intuitionistic and ontological position, in which mathematics is conceived as the central and therefore normative part of our thought (of course : on what there is and how it is). Mathematical science plays the role of logic in relation to other sciences. Its universal applicability and efficiency are consequences of its creating beings on a necessary level, in virtue of the number of its relations (always still by substitution). This highly important philosophy of mathematics (misinterpreted by Crispin Wright) starts with his lectures in Cambridge (in the thirties ) and reaches its culmination in the Remarks on the Foundations of Mathematics and in On Certainty. In a second part this philosophical determination of mathematical reasoning is traced backwards through history. David Hume's contribution is reinterpreted from a new point of view. Inside the total field of our beliefs he distinguishes between different sciences with the critérium of the intricacy of relations between items of our knowledge field. The more and stronger these relations, the more forceful and necessary their influence on the remaining parts of the system of our belief. So mathematics is in the centre, the loose reveries of our fancy on the periphery. Quine's representation of ‘the tribunal of sense experience’, by which the total field should be judged and corrected, must be disqualified. Hume's dictum ‘Whatever we conceive, we conceive it to be existent’ reveals sharply that this evaluative and corrective role is performed by the necessary thoughts (or, if one likes it so, ‘realities’) of mathematical science. That reason and especially mathematical reason is the highest judge on the population and structure of our world and a very precious heritage of Pythagorism and Platonism. From the sources of Sextus Empiricus and Aristotle the author tries to reconstruct exactly the original assertion of Pythagoristic mathematical philosophy, which has nothing to do with a naive hypostazation of numbers or a kabbalistic number mysticism. Philolaos' saying, that some propositions are stronger than we, is demonstrated to refer to mathematical laws. The pythagorical position is fully integrated in Plato's dialectical philosophy. Mathematics is the great mediator towards the intuition of true being, the ‘metaxu’ between sensible phenomena and ideas. This tradition of philosophical taxation of mathematics as the ‘logic of science’ is broken by Aristotle, who didn't use mathematics in his qualitative natural science and considered mathematics as an abstract science (about the quantitative aspect of being). Moreover, he disowned its logical role and created a special science for this task. Human reason is mathematical in so far it is sure of its language and thought, which is excellently expressed by the Greek μαθηματιxα (= what can be understood, learned and taught) and by the Dutch word ‘wiskunde’ (= science of what is certain). The remarks and reflections of Wittgenstein have produced a new perspective on the placevalue (‘Stellenwert’) of mathematics among all possible sciences and beliefs and have proven that its onto-logical purport is an unavoidable implication
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