Talk:Mathematics/Sources

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Popper on Mathematics
Russel's theory of reduction, that is, the theory that mathematics can be reduced to logic, is to be rejected. Mathematics cannot be completely reduced to logic; in fact, it has even led to a considerable refinement in logic and, it may be said, to a critical correction of logic: to a critical correction of our logical intuition and to the critical insight that our logical intuition is not all that reliable. On the other hand, it has also shown that intuition is very important and capable of development. The majority of creative ideas come about througth intuition; and those that do not are the result of the critical refutation of intuitive ideas.

There does not seem to be one system of fundamental principles of mathematics, but different methods of constructing mathematics or the different branches of mathematics. I say 'constructing' and not 'establishing', since there seems to be no ultimate establishment or safeguard for its fundamental principles. Moreover, we can prove the consistency of our construction only in the case of weak systems. And we know from Tarski that important branches of mathematics are fundamentally incomplete, that is to say, these systems may be strengthened, but never to the extent that we can prove within them all true and relevant statements. Most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.

Gödel and Cohen also succeeded in furnishing proofs that the so-called continuum hypothesis can neither be refuted nor proved with the methods of set theory employed so far. This famous hypothesis, which Cantor and Hilber hoped to prove one day, was shown to be independent of current theory. Of course it is possible so to strengthen the theory (by using addictional assumptions) that the hypothesis becomes demonstrabel; but it is equally possible so to strengthen it that the hypothesis can be refuted.