Authors: Thomas Colignatus
> Context • In the philosophy of mathematics there is the distinction between platonism (realism), formalism, and constructivism. There seems to be no distinguishing or decisive experiment to determine which approach is best according to non-trivial and self-evident criteria. As an alternative approach it is suggested here that philosophy finds a sounding board in the didactics of mathematics rather than mathematics itself. Philosophers can go astray when they don’t realise the distinction between mathematics (possibly pure modeling) and the didactics of mathematics (an empirical science). The approach also requires that the didactics of mathematics is cleansed of its current errors. Mathematicians are trained for abstract thought but in class they meet with real world students. Traditional mathematicians resolve their cognitive dissonance by relying on tradition. That tradition however is not targetted at didactic clarity and empirical relevance with respect to psychology. The mathematical curriculum is a mess. Mathematical education requires a (constructivist) re-engineering. Better mathematical concepts will also be crucial in other areas, such as e.g. brain research. > Problem • Aristotle distinguished between potential and actual infinite, Cantor proposed the transfinites, and Occam would want to reject those transfinites if they aren’t really necessary. My book “A Logic of Exceptions” already refuted ‘the’ general proof of Cantor's Conjecture on the power set, so that the latter holds only for finite sets but not for ‘any’ set. There still remains Cantor’s diagonal argument on the real numbers. > Results • There is a bijection by abstraction between N and R. Potential and actual infinity are two faces of the same coin. Potential infinity associates with counting, actual infinity with the continuum, but they would be ‘equally large’. The notion of a limit in R cannot be defined independently from the construction of R itself. Occam’s razor eliminates Cantor’s transfinites. > Constructivist content • Constructive steps S1, ..., S5 are identified while S6 gives non-constructivism (possibly the transfinites). Here S3 gives potential infinity and S4 actual infinity. The latter is taken as ‘proper constructivism with abstraction'. The confusions about S6 derive rather from logic than from infinity.
Comments: 2 Pages. The paper refers to the book FMNAI that supersedes the paper
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