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Extra resources for Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle
Or is mathematics merely a local culture which is propagated in our civilisation more or less in the manner of natural languages? Is there any sense in which anything in mathematics can be said to be certainly true in an absolute sense? Or is all mathematics merely a socially deﬁned behaviour? To what extent is our mathematics a consequence of the peculiar capabilities of the human brain? Would a more advanced species use a totally diﬀerent (and superior) mathematics? Since the real number system seems to be a consequence of our system of physical measurements, what is the signiﬁcance of the absurdly large inﬁnity of elements of the set of real numbers?
7 Remark: The network of mathematical concepts requires coherence. Although the network of mathematical concepts cannot be deﬁned in the sense of deriving all concepts from other concepts in an acyclic manner, the concept network can at least be coherent. 1. Coherence is not the same as logical self-consistency. The latter means that the concept network is tested with respect to a deductive logic framework which is established external to the network. In the situation considered here, the logical framework is part of the concept network which is being tested.
Philosophical considerations The idea that all imperfect physical-world circles are striving towards a single Ideal circle form in a perfect Form-world is quite seductive. 19. Should combine or at least collocate them. 2 Remark: A plausible argument in favour of a Platonic-style ontology for mathematics. A plausible argument may be made in favour of Platonic ontology for the integers in the following form. (1) Numbers written on paper must refer to something. (2) Numbers do not correspond exactly to anything in the sensible world.