Conceptions of Inquiry by Stuart Brown

By Stuart Brown

First released in 1981. Routledge is an imprint of Taylor & Francis, an informa corporation.

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Boole, George (1854) An Investigation into the Laws of Thought, London, Walton & Maberly. 2. N. W. (1910) Principia Mathematica, Cambridge, Cambridge University Press. 4 MATHEMATICS AS LANGUAGE LANCELOT HOGBEN (1895–1975), from Mathematics for the Million, London, George Allen & Unwin, 1936, pp. 26–8 To describe mathematics as a language is sometimes just a way of speaking, with no further implications for mathematical inquiry. When Galileo, for example, said in quite a detailed metaphor: Philosophy is written in this grand book, the universe, which stands continually open to our gaze.

Note 1. Op. , pp. vii–viii. No sharp distinction can be drawn between mathematical theories which apply to external objects, and mathematical inventions which are interesting only in themselves, for there is always a possibility that a mathematical theorem may prove applicable to experience some time. Yet the fact that this is not necessarily true, and indeed appears very unlikely for the far greater part of mathematics, is a distinctive feature of this science. Not being primarily concerned with foretelling what is going to happen, or with contriving what anyone wished to happen, but merely with understanding exactly how alternative aspects of a certain set of conceptions are logically connected, mathematics can extend its subject matter indefinitely by conceiving new problems of this sort, without any reference to experience.

Gottlob Frege in Jena and Giuseppe Peano in Turin worked on combining formal reasoning with the study of sets and numbers. David Hilbert in Göttingen worked on stricter formalizations of geometry than Euclid’s. All of these efforts were directed towards clarifying what one means by ‘proof’. In the meantime, interesting developments were taking place in classical mathematics. A theory of different types of infinities, known as the theory of sets, was developed by Georg Cantor in the 1880s. The theory was powerful and beautiful, but intuition-defying.

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