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Knowledge of Number

A weekly seminar for research students with Keith Hossack, to discuss draft chapters of Keith’s new book.

Thursdays in Term 2, Room 501 30 Russell Square, 1430h-1630h.

Topic If humans were wholly physical beings, how could we apprehend the abstract realm of mathematics? Plato concluded it is the human soul that knows mathematics; Field contraposed to argue that pure maths is pure fiction. Knowledge of Number enters the debate by saying that mathematics is a natural science which studies the laws governing numerical magnitudes, just as physics studies the laws governing magnitudes such as mass, length and time. Aristotle distinguished ‘discrete’ quantity, i.e., multitude or plurality, from ‘continuous’ quantity, i.e., a continuum: to these must be added a third species, serial quantity or series. These three species correspond to the three known species of numbers, i.e., the natural numbers, the reals and the ordinals. When we generalise first-order logic to handle the logic of quantity, we obtain three new logics  – plural logic, mereology (the logic of continua) and serial logic. The properties of the three species of numbers can then all be derived from the laws of the generalised logics. The theory of universals is the background metaphysics, and the theory of a priori knowledge is the background epistemology.

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