Abstract
Abstract Non-Euclidean geometry has occupied a very important position in the discussion about revolutions in mathematics, as the two opposite sides have both used it as a typical case for their views. In Section 9.1, I shall first briefly analyse the discussion so far, and then make my own position clear. In Section 9.2, I shall argue for the revolutionary nature of non-Euclidean geometry, and on this basis, a further analysis of revolutions in mathematics will be given. In Section 9.3, I shall discuss the significance of non-Euclidean geometry and its bearing on problems of the nature of mathematical truth and modes of thought. I shall then present a theory of types of mathematical truth and ‘the harmonious principle of the counter-way thinking’.
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