A History of Mathematics From Mesopotamia to Modernity

206 A History ofMathematics I discovered [about 1890] that in addition to Euclidean geometry there were various non-Euclidean ge ...

Geometries andSpace 207 a strong belief that the way to make mathematics ‘safe’ was through the construction of axiom systems. I ...

208 A History ofMathematics BC E A F D G Fig. 11The picture for Euclid I.16. B C D E A F Fig. 12A ‘large’ triangle on a sphere, ...

Geometries andSpace 209 C A B R R b O c a Fig. 13Solving a spherical triangle. Appendix B. The formulae of spherical and hyperbo ...

210 A History ofMathematics Appendix C. From Helmholtz’s 1876 paper [Reproduced in (1979, pp. 249–50. Helmholtz, to simplify his ...

Geometries andSpace 211 P AQ BC Fig. 14Proof of the ‘angles of a triangle’ theorem. AB CD Fig. 15Figure for Exercise 1(b). B C A ...

212 A History ofMathematics C B Π(a) a Fig. 17Figure for Exercise 7. ∠D′A′F′=∠DAF. (This is where it is essential that the rules ...

9. Modernity and its anxieties 1. Introduction If in summing up a brief phrase is called for that characterizes the life center ...

214 A History ofMathematics tell two knots apart, was their business; one can speculate on how this related to what was going on ...

Modernity and itsAnxieties 215 Mathematicians can only feel flattered by William Everdell’s breathless placing of them as the fo ...

216 A History ofMathematics In Dedekind’s definition, one takes √ 2 to be the cut. This might be considered slightly vague too, ...

Modernity and itsAnxieties 217 and philosophy, and in mathematics it meant a return to doubt, to the search for what was wrong, ...

218 A History ofMathematics work needed to be done on the axioms. There were some disagreements about what should be done, but C ...

Modernity and itsAnxieties 219 to Euclid’s postulate 5: no one liked it, but it became clear (and Zermelo pointed out) that a gr ...

220 A History ofMathematics C D f(A) A g(A) Fig. 2Brouwer fixed point theorem. We have therefore a contradiction from supposing ...

Modernity and itsAnxieties 221 precision and abstraction were seen as impoverished forms of thought, associated with Germany’s d ...

222 A History ofMathematics one which with hindsight is more ambitious than anyone before or since has thought necessary or poss ...

Modernity and itsAnxieties 223 the ruins: Sitting next to the Nazis’ newly appointed minister of education at a banquet, he was ...

224 A History ofMathematics Fig. 3Circle, torus, and sphere. Fig. 4A torus and a knotted torus (which, incidentally are linked), ...

Modernity and itsAnxieties 225 P a a a e f b b b c de c d e g h i j k k f g i k g I II II III IV VI V III IV V VI h i k c d QQR ...