A History of Mathematics- From Mesopotamia to Modernity

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 ...

226 A History ofMathematics Around 1910, Dehn and Wirtinger were aware of the tables of knots (or knot-projections) compiled by ...

Modernity and itsAnxieties 227 Type I Reidemeister move Type II Reidemeister move Type III Reidemeister move Fig. 8The three Rei ...

228 A History ofMathematics 1 2 4 3 Fig. 9Two knots, each with four crossings. The crossings on knot 1 are labelled 1, 2, 3, 4 ( ...

Modernity and itsAnxieties 229 To take a ‘proper’ historical viewpoint one needs to have some understanding of what his work mea ...