Exploring University Mathematics. Lectures Given at Bedford by D. M. Burley, J. S. Griffith, J. H. E. Cohn and N. J.

By D. M. Burley, J. S. Griffith, J. H. E. Cohn and N. J. Hardiman (Auth.)

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Further we suppose that the assignment of a time t2 to the event is by a rule which determines t2 uniquely, from tx and t3. This is expressed by Assumption!. t2=f(tut3). t G. J. Whitrow, The Natural Philosophy of Time. 52 EXPLORING UNIVERSITY MATHEMATICS 2 Here the mathematical notation simply implies the existence of a rule by which, when tx and t3 are given, t2 is specified. The problem, which we may call Einstein's problem, is to determine exactly what this rule is, or rather to formulate FIG.

We shall use the symbol (a, /;), called a "pair", to denote the solution, whatever it is, and try to find the sort of rules that pairs must obey. + / ) . -l·/ clearly a + d=a + b + \=b + a + \=b + c, and, conversely, if a + d= b + c, (E) NUMBERS MADE TO MEASURE 29 we have c= a+(d-bU if d>b, if d^b, so that d — b plays the part of /, and a= c+(b-d) b= d+{b-d) so that we have a = c-r m\ b = d+m\ with m = b — d. Note that equality of pairs as defined by (E) has three impor­ tant properties, without which it would not be very useful.

We want to solve the equation x + b = a, where a and b are natural numbers. If a ^ b then x = a — b is 28 EXPLORING UNIVERSITY MATHEMATICS 2 a natural number and solves the equation. If a < b there is no natural number which solves the equation. We need new numbers, and we shall "make them to measure". § 2. Whatever sort of animals these new numbers are going to be, the one which solves x+b = a in the case b > a must obviously depend on a and b. We shall use the symbol (a, /;), called a "pair", to denote the solution, whatever it is, and try to find the sort of rules that pairs must obey.

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