A History of Mathematics From Mesopotamia to Modernity

Modernity and itsAnxieties 217

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

what Morris Kline many years later was to call the ‘loss of certainty’ [Kline 1980].

Exercise1.Checkthattherelation

√

2.

√

3 =

√

6 doesindeedfollowfromthecutdefinitionbyshowing

(a) that there is a unique positive real number

√

n whose square is n, for any integer n> 0 ;

(b) that if we define a.b for positive real numbers in the obvious way—you may need to take a little care

in doing this—

√

2.

√

3 =

√

6 ;

(c) (Not as difficult as it looks...) Let x 1 ,x 2 ,x 3 ,...be a sequence of real numbers (defined by cuts)

which is bounded above, that is, there exists M such that xi<M for all i. Prove that there exists M 0

(least upper bound) such that:

(1) xi≤M 0 for all i;

(2) if y<M 0 , then for some i,xi>y.

(d) Define a real number to be an infinite decimal, that is, a series of type x=a +.a 1 a 2 a 3 ..., where

a is an integer and the ais are numbers between 0 and 9. In other words, x is the sum of the series

a+

a 1

10

+

a 2

102

+

a 3

103

+···

What problems arise in devising a rule for adding such numbers?

4. Crisis—what crisis?

It seemed unworthy of a grown man to spend his time on such trivialities, but what was I to do? There was something

wrong, since such contradictions were unavoidable on ordinary premisses. (Russell 1967, p. 147)

In [this] light, mathematics appears as a monstrous ‘paper economy’. Real value, comparable to foods in economics, is

only possessed by the singular, the quintessentially singular. Everything general, and all existential statements partake

in it only indirectly. And yet we, as mathematicians, very seldom consider the redemption of this ‘paper money’! It

is not the existential theorem that is the treasure, but the construction carried out in the proof. (Weyl, ‘On the New

Foundational Crisis in Mathematics’ (1921), in Mancosu 1998, p. 98)

The behaviour of many leading mathematicians in the years 1900–1930 is so uncharacteristic
that the reader may feel more in tune with the mathematical aims of the Chinese than with those
of Hilbert, Brouwer, Russell, and their contemporaries. Why was the mathematical enterprise
suddenly seen as so insecure? How did it come about that mathematicians were bitterly divided into
competing schools of thought, who went so far as to call each other ‘Bolshevist’ (Hilbert against
Brouwer and Weyl) or ‘non-Aryan’ (Brouwer against his opponents); and to fight about presence
at conferences and editorship of journals? The year 1900 saw Hilbert’s calm summing-up of the
progress of mathematics, and his famous list of problems awaiting solution in the new century. His
mood was optimistic:

This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within

us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there

is noignorabimus. (Hilbert 1900)

Still, it should be noted that the first two problems deal with foundations: Cantor’s ‘continuum hypo-

thesis’ and the consistency of the axioms for arithmetic (however defined). Already, it appeared,