A History of Mathematics From Mesopotamia to Modernity

Modernity and itsAnxieties 231

The idea ofstructureunified two apparently disparate areas of mathematics. It was to become more
and more important through those who drew inspiration from her writings.
As we have suggested earlier, Noether is too unusual a figure to be in any sense ‘typical’ in
the short list of women in mathematics. Women emerge to take a place as major figures in
this history almost at its last moment. And while one can and should find space for the skilled
amateur Ada Byron, Countess of Lovelace as a ‘joint forerunner’ of the computer, or for Sofia
Kovalevskaya as a high-quality contributor to nineteenth-century analysis, the place of Noether,
like that of Ramanujan, goes well beyond the category of prizes awarded in a special outsiders’
category. Without her, the drive to abstraction which we shall chart later might well have developed
unstoppably, but her work of the 1920s certainly set a particular shape on it.

Appendix A. The cut definition

(From ‘Continuity and irrational Numbers’ reproduced in Fauvel and Gray pp. 575–6.)

From the last remarks it is sufficiently obvious how the discontinuous domain R of rational numbers

may be rendered complete so as to form a continuous domain. [Earlier] it was pointed out that every

rational numberaeffects a separation of the system R into two classes such that every numbera 1

of the first classA 1 is less than every numbera 2 of the second classA 2 ; the numberais either the

greatest number of the classA 1 or the least number of the classA 2. If now any separation of the

systemRinto two classesA 1 ,A 2 , is given which possesses onlythischaracteristic property that every

numbera 1 inA 1 is less than every numbera 2 inA 2 , then for brevity we shall call such a separation

acutand designate it by(A 1 ,A 2 ). We can then say that every rational number produces one cut

or, strictly speaking, two cuts, which, however, we shall not look upon as essentially different; this

cut possesses,besides, the property that either among the members of the first class there exists

a greatest or among the numbers of the second class a least number. And conversely, if a cut

possesses this property, then it is produced by this greatest or least rational number.

But it is easy to see that there are infinitely many cuts not produced by rational numbers.

Appendix B. Intuitionism

(Weyl, in Mancosu 1998, p. 97)

(i) [Weyl] Let us, for example, assume that ‘nhas the propertyE’ means that 2^2

n+ 4

+1 is a prime

number, and that propertyE ̄means the opposite (2^2

n+ 4

+1 is a composite number). Now consider

the following. The view that it is in itself determined whether there is a number with propertyE,or

not, is surely based on the following idea: The numbers 1, 2, 3,...may be tested, one by one, for the

propertyE. If such a number with propertyEis found, the answer isyes. But if such a termination

does not occur, that is to say, after acompletedrunthrough the infinite number sequence, no number

of kindEis found, then the answer isno. Yet this point of view of a completed run through an

infinite sequence is nonsensical.

(ii) [Brouwer] 1. The Axiom of Comprehension, on the basis of which all things with a certain

property are joined into a set...is not acceptable and cannot be used as a foundation of set theory.

A reliable foundation is only to be found in aconstructivedefinition of a set. 2. The axiom of the