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Ladies & Gentlemen: Skip To Vol 2 p. 124

~~As a specimen of mathematical logic in action, I am going to tell you this evening something about the new doctrine of ~~ __maniness__, or __multitude__.

But I must hasten to the subject of numbers. __Whole__ numbers can on the one hand be studied in two ways which are surprisingly different from one another throughout. They can be studied as qualities of collections, making the members of one collection many and those of another few, which is called by the Germans with their usual incapacity for language, the doctrine of __Cardinal Numbers__; but which ought to be called the doctrine of __Multitude__. Or, on the other hand, numbers may be considered simply as objects in a sequence, as __ordinal numbers__. The latter study is a branch of pure mathematics

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-matics, because it makes no difference what kind of objects they are that are in series, nor whether it is a series in time, in space, or in logic. The doctrine of multitude, on the other hand, is not pure mathematics. For the objects it studies, the multitudes are in a linear series exactly as the doctrine of ordinal numbers supposes; and since the doctrine of ordinal numbers permits the members of the series to be objects of any kind, it follows that it permits them to be multitudes. Thus the doctrine of multitude is nothing but a special application of the doctrine of ordinal numbers. But the special objects of its series have a special character which permits them to be studied from a special point of view; and that point of view is a logical point of view. It is not the pure mathematical forms