Bibliography

Lowell Lectures. 1903. Lecture 3

A Collection is anything whose being consists in the existence of whatever there may exist that has any one quality; and if such thing or things exist, the collection is a single thing whose existence consists in the existence of all those very things.

According to this definition, a collection is an ens rationis. [—] A collection has essence and may have existence.

…a necessary reasoning is one which would follow under all circumstances, whether you are talking about the real world or the world of the Arabian Nights’ or what. And that precisely defines mathematical reasoning. It is true that a distinctly mathematical reasoning is one that is so intricate that we need some kind of diagram to follow it out. But something of the nature of a diagram, be it only an imaginary skeleton proposition, or even a mere noun with the ideas of its application and signification[,] is needed in all necessary reasoning.

…if we conceive a science, not as a body of ascertained truth, but, as the living business which a group of investigators are engaged upon, which I think is the only sense which gives a natural classification of sciences, then we must include under mathematics everything that is an indispensible part of the mathematician’s business; and therefore must include the formulation of his hypotheses as well as the tracing out of their consequences.

I shall always use the word multitude to mean the degree of maniness of a collection.

These substantive possibilities, – that is, qualities, relations, and the like, – are prior to existence, in the sense that non-existence is not a necessary proof of non-possibility, but non-possibility is a necessary proof of non-existence. For it is logically impossible that existence should exhaust pure possibilities of any kind. These truths are strictly deducible from the facts of phenomenology, or the analysis of the phenomenon; meaning by the phenomenon whatever is present in the mind in any kind of thought.

…I would define Pure Mathematics as the science of pure hypotheses perfectly definite in all respects which can create or destroy forms of necessary consequences from them and entirely indeterminate in other respects.