Bibliography

On Quantity, with special reference to Collectional and Mathematical Infinity

…the distinguishing characteristic of mathematics is that it is the scientific study of hypotheses which it first frames and then traces to their consequences. Mathematics is either applied or pure. Applied mathematics treats of hypotheses in the forms in which they are first suggested by experience, involving more or less of features which have no bearing upon the forms of deduction of consequences from them. Pure mathematics is the result of an afterthought by which these irrelevant features are eliminated.

Logic and metaphysics make no special observations; but they rest upon observations which have been made by common men. Metaphysics rests upon observations of real objects, while logic rests upon observations of real facts about mental products, such as that, not merely according to some arbitrary hypothesis, but in every possible case, every proposition has a denial, that every proposition concerns some objects of common experience of the deliverer and the interpreter, that it applies to that some idea of familiar elements abstracted from the occasions of the excitation, and that it represents that an occult compulsion not within the deliverer’s control unites that idea to those objects. All these are results of common observation, though they are put into scientific and uncommon groupings.

Each science (except mathematics) rests upon fundamental principles drawn from the truths discovered by the science immediately preceding it in the list, while borrowing data and suggestions from the discoveries of those which follow it.
1. Mathematics, which observes only the creations of the mathematician himself. It borrows suggestions from all other sciences, from philosophy Mathematical Logic, from psychics Mathematical Economics, from physics Mathematical Optics, Metrics, etc.
2. Philosophy …

…the mathematician is not concerned with real truth, but only studies the substance of hypotheses. [—] …the mathematician observes nothing but the diagrams he himself constructs; and no occult compulsion governs his hypotheses except one from the depths of mind itself.

Thus, the distinguishing characteristic of mathematics is that it is the scientific study of hypotheses which it first frames and then traces to their consequences. [—]

Mathematics is […] the study of the substance of hypotheses, or mental creations, with a view to the drawing of necessary conclusions.

… Philosophy, which makes no special observations, but uses facts commonly known. In order to be exact, it must rest on mathematical principles. It divides into Logic, which studies the world of thought, and Metaphysics, which studies the world of being; and the latter must rest upon the principles of the former.

Logic and metaphysics make no special observations; but they rest upon observations which have been made by common men. Metaphysics rests upon observations of real objects, while logic rests upon observations of real facts about mental products, such as that, not merely according to some arbitrary hypothesis, but in every possible case, every proposition has a denial, that every proposition concerns some objects of common experience of the deliverer and the interpreter, that it applies to that some idea of familiar elements abstracted from the occasions of the excitation, and that it represents that an occult compulsion not within the deliverer’s control unites that idea to those objects. All these are results of common observation, though they are put into scientific and uncommon groupings.

… Philosophy, which makes no special observations, but uses facts commonly known. In order to be exact, it must rest on mathematical principles. It divides into Logic, which studies the world of thought, and Metaphysics, which studies the world of being; and the latter must rest upon the principles of the former.

…the distinguishing characteristic of mathematics is that it is the scientific study of hypotheses which it first frames and then traces to their consequences. Mathematics is either applied or pure. Applied mathematics treats of hypotheses in the forms in which they are first suggested by experience, involving more or less of features which have no bearing upon the forms of deduction of consequences from them. Pure mathematics is the result of an afterthought by which these irrelevant features are eliminated.