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Sketch of Dichotomic Mathematics

Any Axiom would be a proposition not deducible from anything asserted in the definitions and postulates, but immediately evident in view of the facts that have been laid down and these are true.

Belief does not principally consist in any particular act of thought, but in a habit of thought and a conduct. A man does not necessarily believe what he thinks he believes. He only believes what he deliberately adopts and is ready to make a habit of conduct.

A Convention would be a proposition concerning a subject which we imagine to exist as an aid in drawing conclusions concerning our main subject; and any such proposition would be a Convention.

Any Corollary (as I shall use the term) would be a proposition deduced directly from propositions already established without the use of any other construction than one necessarily suggested in apprehending the enunciation of the proposition; and any such proposition would be a Corollary.

As Aristotle well says (and his authority is well-nigh absolute upon a question of logical terminology), a definition asserts the existence of nothing. A definition would consist of two members, of which the first should declare that any object to which the definitum, or defined term, should be applicable would possess the characters involved in the definition; while the second should declare that to any object which should possess those characters the definitum would be applicable.

…I call the sign the Entelechy, or perfectionment, of reality.

A Definition is either Nominal or Real. A nominal definition merely explains the meaning of a term which is adopted for convenience.

It may be remarked that if, as I hold, there are three categories, Form, Matter, and Entelechy, then there will naturally be seven schools of philosophy; that which recognizes Form alone, that which recognizes Form and Matter alone, that which recognizes Matter alone (these being the three kinds of nominalism)…

A Postulate would be a proposition necessary as a premiss for a course of deductive reasoning and predicating a contingent character of the hypothetical subject of that course of reasoning; and any proposition of that description would be a Postulate.

Any Problem (as I shall use the term) would be a proposition pronouncing, in effect, that under circumstances known to exist or to be possible, a certain sort of result, described partly, at least, in a general way, exists or is possible, this vague proposition being usually, though not essentially capable of definition, showing under what general conditions it always will be realized, and at any rate, being capable of demonstration from propositions previously established, but only by means of a hypothesis not immediately suggested in the vague enunciation; and any proposition of that description would be a Problem.

A Definition is either Nominal or Real. [—] A Real Definition analyzes a conception. As Aristotle well says (and his authority is well-nigh absolute upon a question of logical terminology), a definition asserts the existence of nothing. A definition would consist of two members, of which the first should declare that any object to which the definitum, or defined term, should be applicable would possess the characters involved in the definition; while the second should declare that to any object which should possess those characters the definitum would be applicable. And any proposition consisting of two members of this description and really contributing to the development of the thought would be a Real Definition.

Any Theorem (as I shall use this term) would be a proposition pronouncing, in effect, that were a general condition which it describes fulfilled, a certain result which it describes in a general way, except so far as it may refer to some object or set of objects supposed in the condition, will be impossible, this proposition being capable of demonstration from propositions previously established, but not without imagining something more than what the condition supposes to exist; and any such proposition would be a Theorem.