XV.
Necessary truths derive their necessity from the Ideas which they involve; and the existence of necessary truths proves the existence of Ideas not generated by experience. (i. 5.)
XVI.
In Deductive Reasoning, we cannot have any truth in the conclusion which is not virtually contained in the premises. (i. 6.)
XVII.
In order to acquire any exact and solid knowledge, the student must possess with perfect precision the ideas appropriate to that part of knowledge: and this precision is tested by the student’s perceiving the axiomatic evidence of the axioms belonging to each Fundamental Idea. (i. 6.)
XVIII.
The Fundamental Ideas which it is most important to consider, as being the Bases of the Material Sciences, are the Ideas of Space, Time (including Number), Cause (including Force and Matter), Outness of Objects, and Media of Perception of Secondary Qualities, Polarity (Contrariety), 8 Chemical Composition and Affinity, Substance, Likeness and Natural Affinity, Means and Ends (whence the Notion of Organization), Symmetry, and the Ideas of Vital Powers. (i. 8.)
XIX.
The Sciences which depend upon the Ideas of Space and Number are Pure Sciences, not Inductive Sciences: they do not infer special Theories from Facts, but deduce the conditions of all theory from Ideas. The Elementary Pure Sciences, or Elementary Mathematics, are Geometry, Theoretical Arithmetic and Algebra. (ii. 1.)