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BOOK II.

THE

PHILOSOPHY

OF THE

PURE SCIENCES.

The way in which we are led to regard human knowledge is like the way in which Copernicus was led to regard the heavens. When the explanation of the celestial motions could not be made to go right so long as he assumed that all the host of stars turns round the spectator, he tried whether it would not answer better if he made the spectator turn, and left the stars at rest. We may make a similar trial in Metaphysics, as to our way of looking at objects. If our view of them must be governed altogether by the properties of the objects themselves, I see not how man can know anything about them à priori. But if the thing, as an object of the senses, is regulated by the constitution of our power of knowing, I can very readily represent to myself this possibility. KANT, Kritik d. R. V. Pref.

BOOK II.

THE PHILOSOPHY OF THE PURE SCIENCES.

[The principal question discussed in the last Book was this (see chaps. v. and VI.): How are necessary and universal truths possible? And the answer then given was: that the necessity and universality of truths are derived from the Fundamental Ideas which they involve. And we proceed in this Book to exemplify this doctrine in the case of the truths of Geometry and Arithmetic, which derive their necessity and universality from the Fundamental Ideas of Space, and Time, or Number.

The question thus examined is that which Kant undertook to deal with in his celebrated work, Kritik der reinen Vernunft (Examination of the Pure Reason): and our solution of the Problem, so far as the Ideas of Space and Time are concerned, agrees in the main with his. The arguments contained in chapters II. and VI. of this Book, are the leading arguments respecting Space and Time, in Kant's Kritik. Kant, however, instead of calling Space and Time Ideas, calls them the necessary Forms of our experience, as I have stated in the text.

But though I have adopted Kant's arguments as to Space and Time, all that follows in the succeeding Books, with regard to other Ideas, has no resemblance to any doctrines of Kant or his school (with the exception, perhaps, of some of the views on the Idea of Cause). The nature and character of the other Scientific Ideas which I have examined, in the succeeding Books, have been established by an analysis of the history of the several Sciences to which those Ideas are essential, and an examination of the writings of the principal discoverers in those Sciences.]

CHAPTER I.

OF THE PURE SCIENCES.

I.

ALL

LL external objects and events which we can contemplate are viewed as having relations of Space, Time, and Number; and are subject to the general conditions which these Ideas impose, as well as to the particular laws which belong to each class of objects and occurrences. The special laws of nature, considered under the various aspects which constitute the different sciences, are obtained by a mixed reference to Experience and to the Fundamental Ideas of each science. But besides the sciences thus formed by the aid of special experience, the conditions which flow from those more comprehensive ideas first mentioned, Space, Time, and Number, constitute a body of science, applicable to objects and changes of all kinds, and deduced without recurrence being had to any observation in particular. These sciences, thus unfolded out of ideas alone, unmixed with any reference to the phenomena of matter, are hence termed Pure Sciences. The principal sciences of this class are Geometry, Theoretical Arithmetic, and Algebra considered in its most general sense, as the investigation of the relations of space and number by means of general symbols.

2. These Pure Sciences were not included in our survey of the history of the sciences, because they are not inductive sciences. Their progress has not consisted in collecting laws from phenomena, true theories from observed facts, and more general from more limited laws; but in tracing the consequences of the ideas themselves, and in detecting the most general and intimate analogies and connexions which prevail

among such conceptions as are derivable from the ideas. These sciences have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming here a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects.

3. The universality of the truths, and the rigour of the demonstrations of these pure sciences, attracted attention in the earliest times; and it was perceived that they offered an exercise and a discipline of the intellectual faculties, in a form peculiarly free from admixture of extraneous elements. They were strenuously cultivated by the Greeks, both with a view to such a discipline, and from the love of speculative truth which prevailed among that people: and the name mathematics, by which they are designated, indicates this their character of disciplinal studies.

4. As has already been said, the ideas which these sciences involve extend to all the objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics.

5. Space, time, and number, may be conceived as forms by which the knowledge derived from our sensations is moulded, and which are independent of the differences in the matter of our knowledge, arising from the sensations themselves. Hence the sciences which have these ideas for their subject may be termed Formal Sciences. In this point of view, they are distinguished from sciences in which, besides these mere formal laws by which appearances are corrected, we endeavour to apply to the phenomena the idea of cause,

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