“Exact sciences are deduced from axiomatic Necessary truths and results in universal propositions, each of which is a Necessary Truth.
“Physical sciences are induced from Experimental Contingent truths, and result in General Propositions, each of which is a contingent truth.
“We obtain our knowledge of the Experimental Contingent Truths from which Physical science is induced, by the united action of our bodily senses and of our minds themselves, which must both in each case contribute their proper quota to make knowledge possible. Every perception necessitates this double element of sensation and intuition,—the objective and subjective factor in combination.
“We obtain our knowledge of the axiomatic Necessary Truths from which Exact science is deduced, by the à priori operation of the mind alone, and (quoad the exact science in question) without the aid of sensation (not, indeed, by à priori operation of a mind which has never worked with sensation, for such a mind would be altogether barren; but of one which has reached normal development under normal conditions; which conditions involve the continual united action productive of perceptions of contingent truths).
“In this distinction between the sources of our knowledge lies the most important discovery of philosophy. Into whatsoever knowledge the element of Sensation necessarily enters as a constituent part, therein there can be no absolute certainty of truth; the fallibility of Sensation being recognised on all hands, and neutralising the certainty of the pure mental element. But when we discover an order of sciences which, without aid from sensation, are deduced by the mind’s own operation from those Necessary truths which we hold on a tenure marking indelibly their distinction from all contingent truths whatsoever, then we obtain footing in a new realm....
“In the ensuing pages I shall endeavour to demonstrate that the science of Morals belongs to the class of Exact sciences, and that it has consequently a right to that credence wherewith we hold the truths of arithmetic and geometry....”
The test which divides the two classes is as follows:—
“What truth soever is Necessary and of universal extent is derived by the mind from its own operation, and does not rest on observation or experience; as, conversely, what truth or perception soever is present to the mind with a consciousness, not of its Necessity, but of its Contingency, is ascribable not to the original agency of the mind itself, but derives its origin from observation and experience.”
After lengthened discussion on this head and on the supposed mistakes of moral intuition, I go on to say:
“The consciousness of the Contingency, or the consciousness of the Necessity (i.e., the consciousness that the truth cannot be contingent, but must hold good in all worlds for ever), these consciousnesses are to be relied on, for they have their origin in, and are the marks of, the different elements from which they have been derived.[[9]] We may apply them to the fundamental truths of any science, and by observing whether the reception of such truths into our minds be accompanied by the consciousness of Necessity or of Contingency, we may decide whether the science be rightfully Exact or Physical, deductive or inductive.