poetic usage may give considerable significance to order of expression. The limited power of our minds prevents our grasping many ideas at once, and thus the order of statement may produce some effect, but not in a simply logical manner. All life proceeds in the succession of time, and we are obliged to write, speak, or even think of things and their qualities one after the other; but between the things and their qualities there need be no such relation of order in time or space. The sweetness of sugar is neither before nor after its weight and solubility. The hardness of a metal, its colour, weight, opacity, malleability, electric and chemical properties, are all coexistent and coextensive, pervading the metal and every part of it in perfect community, none before nor after the others. In our words and symbols we cannot observe this natural condition; we must name one quality first and another second, just as some one must be the first to sign a petition, or to walk foremost in a procession. In nature there is no such precedence.
I find that the opinion here stated, to the effect that relations of space and time do not apply to many of our ideas, is clearly adopted by Hume in his celebrated Treatise on Human Nature (vol. i. p. 410). He says:6—“An object may be said to be no where, when its parts are not so situated with respect to each other, as to form any figure or quantity; nor the whole with respect to other bodies so as to answer to our notions of contiguity or distance. Now this is evidently the case with all our perceptions and objects, except those of sight and feeling. A moral reflection cannot be placed on the right hand or on the left hand of a passion, nor can a smell or sound be either of a circular or a square figure. These objects and perceptions, so far from requiring any particular place, are absolutely incompatible with it, and even the imagination cannot attribute it to them.”
A little reflection will show that knowledge in the highest perfection would consist in the simultaneous possession of a multitude of facts. To comprehend a science perfectly we should have every fact present with every other fact. We must write a book and we must read it successively word by word, but how infinitely higher would be our powers of thought if we could grasp the whole in one collective act of consciousness! Compared with the brutes we do possess some slight approximation to such power, and it is conceivable that in the indefinite future mind may acquire an increase of capacity, and be less restricted to the piecemeal examination of a subject. But I wish here to make plain that there is no logical foundation for the successive character of thought and reasoning unavoidable under our present mental conditions. We are logically weak and imperfect in respect of the fact that we are obliged to think of one thing after another. We must describe metal as “hard and opaque,” or “opaque and hard,” but in the metal itself there is no such difference of order; the properties are simultaneous and coextensive in existence.
Setting aside all grammatical peculiarities which render a substantive less moveable than an adjective, and disregarding any meaning indicated by emphasis or marked order of words, we may state, as a general law of logic, that AB is identical with BA, or AB = BA. Similarly, ABC = ACB = BCA = &c.
Boole first drew attention in recent years to this property of logical terms, and he called it the property of Commutativeness.7 He not only stated the law with the utmost clearness, but pointed out that it is a Law of Thought rather than a Law of Things. I shall have in various parts of this work to show how the necessary imperfection of our symbols expressed in this law clings to our modes of expression, and introduces complication into the whole body of mathematical formulæ, which are really founded on a logical basis.
It is of course apparent that the power of commutation belongs only to terms related in the simple logical mode of synthesis. No one can confuse “a house of bricks” with “bricks of a house,” “twelve square feet”