"The immutability of the same relations between the same immutable things is now the idea that shows him that if the three angles of a triangle were once equal to two right angles, they will always be equal to two right ones. And hence he comes to be certain that what was once true in the case is always true; what ideas once agreed will always agree.... Upon this ground it is that particular demonstrations in mathematics afford general knowledge. If, then, the perception that the same ideas will eternally have the same habitudes and relations be not a sufficient ground of knowledge, there could be no knowledge of general propositions in mathematics.... All general knowledge lies only in our own thoughts, and consists barely in the contemplation of our abstract ideas. Wherever we perceive any agreement or disagreement amongst them, there we have general knowledge; and by putting the names of those ideas together accordingly in propositions, can with certainty pronounce general truths.... What is once known of such ideas will be perpetually and forever true. So that, as to all general knowledge, we must search and find it only in our own minds and it is only the examining of our own ideas that furnisheth us with that. Truths belonging to essences of things (that is, to abstract ideas) are eternal, and are to be found out only by the contemplation of those essences.... Knowledge is the consequence of the ideas (be they what they will) that are in our minds, producing there certain general propositions.... Such propositions are therefore called 'eternal truths,'... because, being once made about abstract ideas so as to be true, they will, whenever they can be supposed to be made again, at any time past or to come, by a mind having those ideas, always actually be true. For names being supposed to stand perpetually for the same ideas, and the same ideas having immutably the same habitudes one to another, propositions concerning any abstract ideas that are once true must needs be eternal verities."
But what are these eternal verities, these 'agreements,' which the mind discovers by barely considering its own fixed meanings, except what I have said?—relations of likeness and difference, immediate or mediate, between the terms of certain series. Classification is serial comparison, logic mediate subsumption, arithmetic mediate equality of different bundles of attention-strokes, geometry mediate equality of different ways of carving space. None of these eternal verities has anything to say about facts, about what is or is not in the world. Logic does not say whether Socrates, men, mortals or immortals exist; arithmetic does not tell us where her 7's, 5's, and 12's are to be found; geometry affirms not that circles and rectangles are real. All that these sciences make us sure of is, that if these things are anywhere to be found, the eternal verities will obtain of them. Locke accordingly never tires of telling us that the
"universal propositions of whose truth or falsehood we can have certain knowledge, concern not existence.... These universal and self-evident principles, being only our constant, clear, and distinct knowledge of our own ideas more general or comprehensive, can assure us of nothing that passes without the mind; their certainty is founded only upon the knowledge of each idea by itself, and of its distinction from others; about which we cannot be mistaken whilst they are in our minds.... The mathematician considers the truth and properties belonging to a rectangle or circle only as they are in idea in his own mind. For it is possible he never found either of them existing mathematically, i.e., precisely true, in his life. But yet the knowledge he has of any truths or properties belonging to a circle, or any other mathematical figure, are nevertheless true and certain even of real things existing; because real things are no farther concerned nor intended to be meant by any such propositions, than as things really agree to those archetypes in his mind. Is it true of the idea of a triangle, that its three angles are equal to two right ones? It is true also of a triangle wherever it really exists. Whatever other figure exists that is not exactly answerable to that idea in his mind is not at all concerned in that proposition. And therefore he is certain all his knowledge concerning such ideas is real knowledge: because, intending things no farther than they agree with those his ideas, he is sure what he knows concerning those figures when they have barely an ideal existence in his mind will hold true of them also when they have a real existence in matter." But "that any or what bodies do exist, that we are left to our senses to discover to us as far as they can."[549]
Locke accordingly distinguishes between 'mental truth' and 'real truth.'[550] The former is intuitively certain; the latter dependent on experience. Only hypothetically can we affirm intuitive truths of real things—by supposing, namely, that real things exist which correspond exactly with the ideal subjects of the intuitive propositions.
If our senses corroborate the supposition all goes well. But note the strange descent in Locke's hands of the dignity of a priori propositions. By the ancients they were considered, without farther question, to reveal the constitution of Reality. Archetypal things existed, it was assumed, in the relations in which we had to think them. The mind's necessities were a warrant for those of Being; and it was not till Descartes' time that scepticism had so advanced (in 'dogmatic' circles) that the warrant must itself be warranted, and the veracity of the Deity invoked as a reason for holding fast to our natural beliefs.
But the intuitive propositions of Locke leave us as regards outer reality none the better for their possession. We still have to "go to our senses" to find what the reality is. The vindication of the intuitionist position is thus a barren victory. The eternal verities which the very structure of our mind lays hold of do not necessarily themselves lay hold on extra-mental being, nor have they, as Kant pretended later,[551] a legislating character even for all possible experience. They are primarily interesting only as subjective facts. They stand waiting in the mind, forming a beautiful ideal network; and the most we can say is that we hope to discover outer realities over which the network may be flung so that ideal and real may coincide.
And this brings us back to 'science' from which we diverted our attention so long ago (see [p. 640]). Science thinks that she has discovered the outer realities in question. Atoms and ether, with no properties but masses and velocities expressible by numbers, and paths expressible by analytic formulas, these at last are things over which the mathematico-logical network may be flung, and by supposing which instead of sensible phenomena science becomes yearly more able to manufacture for herself a world about which rational propositions may be framed. Sensible phenomena are pure delusions for the mechanical philosophy. The 'things' and qualities we instinctively believe in do not exist. The only realities are swarming solids in everlasting motion, undulatory or continued, whose expressionless and meaningless changes of position form the history of the world, and are deducible from initial collocations and habits of movement hypothetically assumed. Thousands of years ago men started to cast the chaos of nature's sequences and juxtapositions into a form that might seem intelligible. Many were their ideal prototypes of rational order: teleological and æsthetic ties between things, causal and substantial bonds, as well as logical and mathematical relations. The most promising of these ideal systems at first were of course the richer ones, the sentimental ones. The baldest and least promising were the mathematical ones; but the history of the latter's application is a history of steadily advancing successes, whilst that of the sentimentally richer systems is one of relative sterility and failure.[552] Take those aspects of phenomena which interest you as a human being most, and class the phenomena as perfect and imperfect, as ends and means to ends, as high and low, beautiful and ugly, positive and negative, harmonious and discordant, fit and unfit, natural and unnatural, etc., and barren are all your results. In the ideal world the kind 'precious' has characteristic properties. What is precious should be preserved; unworthy things should be sacrificed for its sake; exceptions made on its account; its preciousness is a reason for other things' actions, and the like. But none of these things need happen to your 'precious' object in the real world. Call the things of nature as much as you like by sentimental, moral, and æsthetic names, no natural consequences follow from the naming. They may be of the kinds you allege, but they are not of 'the kind's kind': and the last great system-maker of this sort, Hegel, was obliged explicitly to repudiate logic in order to make any inferences at all from the names he called things by.
But when you give things mathematical and mechanical names and call them just so many solids in just such positions, describing just such paths with just such velocities, all is changed. Your sagacity finds its reward in the verification by nature of all the deductions which you may next proceed to make. Your 'things' realize all the consequences of the names by which you classed them. The modern mechanico-physical philosophy of which we are all so proud, because it includes the nebular cosmogony, the conservation of energy, the kinetic theory of heat and gases, etc., etc., begins by saying that the only facts are collocations and motions of primordial solids, and the only laws the changes of motion which changes in collocation bring. The ideal which this philosophy strives after is a mathematical world-formula, by which, if all the collocations and motions at a given moment were known, it would be possible to reckon those of any wished-for future moment, by simply considering the necessary geometrical, arithmetical, and logical implications. Once we have the world in this bare shape, we can fling our net of a priori relations over all its terms, and pass from one of its phases to another by inward thought-necessity. Of course it is a world with a very minimum of rational stuff. The sentimental facts and relations are butchered at a blow. But the rationality yielded is so superbly complete in form that to many minds this atones for the loss, and reconciles the thinker to the notion of a purposeless universe, in which all the things and qualities men love, dulcissima mundi nomina, are but illusions of our fancy attached to accidental clouds of dust which will be dissipated by the eternal cosmic weather as carelessly as they were formed.