CHAPTER III.

MATHEMATICS.

IT has been shown by Kant that the axioms of geometry are synthetic and not analytic judgments.' Thus, in the axiom, "Two straight lines cannot enclose a space," the predication that "they cannot enclose a space," is not contained in the bare notion of "two straight lines." Starting with axioms which involve more than analytic judgments, we are reaching throughout the demonstration more than identical truth. The propositions in the Books of Euclid are all evolved out of the definitions and axioms, but are not identical with them, or with one another.

The question is keenly agitated as to axioms, whether they are or are not the result of the generalizations of experience. It will be found here, as in so many other questions which have passed under our notice, that there is truth on both sides, error on both sides, and confusion in the whole controversy, which is to be cleared up by an exact expression of the mental operation involved in passing the judgment. A mathematic axiom, being a general maxim,

1 Kritik, p. 143. Dr. Mansel (Proleg. Log. 2d ed. p. 103), maintains that such axioms as that "Things which are equal to the same are equal to each other" are analytic. But does not this confound equality with identity? D. Stewart remarks (Elem. Vol. I. chap. ii.) that most of the writers who have maintained that all mathematical evidence resolves ultimately into the perception of identity "have imposed on themselves by using the words identity and equality as literally synonymous and convertible terms. This does not seem to be at all consistent, either in point of expression or fact, with sound logic." Certain modern logicians have fallen into a still greater confusion, when they make the relation between subject and predicate merely one of identity or of equality. The proposition, "Man is mortal," is not interpreted fully when it is said "Man is identical with some mortal," or that, "All men some mortals." By all means let logicians use symbols, but let them devise symbols of their own, and not turn to a new use the symbols of mathematics, which have a meaning, and a well-defined one, simply as applied to quantity, and should not be made to signify the relations of extension and comprehension in logical propositions.

is the result of a process of generalization. If we look to what has passed within our minds, we shall find that it has been by the contemplation of individual instances that the mind has attained to the comprehension and the conviction of the general proposition, that "If equals be added to equals, the sums are equal." The boy understands this best when he is in circumstances to use his marbles, or his apples. The youth who is finding his way through Euclid does not feel that the axiom adds in the least to the cogency of the reasoning; on the contrary, it is rather the case before him that enables him to comprehend the axiom and to acknowledge its truth.

But it does not follow that the axiom is a mere generalization of an outward or a gathered experience. It is not by trying two straight rods, ten, twenty, or a thousand times, that we arrive at the general proposition that two straight lines cannot enclose a space, and thence conclude as to two given lines presented to us that it is impossible they should enclose a space. It is certainly not by placing two rods parallel to each other, and lengthening them more and more, and then measuring their distance to see if they are approaching, that we reach the axiom that two parallel lines will never meet, and thence be convinced as to any given set of like lines that they will never come nearer each other. Place before us two new substances, and we cannot tell beforehand whether they will or will not chemically combine; but on the bare contemplation of two straight lines, we declare they cannot contain a space; and of two parallel lines, that they can never meet.1

1 Mr. Mill maintains (Logic, I. v. 4, 5) that the proposition, "Two straight lines cannot enclose a space," is a generalization from observation, "an induction from the evidence of the senses." That observation is needed I have shown in this treatise; but there is intuition in the observation. That there is generalization in the general maxim I have also shown; but it is not a gathering of outward instances. Observation can of itself tell us that these two lines before us do not enclose a space, and that any other couplets of lines examined by us, twenty, or a hundred, or a thousand, do not enclose a space; but experience can say no more without passing beyond its province. An intellectual generalization of such experience might allow us to affirm that very probably no two lines enclose a space on the earth, but could never entitle us to maintain that two lines could not enclose a space in the constellation Orion. Mr. Mill, in order to account for the necessity which attaches to such convictions, refers to the circumstance that geometrical forms admit of being distinctly painted in the

In mathematical truth, the mind, upon the objects being presented to its contemplation, at once and intuitively pronounces the judgment. It conceives two straight lines, and decides that they cannot be made to enclose a space. But it would pronounce the same decision as to any other, as to every other pair of straight lines, and thus reaches the maxim that what is true of these two lines is true of all. There is thus generalization in the formation of the axiom, but it is a generalization of the individual intuitive judgments of the mind. Hence arises the distinction between the axioms of mathematics and the general laws reached by observation. If we have properly generalized the individual conviction, the necessity that is in the individual goes up into the general, which embraces all the individuals, and the axiom is necessarily true, and true to all beings. But we can never be sure that there may not somewhere be an exception to experiential laws. We are sure that two straight lines cannot enclose a space in any planet, or star, or world, that ever existed or shall exist. But it is quite possible that there may be horned animals which are not ruminant, or white crows in some of the planets; and that there may come a time when the sun shall no longer give heat or light.

In the case of our intuitive convictions regarding space, number, and quantity, the simplicity of the objects makes it easy for us to seize the principle, and to put it in proper formulæ, which can imagination, so that we have "mental pictures of all possible combinations of lines and angles." We might ask him what he makes of algebraic and analytic demonstrations of every kind, where there is no such power of imagination and yet the same necessity. But without dwelling on this I would have it remarked, that in the very theory which he devises to show that the whole is a process of experience, he is appealing to what no experience can ever compass, "to all possible combinations of lines and angles." Intuitive thought, proceeding on intuitive perceptions of space, may tell us the "possible combinations" of geometrical figures; but this cannot be done by observation, by sense, or imagination. Supposing, he says, that two straight lines, after diverging, could again converge, we can transport ourselves thither in imagination, and can frame a mental image of the appearance which one or both of the lines must present at that point, which we may rely on as being precisely similar to the reality." Most freely do I admit all this. We may "rely" on it, but surely it is not experience, nor imagination, but thought which tells us what must be at that point, and that it is a "reality." The very line of remark which he is pursuing might have shown him that the discovery of necessary spatial and quantitative relations is a judgment in which the mind looks upon objects intuitively known, and now presented, or more frequently represented to the mind.

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scarcely fail to be accurately made. Hence these convictions came to be expressed in general forms, in what were then called Common Notions, at a very early age of the history of intellectual culture. The disputes among mathematicians in regard to axioms, relate not to their certainty and universality, but to the forms in which they ought to be put, and as to whether what some regard as first truths may not be demonstrated from prior truths. Such, for instance, is the dispute as to how the axioms and demonstrations as to parallel lines should be best constructed. But in regard to our convictions of extension, number and quantity, it is not difficult to gather the regulating principle out of the individual judgment, and the expression is commonly accurate. It is different with other of our original convictions, such as those which relate to cause and effect; the greater complexity of the objects renders it more difficult to seize on the principle involved, and there is greater room for dispute as to any given formula whether it is an exact expression of the facts.

Another interesting and still disputed topic in the metaphysic of mathematics, relates to the nature and value of Definitions. Mathematical definitions seem to me to be formalized primitive cognitions or beliefs regarding space, number, and quantity. In their formation there is a process of abstraction involved. A point is defined "position, without magnitude;" there is no such point, there can be no such point. A line is length without breadth ;" there was never such a line drawn by pen or diamond point. But the mind in its analysis is sharper than steel or diamond. It can contemplate position without taking extension into view. It can reason about the length of a line without regarding the breadth. In all these definitions there is abstraction, but I must ever protest against the notion that an abstraction is necessarily something unreal. If the concrete be real, the part of it separated by abstraction must likewise be real. The position of the point is a reality, and so also is the length of a line; they are not independent realities, and capable of existing alone and apart, but still they are realities, and when the mind contemplates them separately, it contemplates realities. So far as it reasons about them accurately, according to the laws of thought, the conclusions arrived at will

also relate to realities, not independent realities, but realities of the same nature as those with which we started in our original definitions. Thus, whatever conclusions are arrived at in regard to lines, or circles, or ellipses, will apply to all objects, so far as we consider them as having length, or a circular or elliptic form. We find, in fact, that the conclusions reached in mathematics do hold true of all bodies in earth or sky, so far as we find them occupying space, or having numerical relations.

If this view be correct, we see how inadequate is the representation of those who, like D. Stewart and Mr. J. S. Mill, represent mathematical definitions as merely hypothetical, and represent the whole consistency and necessity as being between a supposition and the consequences drawn from it.' This is to overlook the concrete cognitions or beliefs from which the definition is derived. It is likewise to overlook the fact that these refer to objects, and the further fact that the abstractions from the concretes also imply a reality. This theory also fails to account for the circumstance that the conclusions reached in mathematics admit of an application to the settlement of so many questions in astronomy, and in other departments of natural philosophy. Thus, what was demonstrated of the conic sections by Apollonius, is found true in the orbits of the planets and comets, as revealed by modern discovery. All this can at once be explained if we suppose that the mind starts with cognitions and beliefs, that it abstracts from these, and discovers relations among the things thus abstracted: the reality that was in the original conviction goes on to the farthest conclusion.

I am inclined to look on the primitive cognitions as constituting, properly speaking, the foundation of mathematics. The mind, looking at the things under the clear and distinct aspects in which they are set before it by abstraction, discovers relations between them, and can draw deductions from the combination. In this process the mind proceeds spontaneously, without thinking of the general principle involved in the reasoning. It finds that A is equal to B, and B to C, and it at once concludes that A is equal to C. It does not feel that in order to reach this conclusion it 1 Stewart's Elem. Vol. п. chap. ii. Mill's Logic, î. v. 1.