BOOK II. NUMBER, VARIETY, AND PROBABILITY. CHAPTER VIII. PRINCIPLES OF NUMBER. Nor without much reason did Pythagoras represent the world as ruled by number. Into almost all our acts of clear thought number enters, and in proportion as we can define numerically we enjoy exact and useful knowledge of the Universe. The science of numbers, too, the study of the principles and methods of reasoning in number, has hitherto presented the widest and most practicable training in logic. So free and energetic has been the study of mathematical forms, compared with the forms and laws of logic, that mathematicians have passed far in advance of any pure logicians. Occasionally, in recent times, they have condescended to apply their great algebraic instruments to a reflex advancement of the primary logical science. It is thus that we chiefly owe to profound mathematicians, such as Sir John Herschel, Dr. Whewell, Professor De Morgan or Dr. Boole, the regeneration of logic in the present century, and I entertain no doubt that it is in maintaining a close alliance with the extensive branches of quantitative reasoning that we must look for still further progress in our comprehension of qualitative inference. I cannot assent, indeed, to the common notion that certainty begins and ends with numerical determination. Nothing is more certain and accurate than logical truth. The laws of identity and difference are the tests of all that is true and certain throughout the range of thought, and mathematical reasoning is cogent only when it conforms to these conditions, of which logic is the first development. And if it be erroneous to suppose that all certainty is mathematical, it is equally an error to imagine that all which is mathematical is certain. Many processes of mathematical reasoning are of most doubtful validity. There are many points of mathematical doctrine which are and must long remain matter of opinion; for instance, the best form of the definition and axiom concerning parallel lines, or the true nature of a limit or a ratio of infinitesimal quantities. In the use of symbolic reasoning questions occur at every point on which the best mathematicians may differ, as Bernouilli and Leibnitz differed irreconcileably concerning the existence of the logarithms of negative quantitiesa. In fact we no sooner leave the simple logical conditions of number, than we find ourselves involved in a mazy and mysterious science of symbols. Mathematical science enjoys no monopoly, and not even a supremacy in certainty of results. It is the boundless extent and variety of quantitative questions that surprises. and delights the mathematical student. When simple logic can give but a bare answer Yes or No, the algebraist raises a score of subtle questions, and brings out a score of curious results. The flower and the fruit, all that is attractive and delightful, fall to the share of the mathematician, who too often despises the pure but necessary stem from which all has arisen. But in no part of human thought can a reasoner cast himself free from the prior conditions of logical correctness. The mathematician is a Montucla, 'Histoire des Mathématiques,' vol. iii. p. 373. only strong and true as long as he is logical, and if numbers rule the world, it is the laws of logic which rule number. Nearly all writers have hitherto been strangely content to look upon numerical reasoning as something wholly apart from logical inference. A long divorce has existed between quality and quantity, and it has not been uncommon to treat them as contrasted in nature and restricted to independent branches of human thought. For my own part, I have a profound belief that all the sciences meet somewhere upon common ground. No part of knowledge can stand wholly disconnected from other parts of the great universe of thought; it is incredible, above all, that the two great branches of abstract science, interlacing and co-operating in every discourse, should rest upon totally distinct foundations. I assume that a connection exists, and care only to inquire, What is its nature? Does the science of quantity rest upon that of quality; or, vice versa, does the science of quality rest upon that of quantity? There might conceivably be a third view, that they both rest upon some still deeper set of principles yet undiscovered, but there is an absence of any suggestions to this effect. The late Dr. Boole adopted the second view, and treated logic as a kind of algebra, a special case of analytical reasoning which admits but the two quantities-unity and zero. He proved beyond doubt that a deep analogy does exist between the forms of algebraic and logical deduction; and could this analogy receive no other explanation we must have accepted his opinion, however strange. But I shall attempt to show that just the reverse explanation is the true one. I hold that algebra is a highly developed logic, and number but logical discrimination. Logic resembles algebra, as the mould resembles that which is cast in it. Logic has imposed its own laws upon every branch of mathematical science, and it is no wonder that we ever meet with the traces of those laws from the domain of which we can never emerge. The Nature of Number. Number is but another name for diversity. Exact identity is unity, and with difference arises plurality. An abstract notion, as was pointed out (p. 33), possesses a certain oneness. The quality of justice, for instance, is one and the same in whatever just acts it be manifested. In justice itself there are no marks of difference by which to discriminate justice from justice. But one just act can be discriminated from another just act by many circumstances of time and place, and we can count and number many acts each thus discriminated from every other. In like manner pure gold is simply pure gold, and is so far one and the same throughout. But besides its intrinsic and invariable qualities, gold occupies space and must have shape or size. Portions of gold are always mutually exclusive and capable of discrimination, at least in respect that they must be each without the other. Hence they may be numbered. Plurality arises when and only when we detect difference. For instance, in counting a number of gold coins I must count each coin once, and not more than once. Let C denote a coin, and the mark above it the position in the order of counting. Then I must count the coins I should make the third coin into two, and should imply the existence of difference where there is not difference b. C" and C"" are but the names of one coin named twice b Pure Logic,' Appendix, p. 82, § 192. over. But according to one of the conditions of logical symbols, which I have called the Law of Unity (p. 86), the same name repeated has no effect, and AA=A. We must apply the Law of Unity, and must reduce all identical alternatives before we can count with certainty and use the processes of numerical calculation. Identical alternatives are harmless in logic, but produce deadly error in number. Thus logical science ascertains the nature of the mathematical unit, and the definition may be given in these terms-A unit is any object of thought which can be discriminated from every other object treated as a unit in the same problem. It has often been said that units are units in respect of being perfectly similar to each other; but though they may be perfectly similar in some respects, they must be different in at least one point, otherwise they would be incapable of plurality. If three coins were so similar that they occupied the same space at the same time, they would not be three coins, but one. It is a property of space that every point is discriminable from every other point, and in time every moment is necessarily distinct from any other moment before or after. Hence we frequently count in space or time, and Locke, with some other philosophers, has even held that number arises from repetition in time. Beats of a pendulum might be so perfectly similar that we could discover no difference except that one beat is before and another after. Time alone is here the ground of difference and is a sufficient foundation for the discrimination of plurality; but it is by no means the only foundation. Three coins are three coins, whether we count them successively or regard them all simultaneously. In many cases neither time nor space is the ground of difference, but pure quality alone enters. We can discriminate for instance the weight, inertia, and |