BOOK II.
NUMBER, VARIETY, AND PROBABILITY.
Not without reason did Pythagoras represent the world as ruled by number. Into almost all our acts of 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, 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 of logic, that mathematicians have passed far in advance of pure logicians. Occasionally, in recent times, they have condescended to apply their algebraic instrument to a reflex treatment of the primary logical science. It is thus that we owe to profound mathematicians, such as John Herschel, Whewell, De Morgan, or Boole, the regeneration of logic in the present century. I entertain no doubt that it is in maintaining a close alliance with quantitative reasoning that we must look for 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 than logical truth. The laws of identity and difference are the tests of all that is 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 points of mathematical doctrine which 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. In the use of symbolic reasoning questions occur on which the best mathematicians may differ, as Bernoulli and Leibnitz differed irreconcileably concerning the existence of the logarithms of negative quantities.1 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 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 crowd 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 plain but necessary stem from which all has arisen. In no region of thought can a reasoner cast himself free from the prior conditions of logical correctness. The mathematician is only strong and true as long as he is logical, and if number rules the world, it is logic which rules number.
Nearly all writers have hitherto been strangely content to look upon numerical reasoning as something 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 thought. For my own part, I believe that all the sciences meet somewhere. No part of knowledge can stand wholly disconnected from other parts of the 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