Page images
PDF
EPUB

tion that new events will conform to the conditions detected in our observation of past events. No experience of finite duration can give an exhaustive knowledge of the forces which are in operation. There is thus a double uncertainty; even supposing the Universe as a whole to proceed unchanged, we do not really know the Universe as a whole. We know only a point in its infinite extent, and a moment in its infinite duration. We cannot be sure, then, that our observations have not escaped some fact, which will cause the future to be apparently different from the past; nor can we be sure that the future really will be the outcome of the past. We proceed then in all our inferences to unexamined objects and times on the assumptions—

1. That our past observation gives us a complete knowledge of what exists.

2. That the conditions of things which did exist will continue to be the conditions which will exist.

We shall often need to illustrate the character of our knowledge of nature by the simile of a ballot-box, so often employed by mathematical writers in the theory of probability. Nature is to us like an infinite ballot-box, the contents of which are being continually drawn, ball after ball, and exhibited to us. Science is but the careful observation of the succession in which balls of various character present themselves; we register the combinations, notice those which seem to be excluded from occurrence, and from the proportional frequency of those which appear we infer the probable character of future drawings. But under such circumstances certainty of prediction depends on two conditions:

1. That we acquire a perfect knowledge of the comparative numbers of balls of each kind within the box.

2. That the contents of the ballot-box remain unchanged. Of the latter assumption, or rather that concerning the constitution of the world which it illustrates, the logician or physicist can have nothing to say. As the Creation of the Universe is necessarily an act passing all experience and all conception, so any change in that Universe, or, it may be, a termination of it, must likewise be infinitely beyond the bounds of our mental faculties. No science no

reasoning upon the subject, can have any validity; for without experience we are without the basis and materials of knowledge. It is the fundamental postulate accordingly of all inference concerning the future, that there shall be no arbitrary change in the subject of inference; of the probability or improbability of such a change I conceive that our faculties can give no estimate.

The other condition of inductive inference-that we acquire an approximately complete knowledge of the combinations in which events do occur, is in some degree within our power. There are branches of science in which phenomena seem to be governed by conditions of a most. fixed and general character. We have ground in such cases for believing that the future occurrence of such phenomena can be calculated and predicted. But the whole question now becomes one of probability and improbability. We seem to leave the region of logic to enter one in which the number of events is the ground of inference. We do not really leave the region of logic; we only leave that where certainty, affirmative or negative, is the result, and the agreement or disagreement of qualities the means of inference. For the future, number and quantity will commonly enter into our processes of reasoning; but then I hold that number and quantity are but portions of the great logical domain. I venture to assert that number is wholly logical, both in its fundamental nature and in its developments. Quantity in all its forms is but a development of number. That which is mathematical is not the less logical; if anything it is more logical, in the sense that it presents logical results in a higher degree of complexity and variety.

Before proceeding then from Perfect to Imperfect Induction I must devote a portion of this work to treating the logical conditions of number. I shall then employ number to estimate the variety of combinations in which natural phenomena may present themselves, and the probability or improbability of their occurrence under definite circumstances. It is in later parts of the work that I must endeavour to establish the notions which I have set forth upon the subject of Imperfect Induction, as applied in the investigation of Nature, which notions may be thus briefly stated:

1. Imperfect Induction entirely rests upon Perfect Induction for its materials.

2. The logical process by which we seem to pass directly from examined to unexamined cases consists in an inverse application of deductive inference, so that all reasoning may be said to be either directly or inversely deductive.

3. The result is always of a hypothetical character, and is never more than probable. 4. No net addition is ever made to our knowledge by reasoning; what we know of future events or unexamined objects is only the unfolded contents of our previous knowledge, and it becomes less probable as it is more boldly extended to remote

cases.

BOOK II..

NUMBER, VARIETY, AND PROBABILITY.

CHAPTER VIII.

PRINCIPLES OF NUMBER.

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. 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. 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.

1

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 uncominon 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

1 Montucla. Histoire des Mathématiques, vol. iii. p. 373.

« PreviousContinue »