§1. SCIENCE
CP7-49. What is Science? We cannot define the word with the precision and concision with which we define Circle, or Equation, any more than we can so define Money, Government, Stone, Life. The idea, like these, and more than some of them, is too vastly complex and diversified. It embodies the epitome of man's intellectual development. We can only single out some leading properties of it, and different people will select these differently. To most men, including all who are outside of the world of science, the term means a particular kind of knowledge. Wherein lies the essential peculiarity of this knowledge? Some thinkers agree with the ancient Greeks in making it consist in the Method of knowing, the manner in which the truth is laid hold on. But the majority of modern writers regard the Systematic character of the doctrine itself as more characteristic. Both marks of scientific knowledge are exceedingly important; but the former is deeper cut, and because it is at present less noticed, more needs to be emphasized. Plato is quite right in saying that a true belief is not necessarily knowledge. A man may be willing to stake his life upon the truth of a doctrine which was instilled into his mind before his earliest memories without knowing at all why it is worthy of credence; and while such a faith might just as easily be attached to a gross superstition as to a noble truth, it may, by good luck, happen to be perfectly true. But can he be said to know it? By no means: to render the word knowledge applicable to his belief, he must not only believe it, but must know, -- I will not say, with the ancients, the rationale of the real fact, as a reality, -- but must know what justifies the belief, and just WHY and HOW the justification is sufficient. I beg that the reader will turn this over in his mind and satisfy himself as to how far what I am saying is true. For it is not a very simple point but is one that I intend to insist upon.

50. Before knowledge of any subject can be put to any extensive use, it is almost indispensable that it should be made as thorough and complete as possible, until every detail and feature of the matter is spread out as in a German handbook. But if I am asked to what the wonderful success of modern science is due, I shall suggest that to gain the secret of that, it is necessary to consider science as living, and therefore not as knowledge already acquired but as the concrete life of the men who are working to find out the truth. Given a body of men devoting the sum of their energies to refuting their present errors, doing away with their present ignorance, and that not so much for themselves as for future generations, and all other requisites for the ascertainment of truth are insured by that one. Strictly speaking, one need not ask for so much as that. Given the oxygen, hydrogen, carbon, nitrogen, sulphur, phosphorus, etc., in sufficient quantities and under proper radiations, and living protoplasm will be produced, will develop, will gain power of self-control, and the scientific passion is sure to be generated. Such is my guess. Science was preordained, perhaps, on the Sunday of the Fiat lux.

51. Coming down to the more immediate and more pertinent causes of the triumph of modern science, the considerable numbers of the workers, and the singleness of heart with which, -- (we may forget that there are a few self-seekers who succeed in gaining the power to make themselves more despised than they naturally would be; they are so few,) -- they cast their whole being into the service of science lead, of course, to their unreserved discussions with one another, to each being fully informed about the work of his neighbour, and availing himself of that neighbour's results; and thus in storming the stronghold of truth one mounts upon the shoulders of another who has to ordinary apprehension failed, but has in truth succeeded by virtue of the lessons of his failure. This is the veritable essence of science. It is in the memory of these concrete living gests that we gain the speaking portraiture of true science in all her life and beauty.

52. The point of view just explained enables us to perceive that a particular branch of science, such as Physical Chemistry or Mediterranean Archeology, is no mere word, manufactured by the arbitrary definition of some academic pedant, but is a real object, being the very concrete life of a social group constituted by real facts of inter-relation, -- as real an object as a human carcase, which is made one by the inter-relations of its millions of cells. Any two of these groups (and with them the sciences, which are their lives,) may be related, as to the matter of the groups in either of the three modes of relationship of material wholes; that is, either by Inclusion, one being a part of another; or by Intersection, when each has one part in common with the other, and another part foreign to the other; or by Exclusion, when the two have no part in common. But of greater importance are the dynamical relations between the different sciences, by which I mean that one often acts upon another, not by bringing forward any reason or principle, but as it were with a compulsive quality of action. Thus one group may stimulate another by demanding the solution of some problem. In this way, the practical sciences incessantly egg on researches into theory. For considerable parts of chemical discovery we have to thank the desire to find a substitute for quinine or to make quinine itself synthetically, to obtain novel and brilliant dye-stuffs, and the like. The mechanical theory of heat grew out of the difficulties of steam navigation. For it was first broached by Rankine while he was studying how best to design marine engines. Then again, one group of scientists sometimes urges some overlooked phenomenon upon the attention of another group. It was a botanist who called van't Hoff's attention to the dependence of the pressure of sap in plants upon the strength of the solution, and thus almost instantaneously gave a tremendous impulse to physical chemistry. In 1820, Kästner, a manufacturer of cream of tartar in Mulhouse, called the attention of chemists to the occasional, though rare, occurrence in the wine casks of a modification of tartaric acid, since named racemic acid; and from the impulse so given has resulted a most important doctrine of chemistry, that of the unsymmetric carbon atom, as well as the chief discoveries of Pasteur, with their far-reaching blessings to the human species.

53. It is now time to explain the classification of this chapter, what it aims to be, by what means that aim has been pursued, and how nearly it seems to have been attained. Two questions have to be answered at the outset: What is here meant by science? And what is meant by a science, one of the unit species out of which the system is built up? The spirit of this book is always to look upon those aspects of things which exhibit whatever of living and active there is in them.

54. The prevalent definition of a science, the definition of Coleridge, which influenced all Europe through the Encyclopaedia Metropolitana, that science is systematized knowledge, is an improvement upon a statement of Kant (Metaphysische Anfangsgründe der Naturwissenschaft: 1786): "Eine jede Lehre, wenn sie ein System, dass ist, ein nach Principien geordnetes Ganzes der Erkenntniss sein soll, heisst Wissenschaft."†2 Yet it is to be noted that knowledge may be systematic or "organized," without being organized by means of general principles. Kant's definition, however, is only a modification of the ancient view that science is the knowledge of a thing through its causes, -- the comprehension of it, as we might say, -- as being the only perfect knowledge of it. In short, the Coleridgian definition is nothing but the last development of that sort of philosophy that strives to draw knowledge out of the depths of the Ich-heit. If, on the other hand, one opens the works of Francis Bacon, one remarks that, with all the astounding greenness and inexperience of his views of science, in some respects he is really a scientific man himself. He met his death as the consequence of an experiment. True, it was rather a foolish one; but what a monument to the genuineness of his intelligence, that he, a great legal light, should, at the age of sixty-six, have perished from his zeal in performing disagreeable and dangerous laboratory work that he thought might go toward teaching him something of the nature of true science! For him man is nature's interpreter; and in spite of the crudity of some anticipations, the idea of science is, in his mind, inseparably bound up with that of a life devoted to singleminded inquiry. That is also the way in which every scientific man thinks of science. That is the sense in which the word is to be understood in this chapter. Science is to mean for us a mode of life whose single animating purpose is to find out the real truth, which pursues this purpose by a well-considered method, founded on thorough acquaintance with such scientific results already ascertained by others as may be available, and which seeks coöperation in the hope that the truth may be found, if not by any of the actual inquirers, yet ultimately by those who come after them and who shall make use of their results. It makes no difference how imperfect a man's knowledge may be, how mixed with error and prejudice; from the moment that he engages in an inquiry in the spirit described, that which occupies him is science, as the word will here be used.

55. By a specific science will be meant a group of connected inquiries of sufficient scope and affinity fitly to occupy a number of independent inquirers for life, but not capable of being broken up into smaller coexclusive groups of this description. For since we are to consider science in general as a mode of life, it is proper to take as the unit science the scientific mode of life fit for an individual person. But science being essentially a mode of life that seeks coöperation, the unit science must, apparently, be fit to be pursued by a number of inquirers.

56. It seems plain that, with these definitions, the classification cannot be concerned with all possible sciences, but must be confined to actually realized sciences. If, however, this limitation is to be maintained, the question will arise, To what date or stage of scientific development is the classification to relate? According to the general spirit of this book, which values everything in its relation to Life, knowledge which is altogether inapplicable to the future is nugatory. Consequently, our classification ought to have reference to the science of the future, so far as we are now able to foresee what the future of science is to be. It will therefore be upon the soil of the near future of science that we shall endeavor to plant our flag. If it be objected that we cannot know enough of the science of the future to classify it accurately, the reply would be that even if all faults of classification could be eliminated by remaining on the threshold of the future, it would still be necessary to advance further. For all the applicability of any writing, though it be not (like this,) the fruit of near half a century of study, must evidently be subsequent to its composition, and all its significance for that time has reference to a time still later. But when the objector comes to see the various imperfections that will have to be confessed in that part of the classification which concerns the present state of science, he will probably be disposed himself to acknowledge that its standard will not be much lowered by the danger of mistake about what is likely soon to be discovered.

57. Meantime, let it not be understood that the classification is to ignore the scientific discoveries of the past. For the memoirs of that work are not so poor as not to merit being read critically, precisely as we shall read the memoirs of tomorrow. Such reading is, therefore, of the nature of scientific inquiry. True, it is not original research; but there is original research still to be done in the same specific science. For none of the sciences of the past is finished. If it be one of the positive sciences that is in question, there is not a single conclusion belonging to it which has in the past been made sufficiently precise or sufficiently indubitable. If it be a branch of mathematics, its propositions require to be further generalized, as well as to be more accurately limited. For these reasons all the old science that still stands is to be retained in the classification, but in its most modern forms.

58. The only remaining instinct on our list is the Gnostic Instinct, or curiosity. In one sense, the sciences that are practically ministrant to this are the Theoretical Sciences; but this remark leads us to signalize a distinction the neglect of which is the source of several of the most fatal errors into which philosophers have fallen. It is quite true that the Gnostic Instinct is the cause of all purely theoretical inquiry, and that every discovery of science is a gratification of curiosity. But it is not true that pure science is or can be successfully pursued for the sake of gratifying this instinct. Indeed, if it were so pursued, it would not be true that this instinct was the cause of it. Its motive would then be the Gust- Instinct, or love of pleasure. One wish may be that another wish should be gratified; but no wish can be that that very wish should be gratified. For in that case, the wish would not have any object at all, and having no object it would not be a wish. The case is precisely like that of an assertion which should have no other subject than itself. For a wish is a sort of proposition. To long for anything is to judge it to be good and urgently good. No doubt every assertion implies that it is itself true;†3 but it cannot consist of that alone; and so every wish that is reflective wishes itself gratified; but it must wish something else, besides. Hence, the hedonist, who opines that man can wish for nothing but pleasure, has fallen into a damnable error from a mere confusion of thought. We should commit the same error if we supposed the gratification of curiosity were the sole, or the principal, object of theoretical science. Curiosity is their motive; but the gratification of curiosity is not their aim.

§2. LOGIC AND SCIENTIFIC METHOD
59. It might be supposed that logic taught that much was to be accomplished by mere rumination, though every one knows that experiment, observation, comparison, active scrutiny of facts, are what is wanted, and that mere thinking will accomplish nothing even in mathematics. Logic had certainly been defined as the "art of thinking," and as the "science of the normative laws of thought." But those are not true definitions. "Dyalectica," says the logical textbook of the middle ages, "est ars artium et scientia scientiarum, ad omnium aliarum scientiarum methodorum principia viam habens,"†5 and although the logic of our day must naturally be utterly different from that of the Plantagenet epoch, yet this general conception that it is the art of devising methods of research, -- the method of methods, -- is the true and worthy idea of the science. Logic will not undertake to inform you what kind of experiments you ought to make in order best to determine the acceleration of gravity, or the value of the Ohm; but it will tell you how to proceed to form a plan of experimentation.

60. It is impossible to maintain that the superiority of the science of the moderns over that of the ancients is due to anything but a better logic. No one can think that the Greeks were inferior to any modern people whatever in natural aptitude for science. We may grant that their opportunities for research were less; and it may be said that ancient astronomy could make no progress beyond the Ptolemaic system until sufficient time had elapsed to prove the insufficiency of Ptolemy's tables. The ancients could have no dynamics so long as no important dynamical problem had presented itself; they could have no theory of heat without the steam-engine, etc. Of course, these causes had their influence, and of course they were not the main reason of the defects of the ancient civilization. Ten years' astronomical observations with instruments such as the ancients could have constructed would have sufficed to overthrow the old astronomy. The great mechanical discoveries of Galileo were made with no apparatus to speak of. If, in any direction whatever, the ancients had once commenced research by right methods, opportunities for new advances would have been brought along in the train of those that went before. But read the logical treatise of Philodemus; see how he strenuously argues that inductive reasoning is not utterly without value, and you see where the fault lay. When such an elementary point as that needed serious argumentation it is clear that the conception of scientific method was almost entirely wanting.

61. Modern methods have created modern science; and this century, and especially the last twenty-five years, have done more to create new methods than any former equal period. We live in the very age of methods. Even mathematics and astronomy have put on new faces. Chemistry and physics are on completely new tracks. Linguistics, history, mythology, sociology, biology, are all getting studied in new ways. Jurisprudence and law have begun to feel the impulse, and must in the future be more and more rapidly influenced by it.

62. This is the age of methods; and the university which is to be the exponent of the living condition of the human mind, must be the university of methods.

63. Now I grant you that to say that this is the age of the development of new methods of research is so far from saying that it is the age of the theory of methods, that it is almost to say the reverse. Unfortunately practice generally precedes theory, and it is the usual fate of mankind to get things done in some boggling way first, and find out afterward how they could have been done much more easily and perfectly. And it must be confessed that we students of the science of modern methods are as yet but a voice crying in the wilderness, and saying prepare ye the way for this lord of the sciences which is to come.

64. Yet even now we can do a little more than that. The theory of any act in no wise aids the doing of it, so long as what is to be done is of a narrow description, so that it can be governed by the unconscious part of our organism. For such purposes, rules of thumb or no rules at all are the best. You cannot play billiards by analytical mechanics nor keep shop by political economy. But when new paths have to be struck out, a spinal cord is not enough; a brain is needed, and that brain an organ of mind, and that mind perfected by a liberal education. And a liberal education -- so far as its relation to the understanding goes -- means logic. That is indispensable to it, and no other one thing is.

65. I do not need to be told that science consists of specialties. I know all that, for I belong to the guild of science, have learned one of its trades and am saturated with its current notions.†6 But in my judgment there are scientific men, all whose training has only served to belittle them, and I do not see that a mere scientific specialist stands intellectually much higher than an artisan. I am quite sure that a young man who spends his time exclusively in the laboratory of physics or chemistry or biology, is in danger of profiting but little more from his work than if he were an apprentice in a machine shop.

66. The scientific specialists -- pendulum swingers †7 and the like -- are doing a great and useful work; each one very little, but altogether something vast. But the higher places in science in the coming years are for those who succeed in adapting the methods of one science to the investigation of another. That is what the greatest progress of the passing generation has consisted in. Darwin adapted to biology the methods of Malthus and the economists; Maxwell adapted to the theory of gases the methods of the doctrine of chances, and to electricity the methods of hydrodynamics. Wundt adapts to psychology the methods of physiology;†8 Galton adapts to the same study the methods of the theory of errors; Morgan adapted to history a method from biology; Cournot adapted to political economy the calculus of variations. The philologists have adapted to their science the methods of the decipherers of dispatches. The astronomers have learned the methods of chemistry; radiant heat is investigated with an ear trumpet; the mental temperament is read off on a vernier.

67. Now although a man needs not the theory of a method in order to apply it as it has been applied already, yet in order to adapt to his own science the method of another with which he is less familiar, and to properly modify it so as to suit it to its new use, an acquaintance with the principles upon which it depends will be of the greatest benefit. For that sort of work a man needs to be more than a mere specialist; he needs such a general training of his mind, and such knowledge as shall show him how to make his powers most effective in a new direction. That knowledge is logic.

68. In short, if my view is the true one, a young man wants a physical education and an aesthetic education, an education in the ways of the world and a moral education, and with all these logic has nothing in particular to do; but so far as he wants an intellectual education, it is precisely logic that he wants; and whether he be in one lecture-room or another, his ultimate purpose is to improve his logical power and his knowledge of methods. To this great end a young man's attention ought to be directed when he first comes to the university; he ought to keep it steadily in view during the whole period of his studies; and finally, he will do well to review his whole work in the light which an education in logic throws upon it.

69. I should be the very first to insist that logic can never be learned from logic-books or logic lectures. The material of positive science must form its basis and its vehicle. Only relatively little could be done by the lecturer on method even were he master of the whole circle of the sciences. Nevertheless, I do think that I can impart to you something of real utility, and that the theory of method will shed much light on all your other studies.

70. The impression is rife that success in logic requires a mathematical head. But this is not true. The habit of looking at questions in a mathematical way is, I must say, of great advantage, and thus a turn for mathematics is of more or less service in any science, physical or moral. But no brilliant talent for mathematics is at all necessary for the study of logic.

71. The course which I am to give this year begins with some necessary preliminaries upon the theory of cognition.†9 For it is requisite to form a clear idea at the outset of what knowledge consists of, and to consider a little what are the operations of the mind by which it is produced. But I abridge this part of the course as much as possible, partly because it will be treated by other instructors, and partly because I desire to push on to my main subject, the method of science.

72. I next take up syllogism, the lowest and most rudimentary of all forms of reasoning, but very fundamental because it is rudimentary.†10 I treat this after the general style of De Morgan, with references to the old traditional doctrine. Next comes the logical algebra of Boole, a subject in itself extremely easy, but very useful both from a theoretical point of view and also as giving a method of solving certain rather frequently occurring and puzzling problems. From this subject, I am naturally led to the consideration of relative terms. The logic of relatives, so far as it has been investigated, is clear and easy, and at the same time it furnishes the key to many of the difficulties of logic, and has already served as the instrument of some discoveries in mathematics. An easy application of this branch of logic is to the doctrine of breadth and depth or the relations between objects and characters. I next introduce the conception of number, and after showing how to treat certain statistical problems, I take up the doctrine of chances. A very simple and elegant mathematical method of treating equations of finite differences puts the student into possession of a powerful instrument for the solution of all problems of probability that do not import difficulties extraneous to the theory of probability itself.

73. We thus arrive at the study of that kind of probable inference that is really distinctive; that is to say, Induction in its broadest sense -- Scientific Reasoning. The general theory of the subject is carefully worked out with the aid of real examples in great variety, and rules for the performance of the operation are given. These rules have not been picked up by hazard, nor are they merely such as experience recommends; they are deduced methodically from the general theory.

74. Finally, it is desirable to illustrate a long concatenation of scientific inferences. For this purpose we take up Kepler's great work, De Motu Stellae Martis, the greatest piece of inductive reasoning ever produced. Owing to the admirable and exceptional manner in which the work is written, it is possible to follow Kepler's whole course of investigation from beginning to end, and to show the application of all the maxims of induction already laid down.

75. In order to illustrate the method of reasoning about a subject of a more metaphysical kind, I shall then take up the scientific theories of the constitution of matter.

76. Last of all, I shall give a few lectures to show what are the lessons that a study of scientific procedure teaches with reference to philosophical questions, such as the conception of causation and the like.

77. I will assume, then, that scientific doubt never gets completely set to rest in regard to any question until, at last, the very truth about that question becomes established.†11 Taking the phenomenon as a whole, then, without considering how it is brought about, science is foredestined to reach the truth of every problem with as unerring an infallibility as the instincts of animals do their work, this latter result like the former being brought about by some process of which we are as yet unable to give any account. It is, we will say, the working of the human instinct. It is not (always considering it in its entirety,) of a rational nature, since, being infallible, it is not open to criticism, while "rational" means essentially self-criticizing, self-controlling and self-controlled, and therefore open to incessant question. But this instinctive infallibility is brought about by the exercize of reason, which is all along subject to blunder and to go wrong. The manner in which this comes about may be, I will not quite say illustrated, but may be rendered intelligible, by the following skeletal example. I call it skeletal because it involves the one character of research which is here to be considered, while attempting no representation of it in other respects. Let us suppose, then, that you have a die which may, for all you know, be loaded; and that you proceed to experiment upon it by throwing it repeatedly, counting as you go the total number of throws and also the number of them which turn up the ace side. For the sake of simplicity, I will suppose that the die is really perfect, although you do not know that it is so. After you have thrown it six times, it will be more likely to give either no ace or more than one ace than to give just one. Namely, there is one chance in three that there will be no ace in the first six throws, there are two chances in five that there will be just one, one chance in five that there will be just two; and there will remain one chance in fifteen that there will be more than two aces. Suppose you go on throwing the die a great many times, and after each throw you divide the number of aces that have turned up by the whole number of throws so far. The quotient will be [the] result for the probability of throwing an ace with this die. You will get a new and amended, though not always a really improved, result after every throw. Now although the throws are purely fortuitous, so that to most questions about them only probable answers can be given, yet one thing will certainly happen. Namely, sooner or later, probably very soon, but it may be only very late, yet certainly at length, a time will come after which all your values for the probability of throwing an ace with this die will be correct in the first figure after the decimal point. A later time there will be after which all the successive determinations will be correct in the first two figures, and so on. You will never be certain that that time has come, but it certainly some time will have come. Thus to the question, What is the first figure of the probability?; to the question, What are the first two figures, etc.; all the answer you will obtain will after a time be free from error. This will be the necessary result. Now that which is necessarily inerrant may in a somewhat indefinite sense be fairly called infallible. Thus, a skillful use of fortuitous events will bring infallibly correct replies to an endless series of questions. This kind of infallibility, which may [be], for aught we know, not to say quite probably is, the infallibility of the instinct of animals, is certainly the only kind of infallibility that can be attributed to the results of science, inasmuch as we can so little know when the very truth is reached that even the second law of motion is at this moment under indictment. Moreover, when we come to subject the processes of science to criticism, we shall find it impossible to deny that a conditional form of this kind of infallibility must be attributed to science.

78. In the light of what has been said, what are we to say to that logical fatalism whose stock in trade is the argument that I have already indicated? I mean the argument that science is predestined to reach the truth, and that it can therefore make no difference whether she observes carefully or carelessly nor what sort of formulae she treats as reasons. The answer to it is that the only kind of predestination of the attainment of truth by science is an eventual predestination, -- a predestination aliquando denique. Sooner or later it will attain the truth, nothing more. It means that if you take the most pigheaded and passionate of men who has sworn by all the gods that he never will allow himself to believe the earth is round, and give him time enough, and cram that time with experience in the pertinent sphere, and he will surely come to and rest in the truth about the form of the earth. Such is the infallibility of science. But the secret of the matter is that the man's wilfulness and prejudice will break down before such experience. Such, at least, must be our assumption, if we are to adhere to our faith in the infallibility of science. So far as this assumption goes beyond ordinary everyday experience, it rests on the deeper assumption that that which experience has done for generations of men, who a thousand years ago were substantially in that man's plight, it would do for an individual who were to go through the experiences that those generations have gone through. If one does not believe in this, then the present question does not arise. Our belief in the infallibility of science, which alone prompts the fatalistic suggestion, rests upon our experience of the overwhelming rationalizing power of experience. As long as the man keeps to his determination to exclude from his thoughts whatever might tend to make him assent to the proposition that the earth is round, he certainly will not come to that truth. Granting, therefore, that it is of the nature of experience to develope albuminous matter into rational brain, and to make the mind unceasingly agitate doubt until it finally comes to repose in the true belief, -- which is only a more developed way of formulating our belief in the infallibility of science, it is entirely uncertain when the truth will be reached. It will be reached; but only after the investigator has come, first, to a conception of the nature of truth, and to a worship of it as the purest emanation of That which is creating the universe, and then, to an understanding of the right method to absorb it from the universe of experience. It will infallibly be reached sooner or later, if favorable conditions continue; but man having a short life, and even mankind not a very long one, the question is urgent, How soon? And the answer is, as soon as a sane logic has had time to control conclusions. Everything thus depends upon rational methods of inquiry. They will make that result as speedy as possible, which otherwise would have kicked its heels in the anteroom of chance. Let us remember, then, that the precise practical service of sound theory of logic is to abbreviate the time of waiting to know the truth, to expedite the predestined result. But I here use the words 'abbreviate' and 'expedite' in a peculiar sense. Imagine a derelict wreck to be floating about on the ocean; and suppose that it will be driven hither and thither until it chances to be cast upon a shore. Then, a vessel which should go and take that derelict in tow and deliberately strand it upon the nearest shore, would be "abbreviating" or "expediting" the fulfillment of the destiny of that derelict in the same sense in which I hold that logic "abbreviates" inquiry, and "expedites" its result. It changes a fortuitous event which may take weeks or may take many decennia into an operation governed by intelligence, which will be finished within a month. This is the sense in which logic "abbreviates" and "expedites" the attainment of truth.

§3. SCIENTIFIC METHOD
79. Scientific Method: The general method of successful scientific research. The following are some of its characteristics. Cf. Science.†13

80. (1) The student's first step is to form a perfectly definite and consistent idea of what the problem really is; then he ought to develop the mathematics of the subject in hand as far as possible; and to establish a mathematical method appropriate to the particular problem, if it be one which allows exact treatment. As examples and models of what is meant, may be mentioned Maxwell's researches on colour sensation in the Philos. Trans. for 1860, Flinders Petrie's book Inductive Metrology, the last chapters of Pearson's Grammar of Science. Of course, as the student's understanding of the matter advances, he will return to this first task, and continually improve upon his first essays.

81. The second step will be to consider the logic and methodeutic of the research in hand, unless it is itself a question of pure mathematics, where the logic is inseparable from the mathematics. He will do well to study the manner in which questions somewhat analogous to his own have been successfully resolved in widely different fields; for the greatest advantage has accrued from the extension of methods from one subject to a widely different one, especially from simple to intricate matters.

82. The third step should be to reform his metaphysics, if the question is a broad one. Perhaps he thinks he has no metaphysics, and does not wish to have any. That will be a sure sign that he is badly handicapped with metaphysics of the crudest quality. The only way to disburden himself of it is to direct his attention to it. But he cannot reduce himself to anything like absolute scepticism in metaphysics without arresting his work.

83. The fourth step will be to study the laws of the phenomena dealt with, so far as they can be made out at this stage. The general order of discovery in the nomological sciences is first to pick up the phenomena by excursions in those fields in which they are to be found, with alertness of observation, with those clear ideas that make the new fact instantly recognizable as new, and with the energy that seizes upon the faint trace and follows it up. Witness the manner in which all the new phenomena of radiation have been brought to light during the last generation: cathode rays, X rays, Becquerel rays, etc. After making some acquaintance with the phenomena, the next discovery is of their laws (nomological). In the light of one's metaphysics and general conception of the department of truth dealt with, one considers what different hypotheses have any claims to investigation. The leading considerations here will be those of the 'economics' of research.†14 If, for example, a hypothesis would necessitate an experimental result that can be cheaply refuted if it is not true, or would be greatly at variance with preconceived ideas, that hypothesis has a strong claim to early examination. But one must not give up a hypothesis too readily. Many a discovery has been missed by that fault. Gravitation would have been known a decade earlier if Newton had not hastily thought it refuted, and so set back all the subsequent history of physics by something like that amount of time lost. It is likely that thousands of persons more will die of consumption -- as remote as that may seem -- than would have died if he had not made that error. The testing of the hypothesis proceeds by deducing from it experimental consequences almost incredible, and finding that they really happen, or that some modification of the theory is required, or else that it must be entirely abandoned. The law of the phenomena once made out, it only remains to measure with precision the values of the coefficients in the equation which expresses it.

84. The problem under investigation may not be of a nomological kind. Not that the phenomena are not conceivably subject to law, so that the subject may ultimately be received into the nomological sciences, -- as chemistry, for example, promises some day to mature into a nomological science; but in the present state of knowledge the question, we will suppose, cannot be so studied. Still, a certain amount of nomological study is a necessary preliminary to engaging with the problem itself. Biology calls for aid from physiology. The student who is studying the growth of languages must avail himself of all the knowledge that there is about the physics of speech sounds. In case, then, the question has not yet reached the nomological stage, the sixth step in the work will be of a classificatory nature. Such order, of a more or less imperfect kind, as can be traced in the phenomena must be made out. Students of the classificatory sciences like to call such regularities laws. The tendency is a symptom of health; because it shows that law is their ideal, and that they are striving to bring their sciences to the nomological stage. But such orderlinesses as 'Grimm's Law' (see Gender) and 'Mendeléef's Law' are not laws in the sense in which the association of ideas and the three laws of motion are laws. They are not satisfactory for a minute. They are nothing that can blend with our metaphysics; they are not of a universal kind; and they are not precise. You may imagine that there might be a chain of more and more universal, precise, and reasonable regularities leading from these to those. But there is, in fact, a great gap, which has to be acknowledged. A hypothesis may be made about the cause of the three laws of motion; but we can have no present hopes of satisfactorily proving the truth of such a thing; while we at once set to work with great hopes of making considerable steps towards explaining Mendeléef's Law and Grimm's Law. But the most important distinction between true laws and such regularities lies in the very different way in which we proceed to the discovery of the one and of the other. The whole attitude of mind is so different that it is difficult to believe that the same man would have great success in the two tasks. We have seen in our day the establishment of a grand example of each kind, the Law of the Conservation of Energy (q.v.) and the Periodic Law. The one dealt with a small number of observations. Exactitude was the main thing. The hypothesis itself sprang almost immediately from the natural light of reason. In the other case, it was necessary with a positive effort to put ideas of exactitude aside and to find order in a great tangle of facts.

85. Perhaps the problem in hand relates to one of those sciences basely called descriptive, that is, sciences which study, not classes of facts, but individual facts, such as history, descriptive astronomy, geography. No science is merely descriptive. These sciences are investigations of causes. The historian's facts of observation are not those contained in his text, but those mentioned in the footnotes -- the documents and monuments.†15 It is the supposed causes of these which make the text. Nor is he contented with a mere chronicle of striking public events; he endeavours to show what the hidden causes of them were. So the astronomer's real business is to prove the Nebular Hypothesis (q.v.) or whatever ought to replace it. The geologist does not merely make a geological map, but shows how the existing state of things must have come to pass. To do this the historian has to be a profound psychologist, the geologist a master of physics and dynamics. Just as the classificatory sciences tend to become nomological, so the descriptive, or explanatory, sciences tend to become classificatory. The astronomer finds so many examples of systems in formation, that he can formulate the cycle of events through which they generally pass; as the historian formulates cycles through which communities usually pass, and the geologist formulates cycles through which continents commonly pass. These are analogous to the cyclical laws of the classificatory sciences.

86. But perhaps the problem before the student is not one of theoretical physics or of theoretical psychics, but a practical problem. He wishes to invent. In that case he ought to have a great knowledge both of facts about men's minds and of facts about matter; for he has to adapt the one to the other. He ought to know more than any pure scientist can be expected to know. Of course, as the world goes, he does not.

87. (2) The most vital factors in the method of modern science have not been the following of this or that logical prescription -- although these have had their value too -- but they have been the moral factors. First of these has been the genuine love of truth and conviction that nothing else could long endure. Given that men strive after the truth, and, in the nature of things, they will get it in a measure. The greatest difference between the scientific state of the modern scientific era from Copernicus and the middle ages, is that now the whole concern of students is to find out the truth; while then it was to put into a rational light the faith of which they were already possessed. The chief obstacle to the advance of science among students of science in the modern era has been that they were teachers, and feared the effect of this or that theory. But the salvation from this danger has been the fact that there was no vast institution which anybody for a moment hoped could withstand the mighty tide of fact. The next most vital factor of the method of modern science is that it has been made social. On the one hand, what a scientific man recognizes as a fact of science must be something open to anybody to observe, provided he fulfils the necessary conditions, external and internal. As long as only one man has been able to see a marking upon the planet Venus, it is not an established fact. Ghost stories and all that cannot become the subject of genuine science until they can in some way be welded to ordinary experience.†16 On the other hand, the method of modern science is social in respect to the solidarity of its efforts. The scientific world is like a colony of insects, in that the individual strives to produce that which he himself cannot hope to enjoy. One generation collects premises in order that a distant generation may discover what they mean. When a problem comes before the scientific world, a hundred men immediately set all their energies to work upon it. One contributes this, another that. Another company, standing upon the shoulders of the first, strike a little higher, until at last the parapet is attained. Still another moral factor of the method of science, perhaps even more vital than the last, is the selfconfidence of it. In order to appreciate this, it is to be remembered that the entire fabric of science has to be built up out of surmises at truth. All that experiment can do is to tell us when we have surmised wrong. The right surmise is left for us to produce. The ancient world under these circumstances, with the exception of a few men born out of their time, looked upon physics as something about which only vague surmises could be made, and upon which close study would be thrown away. So, venturing nothing, they naturally could gain nothing. But modern science has never faltered in its confidence that it would ultimately find out the truth concerning any question in which it could apply the check of experiment.

88. These are some of the more vital factors of the method of modern science. For the purely logical elements the reader should consult special topics, e.g. Reasoning,†17 Probable Inference,†18 Psychophysical Methods, Errors of Observation, Empirical Logic, Variation, etc.

89. Verification: It is desirable to understand by a verifiable hypothesis one which presents an abundance of necessary consequences open to experimental tests, and which involves no more than is necessary to furnish a source of those consequences. The verification will not consist in searching the facts in order to find features that accord or disagree with the hypothesis. That is to no purpose whatsoever. The verification, on the contrary, must consist in basing upon the hypothesis predictions as to the results of experiments, especially those of such predictions as appear to be otherwise least likely to be true, and in instituting experiments in order to ascertain whether they will be true or not.

90. These experiments need not be experiments in the narrow and technical sense, involving considerable preparation. That preparation may be as simple as it may. The essential thing is that it shall not be known beforehand, otherwise than through conviction of the truth of the hypothesis, how these experiments will turn out. It does not need any long series of experiments, so long as every feature of the hypothesis is covered, to render it worthy of positive scientific credence. What is of much greater importance is that the experiments should be independent, that is, such that from the results of some, the result of no other should be capable of reasonable surmise, except through the hypothesis. But throughout the process of verification the exigencies of the economy of research should be carefully studied from the point of view of its abstract theory.

91. When, in 1839, Auguste Comte laid down the rule that no hypothesis ought to be entertained which was not capable of verification, it was far from receiving general acceptance. But this was chiefly because Comte did not make it clear, nor did he apparently understand, what verification consisted in. He seemed to think, and it was generally understood, that what was meant was that the hypothesis should contain no facts of a kind not open to direct observation. That position would leave the memory of the past as something not so much as to be entertained as plausible.

§4. SIMPLICITY
92. Parsimony (law of): Ockham's razor, i.e. the maxim 'Entia non sunt multiplicanda praeter necessitatem.' The meaning is, that it is bad scientific method to introduce, at once, independent hypotheses to explain the same facts of observation.

93. Though the maxim was first put forward by nominalists, its validity must be admitted on all hands, with one limitation; namely, it may happen that there are two theories which, so far as can be seen, without further investigation, seem to account for a certain order of facts. One of these theories has the merit of superior simplicity. The other, though less simple, is on the whole more likely. But this second one cannot be thoroughly tested by a deeper penetration into the facts without doing almost all the work that would be required to test the former. In that case, although it is good scientific method to adopt the simpler hypothesis to guide systematic observations, yet it may be better judgment, in advance of more thorough knowledge, to suppose the more complex hypothesis to be true. For example, I know that men's motives are generally mixed. If, then, I see a man pursuing a line of conduct which apparently might be explained as thoroughly selfish, and yet might be explained as partly selfish and partly benevolent, then, since absolutely selfish characters are somewhat rare, it will be safer for me in my dealings with the man to assume the more complex hypothesis to be true; although were I to undertake an elaborate examination of the question, I ought to begin by ascertaining whether the hypothesis of pure selfishness would quite account for all he does.

94. The whole aim of science is to find out facts, and to work out a satisfactory theory of them. Still, a theory does not necessarily lose its utility by not being altogether true. . . .

95. No theory in the positive sciences can be supposed to satisfy every feature of the facts. Although we know that the law of gravitation is one of the most perfect of theories, yet still, if bodies were to attract one another inversely as a power of the distance whose exponent were not 2, but 2.000001, the only observable effect would be a very slow rotation of the line of apsides of each planet. Now the lines of apsides all do rotate in consequence of perturbations, which virtually do alter slightly the sun's attraction, and thus such an effect would probably only produce slight discrepancies in the values obtained for the masses of the planets. In very many cases, especially in practical problems, we deliberately go upon theories which we know are not exactly true, but which have the advantage of a simplicity which enables us to deduce their consequences. This is true of almost every theory used by engineers of all kinds. The most extraordinary departure from the known facts occurs when hydrodynamics is applied, where the theory is in striking opposition to facts which obtrude themselves upon every spectator of moving water. Nevertheless, even in this case, the theory is not useless.

96. In all the explanatory sciences theories far more simple than the real facts are of the utmost service in enabling us to analyse the phenomena, and it may truly be said that physics could not possibly deal even with its relatively simple facts without such analytic procedure. Thus, the kinetical theory of gases, when first propounded, was obliged to assume that all the molecules were elastic spheres, which nobody could believe to be true. If this is necessary even in physics, it is far more indispensable in every other science, and most of all in the moral sciences, such as political economy. Here the sane method is to begin by considering persons placed in situations of extreme simplicity, in the utmost contrast to those of all human society, and animated by motives and by reasoning powers equally unlike those of real men. Nevertheless, in this way alone can a base be obtained from which to proceed to the consideration of the effects of different complications. Owing to the necessity of making theories far more simple than the real facts, we are obliged to be cautious in accepting any extreme consequences of them, and to be also upon our guard against apparent refutations of them based upon such extreme consequences.

§5. KINDS OF REASONING
97. First of all I must establish, as well as I can, the proposition that all Reasoning is either Deduction, Induction, or Retroduction.†21

98. Unfortunately, I am unable to make this as evident as would be desirable, although I think there is very little room for doubting it, since in the course of a long life of active study of reasonings, during which I have never met with any argument not of a familiar type without carefully analyzing and studying it, I have constantly since 1860, or 50 years, had this question prominently in mind, and if I had ever met with an argument not of one of these three kinds, I must certainly have perceived it. But I never have found any such kind of argument except Analogy, which, as I have shown, is of a nature, -- a mixture of the three recognized kinds. Therefore, it may be taken as substantially certain that I have never in 50 years met with a reasoning of any fourth type.

99. Now I have not been the only man whose attention would have been roused by the appearance of any such reasoning; and if anybody in the civilized world had found such an argument, I should have heard of it.

100. Now it is of the nature of a genus of reasoning that it applies to any kind of matter in inexhaustible variety. It is therefore very difficult to believe that there is any kind of reasoning that has not been familiarly employed and known by all the world from time immemorial. On the whole, then, I think my negative experience ought to be pretty convincing, inductively.

101. Though I do not profess to render it strictly speaking, evident that there are but the three types of reasoning, yet it will be interesting to see how nearly I can approach that desideratum.

102. A sound reasoning justifies us in some kind of belief in the truth of a proposition that in the absence of the reasoning we should not have been so much justified in believing.

103. In reasoning, one is obliged to think to oneself. In order to recognize what is needful for doing this it is necessary to recognize, first of all, what "oneself" is. One is not twice in precisely the same mental state. One is virtually (i.e. for pertinent purposes, the same as if one were) a somewhat different person, to whom one's present thought has to be communicated. Consequently, one has to express one's thought so that that virtually other person may understand it. One may, with great advantage, however, employ a language, in thinking to oneself, that is free from much explanation that would be needed in explaining oneself to quite a different person. One can establish conventions with oneself, which enable one to express the essence of what [one] has to communicate free from signs that are not essential. For that reason for example a mathematician has, in thinking of mathematical subjects, an immense advantage. Thus if he has to express to himself a force he will think of D(2/t)S, which, he will remember, or can readily see if he should not remember it, is the same as Ds[1/2(DtS)2].†22 Or he may express the same thing by means of a geometrical diagram, and that in any one of various forms. In like mathematical fashion Existential Graphs †23 enable me here and there greatly to abridge the labor and increase the exactitude of my thought by putting intricate logical relations in the forms that display to me precisely what they involve.

104. In particular, [the system of] Existential Graphs shows clearly that all logical relations are compounds of the relation of consequence, provided we look upon identity as so composed. But Existential Graphs does not so regard Identity. That is, it does not assert that to say that the Battle of Waterloo was the final downfall of Napoleon is precisely the same as to say, that if the Battle of Waterloo was the final downfall of Napoleon then for Napoleon to lose that battle as completely as he did, necessarily involved his final overthrow, while if he had not so lost that battle, he would not then and there have been finally overthrown.

105. My reason in constructing the system of Existential Graphs for not allowing such an identity was that no single actual event can follow as logically consequent upon any other, since if it [were] otherwise in the smallest particular, it would be a different event. If in the Battle of Waterloo one man's wound were shifted a hundredth of an inch, or if it had occurred a tenth of a second earlier or later, the Battle would not have been that actual event that did take place; and we never can be in a situation to affirm that under specified circumstances that which did take place must have taken place with such absolute precision; and it is the merest moonshine to claim to know that only as any describable circumstances had taken place the Battle of Waterloo or any other actual historical event must have taken place precisely as it did. It is a pretty theory although there are grave objections to its precise truth, but to claim to know it is a pretension that I do not think any sober minded man who sufficiently considers the subject will allow himself to make. It has all the ear-marks of the doctrinaire, the man who is willing to accept theories as absolutely true. All the difficulties into which metaphysicians contrive to snarl themselves up are traceable to just that doctrinaire disposition. Certainly, I will take care that my system of logic is not inoculated with that easily avoidable but fatal infection.

106. Therefore, the System of Graphs is so constructed that nothing can be recognized as an apodictic proof that in any circumstances defined in general terms, an event must have happened precisely as it did.

107. But as long as we have to do with general states of things, Existential Graphs analyzes all logical relations into cases of the one relation of consequence, that is the relation between one general description of event, A, an antecedent, and another general description of event, C, a consequent, the relation consisting in the fact that whenever A is realized, C will be realized. All known laws of dynamics as well as all other truths consist of such relations.

108. I will not, therefore, admit that we know anything whatever with absolute certainty.†24 It is possible that twice two is not four. For a computer might commit an error in the multiplication of 2 by 2; and whatever might happen once might happen again. Now 2 has never been multiplied by 2 but a finite number of times; and consequently all such multiplications may have been wrong in the same way. It is true that it would be difficult to imagine a greater folly than to attach any serious importance to such a doubt. Still foolish as that would be, its folly would not be so great as to assert that there is some number of repetitions of a multiplication that renders their result, if all agree, absolutely certain. For if this be the case there is some number which is the least that is sufficient to produce certainty. Let this number be denoted by N. Then N-1 repetitions of the multiplication do not yield an absolutely certain result, but one more, if it agree with all the others, will have that result. Consequently a single multiplication will be sufficient to give us absolute certainty, that the result is the same, unless some other one of N-1 repetitions should give a different result. Thus, disregarding the particular proposition in question one is driven to maintaining that a single experiment is capable of giving us certain knowledge as to the result of any number of experiments. This is sufficient to show that such an assumption is dangerous in the extreme. It is also absurd from various points of view. The only safety is to say that man is incapable of absolute certainty.

109. But some one will ask me, "Do you, then, really entertain any doubt that twice two is four?" To this I must answer, "No, as well as I can perceive, there is not the slightest real doubt of it in my mind."†25 "But," he will say, "how can that be? You say it is not certain. Ought you not then, to entertain a doubt of it; and if you feel that it ought to be doubted, do you not, ipso facto, actually doubt it?" I reply: "Doubt is a certain kind of feeling. It has not only grades of intensity, but also varieties of quality. Now if I were able to modify my state of mind by a sufficiently slight tincture of the right kind of doubt, I ought to do so. But if I were to attempt really to feel any doubt at all, I should certainly either feel none at all or else millions upon millions of times too much. For I could not in the least recognize a tincture so small nor even one that should be millions of times too great. If I were to devote my whole life to the useless task of trying to make such slight distinctions in my feelings, I could not come near to the requisite delicacy. My feeling of doubt is one of the coarser of my sensations; and there would be no practical use in making it more delicate than it is, for it is already so far more delicate than that of almost all the persons with whom I converse, that I often find an insuperable difficulty in making them comprehend the slighter grades of my feeling, and there is no practical difference in my conduct whether, say, 3/8 or 5/13 be the proper degree of doubt about a matter not measurable. It would be a waste of time to adjust my feeling of doubt more accurately, since it neither would have, nor ought to have, any effect upon my scientific conduct. Instead of wasting effort on my feeling, I devote my energies to learning more about the subjects concerning which I have any considerable doubts, while very small doubts I neglect until I can reduce the amount of my doubt concerning subjects of greater importance."

§6. KINDS OF INDUCTION
110. Suppose we define Inductive reasoning as that reasoning whose conclusion is justified not by there being any necessity of its being true or approximately true but by its being the result of a method which if steadily persisted in must bring the reasoner to the truth of the matter or must cause his conclusion in its changes to converge to the truth as its limit. Adopting this definition, I find that there are three orders of induction of very different degrees of cogency although they are all three indispensable.

111. The first order of induction, which I will call Rudimentary Induction, or the Pooh-pooh argument, proceeds from the premiss that the reasoner has no evidence of the existence of any fact of a given description and concludes that there never was, is not, and never will be any such thing. The justification of this is that it goes by such light as we have, and that truth is bound eventually to come to light; and therefore if this mode of reasoning temporarily leads us away from the truth, yet steadily pursued, it will lead to the truth at last. This is certainly very weak justification; and were it possible to dispense with this method of reasoning, I would certainly not recommend it. But the strong point of it is that it is indispensable. It goes upon the roughest kind of information, upon merely negative information; but that is the only information we can have concerning the great majority of subjects.

112. I find myself introduced to a man without any previous warning. Now if I knew that he had married his grandmother and had subsequently buried her alive, I might decline his acquaintance; but since I have never heard the slightest suspicion of his doing such a thing, and I have no time to investigate idle surmises, I presume he never did anything of the sort. I know a great many men, however, whose whole stock of reasoning seems to consist in this argument, which they continue to use where there is positive evidence and where this argument consequently loses all force. If you ask such a man whether he believes in the liquefaction of the blood of St. Januarius, he will say no. Why not? Well, nothing of that kind ever came within the range of my experience. But it did come within the range of Sir Humphrey Davy's experience, who was granted every facility for the thorough investigation of it. His careful report simply confirms the usual allegations with more circumstantial details. You are not justified in poohpoohing such observations; and that the fact is contrary to the apparent ordinary course of nature is no argument whatever. You are bound to believe it, until you can bring some positive reason for disbelieving it.

113. In short this rudimentary kind of induction is justified where there is no other way of reasoning; but it is of all sound arguments the very weakest and must disappear as soon as any positive evidence is forthcoming.

114. The second order of induction consists in the argument from the fulfillment of predictions. After a hypothesis has been suggested to us by the agreement between its consequences and observed fact, there are two different lines that our further studies of it may pursue. In the first place, we may look through the known facts and scrutinize them carefully to see how far they agree with the hypothesis and how far they call for modifications of it. That is a very proper and needful inquiry. But it is Abduction, not Induction, and proves nothing but the ingenuity with which the hypothesis has been adapted to the facts of the case. To take this for Induction, as a great proportion of students do, is one of the greatest errors of reasoning that can be made. It is the post hoc ergo propter hoc fallacy, if so understood. But if understood to be a process antecedent to the application of induction, not intended to test the hypothesis, but intended to aid in perfecting that hypothesis and making it more definite, this proceeding is an essential part of a well-conducted inquiry.

115. The other line which our studies of the relation of the hypothesis to experience may pursue, consists in directing our attention, not primarily to the facts, but primarily to the hypothesis, and in studying out what effect that hypothesis, if embraced, must have in modifying our expectations in regard to future experience. Thereupon we make experiments, or quasi-experiments,†27 in order to find out how far these new conditional expectations are going to be fulfilled. In so far as they greatly modify our former expectations of experience and in so far as we find them, nevertheless, to be fulfilled, we accord to the hypothesis a due weight in determining all our future conduct and thought. It is true that the observed conformity of the facts to the requirements of the hypothesis may have been fortuitous. But if so, we have only to persist in this same method of research and we shall gradually be brought around to the truth. This gradual process of rectification is in great contrast to what takes place with rudimentary induction where the correction comes with a bang. The strength of any argument of the Second Order depends upon how much the confirmation of the prediction runs counter to what our expectation would have been without the hypothesis. It is entirely a question of how much; and yet there is no measurable quantity. For when such measure is possible the argument assumes quite another complexion, and becomes an induction of the Third Order. Inductions of the second order are of two varieties, that are logically quite distinct.

116. The weaker of these is where the predictions that are fulfilled are merely of the continuance in future experience of the same phenomena which originally suggested and recommended the hypothesis, expectations directly involved in holding the hypothesis. Even such confirmation may have considerable weight. This, for example, is the way in which the undulatory theory of light stood before Maxwell. The phenomena of interference suggested undulations, which measures of the velocity of light in different media confirmed; and the phenomena of polarization suggested transverse vibrations. All the direct expectations involved in the hypothesis were confirmed, except that there no phenomena due to longitudinal vibrations were found. But all physicists felt that it was a weakness of the theory that no unexpected predictions occurred. The rotation of the plane of polarization was an outstanding fact not accounted for.

117. The other variety of the argument from the fulfillment of predictions is where truths ascertained subsequently to the provisional adoption of the hypothesis or, at least, not at all seen to have any bearing upon it, lead to new predictions being based upon the hypothesis of an entirely different kind from those originally contemplated and these new predictions are equally found to be verified.

118. Thus Maxwell, noticing that the velocity of light had the same value as a certain fundamental constant relating to electricity, was led to the hypothesis that light was an electromagnetic oscillation. This explained the magnetic rotation of the plane of polarization, and predicted the Hertzian waves. Not only that, but it further led to the prediction of the mechanical pressure of light, which had not at first been contemplated.

119. The second order of induction only infers that a theory is very much like the truth, because we are so far from ever being authorized to conclude that a theory is the very truth itself, that we can never so much as understand what that means. Light is electro-magnetic vibrations; that is to say, it [is] something very like that. In order to say that it is precisely that, we should have to know precisely what we mean by electro-magnetic vibrations. Now we never can know precisely what we mean by any description whatever.

120. The third order of induction, which may be called Statistical Induction, differs entirely from the other two in that it assigns a definite value to a quantity. It draws a sample of a class, finds a numerical expression for a predesignate character of that sample and extends this evaluation, under proper qualification, to the entire class, by the aid of the doctrine of chances. The doctrine of chances is, in itself, purely deductive. It draws necessary conclusions only. The third order of induction takes advantage of the information thus deduced to render induction exact.

121. This family of inductions has three different kinds quite distinct logically. Beginning with the lowest and least certain, we have cases in which a class of individuals recur in endless succession and we do not know in advance whether the occurrences are entirely independent of one another or not. But we have some reason to suppose that they would be independent and perhaps that they have some given ratio of frequency. Then what has to be done is to apply all sorts of consequences of independence and see whether the statistics support the assumption. For instance, the value of the ratio of the circumference of a circle to its diameter, a number usually called π has been calculated in the decimal notation, to over seven hundred figures. Now as there is not the slightest reason to suppose that any law expressible in a finite time connects the value of π with the decimal notation or with any whole number, we may presume that the recurrences of any figure say 5 in that succession are independent of one another and that there is simply a probability of 1/10 that any figure will be a 5.

122. In order to illustrate this mode of induction, I have made a few observations on the calculated number. There ought to be, in 350 successive figures, about 35 fives. The odds are about 2 to 1 that there will be 30-39 [and] 3 to 1 that there will be 29-41. Now I find in the first 350 figures 33 fives, and in the second 350, 28 fives, which is not particularly unlikely under the supposition of a chance distribution. During the process of counting these 5's, it occurred to me that as the expression of a rational fraction in decimals takes the form of a circulating decimal in which the figures recur with perfect regularity, so in the expression of a quantity like π, it was naturally to be expected that the 5's, or any other figure, should recur with some approach to regularity. In order to find out whether anything of this kind was discernible I counted the fives in 70 successive sets of 10 successive figures each. Now were there no regularity at all in the recurrence of the 5's, there ought among these 70 sets of ten numbers each to be 27 that contained just one five each; and the odds against there being more than 32 of the seventy sets that contain just one five each is about 5 to 1. Now it turns out upon examination that there are 33 of the sets of ten figures which contain just one 5. It thus seems as if my surmise were right that the figures will be a little more regularly distributed than they would be if they were entirely independent of one another. But there is not much certainty about it. This will serve to illustrate what this kind of induction is like, in which the question to be decided is how far a given succession of occurrences are independent of one another and if they are not independent what the nature of the law of their succession is.

123. In the second variety of statistical induction, we are supposed to know whether the occurrences are independent or not, and if not, exactly how they are connected, and the inquiry is limited to ascertaining what the ratio of frequency is, after the effects of the law of succession have been eliminated. As a very simple example, I will take the following. The dice that are sold in the toy shops as apparatus for games . . . are usually excessively irregular. It is no great fault, but rather enhances the Christmas gaiety. Suppose, however, that some old frump with an insatiable appetite for statistics [were to] get hold of a die of that sort, and he will spend his Christmas in throwing it and recording the throws in order to find out the relative frequency with which the different faces turn up. He assumes that the different throws are independent of one another and that the ten thousand or so which he makes will give the same relative frequencies of the different faces as would be found among any similar large number of throws until the die gets worn down. At least he can safely assume that this will be the case as long as the die is thrown out of the same box by the same person in the same fashion.

124. This second variety is the usual and typical case of statistical induction. But it occasionally happens that we can sample a finite collection of objects by such a method that in the long run any one object of the collection would be taken as often as every other and any one succession as often as any other. This may [be] termed a random selection. It is obviously possible only in the case of an enumerable collection. When this sort of induction is possible it far surpasses every other in certainty and may closely approach that of demonstration itself.

125. I have now passed in review all the modes of pure induction with which I am acquainted. Induction may, of course, be strengthened or weakened by the addition of other modes of argument leading to the same conclusion or to a contrary conclusion. It may also be strengthened or weakened by arguments which do not directly affect the conclusion of the induction but which increase or diminish the strength of its procedure. There are in particular four kinds of uniformities which may greatly affect an induction.

126. In the first place the members of a class may present a greater or less general resemblance as regards certain kinds of characters. Birds for example are, generally speaking, much more alike than are fishes or mammals; and that will strengthen any induction about birds. Orchids, on the other hand, are extraordinarily various.

127. In the second place a character may have a greater or less tendency to be present or absent throughout the whole of certain kinds of groups. Thus, coloration often differs within one species, while the number of the principal bones of the skeleton, and almost all characters which are developed early in individual life and which persist to maturity are common to all the members of large classes.

128. In the third place, a certain set of characters may be more or less intimately connected, so as probably to be present or absent together in certain kinds of objects. Thus, we generally associate insistency upon minute forms with narrowness of mind, cleanliness with godliness, and so on.

129. In the fourth place, an object may have more or less tendency to possess the whole of certain sets of characters when it possesses any of them. Thus, one meets one man whose views whatever they may be are extreme, while the opinions of another form a strange mosaic.

130. From the knowledge of a uniformity of any one of these four classes or from the knowledge of the lack of such uniformity it may be deductively inferred that a given induction is either stronger or weaker than it otherwise would be.

§7. UNIFORMITY OF NATURE
131. There is still another sense in which we might speak of the uniformity of nature. If we select a good many objects on the principle that they shall belong to a certain class and then find that they all have some common character, pretty much the whole class will generally be found to have that character. Or if we take a good many of the characters of a thing at random, and afterwards find a thing which has all these characters, we shall generally find that the second thing is pretty near the same as the first.

132. It seems to me that it is this pair of facts rather than any others which are properly expressed by saying that nature is uniform. We shall see that it is they which are the leading principles of scientific inference. Peirce: CP 7.132 Cross-Ref:†† Let us ask, then, whether these facts are statements of a particular constitution of the world so as to be properly speaking matters of fact or whether they are purely formal propositions, laws of logic, having no more application to one state of things than they would have to any other.

133. In the first place, I would call your attention to the quantitative indeterminateness of both propositions. The first speaks of a good many samples being selected, and of pretty much all the things in the class from which they are taken being like them, and of this occurring almost always. The second speaks of a good many characters of a thing being taken, and of any thing found to have them being pretty near the same thing, and of this happening almost always. We have no means whatsoever of defining the propositions in either of the three respects in which they are thus seen to be so utterly vague.

134. Now you know how a malicious person [who] wishes to say something ill of another, prefers insinuation; that is, he speaks so vaguely that he suggests a great deal while he expressly says nothing at all. In this way he avoids being confronted by fact. It is the same way with these principles of scientific inference. They are so vague that you cannot bring them to any touch-stone of experience. They rather insinuate a uniformity in nature than state it. And as insinuation always expresses the state of feeling of the person who uses it rather than anything concerning its object, so we may suppose these principles express rather the scientific attitude than a scientific result.

135. But what if we were in a world of chance? How would it be with these principles then, or, to simplify the matter, with the first principle? In that case, it would be extremely seldom that, having selected a number of objects as having certain characters, we should find that they had any other common character; and thus there would be very little applicability for this principle. But, we have seen that the proportion of cases where this principle applies is indefinitely small in our present world. Cases might occur, doubtless would in a world of chance and when they did occur the principle doubtless would hold true.

136. It is a mistake to suppose that there would be no laws in a world of chance. At least, so I should think. Suppose we were to throw a die any number of times and set down the numbers thrown in a column. I could show you that there would be some very curious laws in reference to those numbers. They would appear quite surprizing. So that chance is not the abrogation of all laws.

137. But there is a peculiarity about those laws that chance does not abrogate; suppose that in throwing the die other numbers had turned up from those which actually turned up, so that the row of numbers would have been somewhat different; still the laws would have held; they would hold with one set of numbers as well as with another. Whereas if we were to give a whale legs or a woman wings, the laws of the animal kingdom would be interfered with. So that there are two kinds of laws, those which in a different state of things would continue to hold good and those which in a different state of things would not hold good. The former we call formal laws, the latter material laws. The formal laws do not depend on any particular state of things, and hence we say we have not derived them from experience; that is to say, any other experience would have furnished the premisses for them as well as that which we have experienced; while to discover the material laws we require to have known just such facts as we did. But as the laws which we have mentioned, that as is sample so is the whole and that the sameness of a number of characters manifests identity, are laws which would hold so long as there were any laws, though only formal ones, it is plain that no alteration in the constitution of the world would abrogate them, so that they are themselves formal laws, and therefore not laws of nature but of the conditions of knowledge in general.

138. Two classes of thinkers wish to make the difference between formal and material laws merely relative; namely, those who would reduce all formal laws to material laws, and those who would reduce all material laws to formal laws. But neither can deny that there is a great difference between what we must consider formal and what we must consider material laws. Those who would reduce all material laws to formal laws, have indeed shown that what we call material laws are only those which we cannot discover to be formal; and thus that all material laws may be formal; and in so doing they have cut anyone off from saying that there is a peculiar uniformity of nature consisting in its material laws. On the other hand, those who would reduce formal laws to material laws, among whom is Mr. Mill, have shown that laws may be thought to be formal, that is to be such that a violation of them is unimaginable, owing to a want of imaginative power in us arising from a defective experience, and they infer from that that all formal laws may be material. But so long as there are any laws whatsoever, these laws that the whole is as the sample and that identity goes with similarity in respects [not] chosen to make out the similarity, these laws I say must exist. For these are but as much as to say that there is law. That we shall see in future lectures. Now all law may, in one sense, be contingent. But that there should be knowledge without the existence of law, that there should be intelligence without anything intelligible, all admit to be impossible. These laws therefore cannot be abrogated without abrogating knowledge; and thus are the formal conditions of all knowledge.

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Footnotes
Peirce: CP 7.49 Fn 1 p 37
†1 (Ed.) Paragraphs 49-52 are from manuscript L, undated (but cf. 59n4),Widener IB2-9.
Paragraphs 53-58 are from "Of the Classification of the Sciences. Second Paper. Of the Practical Sciences," Widener II. Paragraphs 53-57, 381n19 and 58 come from the manuscript in that order. This manuscript is dated c.1902 on the basis of references in it.

Peirce: CP 7.54 Fn 2 p 40
†2 (Ed.) This statement appears on page 3 in the edition edited by Alois Höfler, published by C. E. M. Pfeffer, Leipzig, 1900.

Peirce: CP 7.58 Fn 3 p 43
†3 (Ed.) Cf. 5.340.

Peirce: CP 7.59 Fn 4 p 43
†4 (Ed.) Paragraphs 59-76 are "Introductory Lecture on the Study of Logic," [JHUC] 2(Nov 1882)11-12, with two preliminary paragraphs omitted.
Paragraphs 77-78 are from manuscript N, Widener IB2-9, undated, but the manuscript contains results from the census of 1900. This manuscript and manuscript L (cf. 49n1) are probably parts of the same work.

Peirce: CP 7.59 Fn 5 p 44
†5 (Ed.) Orbellis (Nicholaus de), Expositio super textu Petri Hispani, Super libro Peryhermenias, Venice, 1500, fol. a3v.

Peirce: CP 7.65 Fn 6 p 46
†6 (Ed.) See Book I of the present volume.

Peirce: CP 7.66 Fn 7 p 46
†7 (Ed.) Peirce's main task in the United States Coast Survey was to measure the force of gravity by swinging a pendulum.

Peirce: CP 7.66 Fn 8 p 46
†8 (Ed.) See the review of Wundt's book in [CP] VIII, Book I, Review 14.

Peirce: CP 7.71 Fn 9 p 47
†9 (Ed.) See [CP] V.

Peirce: CP 7.72 Fn 10 p 48
†10 (Ed.) See the rest of the present book and [CP] II for discussions of most of the topics mentioned in this and the following paragraphs.

Peirce: CP 7.77 Fn 11 p 49
†11 (Ed.) Cf. 5.383ff.

Peirce: CP 7.79 Fn 12 p 52
†12 (Ed.) Paragraphs 79-88 are "Scientific Method," Dictionary of Philosophy and Psychology (edited by James Mark Baldwin), Vol. II, 1902, pp. 500-503. Paragraphs 89-91 are "Verification," ibid., pp. 761-762.

Peirce: CP 7.79 Fn 13 p 52
†13 (Ed.) Peirce did not define this term for Baldwin's Dictionary, but see Section 1 of the present chapter (49ff.).

Peirce: CP 7.83 Fn 14 p 54
†14 (Ed.) See Chapter 2, "Economy of Research," in the present book.

Peirce: CP 7.85 Fn 15 p 55
†15 (Ed.) See Chapter 3, "The Logic of Drawing History from Ancient Documents," in the present book.

Peirce: CP 7.87 Fn 16 p 57
†16 (Ed.) Cf. Chapter 5, "Telepathy and Perception," in Book III of the present volume.

Peirce: CP 7.88 Fn 17 p 57
†17 (Ed.) 2.773-778.

Peirce: CP 7.88 Fn 18 p 58
†18 (Ed.) 2.783-787.

Peirce: CP 7.92 Fn 19 p 59
†19 (Ed.) Paragraphs 92-93 are "Parsimony (law of)," Dictionary of Philosophy and Psychology (edited by James Mark Baldwin), Vol. II, 1902, p. 264. Paragraphs 94-96 are from "Theory," ibid., pp. 693-694.

Peirce: CP 7.97 Fn 20 p 61

†20 (Ed.) "Notes for my Logical Criticism of Articles of the Christian Creed," Widener IB3. Judging by the reference to 1860 in the second paragraph, this is to be dated c.1910.

Peirce: CP 7.97 Fn 21 p 61
†21 (Ed.) Peirce also uses "Abduction" and "Hypothesis" for what he here calls "Retroduction."

Peirce: CP 7.103 Fn 22 p 62
†22 (Ed.) That is, in the case of a unit mass, the force is equal to the acceleration, and it is also equal to the derivative of the energy with respect to distance.

Peirce: CP 7.103 Fn 23 p 62
†23 (Ed.) See [CP] IV, Book II.

Peirce: CP 7.108 Fn 24 p 64
†24 (Ed.) Peirce's fallibilism, the doctrine that there is no absolute certainty in knowledge, is discussed at 1.8ff., and elsewhere in [CP] I.

Peirce: CP 7.109 Fn 25 p 64
†25 (Ed.) Cf. the discussion of unreal doubt as contrasted to genuine doubt at 5.265 and elsewhere in [CP] V.

Peirce: CP 7.110 Fn 26 p 65
†26 (Ed.) From Vol. I of Lecture 7 of the Lowell Lectures of 1903, Widener IB2-4.
Cf. 2.755-760 and 7.208-217 for related treatments of the same topic.

Peirce: CP 7.115 Fn 27 p 67
†27 (Ed.) "The Deductions which we base upon the hypothesis which has resulted from Abduction produce conditional predictions concerning our future experience. That is to say, we infer by Deduction that if the hypothesis be true, any future phenomena of certain descriptions must present such and such characters. We now institute a course of quasi-experimentation in order to bring these predictions to the test, and thus to form our final estimate of the value of the hypothesis, and this whole proceeding I term Induction. I speak of quasiexperimentation because the term experiment is, according to the usage of scientific men, restricted to the operation of bringing about certain conditions. The noting of the results of experiments or of anything else to which our attention is directed in advance of our noting it, is called Observation. But by quasi-experimentation I mean the entire operation either of producing or of searching out a state of things to which the conditional predictions deduced from hypothesis shall be applicable and of noting how far the prediction is fulfilled." From an earlier passage of the same lecture (110n26). Peirce: CP 7.131 Fn 28 p 72
†28 (Ed.) From Lecture IV (c.1866) of the same series from which 7.579- 596 are taken, Widener IB2-10; cf. 7.579n34.