Dado

“ If every term of a collection can be hooked on to a number, and all the finite numbers are used once, and only once, in the process, then our collection must have just as many terms as there are finite numbers. ”

“ Further, when a proposition which is a datum asserts the existence of something, we say that the something is a datum, as well as the proposition asserting its existence. Thus those objects of whose existence we become certain through perception are said to be data. ”

“ In any science other than psychology the datum is primarily a perception, in which only the sensational core is ultimately and theoretically a datum, though some such accretions as turn the sensation into a perception are practically unavoidable. But if we postulate an ideal observer, he will be able to isolate the sensation, and treat this alone as datum. There is, therefore, an important sense in which we may say that, if we analyse as much as we ought, our data, outside psychology, consist of sensations, which include within themselves certain spatial and temporal relations. ”

“ The object of the Data is to show that when certain things—lines, angles, spaces, ratios, &c.—are given by hypothesis, certain other things are given, that is, are determinable. The book, as we are expressly told, and as we may gather from its contents, was intended for the investigation of problems; and it has been conjectured that Euclid must have extended the method of the Data to the investigation of theorems. ”

“ Is it a breath emanating from God to organize the material universe where nothing obtains form except by the Divinity which is an effect of Number? The least as well as the greatest of creations are distinguishable from each other by quantities, qualities, dimensions, forces,—all attributes created by Number. The infinitude of Numbers is a fact proved to your soul, but of which no material proof can be given. The mathematician himself tells you that the infinite of numbers exists, but cannot be proved. ”

“ And perhaps even those who call them sense-data, may, in part, be attributing to them properties which it may be doubtful whether they possess. If we want to define a sensation or a sense-datum, in a manner which will leave it not open to doubt what sort of things we are talking of, and that there are such things, I do not know that we can do it better than by saying that sense-data are the sort of things, about which such judgments [Pg 232] as these always seem to be made—the sort of things which seem to be the real or ultimate subjects of all such judgments. ”

“ To illustrate, — suppose that a mathematician wants to prove something about the value of π, or about the universal laws of Arithmetic, or about the properties of a continuous function, or about the sum of an infinite series, or about the mathematical relationships of two infinite collections of ideal objects. What he is concerned to demonstrate, lies in the realm of the infinite, and of the eternally valid. And our direct experience gives us only the passing data and the fragmentary ideas of the moment. ”

“ When we ask what are the kinds of objects with which we are acquainted, the first and most obvious example is sense-data. When I see a colour or hear a noise, I have direct acquaintance with the colour or the noise. The sense-datum with which I am acquainted in these cases is generally, if not always, complex. This is particularly [211] obvious in the case of sight. I do not mean, of course, merely that the supposed physical object is complex, but that the direct sensible object is complex and contains parts with spatial relations. ”

“ I shall give the name sensibilia to those objects which have the same metaphysical and physical status as sense-data, without necessarily being data to any mind. Thus the relation of a sensibile to a sense-datum is like that of a man to a husband: a man becomes a husband by [149] entering into the relation of marriage, and similarly a sensibile becomes a sense-datum by entering into the relation of acquaintance. It is important to have both terms; for we wish to discuss whether an object which is at one time a sense-datum can still exist at a time when it is not a sense-datum. ”

“ Classification, logic, and mathematics all result, then, from the mere play of the mind comparing its conceptions, no matter whence the latter may have come. The essential condition for the formation of all these sciences is that we should have grown capable of apprehending series as such, and of distinguishing them as homogeneous or heterogeneous, and as possessing definite directions of what I have called 'increase.' This consciousness of series is a human perfection which has been gradually evolved, and which varies greatly from man to man. ”

“ We may succeed in actually defining the objects of physics as functions of sense-data. Just in so far as physics leads to expectations, this must be possible, since we can only expect what can be experienced. And in so far as the physical state of affairs is inferred from sense-data, it must be capable of expression as a function of sense-data. ”

“ Thus, in seeking our definition of number we have arrived so far at the result that numbers are properties of general terms or general descriptions, not of physical things or of mental occurrences.
Instead of speaking of a general term, such as “man,” as the subject of which a number can be asserted, we may, without making any serious change, take the subject as the class or collection of objects—i.e. “mankind” in the above instance—to which the general term in question is applicable. ”

“ In almost all our previous discussions we have been concerned in the attempt to get clear as to our data in the way of knowledge of existence. What things are there in the universe whose existence is known to us owing to our being acquainted with them? So far, our answer has been that we are acquainted with our sense-data, and, probably, with ourselves. These we know to exist. And past sense-data which are remembered are known to have existed in the past. This knowledge supplies our data. ”

“ If instead of drawing one bean we draw a handful at random and conclude that about of the handful are probably white, the reasoning is of the same sort. If, however, not knowing what proportion of white beans there are in the bag, we draw a handful at random and, finding 23 of the beans in the handful white, conclude that about 23 of those in the bag are white, we are rowing up the current of deductive sequence, and are concluding a rule from the observation of a result in a certain case. ”

“ A complete application of the method which substitutes constructions for inferences would exhibit matter wholly in terms of sense-data, and even, we may add, of the sense-data of a single person, since the sense-data of others cannot be known without some element of inference. ”

“ The judgment of number is verified by enumeration or counting.
We have found five classific properties: kind, form, force, causation, and conception, derived from the essentials by incorporation, and that the kind is a relative unit, the form a relative extension, the force a relative speed, the causation a relative persistence and the conception a relative consciousness.
There are no particles which are not found in bodies, and all bodies are composed of particles. The quantitative properties are found when we consider particles. Classific properties are found when we consider bodies. ”

“ A tree, a house or a church, near at hand, might possibly be distinguished, on account of its elevation above the general surface of the ground, and the bed of a river by reason of its being depressed below it. But we should be obliged rather to guess, and to form a conjecture as to the particular object we wished to distinguish, than to arrive at any certain conclusion respecting it; and if it lay at a considerable distance, it would be impossible, with any degree of probability, to discriminate any one object from another. ”

“ When several stars are taken, we may conceive an infinite number of ways into which the movements of each star can be so decomposed. Each one of these conceivable divisions will have a certain element of probability in its favour. It is the business of the mathematician to determine the amount of that probability. ”

“ What kind of object is it that number can properly be ascribed to? He points out that physical things may be regarded as one or many: for example, if a tree has a thousand leaves, they may be taken altogether as constituting its foliage, which would count as one, not as a thousand; and one pair of boots is the same object as two boots. It follows that physical things are not the subjects of which number is properly predicated; for when we have discovered the proper subjects, the number to be ascribed must be unambiguous. ”

“ In this way all problems about dice, etc., may be solved. When the number of dice thrown is supposed very large, mathematics (which may be defined as the art of making groups to facilitate numeration) comes to our aid with certain devices to reduce the difficulties.
II The conception of probability as a matter of fact, i.e., as the proportion of times in which an occurrence of one kind is accompanied by an occurrence of another kind, is termed by Mr. Venn the materialistic view of the subject. ”

“ It is said in the Bible that God has ordered all things according to weight, number and measure. Omnia in pondere et numero et mensura disposuit Deus. Weight is equilibrium, number is quantity, measure is proportion—these three, and these are the eternal or divine basis of the science of Nature. Here now is the formula of equilibrium: Harmony results from the analogy of contraries. Number is the scale of analogies, the proportion of which is measure. The entire occult philosophy of the Zohar might be termed the science of equilibrium. ”

“ And since we are assured that the all-wise Creator has observed the most exact proportions of number, weight, and measure in the make of all things, the most likely way, therefore, to get any insight into the nature of those parts of the creation which come within our [Pg 30] observation must in all reason be to number, weigh, and measure. ”

“ It is obvious, too, that even when the probabilities are derived from observation and experiment, a very slight improvement in the data, by better observations, or by taking into fuller consideration the special circumstances of the case, is of more use than the most elaborate application of the calculus to probabilities founded on the data in their previous state of inferiority. ”

“ Thus, for example, an infinite series of numbers, which progresses according to a definite law, is for the understanding an object, to which by this law definite qualities are attached, and an object just as good as a finite series, which progresses according to the same law. ”

“ If we compare successively the readings of two documents in all their variations, we have ample materials for ascertaining the leading merits and defects of each. Readings authenticated by the coincidence of strong Intrinsic and strong Transcriptional Probability, or it may be by one alone of these Probabilities in exceptional strength and clearness and uncontradicted by the other, are almost always to be found sufficiently numerous to supply a solid basis for inference. ”

“ And now, as to the more universal meaning of this serial structure of our world of facts, we may note in passing still one further consideration. It is a commonplace that the exact sciences of Nature have owed their exactness, up to the present time, and in the main, to their quantitative treatment of facts. The logic of the Conception of Quantity has its own very complex problems; but thus much is clear: Any system of quantities, such as distances, times, masses, temperatures, pressures, is a serial order-system of facts, or is a complex of such serial order-systems. ”

“ To examine the principles of all the cases in which chances for and against an occurrence have been calculated from real or hypothetical data, would be to trespass into the province of Mathematics, but a few simple cases will serve to show what it is that the calculus attempts to measure, and what is the practical value of the measurement as applied to the probability of a single event. ”

“ The cause of a determined event might either be the total of its conditions, or one antecedent event. This last conception prevailed, as being the more favorable to the application of mathematics. The eighteenth century marks the genesis of the biological sciences. The growing importance of observation and experimental research made against the preponderance of mathematics. The facts of experience were held more solid than the conclusions of reason. The type of causality is placed no longer in deduction, but in sensory intuition; it is the residuum of experience. ”

“ I apprehend the account that hath been given of the nature of proper and improper quantity, may also throw some light upon the controversy about the force of moving bodies, which long exercised the pens of many mathematicians, and, for what I know, is rather dropped than ended; to the no small scandal of mathematics, which hath always boasted of a degree of evidence, inconsistent with debates that can be brought to no issue. ”

“ So long as we confine our attention to one or two individuals, we fail to detect any order in the occurrence of variations; but when we examine large numbers we find that it is possible to arrange them in an orderly series, which can be easily and simply described. The series into which we can arrange the results of observing phenomena of complex causation, whether exhibited by living organisms or not, have certain properties in common, which are dealt with by the theory of chance. ”