Bertrand Russell

“ We can prove that a finite number is always multipliable, but we cannot prove that any infinite number is so. The multiplicative axiom is equivalent to the assumption that all cardinal numbers are multipliable. But in order to identify the reflexive with the non-inductive, or to deal with the problem of the boots and socks, or to show that any progression of numbers of the second class is of the second class, we only need the very much smaller assumption that is multipliable. ”

“ For want of the apparatus of propositional functions, many logicians have been driven to the conclusion that there are unreal objects. It is argued, e.g. by Meinong, [40] that we can speak about "the golden mountain," "the round square," and so on; we can make true propositions of which these are the subjects; hence they must have some kind of logical being, since otherwise the propositions in which they occur would be meaningless. In such theories, it seems to me, there is a failure of that feeling for reality which ought to be preserved even in the most abstract studies. ”

“ Since all terms that are defined are defined by means of other terms, it is clear that human knowledge must always be content to accept some terms as intelligible without definition, in order [Pg 3] to have a starting-point for its definitions. ”

“ And more generally, to suppose that one class of individuals is a member, or is not a member, of another class of individuals will be to suppose nonsense; and to construct symbolically any class whose members are not all of the same grade in the logical hierarchy is to use symbols in a way which makes them no longer symbolise anything.
Thus if there are individuals in the world, and classes of individuals, we cannot form a new class, consisting of both individuals and classes and having members. In this way the attempt to escape from the need for the axiom of infinity breaks down. ”

“ Traditional logic says: "All men are mortal, Socrates is a man, therefore Socrates is mortal." Now it is clear that what we mean to assert, to begin with, is only that the premisses imply the conclusion, not that premisses and conclusion are actually true; even the most traditional logic points out that the actual truth of the premisses is irrelevant to logic. Thus the first change to be made in the above traditional syllogism is to state it in the form: "If all men are mortal and Socrates is a man, then Socrates is mortal." ”

“ The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid. Some believed it to be really a case of induction, in the sense in which that word is used in logic. Poincaré [9] considered it to be a principle of the utmost importance, by means of which an infinite number of syllogisms could be condensed into one argument. We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle. ”

“ Progressions are of great importance in the principles of mathematics. As we have just seen, every progression verifies Peano's five axioms. It can be proved, conversely, that every series which verifies Peano's five axioms is a progression. Hence these five axioms may be used to define the class of progressions: "progressions" are "those series which verify these five axioms." Any progression may be taken as the basis of pure mathematics: we may give the name "0" to its first term, the name "number" to the whole set of its terms, and the name "successor" to the next in the progression. ”

“ But I think the word "proposition" should be limited to what may, in some sense, be called "symbols," and further to such symbols as give expression to truth and falsehood. Thus "two and two are four" and "two and two are five" will be propositions, and so will "Socrates is a man" and "Socrates is not a man." The statement: "Whatever numbers and may be, " is a proposition; but the bare formula "" alone is not, since it asserts nothing definite unless we are further told, or led to suppose, that and are to have all possible values, or are to have such-and-such values. ”

“ If we had been embarking upon mathematical philosophy in earlier days, we should have had to start with something less abstract than the series of natural numbers, which we should reach as a stage on our backward journey. When the logical foundations of mathematics have grown more familiar, we shall be able to start further back, at what is now a late stage in our analysis. But for the moment the natural numbers seem to represent what is easiest and most familiar in mathematics. ”

“ If there were no polygamy or polyandry anywhere in the world, it is clear that the number of husbands living at any moment would be exactly the same as the number of wives. We do not need a census to assure us of this, nor do we need to know what is the actual number of husbands and of wives. We know the number must be the same in both collections, because each husband has one wife and each wife has one husband. The relation of husband and wife is what is called "one-one." ”

“ This is enough to show that definition by extension is not necessary to knowledge about a class. But when we come to consider infinite classes, we find that enumeration is not even theoretically possible for beings who only live for a finite time. We cannot enumerate all the natural numbers: they are 0, 1, 2, 3, and so on. At some point we must content ourselves with "and so on." We cannot enumerate all fractions or all irrational numbers, or all of any other infinite collection. Thus our knowledge in regard to all such collections can only be derived from a definition by intension. ”

“ The whole trend of modern mathematics, with its increased pursuit of rigour, has been against this Kantian theory. The things in the mathematics of Kant's day which cannot be proved, cannot be known—for example, the axiom of parallels. What can be known, in mathematics and by mathematical methods, is what can be deduced from pure logic. What else is to belong to human knowledge must be ascertained otherwise—empirically, through the senses or through experience in some form, but not a priori. ”

“ From the fact that the infinite is not self-contradictory, but is also not demonstrable logically, we must conclude that nothing can be known a priori as to whether the number of things in the world is finite or infinite. The conclusion is, therefore, to adopt a Leibnizian phraseology, that some of the possible worlds are finite, some infinite, and we have no means of knowing to which of these two kinds our actual world belongs. The axiom of infinity will be true in some possible worlds and false in others ”

“ We know that certain scientific propositions—which, in the most advanced sciences, are expressed in mathematical symbols—are more or less true of the world, but we are very much at sea as to the interpretation to be put upon the terms which occur in these propositions. We know much more (to use, for a moment, an old-fashioned pair of terms) about the form of nature than about the matter. Accordingly, what we really know when we enunciate a law of nature is only that there is probably some interpretation of our terms which will make the law approximately true. ”

“ By the general definition of number, the number of terms in the null-class is the set of all classes similar to the null-class, i.e. (as is easily proved) the set consisting of the null-class all alone, i.e. the class whose only member is the null-class. (This is not identical with the null-class: it has one member, namely, the null-class, whereas the null-class itself has no members. A class which has one member is never identical with that one member, as we shall explain when we come to the theory of classes.) ”

“ We may say, without entering upon the grounds for the assertion, that a relation only has a "field" when it is what we call "homogeneous," i.e. when its domain and converse domain are of the same logical type; and as a rough-and-ready indication of what we mean by a "type," we may say that individuals, classes of individuals, relations between individuals, relations between classes, relations of classes to individuals, and so on, are different types. Now the notion of likeness is not very useful as applied to relations that are not homogeneous ”

“ It is clear that an irrational Dedekind cut in some way "represents" an irrational. In order to make use of this, which to begin with is no more than a vague feeling, we must find some way of eliciting from it a precise definition; and in order to do this, we must disabuse our minds of the notion that an irrational must be the limit of a set of ratios. Just as ratios whose denominator is 1 are not identical with integers, so those rational [Pg 71] numbers which can be greater or less than irrationals, or can have irrationals as their limits, must not be identified with ratios. ”

“ If the phenomenal world has three dimensions, so must the world behind phenomena; if the phenomenal world is Euclidean, so must the other be; and so on. In short, every proposition having a communicable significance must be true of both worlds or of neither: the only difference must lie in just that essence of individuality which always eludes words and baffles description, but which, for that very reason, is irrelevant to science. ”

“ The infinity of rationals does not demand, for its definition or use, any infinite classes or infinite integers. It is not, in actual fact, a very important notion, and we could dispense with it altogether if there were any object in doing so. The Cantorian infinite, on the other hand, is of the greatest and most fundamental importance; the understanding of it opens the way to whole new realms of mathematics and philosophy. ”

“ They state that there is to be one class, no more and no less, for each group of formally equivalent propositional functions; e.g. the class of men is to be the same as that of featherless bipeds or rational animals or Yahoos or whatever other characteristic may be preferred for defining a human being. Now, when we say that two formally equivalent propositional functions may be not identical, although they define the same class, we may prove the truth of the assertion by pointing out that a statement may be true of the one function and false of the other ”

“ The notion of order is one which has enormous importance in mathematics. Not only the integers, but also rational fractions and all real numbers have an order of magnitude, and this is essential to most of their mathematical properties. The order of points on a line is essential to geometry; so is the slightly more complicated order of lines through a point in a plane, or of planes through a line. Dimensions, in geometry, are a development of order. The conception of a limit, which underlies all higher mathematics, is a serial conception. ”

“ We can all sympathise with their wish to prove such a very desirable proposition, but we cannot congratulate them on their success. It is true that many of them do not assert objective counterparts to phenomena, and these escape from the above argument. Those who do assert counterparts are, as a rule, very reticent on the subject, probably because they feel instinctively that, if pursued, it will bring about too much of a rapprochement between the real and the phenomenal world. ”

“ But without even one we should find ourselves in difficulties. This, however, is not the point that is important for our present purpose. What is important for us is to observe that form may be the one concern of a general proposition, even when no word or symbol in that proposition designates the form. If we wish to speak about the form itself, we must have a word for it; but if, as in mathematics, we wish to speak about all propositions that have the form, a word for the form will usually be found not indispensable; probably in theory it is never indispensable. ”

“ There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. There is no lower limit short of zero to the difference between the numbers whose square is a little less than 2 and the numbers whose square is a little greater than 2. We can, in short, divide all ratios into two classes such that all the terms in one class are less than all in the other, there is no maximum to the one class, and there is no minimum to the other. ”

“ If is any number, the relation of to is one-one; so is the relation of to or to . When we are considering only positive numbers, the relation of to is one-one; but when negative numbers are admitted, it becomes two-one, since and have the same square. These instances should suffice to make clear the notions of one-one, one-many, and many-one relations, which play a great part in the principles of mathematics, not only in relation to the definition of numbers, but in many other connections. ”

“ We thus arrive at a class of representatives, who make up our Parliament, one being selected out of each constituency. How many different possible ways of choosing a Parliament are there? Each constituency can select any one of its voters, and therefore if there are voters in a constituency, it can make choices. The choices of the different constituencies are independent; thus it is obvious that, when the total number of constituencies is finite, the number of possible Parliaments is obtained by multiplying together the numbers of voters in the various constituencies. ”

“ In seeking a definition of number, the first thing to be clear about is what we may call the grammar of our inquiry. Many philosophers, when attempting to define number, are really setting to work to define plurality, which is quite a different thing. Number is what is characteristic of numbers, as man is what is characteristic of men. A plurality is not an instance of number, but of some particular number. A trio of men, for example, is an instance of the number 3, and the number 3 is an instance of number ”

“ The statement that mathematics is the science of number would be untrue in two different ways. On the one hand, there are recognised branches of mathematics which have nothing to do with number—all geometry that does not use co-ordinates or measurement, for example: projective and descriptive geometry, down to the point at which co-ordinates are introduced, does not have to do with number, or even with quantity in the sense of greater and less. ”

“ To say that a class has a number which is not altered by the addition of 1 is the same thing as to say that, if we take a term which does not belong to the class, we can find a one-one relation whose domain is the class and whose converse domain is obtained by adding to the class. ”

“ On the other hand, it is also possible that, when analysis has been pushed far enough, we can reach terms that really are simple, and therefore logically incapable of the sort of definition that consists in analysing. This is a question which it is not necessary for us to decide; for our purposes it is sufficient to observe that, since human powers are finite, the definitions known to us must always begin somewhere, with terms undefined for the moment, though perhaps not permanently. ”

“ Take, for example, the series of real numbers. Each is what it is, quite definitely and uncompromisingly; it does not pass over by imperceptible degrees into another; it is a hard, separate unit, and its distance from every other unit is finite, though it can be made less than any given finite amount assigned in advance. The question of the relation between the kind of continuity existing among the real numbers and the kind exhibited, e.g. by what we see at a given time, is a difficult and intricate one. ”

“ Broadly speaking, we may say that, if we wished as far as possible to dispense with relational propositions and replace them by such as ascribed predicates to subjects, we could succeed in this so long as we confined ourselves to symmetrical relations: those that do not imply diversity, if they are transitive, may be regarded as asserting a common predicate, while those that do imply diversity may be regarded as asserting incompatible predicates. For example, consider the relation of similarity between classes, by means of which we defined numbers. ”

“ The forms which are taken as simplest in traditional formal logic are really far from being so, and all involve the assertion of all values or some values of a compound propositional function. Take, to begin with, "all is ." We will take it that is defined by a propositional function , and by a propositional function . E.g., if is men, will be " is human"; if is mortals, will be "there is a time at which dies." ”

“ Examples of propositional functions are easy to give: " is human" is a propositional function; so long as remains undetermined, it is neither true nor false, but when a value is assigned to it becomes a true or false proposition. Any mathematical equation is a propositional function. So long as the variables have no definite value, the equation is merely an expression awaiting determination in order to become a true or false proposition. If it is an equation containing one variable, it becomes true when the variable is made equal to a root of the equation, otherwise it becomes false ”

“ Nevertheless, in so far as rigour is lacking in a mathematical proof, the proof is defective; it is no defence to urge that common sense shows the result to be correct, for if we were to rely upon that, it would be better to dispense with argument altogether, [Pg 144] rather than bring fallacy to the rescue of common sense. No appeal to common sense, or "intuition," or anything except strict deductive logic, ought to be needed in mathematics after the premisses have been laid down. ”

“ But when, as above, the argument is formal, nothing depends upon the terms that occur in it. Thus we may substitute for men, for mortals, and for Socrates, where and are any classes whatever, and is any individual. We then arrive at the statement: "No matter what possible values and and may have, if all 's are 's and is an , then is a "; in other words, "the propositional function 'if all 's are and is an , then is a ' is always true." Here at last we have a proposition of logic—the one which is only suggested by the traditional statement about Socrates and men and mortals. ”