Bertrand Russell

“ All those properties of numbers which are proved by an imaginary step-by-step process from one number to the next are liable to fail when we come to infinite numbers. The first of the infinite numbers has no immediate predecessor, because there is no greatest finite number; thus no succession of steps from one number to the next will ever reach from a finite number to an infinite one, and the step-by-step method of proof fails. ”

“ The fact that, so long as we confine ourselves to numbers less than some given finite number, the proportion of squares tends towards zero as the given finite number increases, does not contradict the fact that the number of all finite squares is the same as the number of all finite numbers. ”

“ That is to say, if, in any sentence in which a thing is mentioned, you substitute a class for the thing, you no longer have a sentence that has any meaning: the sentence is no longer either true or false, but a meaningless collection of words. Appearances to the contrary can be dispelled by a moment’s reflection. For example, in the sentence, “Adam is fond of apples,” you may substitute mankind, and say, “Mankind is fond of apples.” ”

“ Mathematics and philosophy differ, however, in their manner of treating the general properties in which all possible worlds agree; for while mathematics, starting from comparatively simple propositions, seeks to build up more and more complex results by deductive synthesis, philosophy, starting from data which are common knowledge, seeks to purify and generalise them into the simplest statements of abstract form that can be obtained from them by logical analysis.
The difference between philosophy and mathematics may be illustrated by our present problem, namely, the nature of number. ”

“ This property of infinite numbers was always thought, until recently, to be self-contradictory; but through the work of Georg Cantor it has come to be recognised that, though at first astonishing, it is no more self-contradictory than the fact that people at the antipodes do not tumble off. In virtue of this property, given any infinite collection of objects, any finite number of objects can be added or taken away without increasing or diminishing the number of the collection. Even an infinite number of objects may, under certain conditions, be added or taken away without altering the number. ”

“ It will be seen that Leibniz regards it as self-contradictory to maintain that the whole is not greater than its part. But the word “greater” is one which is capable of many meanings; for our purpose, we must substitute the less ambiguous phrase “containing a greater number of terms.” In this sense, it is not self-contradictory for whole and part to be equal; it is the realisation of this fact which has made the modern theory of infinity possible.
There is an interesting discussion of the reflexiveness of infinite wholes in the first of Galileo’s Dialogues on Motion. ”

“ We may illustrate this characteristic by means of Leibniz’s conception of many possible worlds, of which one only is actual. In all the many possible worlds, philosophy and mathematics will be the same; the differences will only be in respect of those particular facts which are chronicled by the descriptive sciences. Any quality, therefore, by which our actual world is distinguished from other abstractly possible worlds, must be ignored by mathematics and philosophy alike. ”

“ The unity which philosophers rightly feel to be necessary for the assertion of a number is the unity of the general term, and it is the general term which is the proper subject of number. And this applies equally when there is one object or none which falls under the general term. ”

“ It is this view of number as generated by counting which has been the chief psychological obstacle to the understanding of infinite numbers. Counting, because it is familiar, is erroneously supposed to be simple, whereas it is in fact a highly complex process, which has no meaning unless the numbers reached in counting have some significance independent of the process by which they are reached. And infinite numbers cannot be reached at all in this way. ”

“ But owing to the falsehood of the first proposition where infinite numbers are concerned, counting, even if it were practically possible, would not be a valid method of discovering the number of terms in an infinite collection, and would in fact give different results according to the manner in which it was carried out.
There are two respects in which the infinite numbers that are known differ from finite numbers: first, infinite numbers have, while finite numbers have not, a property which I shall call reflexiveness ”

“ Frege has the merit of accepting both denials, and finding a third assertion by recognising the world of logic, which is neither mental nor physical.
The fact is, as Frege points out, that no number, not even 1, is applicable to physical things, but only to general terms or descriptions, such as “man,” “satellite of the earth,” “satellite of Venus.” The general term “man” is applicable to a certain number of objects: there are in the world so and so many men. ”

“ But obviously you do not mean that there is one individual, called “mankind,” which munches apples: you mean that the separate individuals who compose mankind are each severally fond of apples.
Now, if nothing that can be said significantly about a thing can be said significantly about a class of things, it follows that classes of things cannot have the same kind of reality as things have; for if they had, a class could be substituted for a thing in a proposition predicating the kind of reality which would be common to both. ”

“ The personages in the dialogue are Salviati, Sagredo, and Simplicius, and they reason as follows:
“Simp. Here already arises a Doubt which I think is not to be resolv’d; and that is this: Since ’tis plain that one Line is given greater than another, and since both contain infinite Points, we must surely necessarily infer, that we have found in the same Species something greater than Infinite, since the Infinity of Points of the greater Line exceeds the Infinity of Points of the lesser. But now, to assign an Infinite greater than an Infinite, is what I can’t possibly conceive. ”

“ It will be seen that a hereditary property, though it must belong to all the finite numbers greater than a given number possessing the property, need not belong to all the numbers less than this number. For example, the hereditary property of being greater than 99 belongs to 100 and all greater numbers, but not to any smaller number. ”

“ The infinite numbers actually known, however, are all reflexive as well as non-inductive; thus, in mathematical practice, if not in theory, the two properties are always associated. For our purposes, therefore, it will be convenient to ignore the bare possibility that there may be non-inductive non-reflexive numbers, since all known numbers are either inductive or reflexive. ”

“ In the third place, the real desideratum about such a definition as that of number is not that it should represent as nearly as possible the ideas of those who have not gone through the analysis required in order to reach a definition, but that it should give us objects having the requisite properties. Numbers, in fact, must satisfy the formulae of arithmetic ”

“ These are some of those Difficulties which arise from Discourses which our finite Understanding makes about Infinites, by ascribing to them Attributes which we give to Things finite and terminate, which I think most improper, because those Attributes of Majority, Minority, and Equality, agree not with Infinities, of which we can’t say that one is greater than, less than, or equal to another. ”

“ In the first place, it will be found that when an idea which has grown familiar as an unanalysed whole is first resolved accurately into its component parts—which is what we do when we define it—there is almost always a feeling of unfamiliarity produced by the analysis, which tends to cause a protest against the definition. ”

“ I do not know how many married men there are in England, but I do know that the number is the same as the number of married women. The reason I know this is that the relation of husband and wife relates one man to one woman and one woman to one man. A relation of this sort is called a one-one relation. The relation of father to son is called a one-many relation, because a man can have only one father but may have many sons ”

“ And in the case of an infinite class, enumeration is impossible, so that description by a general characteristic common and peculiar to the members of the class is the only possible description. Here, as we see, the theory of number to which Frege was led by purely logical considerations becomes of use in showing how infinite classes can be amenable to number in spite of being incapable of enumeration.
Frege next asks the question: When do two collections have the same number of terms? In ordinary life, we decide this question by counting ”

“ He concludes that number is neither spatial and physical, nor subjective, but non-sensible and objective. This conclusion is important, since it applies to all the subject-matter of mathematics and logic. Most philosophers have thought that the physical and the mental between them exhausted the world of being. Some have argued that the objects of mathematics were obviously not subjective, and therefore must be physical and empirical ”

“ Such a child may count correctly from the point of view of a grown-up listener, without having any idea of numbers at all. The operation of counting, in fact, can only be intelligently performed by a person who already has some idea what the numbers are; and from this it follows that counting does not give the logical basis of number. ”

“ Every subject-matter, it would seem, can give rise to philosophical investigations as well as to the appropriate science, the difference between the two treatments being in the direction of movement and in the kind of truths which it is sought to establish. In the special sciences, when they have become fully developed, the movement is forward and synthetic, from the simpler to the more complex. ”

“ The two properties of reflexiveness and non-inductiveness, which we have considered as characteristics of infinite numbers, have not so far been proved to be always found together. It is known that all reflexive numbers are non-inductive, but it is not known that all non-inductive numbers are reflexive. ”

“ This will be best explained by defining the positive property of inductiveness which characterises the finite numbers, and which is named after the method of proof known as “mathematical induction.”
Let us first consider what is meant by calling a property “hereditary” in a given series. Take such a property as being named Jones. If a man is named Jones, so is his son; we will therefore call the property of being called Jones hereditary with respect to the relation of father and son. If a man is called Jones, all his descendants in the direct male line are called Jones ”

“ 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. This leads to a discussion of the very prevalent view that number is really something psychological and subjective, a view which Frege emphatically rejects. ”