E C Titchmarsh on Counting

In 1948 E C Titchmarsh published Mathematics for the General Reader. The book covers Counting, Arithmetic, Algebra, The use of numbers in geometry, Irrational numbers, Indices and logarithms, Infinite series and e, The square root of minus one, Trigonometry, Functions, The differential calculus, The integral calculus and Aftermath (see this link).

We give below his first chapter on Counting:-


Counting

Little children easily learn to count. Very early in their lives, they notice the existence around them of recognizable objects. As soon as they can speak, they learn to say the names of some of these. Almost at once they notice that some objects may be classed together, as being obviously of the Same kind. In particular they notice the existence of pairs of objects, and learn to use the word "two". When I speak of two hands and two feet, the child realizes that the set of my hands has something in common with the set of my feet. When I turn on a light and then another light, the child says "two lights". This is the beginning of counting.

Soon other numbers three, four, five and so on are learnt. The use of the word "one" probably comes later, the existence of single objects, being at first too obvious to call for a special name. "Nought", the negation of the existence of any objects of a particular class, is a comparatively abstract idea, which only occurs to us when we are used to counting. Some ancient races had no symbol for "nought", which they did not think of as the same sort of thing as "'one" or "two".

Older children learn the routine of counting up to quite large numbers. Beyond thirty or forty this must soon cease to have any particular meaning for them, but the rhythm of counting (twenty-one, twenty-two, twenty-three) makes it rather like saying very easy poetry. Children sometimes even count backwards to amuse themselves.

It soon becomes obvious that the process of counting can go on a very long way. I once overheard my children discussing the question, "What is the largest number?" One of them thought that it must consist entirely of 9's. The second thought that it must be possible to get it by using all the words "hundred", "thousand", "million", and whatever else there might be, in the most favourable way (the idea of repetition not being thought of). The third objected that one could never count as far as that, supposing apparently that to make it fair one ought to be able to count through all the numbers up to the largest. They all agreed that the subject presented serious difficulties, and passed on to other topics.

They did not ask me what the largest number was. In this they were undoubtedly wise, because I should not have been able to tell them. I should have been faced, like any other mathematician, with a serious dilemma. Either there is a largest number, and when we get to it we must stop; or we go on for ever, and the set of numbers is endless, or, as we say, infinite.

It might be said that, as all the numbers which are ever actually used or thought of individually form a finite set, we might as well confine our attention to such a set, and avoid the necessity of trying to think about infinite classes of numbers. Perhaps it would be possible to do this, but it would really make the practice of mathematics more difficult. Not only should we be condemned for ever to the trivialities of finite arithmetic, but almost every statement in mathematics would be limited by a condition that the numbers involved must not be too large. Of course in our minds there is no barrier to endless counting. However far we have got, we can always count one more.

Practically all mathematicians agree that there is no upper limit beyond which counting must cease; that is, they agree to regard the numbers which begin with one, two, three, ... , the primal elements of mathematics, as an infinite class. Such an agreement, or declaration, which is itself incapable of proof, but which is a necessary starting point for further thinking, is called an axiom. The axiom about the set of numbers going on for ever is called the axiom of infinity.

What are numbers?

To children, and probably to most other people, numbers are just the things we count with. They are words such as "two" or "five", which call up in our minds a familiar set of objects, such as the set of my hands, or the set of fingers on one hand. The number spoken relates a named set to one of these familiar sets; that is, it asserts that we could pair off each object of the named set with one of the objects of the familiar set. "Two lights" might mean that there is a light on my right hand and a light on my left hand. But couples are so often met with that the set of hands may be forgotten, and "two" just relates a new couple to all those which we have met before.

Generally, if we can pair off the members of one set with the members of another set, so that none of either set is left over, then the two sets must have the same number, whatever that may mean. Number must have a meaning such that it is true that I have the same number of fingers on each hand, and the same number of buttons as buttonholes on my waistcoat (with coats the situation does not seem to be so simple).

The question what numbers are has been much debated by philosophers, and they do not seem to have reached any agreement about it. There is nothing particularly surprising or distressing about this. It has been said that mathematicians are happy only when they agree, and philosophers only when they disagree. Philosophic doubts about the nature of number have never prevented mathematicians from getting on with their calculations, or from agreeing when they have got the right answer. So perhaps the situation is satisfactory to all parties.

One of the most famous attempts to define number was made by Bertrand Russell. I will quote, for example, what he says in his book An Introduction to Mathematical Philosophy. "We naturally think that the class of couples is something different from the number 2. But there is no doubt about the class of couples: it is indubitable and not difficult to define, whereas the number 2, in any other sense, is a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples, which we are sure of, than to hunt for a problematical number 2 which must always remain elusive. Thus the number of a couple will be the class of all couples. In fact, the class of all couples will be the number 2, according to our definition. At the expense of a little oddity, this definition secures definiteness and indubitableness; and it is not difficult to prove that numbers so defined have all the properties that we expect numbers to have."

This plausible sounding definition of the number 2 actually raises many difficulties. For example, are we really sure what we mean by "the class of all couples"? Are we to admit physical objects only, or "all objects of all thought" (in pairs) as members of the class? If, as we must suppose, it is the latter, it seems that I can always add to the class by thinking of a fresh couple, and thus that I can create couples which you know nothing about. It is true that you can always test any couple, the existence of which I announce, to see whether it is one; but that removes the ultimate "2" from the class of couples to some test for couples, which is just what Russell seems to wish to avoid.

Another objection is that arguments based on the supposed existence of classes such as the class of all couples lead to certain famous paradoxes which appear amusing, but which are rather destructive to theories of this kind. One of these runs as follows. Some classes are members of themselves; for example, the class of all classes is a class, and so is a member of itself. Others, such as the class of all men, are not members of themselves (since a class is not a man). Consider now the class of all classes which are not members of themselves. Is it a member of itself or not? If it is a member of itself, then by the definition of the class to which it belongs, it is not a member of itself. This is a complete contradiction, which shows that there is something unsound in the attempt to manipulate classes in this way.

Arguments of this kind, in which we seem logically to go round in circles, suggest awkward questions about the class of couples. The class of couples, together with the class of triplets, are two classes, and so should belong to the class of all couples. In fact, classes seem in this way to breed in an alarming manner.

Russell's definition of a number as a class of similar classes is very ingenious, but the difficulties which it involves have never been entirely cleared up. There are other schools of mathematical philosophers, known as Formalists and Intuitionists, who have put forward rival theories to explain what mathematics really is. No doubt this problem will be much studied in the future.

The conclusion of all this seems to be that we must do without a simple and direct answer to the question, "What is a number?" This will not prevent us from doing mathematics. I am all in favour of an intelligent theory of number. It should add to the pleasure of mathematics, just as an intelligent theory of rigid dynamics should add to the pleasure of bicycling. But it is possible to pedal along without it.

Most mathematicians feel that mathematics does not really rest on what philosophers define it to be, but that it has in it a harmony which somehow carries it along. This view seems to be supported by M Black, in his book The Nature of Mathematics. He says: "The title of 'The Foundations of Mathematics' which the philosophical analysis of mathematics has often received is therefore a misleading one if, taken in conjunction with these contradictions, it suggests that the traditional certainty of mathematics is in question. It is a fallacy to which the philosopher is particularly liable to imagine that the mathematical edifice is in danger through weak foundations, or that philosophy must be invited like a newer Atlas to carry the burden of the disaster on its shoulders."

The view put forward by some philosophers, particularly the Intuitionists, that large parts of mathematics rest on insecure foundations and should therefore be abandoned, has never been accepted by the general run of mathematicians. It is no doubt a mistake to regard philosophers as enemies who would destroy our precious possessions. We may take comfort from the following sentence in the same book by Black. "Philosophic analysis of mathematical concepts therefore tends to become a synthetic constructive process, providing new notions which are more precise and clearer than the old notions they replace, and so chosen that all true statements involving the concepts inside the mathematical system considered shall remain true when the new are substituted."

Perhaps we could regard numbers as a sort of medium of exchange, like money. Most people are really interested in the goods and services which the world offers, and to them money is only a symbol for these. But it is not a meaningless symbol. A system of barter, in which we do without money and merely exchange goods, would be very inconvenient, and practically impossible in a complicated society. So a system in which we reduce all mathematics to statements such as "I have more fingers than you have noses" would be too cumbrous to contemplate seriously. Numbers are symbols, and very useful and interesting ones. To mathematicians who work with them every day they acquire a reality at least equal to that of anything else.


JOC/EFR March 2006

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