Rozsa Peter - Playing with Infinity: Mathematical Explorations and Excursions

Should the reader wish to refer to, for instance, the integral he will find in the list of Contents only the title many small make a great. For this reason I have added here as a postscript what mathematical concepts are to be found in the various chapters. (Dont be put off by them!)

LET us begin at the beginning. I am not writing a history of mathematics; this could be done only on the basis of written evidence, and how far from the beginning is the first written evidence! We must imagine primitive man in his primitive surroundings, as he begins to count. In these imaginings, the little primitive man, who grows into an educated human being before our eyes, will always come to our aid; the little baby, who is getting to know his own body and the world, is playing with his tiny fingers. It is possible that the words one, two, three and four are mere abbreviations for this little piggie went to market, this little piggie stayed at home, this one had roast beef, this one had none and so on; and this is not even meant to be a joke: I heard from a medical man that there are people suffering from certain brain injuries who cannot tell one finger from another, and with such an injury the ability to count invariably disappears. This connexion, although unconscious, is therefore still extremely close even in educated persons. I am inclined to believe that one of the origins of mathematics is mans playful nature, and for this reason mathematics is not only a Science, but to at least the same extent also an Art.

We imagine that counting was already a purposeful activity in the beginning. Perhaps primitive man wanted to keep track of his property by counting how many skins he had. But it is also conceivable that counting was some kind of magic rite, since even today compulsion-neurotics use counting as a magic prescription by means of which they regulate certain forbidden thoughts; for example, they must count from one to twenty and only then can they think of something else. However this may be, whether it concerns animal skins or successive time-intervals, counting always means that we go beyond what is there by one: we can even go beyond our ten fingers and so emerges mans first magnificent mathematical creation, the infinite sequence of numbers,

1, 2, 3, 4, 5, 6,...

the sequence of natural numbers. It is infinite, because after any number, however large, you can always count one more. This creation required a highly developed ability for abstraction, since these numbers are mere shadows of reality. For example, 3 here does not mean 3 fingers, 3 apples or 3 heartbeats, etc., but something which is common to all these, something that has been abstracted from them, namely their number. The very large numbers were not even abstracted from reality, since no one has ever seen a billion apples, nobody has ever counted a billion heartbeats; we imagine these numbers on the analogy of the small numbers which do have a basis of reality: in imagination one could go on and on, counting beyond any so-far known number.

Man is never tired of counting. If nothing else, the joy of repetition carries him along. Poets are well aware of this; the repeated return to the same rhythm, to the same sound pattern. This is a very live business; small children do not get bored with the same game; the fossilized grown-up will soon find it a nuisance to keep on throwing the ball, while the child would go on throwing it again and again.

We go as far as 4? Let us count one more, then one more, then one more! Where have we got to? To 7, the same number that we should have got to if we had straight away counted 3 more. We have discovered addition

4 + 1 + 1 + 1 = 4 + 3 = 7

Now let us play about further with this operation: let us add to 3 another 3, then another 3, then another 3! Here we have added 3s four times, which we can state briefly as: four threes are twelve, or in symbols:

3 + 3 + 3 + 3 = 4 x 3 = 12

and this is multiplication.

We may so enjoy this game of repetition that it might seem difficult to stop. We can play with multiplication in the same way: let us multiply 4 by 4 and again by 4, then we shall get

4 x 4 x 4 = 64

This repetition or iteration of multiplication is called raising to a power. We say that 4 is the base, and we indicate by means of a small number written at the top right-hand corner of the 4 the number of 4s that we have to multiply; i.e. the notation is this:

43 = 4 x 4 x 4 = 64

As is easily seen, we keep getting larger and larger numbers: 4 x 3 is more than 4 + 3, and 43 is a good deal more than 4 x 3. This playful repetition carries us well up amongst the large numbers; even more so, if we iterate raising to a power itself. Let us raise 4 to the power which is the fourth power of four:

44 = 4 x 4 x 4 x 4 = 64 x 4 = 256

and we have to raise 4 to this power:

I have no patience to write any more, since I should have to put down 256 4s, not to mention the actual carrying out of the multiplication! The result would be an unimaginably large number, so that we use our common sense, and, however amusing it would be to iterate again and again, we do not include the iteration of powers among our accepted operations.