New Frontiers In Science, Chapter XVIIIIt is natural to assume that what we have once done we can always do. Yet we always find in practice that there are limits, that nothing can be repeated forever. We have always known this to be true in the real world, but it is only recently that it has been realized that this is also true in mathematics. It has long been known that only a small number of objects can be held in mind at the same time. It has also long been a common observation of physicists that if an integer occurs in a formula it is a small integer. The is, in fact, an old saying among physicists that “Seventeen is the largest finite integer.” These common observations were long ignored by mathematicians. However in the never ending quest for new fields of investigation, these long neglected but very suggestive observations were investigated in depth. Very quickly the remarkable and disturbing conclusion was drawn – seventeen is the largest finite integer. This result is suggested by many fields of mathematics. For example, in geometry we can construct a regular polygon of seventeen sides [1] by rule and compass. If there were such a number as eighteen it would be impossible to construct a regular polygon of eighteen sides by rule and compass alone. From this we can draw one of two conclusions: either a regular polygon of eighteen sides cannot be constructed by rule and compass, or there is no such number as eighteen. Similarly in number theory we can divide the numbers from one to seventeen into four classes: odd primes, powers of two, the product of two odd primes, and the product of two and an odd prime. If there were such a number as eighteen it would not be a member of any of these four classes. Again we have two alternatives: either there are more than four classes of numbers or there is no such number as eighteen. [2] These examples might be multiplied indefinitely; in fact there are seventeen of them. For those who are not satisfied with observation we will sketch a proof that seventeen is indeed the largest finite number. Part IWe shall now show, in two different ways, that there are only a finite number of integers.1. The rational number proofWe know that there are more rational numbers than there are integers. Now suppose that the number of integers were infinite; then the number of rational numbers would also be infinite. Thus the number of integers and the number of rational numbers would be the same. But we know that there are more rational numbers than integers. Hence there cannot be an infinite number of integers. 2. The interesting number proofDivide all numbers into two classes: interesting numbers and uninteresting numbers. The class of uninteresting numbers must (if it is not empty) have a smallest member. But this smallest uninteresting number is interesting on that account. Hence there are no uninteresting numbers. But, being finite, we can only be interested in a finite number of numbers. Hence there are only a finite number of integers. Part IIWe have shown that there are only a finite number of integers. But that is only half the problem. The question remains: Which is the largest integer? We give two proofs that this number is seventeen.Proof 1. (by induction)Suppose that there were a number larger than seventeen – for the sake of argument call it eighteen. Then we could add one to it to get a new number, nineteen. Similarly we could keep adding one until we exceeded any preassigned finite number. But this would mean that there would be an infinite number of integers. This we know not to be so. Hence there is no such number as eighteen.Proof 2.Consider all fractions of the form j/k where k runs from 1 to 17 and j runs from 1 to n where n is the largest finite integer. There will be an infinite number of these fractions. But there are an infinite number of rational numbers. Hence [3] these fractions are all of the rational numbers. The smallest fraction in the set is 1/17. Therefore the largest integer is 17.Part IIIWe conclude with the simplest and most elegant proof, reputedly due to the Mongolian sorcerer, Arluis. It is recommended to anyone who can understand Russell’s definition of a number. [4]
Proof 1. (by induction)Suppose there is a number greater than seventeen. For the sake of argument call it eighteen. But that is absurd.
Notes1: This result is due to Gauss, a noted German mathematician. It is true that he work also tended to indicate that a regular polygon of 257 sides could be constructed by rule and compass alone. However his work must have been in error because no one has ever done so.back
2: Although there is no such number as eighteen, it is as legitimate to
discuss its properties as it is to discuss any other non-exist thing,
such as, for example, political honesty.
3: If A and B are two sets having the same number of elements and
if every element of A is an element of B then A and B are identical.
4. “A cardinal number is the set of all sets that are equivalent to the
same set.”
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