FASTER THAN THOUGH. B. V. Bowder.

FASTER THAN THOUGH
B. V. Bowder. (pp. 311-315)

Chapret 26 Thought and machine processes Cogito, ergo sum - DESCARTES
I do not think, therefore I am not - DR. STRABISMUS (whom God preserve) of Utrecht. President of the Anti-cartesian Society

SO FAR WE HAVE DESCRIBED the construction of these new digital computers, and tried to show how useful they can become by doing routine computations. If we are to complete the story we must also try to assess the limitations of the machines which we can build today, and, if possible, to discuss any limits to the performance of machines which may be built in the future. We shall try to compare the processes which go on inside them with those which are responsible for the thoughts in our own minds. This subject is far too complicated to be dealt with in a single chapter, but we shall try to describe some of its more important aspects. We shall begin by giving an account of some of the astonishing feats of mental arithmetic which are demonstrated by those rare individuals who are known as "calculating prodigies." Many accounts of these men have appeared, of which one of the best known is to be found in W. W, R. Ball's book,* to which the reader is referred for a comprehensive historical account of the subject.

At rare intervals there have appeared men and boys who display extraordinary powers of mental arithmetic. In a few seconds they can give the answer to questions which an expert mathematician could obtain only in a much longer time with the aid of pencil and paper. Some of them have remained otherwise illiterate; others, such as Gauss, Ampere, and Bidder, have risen to positions of eminence as mathematicians, physicists, or engineers. Many of them seem to have taught themselves the rules of arithmetic in their childhood, and to have learnt the multiplication table by playing with pebbles. Few of these prodigies have been able to explain in detail how they achieve their apparently miraculous results, but two of the most remarkable of them have been kind enough to discuss their methods with us. We are therefore much indebted to Dr. A. C. Aitken, F.R.S., Professor of Mathematics in Edinburgh University, to several of his old students, to Dr. Stokvis, and to Dr. van Wijngaarden and Mr. William Klein of the Mathematisch Centrum, Amsterdam, for the information which we have used in the following pages.

As far as we can judge, Professor Aitken and Mr. Klein use very similar methods in their mental computations; their speeds are quite comparable, and they are at least as fast as, and probably faster than, any of the prodigies whose performances have been described in the past.

Both men have most remarkable memories - they know by heart the multiplication table up to 100 x 100, all squares up to 1000 x 1000, and an enormous number of odd facts, such as 3937 x 127 = 499999, which are very useful to them, and seem to arise instantaneously in their minds when they are needed. In addition Mr. Klein knows by heart the logarithms of all numbers less than 100, and of all prime numbers less than 10,000 (to twenty decimal places) so that he can work out sums like compound interest by "looking up" the logs in his head, after factorizing the numbers he is using, if need be. He has also learnt enough about the calendar to be able to give the day of the week corresponding to any specified date in history; he learnt most of the Amsterdam telephone directory for fun. Professor Aitken has neglected logarithms in favour of mathematical formulae and the piano music and violin sonatas of Bach and Beethoven, but nevertheless he learnt 802 places of p by heart in about fifteen minutes, an operation which to him was comparable in difficulty to learning a Bach fugue.

If one realizes that in addition to their phenomenal memories both men possess an equally phenomenal ability at mental arithmetic, one can begin to understand some of the feats which they perform every day. Mr. Klein multiplies together numbers of up to six digits by six digits faster in his head than an ordinary man can do by using a desk calculating machine. For example, he wrote down the products of six pairs of three-digit numbers in nine seconds; an experienced calculating-machine operator took a minute to do the same calculations.*

Mr. Klein multiplied 1388978361 x 5645418496 = 7841364129733165056 completely in his head, a calculation which involved twenty-five multiplications each of two two-digit numbers and twenty-four additions of four-digit numbers - forty-nine operations in all - in sixty-four seconds. The reader might care to try it himself with pencil and paper. A dozen of us tried it here in Manchester; the times we took varied between six and sixteen minutes, and all our answers were wrong excepting one.

Professor Aitken's students tell many stories of the prodigious ability in mental arithmetic which he demonstrates in his lectures. For example, he is accustomed to ask members of his class to give him at random nine numbers, each of two or three figures, which will form the elements of a 3 x 3 matrix. He then mentally evaluates the nine co-factors and the determinant, thus obtaining the adjugate and the inverse matrix. He also works out all four roots of a quartic equation with real roots, the coefficients of which have been given to him by his class.

As an example of Professor Aitken's methods we shall describe the operations which he performed while he was mentally evaluating the square root of 567, which he finally checked by comparing his answer with 9V7. His method is based upon the fact that if a is a first approximation to Vn then ½(a + n/a) is considerably closer, but his calculations are facilitated by his astonishing familiarity with tables of reciprocals.

He noted 24 as a first approximation, and 23,8125 as a second (23,8125 = ½(24 + 567/24)). At the same moment he recalled that 1000/42 = 23,809523 the digits of which are close to 23,8125. He performed 567 x 42 = 23814 almost before he had thought what he was doing. Averaging 23,809523 ... and 23,8140 he had as a third approximation 23,81176190476. He recalled simultaneously that 1/84 = 0,0117619047619 ... and in extraordinarily less time than it takes to describe it, perhaps in three seconds at most, he registered 23,811762 as the square root he wanted. But now, how many places are right? He noted that the mid-point of 23-8095 and 23,8140 commits an error of deviation of 0,00225 or relatively, 1/10583. Like lightning he squared this and halved it, and reduced his first answer by one part in 224 million, obtaining 23,8117617985, and announcing as a second result 23,81176180 (in fact it is 23,8117617996). "It would be unreasonable," says Professor Aitken, "to ask for anything more accurate. Words cannot describe the speed of association in these matters, and the resources upon which the memory and the calculative faculty draw. The will rises and makes a most powerful imperative; brain and memory obey like an electric switch."

All calculating prodigies acquire, with long experience, an astonishing familiarity with the properties of numbers. For example, Professor Aitken was once asked to multiply 123456789 by 987654321; he immediately remarked to himself that 987654321 is 80000000001/81, thereby converting a tedious sum into a "gift." Asked for the recurring decimal form of 41/67, he multiplied numerator and denominator by 597, obtaining 24477/39999, and writing down immediately 0,611940298507462686567164179104477.

Even as a schoolboy he was able to astonish his fellows by squaring 57586 in his head in two seconds. He worked it out as follows, using the formula a² = (a - b) (a + b) + b².

57586²	= 57500 x 57672 + 86²
	= 23 x 144180000 + 86²
	= 3316147396.

These short cuts, which are an essential part of the repertoire of all mental prodigies, are quite beyond the scope of a machine, which makes much better time by using straightforward methods once the problem has been explained to it; but it is in this sort of way that Mr. Klein performs a type of computation in which he has a most unusual skill. He can express prime numbers of the form 4n + 1 as the sum of two squares; if they can be expressed as 8n + 1, in the form 2c² + d², etc. For example -

5881	= 75² + 16²
	= 2 x 54² + 7²
	= 3 x 32² + 53²
	= 5 x 9² + 74²
	= 7 x 24² + 43²

all of which he did in 100 seconds. Any machine would take a relatively long time to do such a computation, as it would have to work by a tedious process of trial and error. A mathematician would probably take an hour or so to prepare a tape of instructions for a machine which was to handle the general case, a second or so to feed in the particular number which he wanted to investigate, and the machine would take two or three seconds at most to produce the whole series of squares once it had started to work. The point is, of course, that the machine has to be told everything it needs to know for this particular problem, and even when it knows how to proceed it may well take so long for a man to pose a problem to the machine that a human calculator may have done the sum long before his colleague has had time to punch a tape with which to feed the numbers into the machine.

A calculating prodigy draws continually on the accumulated experience of a lifetime's arithmetic and both his "strategy" and his "tactics" are opportunist. Professor Aitken says: "Though these processes take time to describe, they pass in the mind with prodigious speed, though with the ease and relaxation of a good violinist playing a scale passage. Often the mind is so automatic that it anticipates the will.

"The power of numerical memorizing came to me later than verbal, but rapidly improved with mental calculation; and soon all three kinds, verbal, numerical and musical advanced equally. If the number to be scanned had a strong mathematical interest, like p or e or Euler's constant y, or those almost uncanny numbers like e ^pV163 which is 262537412640768743,999999999999250 (incredibly close to a whole number), then I could hardly help absorbing them to very many decimal places.

"The numbers come into view as one needs them, but even to say that they come into 'view' gives a false impression. It is not 'seeing' in the ordinary sense; it is a compound faculty that has never yet been accurately described. The analogy of music will throw light on calculation. The violinist (unless he is momentarily in a difficulty) docs not need to visualize the notes on the stave, or the fingering or the bowing; the melody is everything - he is caught up in what he is playing. So it is with the mental calculator; visualizing occurs last of all, and only as required when all else has been done."

The rest of us must be content to marvel.

Mr. William Klein's brother Leo, who died at the hands of the Gestapo during the war, was almost as good a computer as William, and a better mathematician. Dr. Stokvis, of Amsterdam, made a psychological study of the brothers;⁽¹⁾ he found that although their performances were very similar, their methods of operation were quite different. For example, Mr. William Klein remembers numbers "audibly"; he mutters to himself as he computes, he can be interrupted by loud noises, and if he ever does make a mistake it is by confusing two numbers which sound alike. Leo, on the other hand, remembered things "visually"; and if he made a mistake it was by confusing digits which look alike. Both brothers were fascinated by numbers from their earliest childhood; William practised arithmetic almost all the time, but Leo hardly ever. Leo studied mathematics at the university, but William read medicine, took a medical degree, and had "walked the hospitals" before he finally decided to earn his living as a computer. Dr. Stokvis investigated the effect of drugs and of hypnosis on Mr. William Klein, and found that neither improved his performance as a computer if he was using methods in which he was experienced and in which he had already achieved an "optimum" performance. Apparently Mr. Klein forgot to go to Dr. Stokvis's lecture so that the public demonstration of his talent which had been arranged had to be indefinitely postponed.

Many other mathematicians whose skill in arithmetic was much less than that of Professor Aitken or Mr. Klein have nevertheless been fascinated by the properties of numbers. Professor Hardy once visited the Indian mathematician Ramanujan, who was lying ill in hospital. To make interesting bed-side conversation Professor Hardy remarked that the number of his taxicab was 1729, which is a multiple of 13, and said he hoped that this was not an ill-omen. "On the contrary," said the sick man, brightening up at once, "1729 is a beautiful number; it is the smallest integer which can be expressed in two different ways as the sum of two cubes." That it can be so expressed is fairly obvious; to prove that it is the smallest such number may occupy the reader in leisure moments for some time, but he may derive, in the process, some idea what mathematicians with a gift for arithmetic talk about in their spare time.

The reader may feel that he is overwhelmed by the possibility of this kind of calculation, but before he decides to take up farming instead of arithmetic let us for one moment consider the mental arithmetic which is sometimes done by a certain Lakeland shepherd. During the course of a day his dog may drive past him a flock of perhaps two thousand sheep. At the end of the day he knows not only how many sheep are missing, but which sheep are missing. Now even if one assumes for purposes of argument that a man can learn to tell the difference between one sheep and another, one must admit that even a shepherd requires and can exploit a skill in mental arithmetic which few of us could ever hope to achieve.

. . .

* Mathematical Recreations and Essays. See also Common Sense and Its Cultivation, by Hanbury Hankin, and Mental Prodigies, by Fred Barlow.

* Kiyoshi Mastuzaki may have been a calculating prodigy, using his abacus as Mr. Klein might use his pipe - to occupy his fingers (see page 6).