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| ISBN-13: | 9780486167404 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 06/10/2014 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 432 |
| File size: | 1 MB |
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The Magic of Numbers
By Eric Temple Bell
Dover Publications, Inc.
Copyright © 1974 Taine T. BellAll rights reserved.
ISBN: 978-0-486-16740-4
CHAPTER 1
The Past Returns
The hero of our story is Pythagoras. Born to immortality five hundred years before the Christian era began, this titanic spirit overshadows western civilization. In some respects he is more vividly alive today than he was in his mortal prime twenty-five centuries ago, when he deflected the momentum of prescientific history toward our own unimagined scientific and technological culture.
Mystic, philosopher, experimental physicist, and mathematician of the first rank, Pythagoras dominated the thought of his age and foreshadowed the scientific mysticisms of our own. So varied was his genius that the crassest superstitions and the most uncompromising rationalisms might appeal to his authority—"Himself said it"—all down the Middle Ages. The essence of his teaching was the mystic doctrine that "Everything is number," With Galileo's revival in the late sixteenth century of the experimental method in the physical sciences, a method in which Pythagoras had pioneered nearly twenty-two centuries earlier, number mysticism passed out of science.
The seventeenth century saw the creation by Newton and Leibniz of a new mathematics, devised to bring the continuously varying flux of nature under the domination of rigorous reasoning. Combining this dynamic mathematics with precise observation and purposeful experiment, Newton and his followers in the seventeenth and eighteenth centuries fixed the modern scientific method for the astronomical and physical sciences. The form they gave it was to stand unchallenged till the third decade of the twentieth century.
The aim of this method was twofold: to sum up the observable phenomena of the physical universe in readily apprehended generalizations—called by their inventors or discoverers "laws of nature"; to enable human beings in some degree to predict the course of nature. Always observation and experiment were the first and last court of appeal. However reasonable or however inevitable the verdict of mathematics or other strict deductive reasoning might appear, it alone was not accepted without confirmation by this final authority.
The successes of the method heavily overbalanced its failures all through the nineteenth century and well into the twentieth. In less than two centuries the application of scientific technology to industry wrought a profounder transformation of western civilization than had all the wars and social upheavals of the preceding four thousand years.
Concurrently with this vast revolution in the material world, equally subversive changes from time to time overthrew established creeds that had possessed the thoughts of men for scores of generations. The universe disclosed by science was not always that of revelation and tradition, nor even that which a supposedly infallible logic insisted must be the fact. Here also the absolutes of more than two thousand years were impartially scrutinized. Those that had proved barren of positive knowledge were ruthlessly abandoned. The unaided reason as an implement of discovery and understanding in the exploration of the material universe dropped out of use. Its sterility in science then cast suspicion on it in its own traditional domain. Of what human value were truths immune to any objective test that human beings might invent? Protests that truths other than those of science exist tunelessly in a realm of Eternal Being, and are forever inaccessible to the finite reach of science, were silenced by the dictum, "Experiment answers all," Then, quite suddenly, about the year 1920, the most positive of all the sciences began to hesitate.
By the middle 1930's a few prominent and respected physicists and astrophysicists had reversed their position squarely. Facing the past unafraid, they strode boldly back to the sixth century B.C. to join their master. Though the words with which they greeted him were more sophisticated than any that Pythagoras might have uttered, they were still in his ancient tongue. The meaning implicit in their refined symbolisms and intricate metaphors had not changed in twenty-five centuries: "Everything is number," He understood what they were saying.
The retreat from experiment to reason was applauded by some philosophers and scientists, deplored by others. But the fact in the new movement was beyond dispute. Either the leaders had gone back to Pythagoras to acknowledge that he had been right all these centuries, or he had come forward to convince them that the modern scientific method of Galileo and Newton is a delusion.
On a first, exploratory pilgrimage to the past the daring ultramoderns had lingered for a few moments in the shadow of Plato. Quickly realizing that in all matters pertaining to the mysteries of number here was only the pupil, they sought his master. Two centuries before Plato was born, Pythagoras had believed and taught that the pure reason alone can reveal the truth of anything; observation and experiment are snares to trap and betray the unwary senses. And of all languages in which constant knowledge as opposed to variable opinion may be described, that of number is the only one on which the pure reason may safely rely. "Himself said it," and now, twenty-five centuries after his historic death, he was repeating himself in the language of a nascent science.
A devout believer in the doctrines of reincarnation and the transmigration of souls, Pythagoras may at last have found a congenial habitation in the sheer abstractions of twentieth century theoretical physics. "For my own lapses from the one true faith," he may now reflect, "I was condemned to spend life after life in the vile dogmas of false philosophers and in the viler imaginings of base numerologists. But now I am unbound from the Wheel of Birth. When I experimented with my hands and my hearing to discover the law of musical intervals, I sinned against the eternal spirit of truth, defiling my soul with the unclean things of the senses. Then I beheld the vision of Number, and knew that I had betrayed my better part. By proclaiming the truth that everything is number, I sought to cleanse my soul and gain release from the Wheel. But it was not enough. Few believed and many misunderstood. To expiate my transgression I passed through that purgatory of error and falsehood, a name honored in the mouths of fools. Now I discern an end to my torment in the dawn of a new enlightenment which was already old ages before I was Pythagoras. The deceptions of the senses shall mislead mankind no more. Observation and experiment, the deceitful panders of sensory experience, shall pass from human memory and only the pure reason remain. Everything is number,"
The master's prophecy becomes less abstract and closer to the scientific actualities of the twentieth century. Speaking as a mathematical physicist and mathematical astrophysicist he proceeds to details. "I believe ... that all the laws of nature that are usually classed as fundamental can be foreseen wholly from epistemological considerations," In a brief aside he reminds us that epistemology is that department of metaphysics which deals with the theory of human knowledge. To preclude any possible misunderstanding of his meaning he elaborates his heretical creed. "An intelligence unacquainted with our universe, but acquainted with the system of thought by which the human mind interprets to itself the content of its sensory experience, should be able to attain all the knowledge of physics that we have attained by experiment. He would not deduce the particular events and objects of our experience, but he would deduce the generalizations we have based on them. For example, he would infer the existence and properties of sodium, but not the dimensions of the earth,"
If Pythagoras—ventriloquizing thus in 1935 through Sir Arthur Eddington, a leader in the retreat to the past—should be right, it would seem that the experimental scientists since Galileo and Newton have gone to much unnecessary labor to discover the obvious and proclaim it in truisms. If it is false that experiment answers all, it may be true, as some of the ancients believed, that reason answers all, or, as the successors of Pythagoras seem to believe, nearly all. For, as we have just been cautioned, reason may be unable to deduce the diameter of the earth from any data wholly within the human mind. But this defect is entirely negligible in comparison with the ability to foresee the existence and properties of the chemical elements "wholly from epistemological considerations,"
By taking sufficient thought the scientific epistemologist may rediscover for himself, without once rising from his chair in an otherwise empty room, all that three centuries of observation and experiment since Galileo and Newton have taught us of the "fundamental laws" of mechanics, heat, light, sound, electricity and magnetism, electronics, the constitution of matter, chemical reactions, the motions of the heavenly bodies, and the distribution in space of stellar systems. And by the same purely abstract considerations the thoughtful epistemologist may attain verifiable knewledge of natural phenomena which are still obscure to science, for example, the internal motions of the spiral nebulae.
Should only some of these impressive claims be sustained, the twentieth century return to Pythagoreanism may be remembered ten thousand years hence as the dawn of a lasting enlightenment and the end of the long night of error which descended on western civilization in the seventeenth century. The costly apparatus of our laboratories and observatories will have crumbled and rusted away, except possibly for a few relics fearfully preserved in the World Museum of Human Error. Above the entrance the guardians of public sanity will have inscribed the truths that liberated mankind: "Experiment answers nothing. Reason answers all," To balance these, the same guardians will have embellished the pediment of the Temple of Knowledge and Wisdom with the summary of the universe and a solemn warning: "Everything is Number. Let no man ignorant of Arithmetic enter my doors,"
But all this is in the calm certainty of a golden age of the future while we, unhappily, must endure the steel and errors of the present. To mitigate our lot we may return to the past for an hour or two, to read in it the certainty of our present and the hope of our future.
What shall we ask the past? Numerous interesting questions suggest themselves. How did human beings like our–selves ever come to believe the nonsense they did about numbers? And what can have induced reputable scientists of the twentieth century A.D. to fetch their philosophy of science from the sixth century B.C.? Have the numerologists —the number mystics—been right all these centuries and the majority of thinking men wrong?
As to the origin of it all, it began some twenty-six centuries ago in the simplest arithmetic and the most elementary school geometry. None of this is beyond the understanding of a normal child of twelve. As for who may be right and who wrong, a physicist or an engineer usually is more easily seduced than a mathematician or a logician by a mathematical demonstration. Few engineers or physicists would devote their best thought to a small but incisive treatise on the unreliability of the principles of logic. It took a mathematician to do that. Logic in its most reliable form is called pure mathematics; and though mathematical reasoning, like any other, has its drastic limitations, it is still the most powerful known. But because mathematics seems to create something out of nothing, whereas it does not, superhuman powers have been ascribed to it, even by logicians and mathematicians.
When a complicated mathematical argument ends in a spectacular prediction, subsequently verified by observation or experiment, a physicist may be excused for feeling that he has participated in a miracle. And when a skilled mathematician astounds himself with a discovery he had no conscious intention of striving after, he may well believe for a few moments as Pythagoras believed all his life, and may even repeat—after the eminent English mathematician, G. H. Hardy—the following confession of faith. "I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations,' are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards ...,"
On coming out of the daze at his own brilliance the average twentieth-century pure mathematician might begin to doubt at least the practicality of this Platonic creed, especially if he happened to be aware of what has taken place in the philosophy of mathematics since the close of the nineteenth century. The doubter might even agree with the distinguished American geometer, Edward Kasner, that the "Platonic reality" of mathematics was abandoned long ago by unmystical mathematicians, and marvel with him that rational human beings could ever have believed anything of the kind. As he puts it, "We have overcome the notion that mathematical truths have an existence independent and apart from our minds. It is even strange to us that such a notion could ever have existed. Yet this is what Pythagoras would have thought—and Descartes, along with hundreds of other great mathematicians before the nineteenth century. Today mathematics is unbound; it has cast off its chains. Whatever its essence, we recognize it to be as free as the mind, as prehensile as the imagination,"
It is not for us to judge between the two schools of thought. We note only that each of the authorities cited as a witness to the truth of mathematics published his conclusions in 1940. Even in a court of law it would be difficult to find a sharper disagreement between competent experts. A like irreconcilable difference of opinion severs the modern Pythagorean scientists from those of the older school, who still believe that reliable knowledge of the physical universe cannot be attained without observation and experiment.
My sole objective in the following chapters will be to see how these differences of opinion came about. Though the theme is number, no mathematics beyond the simplest arithmetic is required for following the story. An occasional allusion to some obvious statement about straight lines, such as young children are taught in school, need not terrify anyone if it is called geometry. The important things are not these trivialities of a grade-school education. What matters is the weird nonsense people no less intelligent than ourselves inferred from these trivialities. To prevent our excursion into the past from degenerating into a journey through a valley of dry bones, we shall become as well acquainted as we may with the great men primarily responsible for our present widely divergent opinions. The majority of the men cited are famous and their major contributions to civilization well known. The aspect of their work which is of interest here may be less familiar, though it was no less important for them than the things for which they are commonly remembered. A few names may be new to some. They are only about ten out of hundreds who left their mark on number mysticism and all that it implies for our own attempt to think straight.
Those who have had no occasion to examine for themselves what the ancient lore of numbers has done, and is doing, for their thinking habits, may be interested to linger for a while at the principal shrines where the magic of numbers paused on its way from the past to the present. Time and the continual changes in the meanings of words have confused the historical record till the hard core of arithmetical fact at the center of some ancient wisdoms is not always evident at a casual glance. Much of the influence of such apparently trivial statements as "three and seven make ten" on philosophic, religious, and scientific thinking is crusted over with the symbolisms of outmoded attempts to fabricate a meaningful image of the material universe. Ambitious and inspiring as such efforts may have been, they are far surpassed—at least in ambition—by some of the earlier struggle to explain human values in terms of numbers. Virtue to the highly imaginative Pythagoreans of antiquity was one number, vice another; and the elusive concepts of the True, the Beautiful, the Good were sublimated into "Ideal Numbers" by no lesser a metaphysician than Plato. And if it seems strange that Pythagoras should have believed that love and marriage are governed by numbers, we have but to observe the like today.
Step by step the immemorial magic of numbers has kept pace with unmystical science all down the centuries. If the patient investigation of numbers has aided the development of science and furthered such enlightenment as science can give, it has also perpetuated older beliefs that but few tolerant men would call enlightened. Once of scientific certitude, these stubborn superstitions long ago ceased to have any meaning for the literate. But the belief that number is the ultimate answer to all the riddles of the physical universe, though subtly disguised, is still recognizable in the refined mathematical mysticism of the modern Pythagoreans. Our principal concern will be to retrace the main steps by which this overwhelming conclusion has reached the living present from a past so remote that only rumors of its existence survive.
(Continues...)
Excerpted from The Magic of Numbers by Eric Temple Bell. Copyright © 1974 Taine T. Bell. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Contents
1. The Past Returns,2. A Royal Mace,
3. For Their Own Sake,
4. The Decisive Century,
5. A Difference of Opinion,
6. Wisdom as a Profession,
7. Not Much, But Enough,
8. One or Many?,
9. A Dream and a Doubt,
10. Half Man, Half Myth,
11. Discord and Harmony,
12. Harmony and Discord,
13. Mythology Transformed,
14. The Cosmos as Number,
15. Himself Made It?,
16. Intimations of the Infinite,
17. A Miscarriage of Reason,
18. Politics and Geometry,
19. "Another I",
20. Number Deified,
21. Pythagoras in Purgatory,
22. Saints and Heretics,
23. A Turning Point,
24. The Skeptical Bishop,
25. Believer and Disbeliever,
26. Changing Views,
27. Return of the Master,