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by
Morris Kline
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June 15 - June 28, 2019
A famous phrase, Mathematics, Queen of the Sciences, is due to him.
Gauss was too brilliant a man to overlook the implication of this fact. If there was some freedom in the choice of a parallel axiom, then one might choose an axiom different from Euclid’s and build a new kind of geometry. Gauss did just this. He pursued the logical implications of a system of axioms which included the assumption that more than one parallel to a given line passed through a given point, and thus created non-Euclidean geometry.
(Gauss himself finally adopted the term non-Euclidean after having called his system anti-Euclidean geometry, and later astral geometry.)
Gauss was far ahead of his times in concluding that Euclidean geometry was not necessarily the correct description of physical space and that some non-Euclidean geometry might prove as accurate.
He said later that two thousand years of fruitless attempts to put the parallel axiom on an unquestionable basis had led him to suspect that it could not be done. From 1829 on Lobachevsky published books and papers in which he expounded the theorems that hold in his non-Euclidean geometry.
The mathematician Georg Cantor spoke of a law of conservation of ignorance.
:-)
NON-EUCLIDEAN GEOMETRY AND THE NATURE OF MATHEMATICS
The existence of non-Euclidean geometries which can fit physical space, to say nothing of the actual use of one of these non-Euclidean geometries in the theory of relativity, has had profound implications for mathematics itself, for science, and for some segments of our culture.
The most important effect of this creation has been the realization that mathematics does not offer truths.
Insofar as the study of the physical world is concerned, mathematics has the same character as any of the sciences. It offers nothing but theories. And, as in science, new mathematical theories may replace older ones when experience or experiment shows that a new theory provides closer correspondence than an older one.
As Einstein put it, So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality.
“The essence of mathematics is its freedom.”
Men such as Plato and Descartes were convinced that mathematical truths were innate in human beings. Kant based his entire philosophy on the existence of mathematical truths. But now philosophy is haunted by the specter that the search for truths may be a search for phantoms.
Apparently the intellectual process does not lead to certainties. In Henri Bergson’s words, “One can always reason with reason.”
It is very likely that the abandonment of absolutes has seeped into the minds of all intellectuals.
It would be more appropriate to say of man that he is surest of what he believes, than to claim that he believes what is sure.
RUSSELL, BERTRAND: The ABC of Relativity, Harper and Bros., New York, 1926.
SOMMERVILLE, D. M. Y.: The Elements of Non-Euclidean Geometry, Dover Publications, Inc., New York, 1958.
WOLFE, HAROLD E.: Introduction to Non-Euclidean Geometry, The Dryden ...
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Mathematicians, notably Descartes and Leibniz, were so much impressed by the usefulness of ordinary algebra that they conceived the idea of inventing an algebra for reasoning in all fields of thought.
The social scientists’ inability to find fundamental principles is undoubtedly due to the immense complexity of the phenomena that they wish to study. Human nature is a more complicated structure than a mass sliding down an inclined plane or a bob vibrating on a spring.
If one attempts to simplify these problems by making assumptions about some of the factors involved or by neglecting what appear to be minor factors, just as Galileo, for example, neglected air resistance, one is likely to make the problem so artificial that its solution no longer has any bearing on real situations.
Life is the art of drawing sufficient conclusions from insufficient premises. SAMUEL BUTLER
Mathematics has been created by man to help him understand the universe and utilize the resources of the physical world.
“When it is not in our power to determine what is true we ought to act in accordance with what is most probable.”
Thus the theory of probability, which was first developed to solve problems of gambling, takes the gamble out of the insurance business.
BOHM, DAVID: Causality and Chance in Modern Physics, Routledge & Kegan Paul Ltd., London, 1957.
BORN, MAX: Natural Philosophy of Cause and Chance, Oxford University Press, New York, 1949.
LEVINSON, HORACE C.: The Science of Chance, Holt, Rinehart and Winston, Inc., New York, 1950.
SCHRÖDINGER, ERWIN: Science and the Human Temperament, W. W. Norton & Co., New York, 1935. Reprinted under the title Science, Theory and Man, Dover Publications, Inc., New York, 1957.
This, therefore, is Mathematics: she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings to light our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth. . . PROCLUS DIADOCHUS
“The science of pure mathematics may claim to be the most original creation of the human spirit.”
THE VALUES OF MATHEMATICS FOR THE STUDY OF NATURE
However, the primary value of mathematics is not so much what the subject itself offers but what it helps man to achieve in the study of the physical world.
Actually many of the greatest mathematicians were also the greatest physicists and astronomers of their ages, and, until very recent times when the increase in knowledge forced specialization, almost all mathematicians contributed to science.
A more powerful mathematical approach to nature was forged by Galileo when he decided that science must seek to establish quantitative laws.
Since persistence is required for all creative work, the mathematician, too, must have the stamina to wrestle with a problem until he has succeeded in solving it. He must have confidence in his powers. He may be driven to creative activity, as is the poet or painter, by pride in his reasoning faculty, the spirit of exploration, and the desire to express himself, but he must persist.
The greatest mathematicians have stressed the concentration and time they have devoted to problems.
Gauss said, perhaps over-modestly but sincerely, “If others would but reflect on mathematical truths as deeply and as continuously as I ...
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Perhaps the best reason for regarding mathematics as an art is not so much that it affords an outlet for creative activity as that it provides spiritual values. It puts man in touch with the highest aspirations and loftiest goals. It offers intellectual de...
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As man’s greatest and most successful intellectual experiment, mathematics demonstrates manifestly how powerful our rational faculty is. It is the finest expression of man’s intellectual strength. His reason has, for example, far outstripped his imagination. He can think about stars so distant that only numbers convey any meaning, about spaces which cannot be pictured, and about electrons too small to be seen with the most powerful microscopes.
For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either. The end of things and their beginnings are impregnably concealed from him in an impenetrable secret. He is equally incapable of seeing the nothingness out of which he was drawn and the infinite in which he is engulfed.
The life of man is solitary, poor, nasty, brutish, and short. He is the prey of trivial happenings.
This is it + computers and networks to broadcast trivialities .
Amid the chaos of life and his environment, he has sought patterns of explanation and systems of knowledge that might help him to attain some mastery over his environment. The chief tool proved to be the product of man’s own reason, and its accomplishments were described by Fourier.
Man’s mathematics may be no more than a workable scheme. Nature itself may be far more complex or have no inherent design. Nevertheless, mathematics remains the method par excellence for the investigation, representation, and mastery of nature.
Mathematics then is a formidable and bold bridge between ourselves and the external world. Though it is a purely human creation, the access it has given us to some domains of nature enables us to progress far beyond all expectations.
The knowledge so gleaned filters through philosophy, literature, religion, the arts, and social thought. It thereby fashions the whole culture and provides whatever answers man has to the major questions he raises about his own life.
BURY, J. B.: The Idea of Progress, Dover Publications, Inc., New York, 1955.
ELLIS, HAVELOCK: The Dance of Life, Chap. 3, The Modern Library, New York, 1929.
POINCARÉ, HENRI: The Value of Science, Chaps. 1 to 3, Dover Publications, Inc., New York, 1958.
