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The Study of the History of Mathematics 2, It is a pity that this should be so for the history of mathematics should really be the. kernel of the history of culture Take the mathematical developments out of the history of. science and you suppress the skeleton which supported and kept together all the rest. Mathematics gives to science its innermost unity and cohesion which can never be. entirely replaced with props and buttresses or with roundabout connections no matter. how many of these may be introduced, On the other hand the historian of mathematics remembering that his activity is. complementary to that of the historian of science will not attempt to do over agin the. latter s task and he may even feel inclined to take of his own subject too technical and. too narrow a view Therefore it is well to insist that he should seize every occasion to. indicate the relationships between mathematics and other sciences and to insist that these. relationships have always been reciprocal mathematical problems being often the result. of physical needs while mathematical elaboration gave physics and gradually other. sciences not only means of discovery of almost miraculous potency but also perfect. models of analysis and synthesis, Some historians of mathematics with a strong bent for humanism are willing to. consider not only other scientific activities than the purely mathematical but the whole. gamut of life So much the better Others moving in the opposite direction feel that the. history of mathematics itself not to speak of the history of science is too complicated. a subject and wishing to avoid the endless intricacies of the mathematical tree they. select one branch of it and study its development in more or less complete isolation from. the others Thus the historian may be led to investigate the development of algebra across. the ages or the amplifications of a single idea like the idea of number function or. Such abstraction in historical research as opposed to the more natural procedure of. considering each fact as it occurs in due chronological order is very arbitrary It is. perhaps worth while to examine the matter a little more carefully The filiation of ideas is. somewhat like the filiation of individuals except that the intricacy is even greater Any. individual A thinking only of his own genealogy has a simple pattern in his mind like our. figure I but that pattern is obviously a false one from every point of view except that of. his own unimportant personality In reality the pattern is enormously more complicated. for each couple may have had more than one child each person may have married more. than once and marriages between cousins have introduced new cross relationships The. complete picture of a man s family is like a network which if it be drawn completely. even for only a few generations is almost inextricable Of course there is nothing to. prevent any individual from selecting in that network and drawing more heavily the lines. which concern him immediately the blood lines but the personal pattern thus abstracted. from the whole network is of no interest except to himself. The Study of the History of Mathematics 3, Now the filiation of ideas is necessarily more complicated for the biological pattern. is rigorously limited by the rule that each individual has two parents neither more nor. less while each idea may result from the fusion of more than two others or on the. contrary may be the fruit of a kind of parthenogenesis Whenever the historian tries to. relate the history of a single group of ideas he is obliged to abstract one pattern from a. network of endless complexity and such an abstraction however interesting it may be is. always arbitrary to a degree, The study of special branches of mathematics or of special mathematical ideas is very.

useful for it helps one to understand those particular ideas more deeply but it should not. be allowed to confuse our historical perspective The historian must try to keep in mind. the chronological succession not of this or that idea abstracted from the rest but of the. main ideas all of them in their mutual relationships and in their diverse connections with. the rest of life, Many times have I compared the history of science with a secret history the account. of a development taking place mysteriously in the darkness while the majority of people. are more interested and more immediately affected by the events happening on the. battlefield or the forum or by the vicissitudes of their own selves and families For. societies even as for individuals one must make a sharp distinction between the things. which are the most urgent and those which are the most important These things are not. by any means the same The most urgent necessity is to live to remain alive that is to. eat sleep to be happy to procreate children and obtain security for one s family That. means physiology business and sport and often enough war However the most. important things are not to satisfy one s physiological needs but to increase the cultural. The Study of the History of Mathematics 4, heritage which has been bequeathed to us The urgent things are obvious enough and. men s efforts to obtain them fill the whole historical picture one can hardly see anything. else Yet all the time some men pursue in the darkness secretly the fulfilment of their. intellectual desires and of humanity s highest purpose. If the history of science is a secret history then the history of mathematics is doubly. secret a secret within a secret for the growth of mathematics is unknown not only to the. general public but even to scientific workers It is true engineers may be found from. time to time employing a new formula but this does not imply any knowledge or. understanding of the process which led to it Even so the average citizen uses every day. more and more complicated and marvelous machines about which he knows less and less. Yet that secret activity is fundamental it is all the time creating new theories which. sooner or later will set new wheels moving new machines working or better still will. enable us to obtain a deeper understanding of the mechanism of the universe. The practical man may neglect those secreta secretorum but the philosopher cannot. neglect them without loss and without disgrace The practical and hard headed. mathematician bent on his own investigations and nothing else may neglect them too. but he will be a poorer man for doing so Indeed one may claim that the history of. mathematics provides for him the very best education the best humanistic initiation one. especially adapted to his own needs, Let us contemplate for a moment the magnificent panorama of mathematical history. as it unfolds itself before us when we evoke the past First millenaries of preparation. during which some fundamental discoveries are already adumbrated the idea of number. slowly emerges from the darkness the idea of fraction the idea of periodicity in. geometrical patterns and others By the middle of the fourth millennium before Christ. the Egyptians were already acquainted with large numbers of the order of millions and. with a decimal system of numeration Before the middle of the second millennium they. had already attained sufficient geometrical insight to determine the area of any triangle as. we do it ourselves and to solve more difficult problems such as finding the volume of a. frustum of a square pyramid To measure the area of a circle they squared eight ninths of. its diameter which was a remarkably good approximation During all that time the. people of Mesopotamia had been developing a mathematics of their own which was as. admirable as the Egyptian In the fourth millennium the Sumerians had already some kind. of position concept in the writing of numbers and had learned to treat submultiples in. the same way as multiplies an idea which the Western world did not recapture until fifty. centuries later The geometry of the Babylonians did not reach the same level as that of. the Egyptians but on the other hand their resourcefulness in algebra was astounding for. they succeeded in solving not only quadratic but even cubic equations To appreciate the. relative importance of these achievements it is well to remember that we are much closer. to Euclid often called the father of geometry than Euclid was to the unknown Egyptian. and Mesopotamian mathematicians, In reality the way for Euclidean mathematics was very gradually and thoroughly. prepared not only by the millenary efforts of Africans and Asiatics but by three. centuries of persistent investigations by the most gifted people among our ancestors the. Greeks of the golden age The historian is made to witness the building up as it were. stone by stone of that wonderful monument geometry as it was finally transmitted to us. in the Elements The Greek miracle continued for at least six more centuries after. The Study of the History of Mathematics 5, Euclid but with less and less intensity and with longer intervals of sleep between the.

periods of creation In the meanwhile the centre of mathematical light had moved from. Athens for a brief interval to Syracuse and then to the Greco oriental city of Alexandria. where it remained for centuries Thus was their debt to Egypt abundantly repaid by the. Greek masters and the Roman disciples, After the Romans come the barbarians and ancient wisdom was in danger of. complete oblivion when it was unexpectedly rescued by the Arabs These were also. barbarians but barbarians redeemed by an intense faith and for a few centuries at least. by an unquenchable curiosity The masterpieces of Greek mathematics were translated. into Arabic and thus transmitted to the West If we call the Greek astounding. rationalization of geometrical thought a miracle by means of which word we simply. mean to convey that we cannot account for the achievement but only marvel at it then. the Arabic rescue and renaissance was another miracle that is a series of events which. nobody could have foreseen and which nobody can completely explain. The Arabs were mainly transmitters and brokers but their brokerage in a period of. crisis was almost providential They brought together Hindu and Greek ideas fertilizing. the ones with the others and revolutionizing arithmetic algebra and trigonometry Their. own contributions in these branches of mathematics were considerable and in geometry. they were sufficiently good pupils of the Greeks to discuss the postulates of Euclid and. solve the most difficult problems of Archimedean and Apollonian geometry at a time. when Latin knowledge had sunk below the Egyptian or the Babylonian level After five. centuries of leadership the Arabic culture succumbed under the stress of political. vicissitudes and Muslim obscurantism and a new renaissance of mathematics began in. Western Europe, That Renaissance slowly prepared by Christian and Jewish mathematicians. blossomed first as we should expect in Italy then in the Netherlands England and the. other countries of Europe where trade was flourishing and new cities rapidly growing. where universities vied with one another and emulation was excited by proud challenges. from some of the mathematicians to their rivals Thus was gradually introduced a second. golden age almost as brilliant as the first Just think of this array of men the children of a. single century Kepler Napier Briggs Fermat Descartes Desargues Pascal Huygens. Newton Leibniz Seki K wa What could we say of those giants in so brief a sketch as. this except that the glory of Greece so well known to all of them except the last was. resurrected in them In a way they continued the Greek tradition and they did it with so. much fervour that they almost forgot their humbler but very real debts to the Middle. Ages This golden age was not transitory like the Greek one it continued with less. splendor perhaps but with equal greatness until our own days The immense prestige of. the seventeenth century mathematics is partly due to the effect of contrast The giants of. those golden days seem more gigantic because they rose so near the mediaeval plains. We are startled when we think of the close succession of their achievements and the. cumulative effect upon us is similar to that of the mountains which we see in the course. of a journey As we come from the lowlands the first snowy peak amazes us and if many. such giants of nature follow each other within a relatively short time we may be. completely overwhelmed There were a number of mathematical giants in the eighteenth. and nineteenth centuries but by that time a new pace had already been set and one. almost expected mathematical progress to continue indefinitely as the same rate. The Study of the History of Mathematics 6, Will it continue It is too early to know but the twentieth century has strongly. accentuated the spirit of criticism which characterized the end of the last century and. created in mathematics as well as in other fields of science a period of examination. experimentation yes even in mathematics 2 and revolutionary thought which may be. the best preparation for new adventures and new discoveries We cannot tell what will. happen because such fermentation of mathematical ideas has not occurred before on the. same scale It may be a good omen or the outcome may be smaller than the bustle Let us. remember the secrecy of mathematics Great discoveries are not made without. preparation far from it but they are likely to come in very quietly without drums and. The history of mathematics is essentially different from the history of other sciences in its relationship with the history of science because it never was an integral part of the latter in the Whewellian sense The reason for this is obvious mathematics being far more esoteric than the other sciences its history can only be told to a select group of initiates It is true that there are in

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