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The History of the Calculus and Its Conceptual Development
Fluent description of the development of both the integral and differential calculus. Early beginnings in antiquity, medieval contributions, and a century of anticipation lead up to a consideration of Newton and Leibniz, the period of indecison that followed them, and the final rigorous formulation that we know today.
368 pages, Paperback
First published June 1, 1959
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April 4, 2021
Without a doubt, the differential and integral calculus marks the watershed between ancient and modern mathematics. For this reason, and not solely because of its crucial relevance to the revolutionary mathematical physics of the seventeenth century, the philosopher of mathematics and no less the enterprising researcher will want to command a synthetic view of its history and conceptual development, just what the prominent and able historian Carl Boyer promises us here!
What are the main conceptual issues at play?
1) Status of the infinite/infinitesimal. During the century leading up to the epochal works of Newton and Leibniz, mathematicians who held the continuum to be divisible into infinitely many points began to tackle increasingly difficult problems of quadrature, adding to the isolated but impressive examples known since antiquity (i.e., they view the continuum as an aggregate of indivisibles, which Aristotle does not allow). Cavalieri updated the ancients’ powerful but unwieldy method of exhaustion and forged an adaptable, though seemingly less rigorous technique for carrying out of quadratures. The mathematical community divided into warring camps, pitting those who continued to insist upon the full rigor of the method of exhaustion against the looser pragmatists, who justified the newer methods in light of their undeniable successes. The underlying bone of contention has to do with whether it is valid to think in terms of infinitesimals in the first place; the ancients did not. But both Newton and Leibniz did in their early work. In the absence of a formal concept of limit, they performed calculations to evaluate a derivative in which the last step would be to set the terms of higher order in the differences to zero. The procedure, while it leads to correct results, is evidently somewhat slippery and Berkeley could mock it in his Analyst. Nobody had a satisfactory answer to Berkeley’s charges until Cauchy and Weierstrass a century and more later. Boyer recounts the justificatory stratagems various parties to the controversy resorted to, not all being all that convincing.
2) Is there a temporal dimension to mathematics’ supposed eternal truths, at least as far as we can conceive them? Consider Newton’s fluxions and fluents, Leibniz’ moments or striving [nisus]. Boyer sticks to the descriptive here; he does not enter into the metaphysical question with a position of his own to defend at all. But the issue is perplexing. For indeed, mathematical truth as we in the train of the Greeks like to imagine it is static, eternally unchanging. Couldn’t one interrogate our inherited sense of mathematical truth? Granted, and Boyer does appear so to grant, it may serve a useful psychological function to picture the tangent to a curve being generated by the instantaneous state of motion of a point traveling along the curve, or the integral of a function as the summation of a the continually occurring increments along the way. Nobody denies this. If we pay attention to genealogy, our mathematical concepts are nothing but convenient condensations of psychological experience. And thus, transcendentally speaking, temporal existence is the condition of the possibility of our imagining of mathematical truth – so the latter need not necessarily be construed as eternal (in the customary sense of the term).
3) Concept of number and function. The geometrical orientation of the Greeks prevented them from arriving at the concept of function at all. But once one has become accustomed to thinking algebraically, the function concept arises naturally, whether implicitly or explicitly. Thus, the real issue comes down to how one thinks about the numbers upon which algebraic operations are to be performed. A glance at the index shows that Boyer treats of this extensively throughout. As for Boyer’s understanding of the modern view towards which everything is headed (all around, Boyer tends to be pretty Whiggish in his approach to history!), his tack stresses the ordinal concept not arithmetization per se: ‘We shall find that the history of the calculus affords an unusually striking example of the slow formation of mathematical concepts by the emancipation from all sense data of ideas born of our primary intuitions. The derivative and the integral are, in the last analysis, synthetically defined in terms of ordinal considerations and not of those of continuous quantity and variability’ (p. 11). Is he right in this? How else could one identify gaps between the rationals? This reviewer’s idea: let us ask why can we be justified in applying an ordinal concept of the number line? Isn’t this to be set down to spatial experience? But then ordinality would be secondary to a primitive notion of the continuum. For reasons such as this, the reviewer judges it better to speak of us arriving at real numbers by a process of arithmetization guided by spatial intuition rather than through an ordinal concept eo ipso. At any rate, Boyer’s take on what constitutes the modern terminus ad quem is debatable!
4) Concepts of limit, continuity, epsilontics, rigor.
Some fine-grained points in Boyer’s exposition: first, he takes pains to spell out why, contrary to the common and most immediate view, the method of exhaustion originally due to Eudoxus and perfected by Archimedes is not truly an anticipation of the calculus. Roughly speaking, nowhere do infinitesimals properly speaking enter the problem; one never goes all the way to the limit but always stops after a finite number of steps.
Second, Boyer attributes a major role to the medieval computists in breaking the ground for the eventual emergence of the calculus—what is new to the reviewer. Yes, one had heard of them, but Boyer does bring out their significance very well: ‘The blending of theological, philosophical, mathematical and scientific considerations which has so far been evident in Scholastic thought is seen to even better advantage in a study of what was perhaps the most significant contribution of the fourteenth century to the development of mathematical physics. It has commonly been protested that additions, if any, which were made to scientific knowledge in the medieval period lay solely in the field of practical discoveries and applications; and that the only mathematical achievement during this time was the simplification of the rules of operation for the Hindu-Arabic numerals, the latter having been made known to Europe by Leonardo of Pisa and other men of the thirteenth century. There is at least one exception to such an assertion, for it was precisely during this interval, and particularly in the fourteenth century, that a theoretical advance was made which was destined to be remarkably fruitful in both science and mathematics, and to lead in the end to the concept of the derivative. This consisted in the idea—often expressed, to be sure, in terms of dialectical rather than mathematical method—of studying change quantitatively, and thus admitting into mathematics the concept of variation’ (pp. 70-71).
Third, Leibniz and Cauchy on differential versus derivative. Leibniz thinks in terms of differentials, possibly infinitely small. Boyer is unambiguous that Cauchy made the derivative the fundamental object and that this continues to be the case today. There is nothing on hyperreals in non-standard analysis (Robinson had not published as of the time of writing, or perhaps Boyer felt he lacked the resources to undertake such a major endeavor).
Advantages of Boyer’s presentation: he is conceptually clear; regales us with many an aperçu that neatly encapsulates his understanding of a perhaps subtle conceptual or philosophical point – would that every author strove for such clarity and concision! As for disadvantages: there is nothing very technical here and no scholarly apparatus. Tangential philosophical remarks are tossed out from time to time but never followed up on; that is, one can fasten upon no systematic philosophical perspective on the part of the author, per se (all right, Boyer doesn’t pretend to purvey such either).
The exposition simply skips over Arabic and Hindu mathematics, aside from a handful of minor notices. The omission of the former is puzzling, given that undoubtedly the single greatest appanage with which European mathematicians of the seventeenth century were endowed and which propelled them past the achievement of the ancient Greeks was a ready facility with algebraic operations. The Greeks were stymied by their insistence on purely geometrical construction, which is stationary and tends very much to militate against a concept of a function depending on a variable quantity. But the latter is key to the notion of velocity (itself nothing but a ratio between the rates at which the ordinate and the abscissa vary). Unless one be a Newton, geometrical construction is too cumbersome to get one very far, or very fast, in the calculus. Indeed, Newton himself made his discoveries as a young man largely through algebraic means and only later in life, as a result of an all but fanatical penchant to recover the rigor of Archimedes, sought to recast everything in geometrical terms—what is, in the main, responsible for making the Principia hard going. So: the calculus emerged from the fruitful conjunction of algebraical and infinitesimal methods. But, isn’t algebra precisely what we owe to the great Arabic mathematicians? We sure did not inherit it from the Greeks, or from anyone else. How then could Boyer content himself with a treatment of Arabic mathematics that cannot even be called perfunctory? Al-Khwārizmī does not even appear in the index. Meanwhile, the other half of the recipe, the disposition towards infinitesimals, we derive from the medieval Latin West. Boyer does not skimp on this development, but devotes a thirty-five page chapter to it.
As for Hindu mathematics, Boyer’s attitude is unfairly dismissive. In part, this could be a function of the circumstance that the Indian genius for mathematics seems to be partial towards number theory. Certainly, Brahmagupta in the seventh century was versed in theory and computations with numbers that range far beyond what most everyone today, even professional mathematicians, could honestly claim to be capable of. But, here’s the rub for Boyer: number theory, with its orientation to discrete structures such as the integers, has little, on the surface of things, to contribute to a calculus of continuously varying real quantities. After the advent of the classical function theory of the nineteenth century, all this changes dramatically, but clearly later developments such as that of the application of analytical methods in number theory would have to wait upon the prior development of a calculus of real quantities in their own right. So, the strengths of Hindu mathematics were not well suited for it to play a material role in the origins of the calculus itself. Moreover, Boyer’s criticism of Hindu mathematics as too computational for the Greek stress on the logical derivation of concepts to have gained much purchase there, may have an element of justice to it. If this were so, one could understand how another pinnacle of Hindu mathematics, the discovery of what we would call Taylor series expansions for the trigonometrical functions centuries before the Europeans got there, failed to nourish an environment hospitable to the full elaboration of the logical notions of the calculus, to wit, the limit, the derivative, the antiderivative etc. One is left to wonder whether the Hindu mathematicians, if left to themselves, could eventually have reached something like the modern concept of real number, and from this the calculus, by wielding appropriate infinitary procedures with the rational numbers, themselves constructed from the integers (as did Dedekind, long after the calculus was known at the heuristic level). It would be an interesting and highly rewarding intellectual exercise to compare the conditions of discovery of Taylor series in the Occident versus the Orient.
The ninety-page chapter entitled ‘A century of anticipation’ is poorly organized. A lot on Stevin, Valerio, Kepler, Galileo, Cavalieri, Cusanus, Torricelli, Fermat, Wallis, Hobbes, Barrow, but little on Descartes or Rolle – Struik’s brief outline of the history of mathematics is notably clearer on adjudicating the roles of the respective players than Boyer’s. Lastly, there is little of substance on the eighteenth century though it has much material to offer. The chapter entitled ‘A period of indecision’ is geared mainly to showing how uncomfortable people were with the concept and use of infinitesimals, but has no coverage of why Leibnizian formalism prevailed over Newtonian on the continent (and the reverse in England) or of the spectacular consolidation of the methods of the calculus and its applications to numerous remarkable problems in mathematical physics.
Closing remarks: Boyer is not by any means overly philosophical or recondite, as many other reviewers suggest; he remains consistently appropriate and not too technical in his comparatively sparse philosophical remarks, a sign of a skilled writer who knows what is better left out. Asides such as these, when indulged in, are good in order to help the reader to see what is really at issue in the mathematics; they are included for sake of what professional mathematicians call ‘culture’. This recensionist will contend that such obiter dicta do in fact contribute to the understanding of the mathematics itself. For mathematics consists in much more than merely having the definitions; its lifeblood, rather, lies in knowing, or suspecting, what original things one can do with the concepts thus defined. Not only this, an adroit and useful definition will itself always be the product and condensation of a series of reasonings, once one has penetrated to the heart of the matter and understood what its necessary and sufficient conditions are. Boyer is certainly a competent scholar and his individual judgments may often be fine, but no overall thesis stands out, just an array of position taking with respect to others.
To be read through in a sitting or two, to alert oneself to the current discussion and debates among historians of the calculus, not a lasting scholarly reference because not detailed enough (compare with Otto Neugebauer and Jeremy Gray). Hence, the main function of Boyer’s study is to serve as a primer and entryway into the professional scholarly literature through its extensive bibliography. Now, for the dedicated student of the history of mathematics looking for a solid reference work not subject to the limitations identified above in Boyer’s: what about Moritz Cantor’s mammoth four-volume, four-thousand-page Vorlesungen über Geschichte der Mathematik? We’ll see in due course, but not anytime soon!
What are the main conceptual issues at play?
1) Status of the infinite/infinitesimal. During the century leading up to the epochal works of Newton and Leibniz, mathematicians who held the continuum to be divisible into infinitely many points began to tackle increasingly difficult problems of quadrature, adding to the isolated but impressive examples known since antiquity (i.e., they view the continuum as an aggregate of indivisibles, which Aristotle does not allow). Cavalieri updated the ancients’ powerful but unwieldy method of exhaustion and forged an adaptable, though seemingly less rigorous technique for carrying out of quadratures. The mathematical community divided into warring camps, pitting those who continued to insist upon the full rigor of the method of exhaustion against the looser pragmatists, who justified the newer methods in light of their undeniable successes. The underlying bone of contention has to do with whether it is valid to think in terms of infinitesimals in the first place; the ancients did not. But both Newton and Leibniz did in their early work. In the absence of a formal concept of limit, they performed calculations to evaluate a derivative in which the last step would be to set the terms of higher order in the differences to zero. The procedure, while it leads to correct results, is evidently somewhat slippery and Berkeley could mock it in his Analyst. Nobody had a satisfactory answer to Berkeley’s charges until Cauchy and Weierstrass a century and more later. Boyer recounts the justificatory stratagems various parties to the controversy resorted to, not all being all that convincing.
2) Is there a temporal dimension to mathematics’ supposed eternal truths, at least as far as we can conceive them? Consider Newton’s fluxions and fluents, Leibniz’ moments or striving [nisus]. Boyer sticks to the descriptive here; he does not enter into the metaphysical question with a position of his own to defend at all. But the issue is perplexing. For indeed, mathematical truth as we in the train of the Greeks like to imagine it is static, eternally unchanging. Couldn’t one interrogate our inherited sense of mathematical truth? Granted, and Boyer does appear so to grant, it may serve a useful psychological function to picture the tangent to a curve being generated by the instantaneous state of motion of a point traveling along the curve, or the integral of a function as the summation of a the continually occurring increments along the way. Nobody denies this. If we pay attention to genealogy, our mathematical concepts are nothing but convenient condensations of psychological experience. And thus, transcendentally speaking, temporal existence is the condition of the possibility of our imagining of mathematical truth – so the latter need not necessarily be construed as eternal (in the customary sense of the term).
3) Concept of number and function. The geometrical orientation of the Greeks prevented them from arriving at the concept of function at all. But once one has become accustomed to thinking algebraically, the function concept arises naturally, whether implicitly or explicitly. Thus, the real issue comes down to how one thinks about the numbers upon which algebraic operations are to be performed. A glance at the index shows that Boyer treats of this extensively throughout. As for Boyer’s understanding of the modern view towards which everything is headed (all around, Boyer tends to be pretty Whiggish in his approach to history!), his tack stresses the ordinal concept not arithmetization per se: ‘We shall find that the history of the calculus affords an unusually striking example of the slow formation of mathematical concepts by the emancipation from all sense data of ideas born of our primary intuitions. The derivative and the integral are, in the last analysis, synthetically defined in terms of ordinal considerations and not of those of continuous quantity and variability’ (p. 11). Is he right in this? How else could one identify gaps between the rationals? This reviewer’s idea: let us ask why can we be justified in applying an ordinal concept of the number line? Isn’t this to be set down to spatial experience? But then ordinality would be secondary to a primitive notion of the continuum. For reasons such as this, the reviewer judges it better to speak of us arriving at real numbers by a process of arithmetization guided by spatial intuition rather than through an ordinal concept eo ipso. At any rate, Boyer’s take on what constitutes the modern terminus ad quem is debatable!
4) Concepts of limit, continuity, epsilontics, rigor.
Some fine-grained points in Boyer’s exposition: first, he takes pains to spell out why, contrary to the common and most immediate view, the method of exhaustion originally due to Eudoxus and perfected by Archimedes is not truly an anticipation of the calculus. Roughly speaking, nowhere do infinitesimals properly speaking enter the problem; one never goes all the way to the limit but always stops after a finite number of steps.
Second, Boyer attributes a major role to the medieval computists in breaking the ground for the eventual emergence of the calculus—what is new to the reviewer. Yes, one had heard of them, but Boyer does bring out their significance very well: ‘The blending of theological, philosophical, mathematical and scientific considerations which has so far been evident in Scholastic thought is seen to even better advantage in a study of what was perhaps the most significant contribution of the fourteenth century to the development of mathematical physics. It has commonly been protested that additions, if any, which were made to scientific knowledge in the medieval period lay solely in the field of practical discoveries and applications; and that the only mathematical achievement during this time was the simplification of the rules of operation for the Hindu-Arabic numerals, the latter having been made known to Europe by Leonardo of Pisa and other men of the thirteenth century. There is at least one exception to such an assertion, for it was precisely during this interval, and particularly in the fourteenth century, that a theoretical advance was made which was destined to be remarkably fruitful in both science and mathematics, and to lead in the end to the concept of the derivative. This consisted in the idea—often expressed, to be sure, in terms of dialectical rather than mathematical method—of studying change quantitatively, and thus admitting into mathematics the concept of variation’ (pp. 70-71).
Third, Leibniz and Cauchy on differential versus derivative. Leibniz thinks in terms of differentials, possibly infinitely small. Boyer is unambiguous that Cauchy made the derivative the fundamental object and that this continues to be the case today. There is nothing on hyperreals in non-standard analysis (Robinson had not published as of the time of writing, or perhaps Boyer felt he lacked the resources to undertake such a major endeavor).
Advantages of Boyer’s presentation: he is conceptually clear; regales us with many an aperçu that neatly encapsulates his understanding of a perhaps subtle conceptual or philosophical point – would that every author strove for such clarity and concision! As for disadvantages: there is nothing very technical here and no scholarly apparatus. Tangential philosophical remarks are tossed out from time to time but never followed up on; that is, one can fasten upon no systematic philosophical perspective on the part of the author, per se (all right, Boyer doesn’t pretend to purvey such either).
The exposition simply skips over Arabic and Hindu mathematics, aside from a handful of minor notices. The omission of the former is puzzling, given that undoubtedly the single greatest appanage with which European mathematicians of the seventeenth century were endowed and which propelled them past the achievement of the ancient Greeks was a ready facility with algebraic operations. The Greeks were stymied by their insistence on purely geometrical construction, which is stationary and tends very much to militate against a concept of a function depending on a variable quantity. But the latter is key to the notion of velocity (itself nothing but a ratio between the rates at which the ordinate and the abscissa vary). Unless one be a Newton, geometrical construction is too cumbersome to get one very far, or very fast, in the calculus. Indeed, Newton himself made his discoveries as a young man largely through algebraic means and only later in life, as a result of an all but fanatical penchant to recover the rigor of Archimedes, sought to recast everything in geometrical terms—what is, in the main, responsible for making the Principia hard going. So: the calculus emerged from the fruitful conjunction of algebraical and infinitesimal methods. But, isn’t algebra precisely what we owe to the great Arabic mathematicians? We sure did not inherit it from the Greeks, or from anyone else. How then could Boyer content himself with a treatment of Arabic mathematics that cannot even be called perfunctory? Al-Khwārizmī does not even appear in the index. Meanwhile, the other half of the recipe, the disposition towards infinitesimals, we derive from the medieval Latin West. Boyer does not skimp on this development, but devotes a thirty-five page chapter to it.
As for Hindu mathematics, Boyer’s attitude is unfairly dismissive. In part, this could be a function of the circumstance that the Indian genius for mathematics seems to be partial towards number theory. Certainly, Brahmagupta in the seventh century was versed in theory and computations with numbers that range far beyond what most everyone today, even professional mathematicians, could honestly claim to be capable of. But, here’s the rub for Boyer: number theory, with its orientation to discrete structures such as the integers, has little, on the surface of things, to contribute to a calculus of continuously varying real quantities. After the advent of the classical function theory of the nineteenth century, all this changes dramatically, but clearly later developments such as that of the application of analytical methods in number theory would have to wait upon the prior development of a calculus of real quantities in their own right. So, the strengths of Hindu mathematics were not well suited for it to play a material role in the origins of the calculus itself. Moreover, Boyer’s criticism of Hindu mathematics as too computational for the Greek stress on the logical derivation of concepts to have gained much purchase there, may have an element of justice to it. If this were so, one could understand how another pinnacle of Hindu mathematics, the discovery of what we would call Taylor series expansions for the trigonometrical functions centuries before the Europeans got there, failed to nourish an environment hospitable to the full elaboration of the logical notions of the calculus, to wit, the limit, the derivative, the antiderivative etc. One is left to wonder whether the Hindu mathematicians, if left to themselves, could eventually have reached something like the modern concept of real number, and from this the calculus, by wielding appropriate infinitary procedures with the rational numbers, themselves constructed from the integers (as did Dedekind, long after the calculus was known at the heuristic level). It would be an interesting and highly rewarding intellectual exercise to compare the conditions of discovery of Taylor series in the Occident versus the Orient.
The ninety-page chapter entitled ‘A century of anticipation’ is poorly organized. A lot on Stevin, Valerio, Kepler, Galileo, Cavalieri, Cusanus, Torricelli, Fermat, Wallis, Hobbes, Barrow, but little on Descartes or Rolle – Struik’s brief outline of the history of mathematics is notably clearer on adjudicating the roles of the respective players than Boyer’s. Lastly, there is little of substance on the eighteenth century though it has much material to offer. The chapter entitled ‘A period of indecision’ is geared mainly to showing how uncomfortable people were with the concept and use of infinitesimals, but has no coverage of why Leibnizian formalism prevailed over Newtonian on the continent (and the reverse in England) or of the spectacular consolidation of the methods of the calculus and its applications to numerous remarkable problems in mathematical physics.
Closing remarks: Boyer is not by any means overly philosophical or recondite, as many other reviewers suggest; he remains consistently appropriate and not too technical in his comparatively sparse philosophical remarks, a sign of a skilled writer who knows what is better left out. Asides such as these, when indulged in, are good in order to help the reader to see what is really at issue in the mathematics; they are included for sake of what professional mathematicians call ‘culture’. This recensionist will contend that such obiter dicta do in fact contribute to the understanding of the mathematics itself. For mathematics consists in much more than merely having the definitions; its lifeblood, rather, lies in knowing, or suspecting, what original things one can do with the concepts thus defined. Not only this, an adroit and useful definition will itself always be the product and condensation of a series of reasonings, once one has penetrated to the heart of the matter and understood what its necessary and sufficient conditions are. Boyer is certainly a competent scholar and his individual judgments may often be fine, but no overall thesis stands out, just an array of position taking with respect to others.
To be read through in a sitting or two, to alert oneself to the current discussion and debates among historians of the calculus, not a lasting scholarly reference because not detailed enough (compare with Otto Neugebauer and Jeremy Gray). Hence, the main function of Boyer’s study is to serve as a primer and entryway into the professional scholarly literature through its extensive bibliography. Now, for the dedicated student of the history of mathematics looking for a solid reference work not subject to the limitations identified above in Boyer’s: what about Moritz Cantor’s mammoth four-volume, four-thousand-page Vorlesungen über Geschichte der Mathematik? We’ll see in due course, but not anytime soon!
May 6, 2017
Based on unverified claims taken as the norm. Highly misinforming with false, unresearched and biased attributions. It does not represent a true history of calculus and its development, but only a biased view which propagates the idea of western universalism in the field of mathematics and all aspects of science.
The author only regurgitates the story of Greek origins of mathematics along with doctrine of "independent rediscovery", as the later-day racist and colonial historians who built on the legacy of glorifying themselves and belittling others.
Clearly, this story of the Hellenic origin of all worthwhile secular knowledge is contrary to commonsense: why should all knowledge have originated in one place ? Myth proceeds by linking story to story, and the Hellenic story is linked to the Great Library of Alexandria - most Greek names associated with science are today traced to Alexandria (in Africa).
But what was the source of the Alexandrian library ? Over the years no one seems to have asked this question, thereby promoting the belief as an implicit postulate that this library was of Greek origin. This is the a big lie on which the story of Helenistic origin of science was concocted.
Greeks lacked science until the time of Alexander, while the books in the Alexandrian library were produced by others, which Alexander obtained as part of his war booty. This recognisably similar to the way the Toledo library was obtained-as war-booty by the proto-Crusaders. The older civilisations such as Egypt, Persia, Babylon and India has been around for long enough, and had ample economic surplus to have produced books on the scale required for the Alexandrian library.
Also the tiny Greek city states with small populations of a few thousand citizens, could hardly have produced books on this scale. The book technology then involved papyrus: a material made in Egypt, expensive to import and even more expensive to maintain. Besides, how did the Greeks support the vast leisured class needed to produce and maintain book in such numbers ? The Greek city-states were constantly engaged in petty warfare. Plato points out that prior to the library there was no culture of science in Greece. Socrates was charged with a great crime of declaring the moon to be a clod of earth and a death penalty demanded just on the ground that he did not worship the moon as a divinity. Clearly the Greeks customarily put to death anyone who dared to do anything remotely scientific in astronomy. This situation persisted till after the time of Alexander, for Aristotle too ran away from Athens because he feared death for dabbling in scientific books.
How could such an intolerant and superstitious culture have produced any science, or even during the Crusades when knowledge was theologically sanitised ? yet they are credited with most of it.
Indian planetary models had used epicycles with variable radii, corresponding to elliptic planetary orbits. Calculations related to these orbits used the calculus. This process was initiated by Aryabhata in the 5th century CE using an elegant numerical technique to calculate trigonometric values with great facility. Aryabhata's technique is essentially equivalent to what is today called "Euler's" method of numerically solving ordinary differential equations. Geometry and calculus transmitted from India to the Portuguese Romans, who setup the first Indian Roman Catholic mission in Kerala, India in 1500CE. The school, now a college was taken over by the Jesuits and was made into another Toledo library, acquiring and translating a variety of Indian texts and transmitting them back to Rome. For this they used the Syrian Christians who were already settled in Kerala long before the Jesuits arrived. There was ample motivation for Europeans to acquire the Indian calculus techniques. The European navigation problem was then the foremost scientific problem in Europe, for it held the key to the European dream of wealth through overseas trade.
Many governments offered huge rewards for solutions and European navigators depended on charts, and the construction of the Mercator chart needed a precise take of secants. Precise trigonometric values were of great concern and were already in Indian contexts, derived using Calculus. However, the calculus taken was not readily comprehended by Europeans and this is seen in Descartes quote where he thought calculus required super tasks "beyond the capacity of the human mind".
Many others reproduced the Indian infinite series (always without acknowledging the pagan sources, as was the norm in Europe to avoid death by Church).
Many many points could be discussed, but I suggest reading works of independent scholars who have researched the same but have not received much recognition.
There are a host of other cases, and the discussed cases are not exhaustive. However they only illustrate a general phenomenon: the systematic fabrication of history through claims of "independent rediscovery". These Europeans made "discoveries" in exactly the sense that Vasco de Gama or Columbus made "discoveries"-by the simple process of declaring as non-persons all those who were theologically incorrect.
Although this "Doctrine of Christian Discovery" was initiated by the 15th century papal bulls, it was fully accepted by Protestant countries, as the US Supreme Court pointed out, while granting it legal sanctity. Later day Western historians then made this into a "Doctrine of Independent Rediscovery"
So we can either understand the history of science as a series of systematic miracles, which took place in the West, or accept systemic fraud in Western history-writing.
The author only regurgitates the story of Greek origins of mathematics along with doctrine of "independent rediscovery", as the later-day racist and colonial historians who built on the legacy of glorifying themselves and belittling others.
Clearly, this story of the Hellenic origin of all worthwhile secular knowledge is contrary to commonsense: why should all knowledge have originated in one place ? Myth proceeds by linking story to story, and the Hellenic story is linked to the Great Library of Alexandria - most Greek names associated with science are today traced to Alexandria (in Africa).
But what was the source of the Alexandrian library ? Over the years no one seems to have asked this question, thereby promoting the belief as an implicit postulate that this library was of Greek origin. This is the a big lie on which the story of Helenistic origin of science was concocted.
Greeks lacked science until the time of Alexander, while the books in the Alexandrian library were produced by others, which Alexander obtained as part of his war booty. This recognisably similar to the way the Toledo library was obtained-as war-booty by the proto-Crusaders. The older civilisations such as Egypt, Persia, Babylon and India has been around for long enough, and had ample economic surplus to have produced books on the scale required for the Alexandrian library.
Also the tiny Greek city states with small populations of a few thousand citizens, could hardly have produced books on this scale. The book technology then involved papyrus: a material made in Egypt, expensive to import and even more expensive to maintain. Besides, how did the Greeks support the vast leisured class needed to produce and maintain book in such numbers ? The Greek city-states were constantly engaged in petty warfare. Plato points out that prior to the library there was no culture of science in Greece. Socrates was charged with a great crime of declaring the moon to be a clod of earth and a death penalty demanded just on the ground that he did not worship the moon as a divinity. Clearly the Greeks customarily put to death anyone who dared to do anything remotely scientific in astronomy. This situation persisted till after the time of Alexander, for Aristotle too ran away from Athens because he feared death for dabbling in scientific books.
How could such an intolerant and superstitious culture have produced any science, or even during the Crusades when knowledge was theologically sanitised ? yet they are credited with most of it.
Indian planetary models had used epicycles with variable radii, corresponding to elliptic planetary orbits. Calculations related to these orbits used the calculus. This process was initiated by Aryabhata in the 5th century CE using an elegant numerical technique to calculate trigonometric values with great facility. Aryabhata's technique is essentially equivalent to what is today called "Euler's" method of numerically solving ordinary differential equations. Geometry and calculus transmitted from India to the Portuguese Romans, who setup the first Indian Roman Catholic mission in Kerala, India in 1500CE. The school, now a college was taken over by the Jesuits and was made into another Toledo library, acquiring and translating a variety of Indian texts and transmitting them back to Rome. For this they used the Syrian Christians who were already settled in Kerala long before the Jesuits arrived. There was ample motivation for Europeans to acquire the Indian calculus techniques. The European navigation problem was then the foremost scientific problem in Europe, for it held the key to the European dream of wealth through overseas trade.
Many governments offered huge rewards for solutions and European navigators depended on charts, and the construction of the Mercator chart needed a precise take of secants. Precise trigonometric values were of great concern and were already in Indian contexts, derived using Calculus. However, the calculus taken was not readily comprehended by Europeans and this is seen in Descartes quote where he thought calculus required super tasks "beyond the capacity of the human mind".
Many others reproduced the Indian infinite series (always without acknowledging the pagan sources, as was the norm in Europe to avoid death by Church).
Many many points could be discussed, but I suggest reading works of independent scholars who have researched the same but have not received much recognition.
There are a host of other cases, and the discussed cases are not exhaustive. However they only illustrate a general phenomenon: the systematic fabrication of history through claims of "independent rediscovery". These Europeans made "discoveries" in exactly the sense that Vasco de Gama or Columbus made "discoveries"-by the simple process of declaring as non-persons all those who were theologically incorrect.
Although this "Doctrine of Christian Discovery" was initiated by the 15th century papal bulls, it was fully accepted by Protestant countries, as the US Supreme Court pointed out, while granting it legal sanctity. Later day Western historians then made this into a "Doctrine of Independent Rediscovery"
So we can either understand the history of science as a series of systematic miracles, which took place in the West, or accept systemic fraud in Western history-writing.
June 9, 2015
I love math a lot. Naturally, because I also like history this book was extremely enjoyable.
Carl Boyer writes in such a way that he makes the history of math accessible and enjoyable to the laymen and the expert. Boyer traces the history of the development of calculus from is conceptual roots in Ancient Greek math, in through the debates over the validity of the infinite in the middle ages, to the debate and differences because Newtonian and Leibnizian calculus and their aftermaths.
This was a wonderfully enjoyable books.
Carl Boyer writes in such a way that he makes the history of math accessible and enjoyable to the laymen and the expert. Boyer traces the history of the development of calculus from is conceptual roots in Ancient Greek math, in through the debates over the validity of the infinite in the middle ages, to the debate and differences because Newtonian and Leibnizian calculus and their aftermaths.
This was a wonderfully enjoyable books.
April 4, 2014
Quite dry, and would be improved by having more than a mere 22 figures—several geometric explanations would have been clearer for it. I would have gotten more out of it if I already had a stronger understanding of the contemporary foundations of the calculus, but at the same time, having read this, I will be better prepared to learn those concepts, since I will have some knowledge of the context in which and the difficulties in answer to which they were developed.
January 1, 2017
What was I doing?! Too clever for me by far!
May 31, 2023
TL; DR: Just what you'd expect this to be.
No surprises here. The writing is bone dry, but aside from some poorly explained proofs is generally readable. Not worth your time, though, and that’s coming from someone who loves math. Really this is a 1939 PhD thesis on the history of calculus, so, regardless of your interest in the subject, the writing is about as unsexy and esoteric as it gets. A review is pretty much superfluous here; “don’t judge a book by its cover” does not apply. Take a look at this one, assume what the reading experience will be like for you, and you’ll be exactly right.
Grade: C
No surprises here. The writing is bone dry, but aside from some poorly explained proofs is generally readable. Not worth your time, though, and that’s coming from someone who loves math. Really this is a 1939 PhD thesis on the history of calculus, so, regardless of your interest in the subject, the writing is about as unsexy and esoteric as it gets. A review is pretty much superfluous here; “don’t judge a book by its cover” does not apply. Take a look at this one, assume what the reading experience will be like for you, and you’ll be exactly right.
Grade: C
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February 21, 2024Some of the mathematical explanations could have been expressed more clearly, but otherwise this was an excellent survey of the development of the themes and concepts ( and, as such, of the philosophical underpinnings of mathematics overall). I would recommend to those already well versed in calculus.
April 6, 2018
Not an easy read, but full of insight, thorough, and thought-provoking.
December 10, 2021
Must read for anyone interested in the history of mathematics. Probably my second most favorite book of all time.
July 14, 2013
It's a classic, but some of the organization could be redone to aid in the progression of major concepts.
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