Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2108963
F. Jamil Ragep
ABSTRACT The history of mathematics is replete with multidimensionality and multiculturalism. This essay attempts to use the history of astronomy (one of the mathematical sciences) to emphasize the multiple traditions from various cultural zones that contributed to that history. In doing so, it supplements, but also challenges, the more unidimensional story that Reviel Netz puts forth in his own essay on the importance of the Archimedean tradition. Specifically, the essay uses examples from Babylonian, Greek, Indian, and, especially, Islamic astronomy to show how traditions not tied to Archimedes were of major importance on the often circuitous path to modern science. The notion of contingency is also explored, in particular the idea that without Greek mathematics in general, and Archimedes in particular, early modern and modern science might not have been possible. Counter to this, the essay explores other possible scenarios, whereby different mathematical traditions in other cultural settings could have plausibly led to major breakthroughs associated with the scientific revolution.
{"title":"Mathematics, the mathematical sciences, and historical contingency: Some thoughts on reading Netz","authors":"F. Jamil Ragep","doi":"10.1080/03080188.2022.2108963","DOIUrl":"https://doi.org/10.1080/03080188.2022.2108963","url":null,"abstract":"ABSTRACT The history of mathematics is replete with multidimensionality and multiculturalism. This essay attempts to use the history of astronomy (one of the mathematical sciences) to emphasize the multiple traditions from various cultural zones that contributed to that history. In doing so, it supplements, but also challenges, the more unidimensional story that Reviel Netz puts forth in his own essay on the importance of the Archimedean tradition. Specifically, the essay uses examples from Babylonian, Greek, Indian, and, especially, Islamic astronomy to show how traditions not tied to Archimedes were of major importance on the often circuitous path to modern science. The notion of contingency is also explored, in particular the idea that without Greek mathematics in general, and Archimedes in particular, early modern and modern science might not have been possible. Counter to this, the essay explores other possible scenarios, whereby different mathematical traditions in other cultural settings could have plausibly led to major breakthroughs associated with the scientific revolution.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"464 - 477"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47446577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2108968
G. Lloyd
ABSTRACT The points at which Greek mathematics in general and Archimedes' contributions in particular are exceptional are here assessed by way of a comparison with the extensive evidence from ancient China. While underlining the need for caution concerning the extent to which concrete conclusions are possible, the outcome is broadly to confirm Netz's argument that a key factor in Archimedes' success and influence was the way in which in a social and intellectual environment that favoured debate, he was able to contest an assumption found in both the Platonic and Aristotelian traditions. Where they had imagined a sharp division (albeit differently defined) between what they assigned to ‘physics’ and to ‘mathematics’ respectively, Archimedes showed how those two inquiries could be treated as complementary to one another, thereby opening up the possibility of new styles of physical demonstration.
{"title":"The problems of exceptionality: The case of Archimedes and the Greeks","authors":"G. Lloyd","doi":"10.1080/03080188.2022.2108968","DOIUrl":"https://doi.org/10.1080/03080188.2022.2108968","url":null,"abstract":"ABSTRACT The points at which Greek mathematics in general and Archimedes' contributions in particular are exceptional are here assessed by way of a comparison with the extensive evidence from ancient China. While underlining the need for caution concerning the extent to which concrete conclusions are possible, the outcome is broadly to confirm Netz's argument that a key factor in Archimedes' success and influence was the way in which in a social and intellectual environment that favoured debate, he was able to contest an assumption found in both the Platonic and Aristotelian traditions. Where they had imagined a sharp division (albeit differently defined) between what they assigned to ‘physics’ and to ‘mathematics’ respectively, Archimedes showed how those two inquiries could be treated as complementary to one another, thereby opening up the possibility of new styles of physical demonstration.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"426 - 439"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49444313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2108962
P. D. Napolitani
ABSTRACT This article argues that the recovery of Greek mathematics and particularly the mathematics of Archimedes in the course of the twelfth to sixteenth centuries was not the only factor that brought about the creation of modern mathematics in the course of the seventeenth. On the contrary, the work of mathematicians such as Francesco Maurolico, Luca Valerio, and Bonaventura Cavalieri, or of scientists such as Guidobaldo dal Monte and Galileo, was impeded by their attempts to hold to the paradigm of Greek mathematics.
{"title":"History and mythography: On the role of Archimedean mathematics in the Renaissance","authors":"P. D. Napolitani","doi":"10.1080/03080188.2022.2108962","DOIUrl":"https://doi.org/10.1080/03080188.2022.2108962","url":null,"abstract":"ABSTRACT This article argues that the recovery of Greek mathematics and particularly the mathematics of Archimedes in the course of the twelfth to sixteenth centuries was not the only factor that brought about the creation of modern mathematics in the course of the seventeenth. On the contrary, the work of mathematicians such as Francesco Maurolico, Luca Valerio, and Bonaventura Cavalieri, or of scientists such as Guidobaldo dal Monte and Galileo, was impeded by their attempts to hold to the paradigm of Greek mathematics.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"449 - 463"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46610035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2108965
W. Scheidel
ABSTRACT This paper responds to Reviel Netz’s attempt to link modernizing development in Europe to the legacy of the ancient Greek mathematician Archimedes. It defends the value of the kind of counterfactual reasoning that underpins Netz’s effort and contextualizes his argument within the broader context of the debate about the origins of the ‘Great Divergence’ between Europe and other parts of the world. The nexus between Archimedean insights and the development of the modern steam engine posited by Netz receives critical attention: we must ask to what extent counterfactual alternative modes of modernization might allow us to dissociate modernization from ancient Greek legacies. Finally, this paper draws attention to the political implications of presenting an exceptionalist vision of developmental features that are held to be uniquely associated with ancient Greece and early modern Europe, concluding that special care needs to be taken in employing this contentious perspective.
{"title":"Non-Archimedean modernities","authors":"W. Scheidel","doi":"10.1080/03080188.2022.2108965","DOIUrl":"https://doi.org/10.1080/03080188.2022.2108965","url":null,"abstract":"ABSTRACT This paper responds to Reviel Netz’s attempt to link modernizing development in Europe to the legacy of the ancient Greek mathematician Archimedes. It defends the value of the kind of counterfactual reasoning that underpins Netz’s effort and contextualizes his argument within the broader context of the debate about the origins of the ‘Great Divergence’ between Europe and other parts of the world. The nexus between Archimedean insights and the development of the modern steam engine posited by Netz receives critical attention: we must ask to what extent counterfactual alternative modes of modernization might allow us to dissociate modernization from ancient Greek legacies. Finally, this paper draws attention to the political implications of presenting an exceptionalist vision of developmental features that are held to be uniquely associated with ancient Greece and early modern Europe, concluding that special care needs to be taken in employing this contentious perspective.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"520 - 529"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46550720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2130595
Agathe Keller
ABSTRACT This essay argues against a history of mathematics that celebrates a unique Greek ‘outlier’ in world history, while raising the question of the new type of histories of mathematics that we should write today. Taking examples from the history of mathematical sources in Sanskrit and histories of mathematics written in South Asia, this essay deconstructs some assumptions behind narratives of Greek and European exceptionalism. In particular, it challenges the notion that ancient Greece possessed a unique democratic culture that fostered scientific debate and that only Greece possessed brash authors able to challenge common sense. The essay provides a reflection on the political histories that European exceptionalism – in particular regarding mathematics – has directly or indirectly shaped.
{"title":"Navigating the sea of histories of mathematics","authors":"Agathe Keller","doi":"10.1080/03080188.2022.2130595","DOIUrl":"https://doi.org/10.1080/03080188.2022.2130595","url":null,"abstract":"ABSTRACT This essay argues against a history of mathematics that celebrates a unique Greek ‘outlier’ in world history, while raising the question of the new type of histories of mathematics that we should write today. Taking examples from the history of mathematical sources in Sanskrit and histories of mathematics written in South Asia, this essay deconstructs some assumptions behind narratives of Greek and European exceptionalism. In particular, it challenges the notion that ancient Greece possessed a unique democratic culture that fostered scientific debate and that only Greece possessed brash authors able to challenge common sense. The essay provides a reflection on the political histories that European exceptionalism – in particular regarding mathematics – has directly or indirectly shaped.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"404 - 425"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43389412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2109100
R. Netz
ABSTRACT Did science, as we know it, have to be? The article explores a possible response in the negative, organized around a specific contingency: that of Greek mathematics or, even more specifically, that of the mathematics of the generation of Archimedes. The argument is that (1) Greek mathematics, seen against a cross-cultural comparison, is an anomaly, (2) the scientific revolution, as it in fact unfolded, was directly shaped by the anomaly of Greek mathematics, and (3) it is not clear that, absent Greek mathematics, an equivalent scientific revolution would have taken place. The argument is developed in some detail concerning the history of ancient Greek science, but more is said on the inevitably philosophical questions of counterfactuals in history and on the specific question of the contingency of science.
{"title":"The place of Archimedes in world history","authors":"R. Netz","doi":"10.1080/03080188.2022.2109100","DOIUrl":"https://doi.org/10.1080/03080188.2022.2109100","url":null,"abstract":"ABSTRACT Did science, as we know it, have to be? The article explores a possible response in the negative, organized around a specific contingency: that of Greek mathematics or, even more specifically, that of the mathematics of the generation of Archimedes. The argument is that (1) Greek mathematics, seen against a cross-cultural comparison, is an anomaly, (2) the scientific revolution, as it in fact unfolded, was directly shaped by the anomaly of Greek mathematics, and (3) it is not clear that, absent Greek mathematics, an equivalent scientific revolution would have taken place. The argument is developed in some detail concerning the history of ancient Greek science, but more is said on the inevitably philosophical questions of counterfactuals in history and on the specific question of the contingency of science.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"301 - 330"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43394302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2109858
C. Roby
ABSTRACT For Archimedes’ work to have furnished the ‘key theoretical tools’ for the scientific revolution, the texts must have been comprehensible to early modern readers. Yet as Archimedes’ readers and commentators have observed for centuries, his work can be very difficult going indeed. In this chapter, I explore how commentaries and other explanatory texts, like Guidobaldo dal Monte’s 1588 ‘paraphrase’ commentary to Archimedes’ Planes in Equilibrium, can help unfurl the difficulties of Archimedes’ sparse proofs by means of additional explanations and examples that help the reader develop their own ability to follow Archimedes’ reasoning. I give particular attention to insights from contemporary cognitive science into how strategic repetition of information, the materiality of the text, and highlighting connections between mathematics and the material world can all aid in the learning process.
{"title":"Archimedes for the rest of us: Thinking commentary with Guidobaldo dal Monte","authors":"C. Roby","doi":"10.1080/03080188.2022.2109858","DOIUrl":"https://doi.org/10.1080/03080188.2022.2109858","url":null,"abstract":"ABSTRACT For Archimedes’ work to have furnished the ‘key theoretical tools’ for the scientific revolution, the texts must have been comprehensible to early modern readers. Yet as Archimedes’ readers and commentators have observed for centuries, his work can be very difficult going indeed. In this chapter, I explore how commentaries and other explanatory texts, like Guidobaldo dal Monte’s 1588 ‘paraphrase’ commentary to Archimedes’ Planes in Equilibrium, can help unfurl the difficulties of Archimedes’ sparse proofs by means of additional explanations and examples that help the reader develop their own ability to follow Archimedes’ reasoning. I give particular attention to insights from contemporary cognitive science into how strategic repetition of information, the materiality of the text, and highlighting connections between mathematics and the material world can all aid in the learning process.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"491 - 519"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45599950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2109102
J. Høyrup
ABSTRACT This answer to Reviel Netz’s article at first questions his optimism concerning our technological future. After that, it suggests a different role for Archimedes and the other prominent Greek mathematicians than the one claimed by Netz: as providers not of answers but of problems which called for the transformation of the abbacus and cossist algebra tradition, thereby allowing first Viète and then Descartes to create the first level of the new analysis of the seventeenth century (the second level being infinitesimal analysis).
{"title":"Where and how did Archimedes get in? Oblique and labyrinthine reflections","authors":"J. Høyrup","doi":"10.1080/03080188.2022.2109102","DOIUrl":"https://doi.org/10.1080/03080188.2022.2109102","url":null,"abstract":"ABSTRACT This answer to Reviel Netz’s article at first questions his optimism concerning our technological future. After that, it suggests a different role for Archimedes and the other prominent Greek mathematicians than the one claimed by Netz: as providers not of answers but of problems which called for the transformation of the abbacus and cossist algebra tradition, thereby allowing first Viète and then Descartes to create the first level of the new analysis of the seventeenth century (the second level being infinitesimal analysis).","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"391 - 403"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43870469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2108967
N. Guicciardini
ABSTRACT During the Scientific Revolution, the works of Archimedes played a momentous role. The geometers and natural philosophers of seventeenth-century Europe used Archimedes as a resource for tasks that varied considerably. In this paper, after some introductory remarks, I will consider and contrast the different readings of Archimedes provided by Leibniz and Newton. Leibniz's Archimedes is a precursor of calculus because of his use of exhaustion methods and infinitesimal magnitudes for the calculation of the dimension (area or volume) of curvilinear figures. In Newton's mathematical writings, Archimedes is invoked not to provide a rigorous foundation to the infinitesimal methods of the Moderns, but as an alternative to the symbolic approach to geometry championed by Descartes.
{"title":"The variety of readings of Archimedes in the scientific revolution: Leibniz vs. Newton","authors":"N. Guicciardini","doi":"10.1080/03080188.2022.2108967","DOIUrl":"https://doi.org/10.1080/03080188.2022.2108967","url":null,"abstract":"ABSTRACT During the Scientific Revolution, the works of Archimedes played a momentous role. The geometers and natural philosophers of seventeenth-century Europe used Archimedes as a resource for tasks that varied considerably. In this paper, after some introductory remarks, I will consider and contrast the different readings of Archimedes provided by Leibniz and Newton. Leibniz's Archimedes is a precursor of calculus because of his use of exhaustion methods and infinitesimal magnitudes for the calculation of the dimension (area or volume) of curvilinear figures. In Newton's mathematical writings, Archimedes is invoked not to provide a rigorous foundation to the infinitesimal methods of the Moderns, but as an alternative to the symbolic approach to geometry championed by Descartes.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"376 - 390"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47147280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-02DOI: 10.1080/03080188.2022.2108966
L. Daston
ABSTRACT My commentary on Reviel Netz’s essay ‘The Place of Archimedes in World History’ makes three main claims: first, that the argument concerning the probability and improbability is flawed; second, that the argument nonetheless carries the ring of plausibility because of the enormity of the alleged outcome value of the improbable chain of events Netz traces, namely the origins of modernity; and third, that the notion of modernity, although central to the institutionalization of the history of science as a discipline in the mid-twentieth century, is too amorphous and ideologically laden to be of analytical value for the history of science.
{"title":"Winning the modernity lottery: Commentary on Reviel Netz, ‘The place of Archimedes in world history’","authors":"L. Daston","doi":"10.1080/03080188.2022.2108966","DOIUrl":"https://doi.org/10.1080/03080188.2022.2108966","url":null,"abstract":"ABSTRACT My commentary on Reviel Netz’s essay ‘The Place of Archimedes in World History’ makes three main claims: first, that the argument concerning the probability and improbability is flawed; second, that the argument nonetheless carries the ring of plausibility because of the enormity of the alleged outcome value of the improbable chain of events Netz traces, namely the origins of modernity; and third, that the notion of modernity, although central to the institutionalization of the history of science as a discipline in the mid-twentieth century, is too amorphous and ideologically laden to be of analytical value for the history of science.","PeriodicalId":50352,"journal":{"name":"Interdisciplinary Science Reviews","volume":"47 1","pages":"351 - 359"},"PeriodicalIF":1.1,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42896503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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