The aggregate EIRP of an N-element antenna array is proportional to N 2. This observation illustrates an effective approach for providing deep space networks with very powerful uplinks. The increased aggregate EIRP can be employed in a number of ways, including improved emergency communications, reaching farther into deep space, increased uplink data rates, and the flexibility of simultaneously providing more than one uplink beam with the array. Furthermore, potential for cost savings also exists since the array can be formed using (...) small apertures. (shrink)
The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting (...) with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)
In this book, which is both a philosophical and historiographical study, the author investigates the fallibility and the rationality of mathematics by means of rational reconstructions of developments in mathematics. The initial chapters are devoted to a critical discussion of Lakatos' philosophy of mathematics. In the remaining chapters several episodes in the history of mathematics are discussed, such as the appearance of deduction in Greek mathematics and the transition from Eighteenth-Century to Nineteenth-Century analysis. The author aims at developing a (...) notion of mathematical rationality that agrees with the historical facts. A modified version of Lakatos' methodology is proposed. The resulting constructions show that mathematical knowledge is fallible, but that its fallibility is remarkably weak. (shrink)
A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.
The business of mathematics is definition and proof, and its foundations comprise the principles which govern them. Modern mathematics is founded upon set theory. In particular, both the axiomatic method and mathematical logic belong, by their very natures, to the theory of sets. Accordingly, foundational set theory is not, and cannot logically be, an axiomatic theory. Failure to grasp this point leads obly to confusion. The idea of a set is that of an extensional plurality, limited and definite in size, (...) composed of well defined objects.It is the extension of Greek notion of 'number' (arithmos) into Cantor's 'transfinite'. (shrink)
The thirty year long friendship between Imre Lakatos and the classic scholar and historian of mathematics Árpád Szabó had a considerable influence on the ideas, scholarly career and personal life of both scholars. After recalling some relevant facts from their lives, this paper will investigate Szabó's works about the history of pre-Euclidean mathematics and its philosophy. We can find many similarities with Lakatos' philosophy of mathematics and science, both in the self-interpretation of early axiomatic Greek mathematics as Szabó reconstructs (...) it, and in the general overview Szabó provides us about the turn from the intuitive methods of Greek mathematicians to the strict axiomatic method of Euclid's Elements. As a conclusion, I will argue that the correct explanation of these similarities is that in their main works they developed ideas they had in common from the period of intimate intellectual contact in Hungarian academic life in the mid-twentieth century. In closing, I will recall some relevant features of this background that deserve further research. (shrink)
Euclid discusses the ex aequali relationship twice in the Elements. The first is in Book V (based on definitions 17 and 18, propositions 22 and 23), during his discussion of arithmetical relations between mathematical magnitudes in general. The second is in Books VIIIX (numbers), he was not much troubled by the differences between his treatment of ex aequali ratios in these two contexts. Later generations of mathematicians, however, found these differences less acceptable and tried to minimize them in various ways. (...) This paper summarizes Euclid's use of the ex aequali relation in developing his mathematics. The paper then outlines the fate of the post-Theonine Greek attempts to the Euclidean discussion when the Elements entered the Arabic/Islamic intellectual tradition. The study concludes with the attempts by Ibn al-Hayam and Ibn al-Sarī to improve the parallelism between the discussions of ex aequali ratios in Book V and Book VII. (shrink)
The logic of discovering and that of justifying have been a permanent source of debate in mathematics, because of their different and apparently contradictory features within the processes of production of mathematical sentences. In fact, a fundamental unity appears as soon as one investigates deeply the phenomenology of conjecturing and proving using concrete examples. In this paper it is shown that abduction, in the sense of Peirce, is an essential unifying activity, ruling such phenomena. Abduction is the major ingredient in (...) a theoretical model suitable for describing the tran-sition from the conjecturing to the proving phase. In the paper such a model is introduced and worked out to test Lakatos' machinery of proofs and refutations from a new point of view. Abduction and its categorical counter-part, adjunction, allow to explain within a unifying framework most of the phenomenology of conjectures and proofs, encompassing also the method of Greek analysis-synthesis. (shrink)
The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting (...) with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)
Book description: This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice. Questions concerning concept-formation, understanding, heuristics, changes in style of reasoning, the role of analogies and diagrams, etc. have become the subject of intense interest. The historians and philosophers in this book agree that there is more to understanding mathematics than a study of its logical structure. (...) How are mathematical objects and concepts generated? How does the process tie up with justification? What role do visual images and diagrams play in mathematical activity? What are the different epistemic virtues (explanatoriness, understanding, visualizability, etc.) which are pursued and cherished by mathematicians in their work? The reader will find here systematic philosophical analyses as well as a wealth of philosophically informed case studies ranging from Babylonian, Greek, and Chinese mathematics to nineteenth century real and complex analysis. (shrink)
This anthology looks at the early sages of Western philosophy and science who paved the way for Plato and Aristotle and their successors. Democritus's atomic theory of matter, Zeno's dazzling "proofs" that motion is impossible, Pythagorean insights into mathematics, Heraclitus's haunting and enigmatic epigrams-all form part of a revolution in human thought that relied on reasoning, forged the first scientific vocabulary, and laid the foundations of Western philosophy. Jonathan Barnes has painstakingly brought together the surviving Presocratic fragments in their original (...) contexts, utilizing the latest research and a major new papyrus of Empedocles. Translated and edited by Jonathan Barnes. (shrink)
Locke, Berkeley, Gentzen gave dierent justi cations of universal generalization. In particular, Gentzen's justi cation is the one currently used in most logic textbooks. In this paper I argue that all such justi cations are problematic, and propose an alternative justi cation which is related to the approach to generality of Greek mathematics.
Montague was born September 20, 1930 in Stockton, California and died March 7, 1971 in Los Angeles. At St. Mary’s High School in Stockton he studied Latin and Ancient Greek. After a year at Stockton Junior College studying journalism, he entered the University of California, Berkeley in 1948, and studied mathematics, philosophy, and Semitic languages, graduating with an A.B. in Philosophy in 1950. He continued graduate work at Berkeley in all three areas, especially with Walter Joseph Fischel in Arabic, (...) with Paul Marhenke and Benson Mates in philosophy, and with Alfred Tarski in mathematics and philosophy, receiving an M.A. in mathematics in 1953 and his Ph.D. in Philosophy in 1957. Alfred Tarski, one of the pioneers, with Frege and Carnap, in the model-theoretic semantics of logic, was Montague’s main influence and directed his dissertation (Montague 1957). Montague taught in the UCLA Philosophy Department from 1955 until his death. (shrink)
For Aristotle and other Greek thinkers, philosophy is itself a rethinking. There are other branches of knowledge, like medicine and mathematics, that each grasp some particular subject matter. Since philosophy or, as it has come to be called, metaphysics is the highest science, its job is to grasp somehow all the other sciences and all their subjects. If the science of a subject requires a type of thinking proper to the subject, then the science of that science requires a (...) rethinking of this and all other subjects. In this paper I explore some of Aristotle’s modes of rethinking philosophy. I am interested in the connection between rethinking philosophy and the kinds of philosophical principles that emerge from this rethinking. I argue that reflexive principles are implicit in rethinking but that theyare projected onto things for systematic reasons. Because my time is short, my discussion is limited to broad brush strokes, but there are so many textual details and so much that is contentious about them that a broad sketch may be the best way to set out my point. It is plausible to proceed this way because Aristotle’s main themes are often much clearer than the details of his discussions and my argument relies only on the broad lines of his organization. (shrink)
Adam Smith was born in Kirkcaldy, Scotland, in 1723 (Source on Smith's life: E G West, Adam Smith ). He entered Glasgow University in 1737, aged 14. This university still followed some practices of the medieval universities, for example in admitting students at age 14. Its professors still took fees directly from students: that had been the original practice in medieval universities, but in more famous universities rich people had endowed colleges within the university, which paid lecturers' salaries. The Glasgow (...) timetable was still medieval. The main lecture took place at 7.30 am in the cold and dark, at 11 the students were quizzed on the mornings lecture, at 12 there was a lecture on an optional topic. This was the typical student's day in the thirteenth century. But the curriculum was modern: besides philosophy (the main medieval subject) students took Greek and Mathematics. The philosophy was modern. At Glasgow Adam Smith studied under Francis Hutcheson (see extracts from his works in Raphael British Moralists vol.1, p.261ff.)). Hutchison taught in English (not Latin) and was a vivid lecturer. Moral philosophy, or ethics, was a flourishing subject at the time. The main division was between two schools of 'intuitionists' (as they would now be called). To remind you: Ethics is concerned with what is good and bad, better and worse, in human conduct - in the ends we seek, in the actions in which we seek our ends. Intuitionism is the doctrine that in the last analysis we simply 'see' that some way of acting is good or right, or the opposite: that basic ethical assessments cannot be justified by argument, and do not need to be. 'See' of course is a metaphor. Many 18C moral philosophers held that it is reason that 'sees' what is good and right. Hutchison said that it is a moral sense: not reason, and not the bodily senses of vision, hearing etc., but something more like a bodily sense than like reason. On Hutchison's analysis, ethical judgement is a specific kind of emotional reaction to a comtemplated act.. (shrink)
The University of Warsaw has a splendid modern library with 60,000 m 2 of ﬂoor space. It resembles a shopping centre. The long and elegant modern building on ulica Dobra (a typical Varsovian street-name), on the low ground between the old University and the Vistula, was opened in 1998 replacing the previous hopelessly inadequate facilities. It has an imposing sequence of copper-green “great texts” on its front side in Greek, Arabic, Sanskrit, Hebrew, Latin, Polish, music, and mathematics. These are (...) international symbols, posting Warsaw’s claim to international status. But inside, once one has passed the mall-like coffee-shop. (shrink)
In the first part of chapter 2 of book II of the Physics Aristotle addresses the issue of the difference between mathematics and physics. In the course of his discussion he says some things about astronomy and the ‘ ‘ more physical branches of mathematics”. In this paper I discuss historical issues concerning the text, translation, and interpretation of the passage, focusing on two cruxes, ( I ) the first reference to astronomy at 193b25–26 and ( II ) the reference (...) to the more physical branches at 194a7–8. In section I, I criticize Ross’s interpretation of the passage and point out that his alteration of ( I ) has no warrant in the Greek manuscripts. In the next three sections I treat three other interpretations, all of which depart from Ross's: in section II that of Simplicius, which I commend; in section III that of Thomas Aquinas, which is importantly influenced by a mistranslation of ( II ), and in section IV that of Ibn Rushd, which is based on an Arabic text corresponding to that printed by Ross. In the concluding section of the paper I describe the modern history of the Greek text of our passage and translations of it from the early twelfth century until the appearance of Ross's text in 1936. (Published Online August 10 2006) Footnotes1 This paper was prepared as the basis of a presentation at a conference entitled “Writing and rewriting the history of science, 1900–2000,” Les Treilles, France, September, 2003, organized by Karine Chemla and Roshdi Rashed. I have compared Aristotle's and Ptolemy's views of the relationship between astronomy and physics in a paper called “Astrologogeômetria and astrophysikê in Aristotle and Ptolemy,” presented at a conference entitled “Physics and mathematics in Antiquity,” Leiden, The Netherlands, June, 2004, organized by Keimpe Algra and Frans de Haas. For a discussion of Hellenistic views of this relationship see Ian Mueller, “Remarks on physics and mathematical astronomy and optics in Epicurus, Sextus Empiricus, and some Stoics,” in Philippa Lang (ed.), Re-inventions: Essays on Hellenistic and Early Roman Science, Apeiron 37, 4 (2004): 57–87. I would like to thank two anonymous readers of this essay for meticulous corrections and thoughtful suggestions, almost all of which I readily adopted. (shrink)
A Companion to Ancient Philosophy provides a comprehensive and current overview of the history of ancient Greek and Roman philosophy from its origins until late antiquity. Comprises an extensive collection of original essays, featuring contributions from both rising stars and senior scholars of ancient philosophy Integrates analytic and continental traditions Explores the development of various disciplines, such as mathematics, logic, grammar, physics, and medicine, in relation to ancient philosophy Includes an illuminating introduction, bibliography, chronology, maps and an index.