Download or read book Transfiniteness written by Armen H. Zemanian and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: "What good is a newborn baby?" Michael Faraday's reputed response when asked, "What good is magnetic induction?" But, it must be admitted that a newborn baby may die in infancy. What about this one- the idea of transfiniteness for graphs, electrical networks, and random walks? At least its bloodline is robust. Those subjects, along with Cantor's transfinite numbers, comprise its ancestry. There seems to be general agreement that the theory of graphs was born when Leonhard Euler published his solution to the "Konigsberg bridge prob lem" in 1736 [8]. Similarly, the year of birth for electrical network theory might well be taken to be 184 7, when Gustav Kirchhoff published his volt age and current laws [ 14]. Ever since those dates until just a few years ago, all infinite undirected graphs and networks had an inviolate property: Two branches either were connected through a finite path or were not connected at all. The idea of two branches being connected only through transfinite paths, that is, only through paths having infinitely many branches was never invoked, or so it appears from a perusal of various surveys of infinite graphs [17], [20], [29], [32]. Our objective herein is to explore this idea and some of its ramifications. It should be noted however that directed graphs having transfinite paths have appeared in set theory [6, Section 4.
Download or read book Pristine Transfinite Graphs and Permissive Electrical Networks written by Armen H. Zemanian and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 189 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides a relatively accessible introduction to its subject that captures the essential ideas of transfiniteness for graphs and networks.
Download or read book Graphs and Networks written by Armen H. Zemanian and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 207 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained book examines results on transfinite graphs and networks achieved through continued research effort over the past several years. These new results, covering the mathematical theory of electrical circuits, are different from those presented in two previously published books by the author, Transfiniteness for Graphs, Electrical Networks, and Random Walks and Pristine Transfinite Graphs and Permissive Electrical Networks. Specific topics covered include connectedness ideas, distance ideas, and nontransitivity of connectedness. The book will appeal to a diverse readership, including graduate students, electrical engineers, mathematicians, and physicists working on infinite electrical networks. Moreover, the growing and presently substantial number of mathematicians working in nonstandard analysis may well be attracted by the novel application of the analysis employed in the work.
Download or read book The Future of Post Human Geometry written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2009-05-27 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: Why should some essential properties of geometry (i.e., infinity, symmetry, and dimensionality) be both necessary and desirable in the way that they have been constructed—albeit with different modifications over time—since time immemorial? Contrary to the conventional wisdom in all history hitherto existing, the essential properties of geometry do not have to be both necessary and desirable. This is not to suggest, of course, that one has nothing to learn from geometry. On the contrary, geometry has contributed to the advancement of knowledge in many ways since its inception as a field of knowledge some millennia ago. The point in this book, however, is to show an alternative (better) way to understand the nature of geometry, which goes beyond human conception, intuition, and imagination, together with worldly experience of course, as its foundation, while learning from them all—with theoretical implications for time travel, hyperspace, and other important issues. If true, this seminal view will fundamentally change the way that the nature of abstraction in the thinking process is to be understood, with its enormous implications for the future advancement of knowledge, in a small sense, and what I originally called its “post-human” fate, in a large one.
Download or read book The Future of Post Human Accounting written by Peter Baofu and published by IAP. This book was released on 2014-06-01 with total page 721 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is the invention of accounting so useful that, as Charlie Munger once said, “you have to know accounting. It's the language of practical business life. It was a very useful thing to deliver to civilization. I've heard it came to civilization through Venice which of course was once the great commercial power in the Mediterranean”? (WOO 2013) This positive view on accounting can be contrasted with an opposing view by Paul Browne that “the recent [accounting] scandals have brought a new level of attention to the accounting profession as gatekeepers and custodians of social interest.” (DUM 2013) Contrary to these opposing views (and other ones as will be discussed in the book), accounting (in relation to addition and subtraction) are neither possible (or impossible) nor desirable (or undesirable) to the extent that the respective ideologues (on different sides) would like us to believe. Of course, this reexamination of different opposing views on accounting does not mean that the study of addition and subtraction is useless, or that those fields (related to accounting)—like bookkeeping, auditing, forensics, info management, finance, philosophy of accounting, accounting ethics, lean accounting, mental accounting, environmental audit, creative accounting, carbon accounting, social accounting, and so on—are unimportant. (WK 2013) In fact, neither of these extreme views is plausible. Rather, this book offers an alternative (better) way to understand the future of accounting in regard to the dialectic relationship between addition and subtraction—while learning from different approaches in the literature but without favoring any one of them (nor integrating them, since they are not necessarily compatible with each other). More specifically, this book offers a new theory (that is, the double-sided theory of accounting) to go beyond the existing approaches in a novel way and is organized in four chapters. This seminal project will fundamentally change the way that we think about accounting in relation to addition and subtraction from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what I originally called its “post-human” fate.
Download or read book The Future of Post Human Performing Arts written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2013-01-03 with total page 560 pages. Available in PDF, EPUB and Kindle. Book excerpt: Are the performing arts really supposed to be so radical that, as John Cage once said in the context of music, “there is no noise, only sound,” since “he argued that any sounds we can hear can be music”? (WK 2007a; D. Harwood 1976) This radical tradition in performing arts, with music as an example here, can be contrasted with an opposing view in the older days, when “Greek philosophers and medieval theorists in music defined music as tones ordered horizontally as melodies, and vertically as harmonies. Music theory, within this realm, is studied with the presupposition that music is orderly and often pleasant to hear.” (WK 2007a) Contrary to these opposing traditions (and other views as will be discussed in the book), performing arts, in relation to both the body and its presence, is neither possible nor desirable to the extent that the respective ideologues on different sides would like us to believe. Needless to say, the challenge to these opposing traditions in performing arts does not imply that performing arts are worthless human endeavors, or that those fields of study related to performing arts like aesthetics, acoustics, communication studies, psychology, culture studies, sociology, religion, morality, and so on should be rejected too. Of course, neither of these extreme views is reasonable. Instead, this book provides an alternative, better way of understanding the future of performing arts, especially in the dialectic context of the body and its presence—while learning from different approaches in the literature but without favoring any one of them or integrating them, since they are not necessarily compatible with each other. In other words, this book offers a new theory (that is, the transdisiciplinary theory of performing arts) to go beyond the existing approaches in a novel way. If successful, this seminal project will fundamentally change the way that we think about performing arts, from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what the author originally called its “post-human” fate.
Download or read book The Future of Post Human Culinary Art written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2013-01-03 with total page 615 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is culinary art really so exact that, as Delia Smith once wrote, “cooking is an exact art and not some casual game”? (BQ 2012) This exact view of cooking can be contrasted with an opposing observation by Tom Jaine, when he argued that, “if cooking becomes an art form rather than a means of providing a reasonable diet, then something is clearly wrong.” (BQ 2012a) Contrary to these opposing views (and other ones as will be discussed in the book), culinary art, in relation to both ingredients and techniques, is neither possible or impossible, nor desirable or undesirable, to the extent that the respective ideologues on different sides would like us to believe. Needless to say, this challenge to the opposing views of cooking does not mean that culinary art has no practical value, or that those interdisciplinary fields (related to culinary art) like food science, nutritional economics, food chemistry, food aesthetics, the ethics of killing for food, molecular gastronomy, food rheology, food photography, Shechita, the science of aphrodisiacs, and so on, are unimportant. Of course, neither of these extreme views is reasonable. Rather, this book offers an alternative, better way to understand the future of culinary art, especially in the dialectic context of ingredients and techniques—while learning from different approaches in the literature but without favoring any one of them or integrating them, since they are not necessarily compatible with each other. More specifically, this book offers a new theory (that is, the inquisitive theory of culinary art) to go beyond the existing approaches in a novel way. If successful, this seminal project is to fundamentally change the way that we think about culinary art in relation to ingredients and techniques from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what the author originally called its “post-human” fate.
Download or read book The Future of Post Human Chemistry written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2011-08-08 with total page 515 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is chemistry really so valuable that, as Theodore L. Brown (2011) and his colleagues continue to claim in the twelfth edition of their work in 2011, chemistry is “the central science” in connecting the physical sciences with the life and applied sciences? (WK 2011 & 2011; C. Reinhardt 2001) This crowning of chemistry, however, can be contrasted with an opposing view, as Michael Polanyi once questioned the centrality of chemistry, when he wrote that “[n]o inanimate object is ever fully determined by the laws of . . . chemistry,” so other fields of study are just as important. (BQ 2011) Contrary to these conflicting views about chemistry (and other ones discussed in the book), chemistry, in relation to substances and their changes, is neither possible nor desirable to the extent that the respective ideologues on different sides would like us to believe. This challenge to the conflicting views about chemistry does not mean, however, that chemistry is useless, or that those fields of study related to chemistry like astronomy, physics, geology, mathematics, material science, biology, psychology, computer science, and so on should be ignored too. Of course, neither of these extreme views is reasonable. Instead, this book provides an alternative, better way of understanding the future of chemistry —especially in the dialectic context of substances and their changes—while learning from different approaches in literature but without favoring any one of them or integrating them, since they are not necessarily compatible with each other. This book offers a new theory (that is, the creational theory of chemistry) to go beyond the existing approaches to literature in an original way. If successful, this seminal project will fundamentally change the way that we think about chemistry, from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what the author originally called its “post-human” fate.
Download or read book The Future of Post Human Transportation written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2013-01-03 with total page 615 pages. Available in PDF, EPUB and Kindle. Book excerpt: Can transportation really have such a destructive impact on society that, as Jay Holtz Kay (1998) once forcefully wrote, with the automobile industry as an example, that “the modern consequences of heavy automotive use contribute to the use of non-renewable fuels, a dramatic increase in the rate of accidental death, social isolation, the disconnection of community, the rise in obesity, the generation of air and noise pollution, urban sprawl, and urban decay”? (WK 2012) This negative expectation from transportation, with the automobile industry as an example here, can be contrasted with an opposing (positive) expectation in the old “glory days” when, as Skip McGoun (2012) thus reminded us, “we have sung songs about the glory and wonder that surrounds the very concept of the car. Examples of this range from the 1909 tune, ‘In My Merry Oldsmobile,’ to what is considered to be the first rock and roll song, ‘Rocket 88,’ in 1949. . . . Motion pictures have portrayed . . . expensive sleek sports cars . . . associated with wealth and success. . . . One commercial described Hell as being a place where a teenager would have to drive a minivan!” Contrary to these opposing expectations (and other views as will be discussed in the book), transportation, in relation to both networks and operations, is neither possible or impossible, nor desirable or undesirable, to the extent that the respective ideologues on different sides would like us to believe. This challenge to the opposing expectations from transportation does not mean that transportation is useless, or that those interdisciplinary fields (related to transportation studies) like urban planning, environmental sustainability, migration, tourism, transport economics, traffic engineering, transportation technology, energy efficiency, the tragedy of the commons, and so on are unimportant. Needless to say, neither of these extreme views is reasonable. Rather, this book offers an alternative, better way to understand the future of transportation, especially in the dialectic context of networks and operations—while learning from different approaches in the literature but without favoring any one of them or integrating them, since they are not necessarily compatible with each other. More specifically, this book offers a new theory (that is, the panoramic theory of transportation) to go beyond the existing approaches in a novel way. If successful, this seminal project is to fundamentally change the way that we think about transportation in relation to networks and operations from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what the author originally called its “post-human” fate.
Download or read book Beyond Natural Resources to Post Human Resources written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2014-09-18 with total page 704 pages. Available in PDF, EPUB and Kindle. Book excerpt: Are natural resources really so limited that, as Mahatma Gandhi once famously said, “Earth provides enough to satisfy every man’s need, but not every man’s greed”? (TE 2012) This limiting view of natural resources can be contrasted with an opposing view by John Maynard Keynes, who “summarized Say’s Law as ‘supply creates its own demand’” but then “turned Say’s Law on its head in the 1930s by declaring that demand creates its own supply,” so whenever a demand exists, there will be resources to create the supply. (EN 2012) Contrary to these opposing views (and other ones as will be discussed in the book), natural resources, in relation to both diversity and discontinuity are neither possible or impossible, nor desirable or undesirable to the extent that the respective ideologues on different sides would like us to believe. Needless to say, this challenge to the opposing views of natural resources does not mean that natural resources are unimportant, or that those interdisciplinary fields (related to natural resources) like conservation biology, environmental management, ecological economics, political ecology, environmental ethics, adaptive management, genetic engineering, Malthusianism, and so on are not worth studying. Of course, neither of these extreme views is reasonable. Rather, this book offers an alternative, better way to understand the future of natural resources, especially in the dialectic context of diversity and discontinuity—while learning from different approaches in the literature but without favoring any one of them or integrating them, since they are not necessarily compatible with each other. More specifically, this book offers a new theory (that is, the resilient theory of natural resources) to go beyond the existing approaches in a novel way. If successful, this seminal project is to fundamentally change the way that we think about natural resources in relation to diversity and discontinuity from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what the author originally called its “post-human” fate.
Download or read book The Future of Post Human Migration written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2013-01-03 with total page 630 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is migration really so constructive that, as Ralph Emerson (1909) once wrote, in the context of the New World, “asylum of all nations . . . will construct a new race, a new religion, a new state, a new . . . smelting-pot”? (WK 2012) This noble lie—the “melting pot” in the 20th century—can be contrasted with an opposing noble lie of the “salad bowl” in the 21st century, when those in multiculturalism like Tariq Modood (2007) argue nowadays that multiculturalism “is most timely and necessary, and . . . we need more not less.” (WK 2012a) Contrary to these opposing noble lies (and other views as will be discussed in the book), migration, in relation to both the Same and the Others, is neither possible or impossible, nor desirable or undesirable, to the extent that the respective ideologues on different sides would like us to believe. Surely, this exposure of the opposing noble lies about migration does not mean that the specific field of study on migration is a waste of time, or that those interdisciplinary fields (related to the study of migration) like animal migration, gene migration, diaspora politics, culural assimlation, human trafficking, urbanization, brain drain, tourism, ethnic cleansing, environmental migration, globalization, religious persecution, national identity, gentrification, fifth column, migration art, xenophobia, space colonization, multiculturalism, and so on are worthless. Needless to say, neither of these extreme views is reasonable. Instead, this book offers an alternative, better way to understand the future of migration, especially in the dialectic context of the Same and the Others—while learning from different approaches in the literature but without favoring any one of them or integrating them, since they are not necessarily compatible with each other. More specifically, this book offers a new theory (that is, the theory of the cyclical progression of migration) to go beyond the existing approaches in a novel way. If successful, this seminal project is to fundamentally change the way that we think about migration in relation to Sameness, Otherness, and identity, from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what the author originally called its “post-human” fate.
Download or read book The Future of Post Human Waste written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2013-01-03 with total page 635 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is waste (or trash) really so useless that, as William Faulkner once wrote, “[r]ead everything—trash, classics, good and bad, and see how they do it. . . . If it is good, you’ll find out. If it’s not, throw it out the window”? (TE 2012) Interestingly, this critical view of waste (or trash) can be contrasted with an opposing observation by Isaac Bashevis Singer, who once famously said that “the waste basket is the writer’s best friend.” (TE 2012a) Contrary to these opposing views (and other ones as will be discussed in the book), waste, in relation to both uselessness and usefulness is neither possible or impossible, nor desirable or undesirable to the extent that the respective ideologues on different sides would like us to believe. Of course, this challenge to the opposing views of waste does not imply that waste has no practical value, or that those interdisciplinary fields (related to waste) like epidemiology, global warming, waste management, low-carbon economics, ethical consumerism, resource recovery, freeganism, environmental justice, space debris, and so on are unimportant. Of course, neither of these extreme views is reasonable. Rather, this book offers an alternative, better way to understand the future of waste, especially in the dialectic context of uselessness and usefulness—while learning from different approaches in the literature but without favoring any one of them or integrating them, since they are not necessarily compatible with each other. More specifically, this book offers a new theory (that is, the transfigurative theory of waste) to go beyond the existing approaches in a novel way. If successful, this seminal project is to fundamentally change the way that we think about waste in relation to uselessness and usefulness from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what the author originally called its “post-human” fate.
Download or read book Vaguely Defined Objects written by M. Wygralak and published by Springer. This book was released on 2007-11-23 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years, an impetuous development of new, unconventional theories, methods, techniques and technologies in computer and information sciences, systems analysis, decision-making and control, expert systems, data modelling, engineering, etc. , resulted in a considerable increase of interest in adequate mathematical description and analysis of objects, phenomena, and processes which are vague or imprecise by their very nature. Classical two-valued logic and the related notion of a set, together with its mathematical consequences, are then often inadequate or insufficient formal tools, and can even become useless for applications because of their (too) categorical character: 'true - false', 'belongs - does not belong', 'is - is not', 'black - white', '0 - 1', etc. This is why one replaces classical logic by various types of many-valued logics and, on the other hand, more general notions are introduced instead of or beside that of a set. Let us mention, for instance, fuzzy sets and derivative concepts, flou sets and twofold fuzzy sets, which have been created for different purposes as well as using distinct formal and informal motivations. A kind of numerical information concerning of 'how many' elements those objects are composed seems to be one of the simplest and more important types of information about them. To get it, one needs a suitable notion of cardinality and, moreover, a possibility to calculate with such cardinalities. Unfortunately, neither fuzzy sets nor the other nonclassical concepts have been equipped with a satisfactory (nonclassical) cardinality theory.
Download or read book The Future of Post Human Formal Science written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2010-02-19 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: What exactly is so appealing in formal science, such that its influence can be seen in numerous disciplines nowadays, for practical purposes like better functionality, performance, and so on—as Pythagoras already famously said in antiquity: “Number is the ruler of forms and ideas and the cause of gods and demons”? This contemporary addiction to practical convenience in formal science has turned a blind eye to its other side, which has impoverished both our knowledge of reality and the well-being of our lifeworld. Contrary to conventional wisdom, the other side of this appealing addiction has yet to be comprehensively understood, nor has the fact that its practical convenience is neither possible nor desirable to the extent that the proponents of formal science would like us to believe. Needless to say, this by no means suggests that formal science should not be used for practical purposes, or that the literature in formal science (and other related fields like computer science, information theory, microeconomics, decision theory, statistics, and linguistics, just to cite a few of them) should be dismissed. Of course, neither of these two extreme views is reasonable either. Instead, this book provides an alternative (better) way to understand the nature of formal science, especially in relation to systems theory for practical convenience—while learning from different approaches in the literature but without favoring any one of them (nor integrating them, since they are not necessarily compatible with each other). In the end, this book offers a new theory to transcend the existing approaches in the literature in a new direction not thought of before. This seminal project is to fundamentally alter the way that we think about formal science, from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what I originally called its “post-human” fate.
Download or read book The Future of Post Human War and Peace written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2010-04-16 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is peace really so precious that it is popularly viewed in irenology (peace studies) of our time as “natural” and “a prime force in human behavior”—whereas war, its opposite, is instead condemned as “sinful” and “not” natural? In fact, there is even the prestigious Nobel “peace” prize to be awarded to those who contribute to world peace but not an equivalent Nobel “war” prize to those who do the contrary. This euphoric view of peace is by no means a constant conventional wisdom in human history, as it can be sharply contrasted with an equally seductive view of war in polemology (war studies). For instance, only several decades ago, the well-known writer Thomas Mann once approvingly asked about war: “Is not peace an element of civil corruption and war a purification, a liberation, an enormous hope?” (WK 2009a) This fickleness of conventional wisdom on war and peace has blinded us from the dark sides of both war and peace, with the consequence of impoverishing our understanding of the human condition and its future. Contrary to the two opposing sides of this conventional wisdom in its fickle history, war and peace are neither possible nor desirable to the extent that their respective ideologues would like us to believe. In addition, war and peace cannot exist with each other. Of course, this is not to suggest that irenology (peace studies) and polemology (war studies) are worthless, or that other fields of study (related to war studies and peace studies) like “political science, economics, psychology, sociology, international relations, history, anthropology, religious studies,…gender studies, as well as a variety of others” should be ignored. (WK 2009) Needless to say, neither of these two extreme views is reasonable either. Instead, this book accepts the challenging task to provide an alternative (better) way to understand the nature of war and peace, especially in relation to aggression and pacificity—while learning from different approaches in the literature but without favoring any one of them (nor integrating them, since they are not necessarily compatible with each other). Thus, this book offers a new theory to transcend the existing approaches in the literature on war and peace in a way not conceived before. If successful, this seminal project is to fundamentally change the way that we think about war and peace, from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what I originally called its “post-human” fate.
Download or read book The Future of Post Human Law written by Peter Baofu and published by Cambridge Scholars Publishing. This book was released on 2010-02-19 with total page 474 pages. Available in PDF, EPUB and Kindle. Book excerpt: What makes the rule of law so special that it is to conscientiously punish the “bad” doers and reward the “good” ones—such that, where there is the rule of law, peace and order are to be expected, so that “the rule of law is better than the rule of any individual”? Take the case of international law, as an illustration. While different international courts have been busy going after the killers of innocent victims in Rwanda and Liberia, they have turned a blind eye to the major powers which have killed—on a much larger and more brutal scale, by comparison—innocent civilians in Iraq and Afghanistan, just to cite two current examples. Contrary to the conventional wisdom conveniently held by many in human history, the rule of law has its other side which has not yet been systematically understood, such that the rule of law is neither possible nor desirable to the extent that the defenders of legal institutions in human history would like us to believe. Lest any misunderstanding hastily occur, this is not to imply that the rule of law is absolutely useless, or that the literature in jurisprudence (and other related fields like political philosophy, ethics, law and economics, and the sociology of law) should be dismissed because of its scholarly irrelevance. Of course, neither of these two extreme views is reasonable either. Instead, this book provides an alternative (better) way to understand the nature of law, in relation to its necessity and contingency in the context of justice—while learning from different approaches in the literature but without favoring any one of them (nor integrating them, since they are not necessarily compatible with each other). In the process, this book offers a new theory to transcend the existing approaches in the literature in a new direction—in that, in the end, there is no justice without injustice and that it will be transcended too. This seminal project, if successful, will fundamentally change the way that we think about the nature of law, from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what I originally called its “post-human” fate.
Download or read book The Continuous the Discrete and the Infinitesimal in Philosophy and Mathematics written by John L. Bell and published by Springer Nature. This book was released on 2019-09-09 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.
