The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise (Dover Books on Mathematics)
Average customer rating: 3.5 out of 5 stars
  • good- but not true to title
  • Fascinating Introduction to the Generalized Continuum Hypothesis
  • Finitistic Philosophy, with Some Comments on Cantor
The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise (Dover Books on Mathematics)
Mary Tiles
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486435202

Book Description

This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory, and more. 1989 edition. 32 figures.

Customer Reviews:

3 out of 5 stars good- but not true to title.......2006-08-22

While this book is well-written and appears to provide a sufficient introduction to the concepts of set theory, it approaches but falls short of its titular subject- discussing the impact of set theory on philosophy.

While it addresses Cantor's project of putting analysis on a firm footing stripped of geometrical intuition, it neither addresses what additional benefit is to be gained in pursuit of this project, nor truly addresses the impact of set theory on philosophy. Instead, it makes an effective but somewhat pointless summation of aristotelian philosophical concerns, including also such things as Zeno's paradoxes, but does not address how formal set theory addresses these concerns other than drawing an entirely imaginary universe in which paradoxes are excluded by definition.

As such, set theory is relegated to the same realm as string theory- an interesting religion, entirely believable, but providing no testable benefit. Analysis and the concept of infintesimal limits and calculus provide concrete tools for challenging Zeno, even if they don't provide a definitive footing to formalize themselves.

This book would have benefited greatly from going full circle; returning to the world of philosophy in a post-Cantorian world. It ends, though, satisfied that it has explained set theory without making any attempt to tie it back to the earlier discussion of pre-Cantorian, pre-Russellian philosophy, as though Russell and logical positivism is the end of philosophy. Any book which has a title containing "philosophy" should require a chapter on "next steps", beyond simply suggesting additional reading, to give the reader a desire to pursue the topic further. Tiles' work fails on this account, despite being well written, and as such receives only a mediocre grade.

5 out of 5 stars Fascinating Introduction to the Generalized Continuum Hypothesis.......2005-12-25

The Philosophy of Set Theory - An Historical Introduction to Cantor's Paradise by Mary Tiles is a fascinating mix of mathematics, mathematical logic, and philosophy that should appeal to (and challenge) both mathematics and philosophy majors at the undergraduate and graduate level.

The focus is on the Generalized Continuum Hypothesis (GCH); the reader will meet topics like numbering the continuum, developing Cantor's transfinite ordinal and cardinal numbers, evaluating the ZF axioms underlying set theory, and examining the work of Frege and Russell.

The first four chapters (The Finite Universe; Classes and Aristotelian Logic; Permutations, Combinations, and Infinite Cardinalities; and Numbering the Continuum) provide a historical, philosophical, and mathematical context for the more challenging chapters that follow. Some readers may wish to skip familiar sections although I found these early chapters to be quite engaging.

Chapter 5 - Cantor's Transfinite Paradise is a good, standalone introduction to Cantor's transfinite ordinal and cardinal numbers and to the General Continuum Hypothesis (GCH).

Chapter 6 - Axiomatic Set Theory is another good standalone chapter. Mary Tiles introduces the Zermelo-Fraenkel axioms that underlie modern set theory and develops a restatement of the GCH in the language of the ZF axioms.

Chapter 7 - Logical Objects and Logical Types delves deeply into the work of Frege and Russell. This was not the first time that I had encountered Russell's ramified type hierarchy, but nonetheless I still found this section slow going.

Chapter 8 - Independence Results and the Universe of Sets assumes substantial familiarity with model theory. Specific topics include Godel's constructible sets, cardinals and ordinals in models, inner models, and generic sets. Readers can either browse this technical chapter or omit it if they are willing to accept on trust the independence of the generalized continuum hypothesis and of the axiom of choice from the remaining Zermelo-Fraenkel set theory.

The final chapter, Mathematical Structure - Construct and Reality, summarizes the key philosophic issues underlying not only the generalized continuum hypothesis, but also with set theory in general and with the theory of transfinite numbers in particular.

I thoroughly enjoyed this introduction to Cantor's transfinite numbers. Mary Tiles has created an intriguing examination of the generalized continuum hypothesis.

3 out of 5 stars Finitistic Philosophy, with Some Comments on Cantor.......2005-11-25

When Newton and his successors defined the calculus in the 17th and 18th centuries, they were quite cavalier about infinities. For example, they treated sums of infinitely many numbers essentially the same way they treated sums of finitely many numbers. And when talking about derivatives, they were content to talk of changes over infinitely shrinking intervals without quite saying what they meant by that. As mathematics developed increasingly abstract methods, more divorced from the simple observations of physics, many problems cropped up, mostly having to do with the careless use of infinities. In order to deal with these problems, mathematicians devised precise definitions which made no explicit use of infinities.

But the new methods made it necessary for mathematicians to consider the sets of points where the methods broke down. In investigating them, Cantor had to consider infinite sets and even had to compare different sizes of infinity. Understandably, many mathematicians were upset. But others found Cantor's mathematics useful and worked to put set theory on a solid basis. A new theory, the Zermelo-Frankel theory of sets, was the result. It's not perfect, but it's good enough for most mathematicians. Most mathematicians today are quite comfortable with infinite sets.

I mention all this because Tiles doesn't. Given the subtitle, "An Historical Introduction to Cantor's Paradise", I was expecting to read about the intellectual climate in which Cantor developed his theory. So I am writing this book for the sake of anyone who might like a book putting Cantor's theory into its historical context: this is not the book.

It is a book about the philosophy of finitism, from ancient Greek times to the 20th century. It might be a good book on that topic; I have no way of knowing. For that reason, I can't fairly rate the book and the three stars shouldn't be taken seriously.

(My degrees are in math, not philosophy; for more about that, click on my name at the head of this review. Math texts are strong on theory but, unfortunately, weak on context.)

(Original review 25 Nov 2005; this paragraph 11 Jan 2006) The book makes several references to Zeno's paradoxes. These are based on an assumptiion which is incorrect in a universe, such as ours, where quantum effects are fundamental. Hence they are of purely philosophical interest.
Scientific Method: A Historical and Philosophical Introduction
Average customer rating: 4 out of 5 stars
  • Solid presentation
Scientific Method: A Historical and Philosophical Introduction
Barry Gower
Manufacturer: Routledge
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Binding: Paperback

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ASIN: 0415122821

Book Description

The results, conclusions and claims of science are often taken to be reliable because they arise from the use of a distinctive method. Yet today, there is widespread skepticism as to whether we can validly talk of method in modern science. This outstanding survey explains how this controversy has developed since the 17th century, and explores its philosophical basis.

Customer Reviews:

4 out of 5 stars Solid presentation.......2002-03-30

Gower's survey begins with Galileo and ends with Carnap. In between, he covers a number of prominent scientists - the ones you expect, like Bacon and Newton; and a few you don't, like John Maynard Keynes. He has grouped his subjects according to the nature of their contribution, which means that this reads as the series of essays that it is. Each essay is separately referenced.

There are a few odd omissions, like Descartes (who is nonethless mentioned in passing). But the net result is that, instead of being presented with the historical panorama, one is confronted with a series of ideas - and how they fit into the overall understanding of contemporaries.
Minds, Brains, Computers: An Historical Introduction to the Foundations of Cognitive Science
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    Minds, Brains, Computers: An Historical Introduction to the Foundations of Cognitive Science
    Robert Harnish
    Manufacturer: Blackwell Publishing Limited
    ProductGroup: Book
    Binding: Paperback

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    ASIN: 0631212604

    Book Description

    Minds, Brains, Computers serves as both an historical and interdisciplinary introduction to the foundations of cognitive science.Tracing the history of central concepts from the nineteenth century to the present, this study surveys the significant contributions of philosophy, psychology, neuroscience, and computer science. The volume also investigates the theory of mind from two contrasting approaches: the digital computer vs. neural network models.Authoritative and comprehensive, this is the ideal text for introductory courses in cognitive science as well as an excellent supplementary text for courses in philosophy of mind.
    Judaism and Science: A Historical Introduction (Greenwood Guides to Science and Religion)
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      Judaism and Science: A Historical Introduction (Greenwood Guides to Science and Religion)
      Noah J. Efron
      Manufacturer: Greenwood Press
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      Binding: Hardcover

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      ASIN: 0313330530

      Book Description

      Judaism and Science canvases three millennia of Jewish attitudes towards nature and its study. It answers many questions about the complex relationship of religion and science. How did religious attitudes and dogmas affect Jewish attitudes towards natural knowledge? How was Jewish interest in science reflected, and was facilitated by, links with other cultures - Egypt and Assyria and Babylon in ancient times, Moslem culture in medieval times, and Christian culture during the Renaissance and since? How did science serve as a bridge between religious communities that were otherwise estranged and embattled? How did science serve as a vehicle of assimilation into the wider intellectual culture in which Jews found themselves? The book considers the attitudes and work of particular Jews in different epochs. It takes an "eagle's-eye view" of its subject, considering broad themes from a high vantage, but also swooping down to consider particular individuals at high focus, and in detail. Judaism and Science encompasses the entire history of the interaction of Jews and natural knowledge. BLPart I: The Sages of Israel and Natural Wisdom describes the images of nature and natural philosophy in the two most important sets of books on the Jewish bookshelf: the Biblical corpus and the Talmudic/Early Rabbinic corpus
    • Part II: Jews and Natural Philosophy shows how Jews explained nature, especially the nature of the heavens, or astronomy and astrology, in medieval times and early modern times. BLPart III: Jews and Science -- describes the entry of Jews into modern science, beginning in 19th century Europe and 20th century United States, USSR and Israel, emphasizing the social background of the rapid entry of Jews into modern sciences, and of their remarkable successes. BLThe volume includes annotated primary source documents, a timeline of important events, and an bibliography of essential primary and secondary sources for further research..
      Common Sense, Science and Scepticism: A Historical Introduction to the Theory of Knowledge
      Average customer rating: 5 out of 5 stars
      • fantastic introduction
      • Epistemology and critical rationalism
      • Excellent introduction to the theory of knowledge
      • Compact and Comprehensible
      Common Sense, Science and Scepticism: A Historical Introduction to the Theory of Knowledge
      Alan Musgrave
      Manufacturer: Cambridge University Press
      ProductGroup: Book
      Binding: Paperback

      EpistemologyEpistemology | Philosophy | Nonfiction | Subjects | Books
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      ASIN: 0521436257

      Book Description

      Can we know anything for certain? There are those who think we can (traditionally labeled the "dogmatists") and those who think we cannot (traditionally labeled the "skeptics"). The theory of knowledge, or epistemology, is the great debate between the two. This book is an introductory and historically-based survey of the debate. It sides for the most part with the skeptics. It also develops out of skepticism a third view, fallibilism or critical rationalism, which incorporates an uncompromising realism about perception, science, and the nature of truth.

      Customer Reviews:

      5 out of 5 stars fantastic introduction.......2004-04-15

      Musgraves introduction to epistemology is great! the problems discussed are very well set out. Musgrave writes clearly and is easy to read.
      Plus, hes autographed my book!

      5 out of 5 stars Epistemology and critical rationalism.......2003-04-19

      Alan Musgrave is a good writer and the book is both interesting for the expirienced and those new to philosophy. It should be noted though that the book is written by a follower of Karl Popper's critical rationalism and that it beacause of that sometimes differ in its conclusions from mainstream analytic philosophy, mostly in the last chapter. So it's a quite god introduction to epistemology and its history but also a brilliant introduction to critical rationalist epistemology.

      5 out of 5 stars Excellent introduction to the theory of knowledge.......2001-01-23

      This is the best introduction to epistemology that I have read. Musgrave writes with clarity and grace, taking the reader to the heart the theory of knowledge. Unlike most introductory texts or short histories, which merely summarize the conclusions of great philosophers, "Common Sense, Science, and Scepticism" focuses on the arguments of the philosophers. Thus, the book is an excellent introduction to what it means to "do" philosophy. The text is rewarding, and anyone interested in philosophy can read it with profit.

      5 out of 5 stars Compact and Comprehensible.......1999-11-03

      An excellent introduction for those coming to epistemology for the first time. The author avoids potentially confusing formulations of the issues and generally presents the various positions well.
      Ideas Of Human Nature: An Historical Introduction
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        Ideas Of Human Nature: An Historical Introduction
        Roger Trigg
        Manufacturer: BLACKWELL, B.H.
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        ASIN: 0631214062

        Book Description

        Ideas of Human Nature, now revised and updated in this second edition, presents twelve of the most influential Western thinkers on the topic of human nature. Roger Trigg examines the thinkers in their historical context and discusses their relevance to contemporary controversies.The issues covered include perennial philosophical problems: the connection between mind and body; life after death; the role of reason; free-will and determinism; the relationship between the individual and society; and the problem of relativism.Including new chapters on Locke and Kant, this book is an accessible and key text for anyone interested in the theories that have altered the course of human history, and continue to impact on our lives today.
        A Historical Introduction to the Philosophy of Science (OPUS)
        Average customer rating: 3.5 out of 5 stars
        • Superb HISTORY
        • As the title says
        • over-simplified to the point of uselessness
        • A nice little volume
        A Historical Introduction to the Philosophy of Science (OPUS)
        John Losee
        Manufacturer: Oxford University Press, USA
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        Binding: Paperback

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        ASIN: 0198700555

        Book Description

        Designed for those coming to the subject for the first time, this stimulating introduction offers a historical exposition of differing views on the philosophy of science. With concise profiles introducing the major philosophers whose contributions are discussed, Losee explores the long-argued questions raised by philosophers and scientists about the proper evaluation of science. This new edition incorporates contemporary developments in the discipline, including recent work on theory-appraisal, experimental practice, the debate over scientific realism, and the philosophy of biology. Taking a balanced and informative approach, with a focus on the historical development of the subject, John Losee presents a full and up to date account that is ideal for the introductory reader. NEW TO THIS EDITION: coverage of recent developments in Philosophy of Science, including philosophy of biology, normative naturalism, theory appraisal, experimental practice, theories of explanation, and scientific realism

        Customer Reviews:

        5 out of 5 stars Superb HISTORY.......2005-12-02

        This is NOT a basic introduction to philosophy of science; it is a basic history of the evolution of philosophy of science. As such, it serves its purpose very well.

        4 out of 5 stars As the title says.......2003-11-11

        this is an introduction to the history of philosophical thinking throughout history, not an indepth treatment of ethics throughout history as one reviewer criticised it for not being (something it never stated it was).

        It is thorough without being overwhelming for someone interested in an introduction to philosopphy, and it is interesting not dry and boring or hard to follow as many philosophy books can prove to be.

        It is exactly what it says it is and does a good job at it.

        1 out of 5 stars over-simplified to the point of uselessness.......2002-04-04

        I was interested in this book because its pretty rare to find historical treatments of problems in anglo-saxon or analytic philosophy beyond as a historical curiosity. First of all this book is over-simplified to the point of uselessness for any meaningful understanding beyond even maybe high-school levels at times. The historical orientation here seems to mean chronological- that you start with Aristotle and come down to Mill or Kuhn both without getting anything beyond magazine level information nor some (relevant and important) sense of historical connectedness, which would require a more extensive treatment of each philosopher at any rate.. So the historical thrust fails for the reason that the level of the book is too low to seriously carry it out, and maybe the author's knowledge on past philosophers is not deep enough - as is common in English language philosophy. I want to believe the first reason is truer, but in any case i think this book's level of treatment will be boringly low for most interested adult (over twenty let's say) readers, and its wiser to try a non-historical introduction to the topic.

        4 out of 5 stars A nice little volume.......1998-12-30

        Losse starts with the Pythagoreans, the atomists, and Aristotle. He ends up discussing Popper, Hempel, Kuhn, Feyerband, Lakatos, and a variety of contemporary philosophers. It's densely packed, and for novices to philosophy or logic it might take some effort, but for those with a little background it is easy and, at least for me, fun.
        Science Rules: A Historical Introduction to Scientific Methods
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          Science Rules: A Historical Introduction to Scientific Methods

          Manufacturer: The Johns Hopkins University Press
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          Binding: Paperback

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          5. A Historical Introduction to the Philosophy of Science (OPUS) A Historical Introduction to the Philosophy of Science (OPUS)

          ASIN: 0801879442

          Book Description

          Is there a universal set of rules for discovering and testing scientific hypotheses? Since the birth of modern science, philosophers, scientists, and other thinkers have wrestled with this fundamental question of scientific practice. Efforts to devise rigorous methods for obtaining scientific knowledge include the twenty-one rules Descartes proposed in his Rules for the Direction of the Mind and the four rules of reasoning that begin the third book of Newton's Principia, and continue today in debates over the very possibility of such rules. Bringing together key primary sources spanning almost four centuries, Science Rules introduces readers to scientific methods that have played a prominent role in the history of scientific practice.

          Editor Peter Achinstein includes works by scientists and philosophers of science to offer a new perspective on the nature of scientific reasoning. For each of the methods discussed, he presents the original formulation of the method; selections written by a proponent of the method together with an application to a particular scientific example; and a critical analysis of the method that draws on historical and contemporary sources.

          The methods included in this volume are Cartesian rationalism with an application to Descartes' laws of motion; Newton's inductivism and the law of gravity; two versions of hypothetico-deductivism -- those of William Whewell and Karl Popper -- and the nineteenth-century wave theory of light; Paul Feyerabend's principle of proliferation and Thomas Kuhn's views on scientific values, both of which deny that there are universal rules of method, with an application to Galileo's tower argument. Included also is a famous nineteenth-century debate about scientific reasoning between the hypothetico-deductivist William Whewell and the inductivist John Stuart Mill; and an account of the realism-antirealism dispute about unobservables in science, with a consideration of Perrin's argument for the existence of molecules in the early twentieth century.

          De Motu and The Analyst: A Modern Edition, with Introductions and Commentary (The New Synthese Historical Library)
          Average customer rating: 5 out of 5 stars
          • Very worthwhile.
          De Motu and The Analyst: A Modern Edition, with Introductions and Commentary (The New Synthese Historical Library)
          George Berkeley
          Manufacturer: Springer
          ProductGroup: Book
          Binding: Hardcover

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          ASIN: 0792315200

          Customer Reviews:

          5 out of 5 stars Very worthwhile........2000-11-11

          This is an expensive item, and it obviously is not for everyone. That said, it is a valuable work and should be considered an adjunct to Fraser's "Works of George Berkeley." In Jesseph's book, he presents two of Berkeley's essays - "De Motu", and "The Analyst", to which he provides extensive introductions and references. Both essays (and Jesseph's supporting material) will be reviewed in turn.

          "De Motu" (On Motion) was originally written in Latin. Jesseph's first service is that he provides an English translation along with the Latin version. In this essay, Berkeley described and critiqued then-contemporary theories on the nature of motion. Jesseph does the reader a great service by introducing 17th century physics to the reader, explaining terms, and tracking down Berkeley's references.

          What makes "De Motu" something other than a period piece is Berkeley's methodology. In "A Treatise Concerning the Principals of Human Knowledge", Berkeley laid out an argument against terms denoting entities which could not be experienced or imagined. An example of such a thing was Newton's absolute space. In "De Motu", Berkeley wrote:

          "And so let us imagine that all bodies have been destroyed and reduced to nothing. What remains they call absolute space, all relation which arose from the position and distances of bodies having been removed along with the bodies themselves. Now this space is infinite, immobile, indivisible, insensible, without relation and without distinction. That is, all of its attributes are privative and negative: it seems therefore to be merely nothing. ... Therefore let us take from absolute space just the words, and nothing will remain in the sense, imagination, or intellect; therefore they designate nothing, except pure privation or negation, that is, merely nothing."

          While Berkeley granted that such terms could be useful in calculation, he argued that they led only to meaningless wrangling when imagined as real. He held up a difference between Newton and Torricelli on force as an example:

          "Newton says that impressed force consists solely in action, and it is the action exerted on a body to change its state, nor does it remain after the action. Torricelli contends that a certain accumulation or aggregate of impressed forces is received by percussion in a mobile body, and that the same remains and constitutes impetus. ... And in truth, though Newton and Torricelli seem to disagree, nevertheless, each advances a consistent account, and the matter is adequately explained by both. For all forces attributed to bodies are ... mathematical hypotheses. Mathematical entities, however, have no stable essence in the nature of things: they depend on the notion of the definer: whence the same thing can be explained in different ways."

          In sum, "De Motu" is valuable both as a general critique of science and as a fascinating application of Berkeley's epistimological ideas and is well worth reading on that basis.

          The other Berkeley essay Jesseph covers is "The Analyst". This essay attacked the soundness of the mathematical foundations of Newton's calculus. Because Newton's notation, method, and terminology are no longer in use, the essay is difficult for the modern reader to follow. Jesseph's introduction to "The Analyst" is a fine piece of scholarship and immensely helpful, even necessary, to full understanding of Berkeley's essay.

          "The Analyst" was motivated by apologetic purposes. Berkeley was annoyed at the contrast set up by "free thinkers" between religious belief and math and the sciences, and he sought to demonstrate that mathematics has its mysteries as much as religion. His target was Newton's calculus: in particular, fluxions. Fluxions were infinitesimal quantities, which Berkeley attacked as being literally inconceivable, following his general principals of meaning, and further that Newton did not handle them consistently - sometimes rounding them to zero, and other times not, with the only criterion being whichever was necessary to make the answers come out right.

          "The Analyst" set off a firestorm among mathemeticians. Berkeley's acid style led to angry responses, but the mathematical problems Berkeley had attacked were real, and the defenders of Newton offered very different (and incompatible) approaches to resolving the problems Berkeley had raised, and they soon began attacking each other. It was only in the nineteeth century that the problems surrounding the foundations of Calculus were finally settled.

          Certainly, "The Analyst" is of interest as a part of the history of mathematics, but it is also of interest as an application of Berkeley's general approach. The paragraph below on infinitesmals, for example, clearly follows the same approach as that on absolute space quoted previously:

          "Now to conceive a Quantity infinitely small, that is, infinitely less than any sensible or imaginable Quantity, or than the least finite Magnitude, is, I confess, above my Capacity. But to conceive a Part of such infinitely small Quantity, that shall be infinitely less than it, and consequently though multiply'd infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any man whatsoever...Nothing is easier to devise Expressions or Notations, for Fluxions and Infinitesimals of the first, second, third, fourth and subsequent Orders, proceeding in the same regular form without end or limit ... dx, ddx, dddx, ddddx, &c. These Expressions indeed are clear and distinct, and the Mind finds no difficulty in conceiving them to be continued beyond any assignable Bounds. But if we remove the Veil and look underneath, if laying aside the Expressions we set ourselves attentively to consider the things themselves, which are supposed to be expressed or marked thereby, we shall discover much Emptiness, Darkness, and Confusion..."

          The last thing worth noting about "The Analyst" is that Berkeley wrote two follow-on essays in response to Newton's defenders, both of which are available in Fraser's "Works".
          International Political Thought: A Historical Introduction
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            International Political Thought: A Historical Introduction
            Edward Keene
            Manufacturer: Polity Press
            ProductGroup: Book
            Binding: Hardcover

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            ASIN: 0745623042

            Book Description

            This volume offers an accessible and wide-ranging introduction to the history of international political thought.Taking as its starting-point the various concepts people have used to think about differences between political communities, the book explores changing perceptions of international politics from antiquity to the twentieth century. As well as discussing well-known themes such as relations between independent sovereign states and the tension between raison d'état and a universal code of natural law, it also examines less familiar ideas which have influenced the development of international political thought such as the distinction between civilization, national culture and barbarism, religious attitudes towards infidels, and theories about racial difference and imperialism. Among the key thinkers covered are Thucydides, Machiavelli, Hobbes, Kant, Marx and Morgenthau, alongside less commonly studied figures such as Herodotus, Pope Innocent IV, Herder, Constant and Zimmern. Each chapter concludes with a guide to further reading which will help students to develop a more detailed understanding of the subject.Written with the beginner student in mind, this lively textbook is an ideal introduction for anyone studying international political thought.

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