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This is what our course will be concerned with: mathematical philosophy, that is, philosophy done with the help of mathematical methods. As we will try to show, one can analyze philosophical concepts much more clearly in mathematical terms, one can derive philosophical conclusions from philosophical assumptions by mathematical proof, and one can build mathematical models in which we can study philosophical problems. So, as Leibniz would have said: even in philosophy, calculemus. Let's calculate. Many of my papers are available online at Academia.
Links and Functions www. Further Information Hannes Leitgeb completed a Masters and a PhD degree in mathematics and a PhD degree in philosophy, each at the University of Salzburg, where he later also worked as an Assistant Professor at the Department of Philosophy.
Leitgeb, Hannes - Munich Center for Mathematical Philosophy (MCMP) - LMU Munich
Research Interests Hannes Leitgeb's research interests are in logic theories of truth and modality, paradox, conditionals, nonmonotonic reasoning, dynamic doxastic logic , epistemology belief, inference, belief revision, foundations of probability, Bayesianism , philosophy of mathematics structuralism, informal provability, abstraction, criteria of identity , philosophy of language indeterminacy of translation, compositionality , cognitive science symbolic representation and neural networks, metacognition , philosophy of science empirical content, measurement theory , and history of philosophy Logical Positivism, Carnap, Quine.
Coursera MOOC: Introduction to Mathematical Philosophy with Stephan Hartmann Since antiquity, philosophers have questioned the foundations - the foundations of the physical world, of our everyday experience, of our scientific knowledge, and of culture and society. In what follows, I will avoid taking issue with particular essays since it would be awkward to highlight some but not others , and instead simply sketch the main topics and chapters.
The volume begins with a very useful and equally user-friendly sketch of the target field s. The two fields -- philosophy of mathematics, and of logic -- are very closely related, but distinguishable; the introductory essay, and the remaining chapters themselves, nicely illustrate the relation.
In turn, there are some historical chapters that, while independently interesting, serve to set the stage for contemporary debates, debates that are well represented in subsequent chapters. Three of the best known positions in philosophy of mathematics are logicism , formalism , and intuitionism , positions that dominated the field in the early twentieth century -- the 'big three', as Shapiro calls them. Following the historical chapters, mentioned above, are a handful of chapters treating each of the 'big three'.
For each such position, there is for the most part a sympathetic chapter and a critical chapter these being flagged by titles with the word 'reconsidered'. After the 'big three' come chapters that represent late twentieth-century positions, with for the most part both critical and sympathetic discussions.
After a very informative and quite sympathetic chapter on so-called predicative approaches in philosophy and foundations of mathematics, and a general chapter on the issue of applying mathematics , the last part of the book contains sections on conspicuously logical topics, topics and debates squarely in the philosophy of logic but, of course, nonetheless relevant to philosophy of mathematics. The general structure, as above, is well conceived, and lends itself to a first step into the target fields, as well as affording, as I mentioned, a reasonable view of the current 'state of the art'.
The structure is perhaps easier to appreciate by a quick glance at the individual chapters, which I will here break up into sections even though the book doesn't use these section-titles. Stewart Shapiro. This essay is a broad but informative sketch of the target fields.
The essay nicely sketches the topics and their respective places in the field, and it serves as a sufficient background to the subsequent chapters.
Lisa Shabel. This essay discusses Kant's relevant views, and the views of his predecessors.
John Skorupski. This essay lays out the influence of later empiricism on the target fields, and discusses the impact and ideas of John Stuart Mill and the logical positivists. Juliet Floyd. Like the other chapters in this section and their target figures, Floyd's essay nicely sketches the content and history of Wittgenstein's thought, as well as the ongoing impact that surfaces in different interpretations -- for example, inconsistency approaches to truth, and so on.
William Demopoulos and Peter Clark. This is an excellent introduction to the so-called logicist program. The chapter sketches both the historical roots and ideas of the program as found in Frege, Dedekind, and Russell. Bob Hale and Crispin Wright. Here we meet so-called neologicism , the contemporary offspring of the 'old' logicism discussed in the previous chapter.
This essay nicely lays out the apparent need to go 'neo' and the subsequent routes towards achieving the neo- logicist's aims. This essay focuses on the neologicist project and provides a technical assessment of where the program stands today. While it is slightly more technical than some of the others, the essay is nonetheless very user-friendly, with each technical notion being clearly defined.
Michael Detlefsen. This is the only chapter that focuses entirely on formalism, but the length, breadth, and clarity of the chapter make additional chapters unnecessary. After an introduction, the book begins with a historical section, consisting of a chapter on the modern period, Kant and his intellectual predecessors, a chapter on later empiricism, including Mill and logical positivism, and a chapter on Wittgenstein.
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The next section of the volume consists of seven chapters on the views that dominated the philosophy and foundations of mathematics in the early decades of the 20th century: logicism, formalism, and intuitionism. They approach their subjects from a variety of historical and philosophical perspectives. The next section of the volume deals with views that dominated in the later twentieth century and beyond: Quine and indispensability, naturalism, nominalism, and structuralism.
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The next chapter in the volume is a detailed and sympathetic treatment of a predicative approach to both the philosophy and the foundations of mathematics, which is followed by an extensive treatment of the application of mathematics to the sciences. The last six chapters focus on logical matters: two chapters are devoted to the central notion of logical consequence, one on model theory and the other on proof theory; two chapters deal with the so-called paradoxes of relevance, one pro and one contra; and the final two chapters concern second-order logic or higher-order logic , again one pro and one contra.
Keywords: mathematics , logic , Immanuel Kant , empiricism , John Stuart Mill , logical positivism , Wittgenstein , logicism , formalism , intuitionism , Quine , indispensability , naturalism , nominalism , structuralism , predicativity , logical consequence , paradoxes of relevance , second-order logic , higher-order logic. Forgot password? Don't have an account? All Rights Reserved. OSO version 0.
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