a structural account of mathematics

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A Structural Account Of Mathematics

Author : Charles S. Chihara
ISBN : 9780199267538
Genre : Philosophy
File Size : 67. 49 MB
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Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chiharapresents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings. A Structural Account of Mathematics will be required reading for anyone working in this field.

Rigor And Structure

Author : John P. Burgess
ISBN : 9780191033605
Genre : Philosophy
File Size : 57. 33 MB
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While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.

Structuralism And Structures

Author : Charles Earl Rickart
ISBN : 9810218605
Genre : Mathematics
File Size : 56. 81 MB
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This book is devoted to an analysis of the way that structures must enter into a serious study of any subject, and the term ?structuralism? refers to the general method of approaching a subject from the viewpoint of structure. A proper appreciation of this approach requires a deeper understanding of the concept of structure than is provided by the simple intuitive notion of structures that everyone posseses to some degree. Therefore, a large part of the discussion is devoted directly or indirectly to a study of the nature of structures themselves. A formal definition of a structure, plus some basic general properties and examples, is given early in the discussion. Also, in order to clarify the general notions and to see how they are used, the later chapters are devoted to an examination of how structures enter into some special fields, including linguistics, mental phenomena, mathematics (and its applications), and biology (especially in the theory of evolution). Because the author is a mathematician, certain mathematical ideas have influenced greatly the choice and approach to the material covered. In general, however, the mathematical influence is not on a technical level and is often only implicit. Even the chapter on mathematical structures is nontechnical and is about rather than on mathematics. Only in the last chapter and earlier in three short sections does one find any of the expected ?formal? mathematics. In other words, the great bulk of the material is accessible to someone without a mathematical background.

Proof And Other Dilemmas

Author : Bonnie Gold
ISBN : 0883855674
Genre : Mathematics
File Size : 20. 26 MB
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For the majority of the twentieth century, philosophers of mathematics focused their attention on foundational questions. However, in the last quarter of the century they began to return to basics, and two new schools of thought were created: social constructivism and structuralism. The advent of the computer also led to proofs and development of mathematics assisted by computer, and to questions concerning the role of the computer in mathematics. This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers have not yet discussed. The other half, written by philosophers of mathematics, summarise the discussion in that community during the last 35 years. A connection is made in each case to issues relevant to the teaching of mathematics.

Modern Algebra And The Rise Of Mathematical Structures

Author : Leo Corry
ISBN : 9783034879170
Genre : Mathematics
File Size : 78. 16 MB
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This book describes two stages in the historical development of the notion of mathematical structures: first, it traces its rise in the context of algebra from the mid-1800s to 1930, and then considers attempts to formulate elaborate theories after 1930 aimed at elucidating, from a purely mathematical perspective, the precise meaning of this idea.

The Development Of Mathematics

Author : E. T. Bell
ISBN : 9780486152288
Genre : Mathematics
File Size : 62. 29 MB
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Time-honored study by a prominent scholar of mathematics traces decisive epochs from the evolution of mathematical ideas in ancient Egypt and Babylonia to major breakthroughs in the 19th and 20th centuries. 1945 edition.

The Bulletin Of Symbolic Logic

Author :
ISBN : UOM:39015072636973
Genre : Logic, Symbolic and mathematical
File Size : 52. 35 MB
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Epsa Philosophical Issues In The Sciences

Author : Mauricio Suárez
ISBN : 9048132525
Genre : Science
File Size : 64. 6 MB
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This volume collects papers presented at the Founding Conference of the European Philosophy of Science Association meeting, held November 2007. It provides an excellent overview of the state of the art in philosophy of science in different European countries.

Structural Bioinformatics

Author : Forbes J. Burkowski
ISBN : 9781420011791
Genre : Science
File Size : 48. 27 MB
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The Beauty of Protein Structures and the Mathematics behind Structural Bioinformatics Providing the framework for a one-semester undergraduate course, Structural Bioinformatics: An Algorithmic Approach shows how to apply key algorithms to solve problems related to macromolecular structure. Helps Students Go Further in Their Study of Structural Biology Following some introductory material in the first few chapters, the text solves the longest common subsequence problem using dynamic programming and explains the science models for the Nussinov and MFOLD algorithms. It then reviews sequence alignment, along with the basic mathematical calculations needed for measuring the geometric properties of macromolecules. After looking at how coordinate transformations facilitate the translation and rotation of molecules in a 3D space, the author introduces structural comparison techniques, superposition algorithms, and algorithms that compare relationships within a protein. The final chapter explores how regression and classification are becoming more useful in protein analysis and drug design. At the Crossroads of Biology, Mathematics, and Computer Science Connecting biology, mathematics, and computer science, this practical text presents various bioinformatics topics and problems within a scientific methodology that emphasizes nature (the source of empirical observations), science (the mathematical modeling of the natural process), and computation (the science of calculating predictions and mathematical objects based on mathematical models).

Reasonable Faith 3rd Edition

Author : William Lane Craig
ISBN : 9781433521188
Genre : Religion
File Size : 86. 5 MB
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Perfect as a textbook yet excellent for lay readers, this updated edition builds a positive case for Christianity by applying the latest thought to core theological themes. J. Gresham Machen once said, "False ideas are the greatest obstacles to the reception of the gospel"-which makes apologetics that much more important. Wanting to engage not just academics and pastors but Christian laypeople and seekers, William Lane Craig has revised and updated key sections in this third edition of his classic text to reflect the latest work in astrophysics, philosophy, probability calculus, the arguments for the existence of God, and Reformed epistemology. His approach-that of positive apologetics-gives careful attention to crucial questions and concerns, including: the relationship of faith and reason, the existence of God, the problems of historical knowledge and miracles, the personal claims of Christ, and the historicity of the resurrection of Jesus. He shows that there is good reason to think Christianity is true. As Craig says, "If you have a sound and persuasive case for Christianity, you don't have to become an expert in comparative religions and Christian cults. A positive justification of the Christian faith automatically overwhelms all competing world views lacking an equally strong case."

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