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A Beginner's Guide to Constructing the Universe : The Mathematical Archetypes of Nature, Art, and Science

A Beginner's Guide to Constructing the Universe : The Mathematical Archetypes of Nature, Art, and Science
by Michael S. Schneider

Philosophy of Mathematics

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Social Constructivism as a Philosophy of Mathematics

New Directions in the Philosophy of MathematicsNew Directions in the Philosophy of Mathematics by Thomas Tymoczko (Editor)

The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians, and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics, and what processes a proof undergoes before it reaches publishable form. This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.

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The Diamond Theory of Truth

Steven Cullinane

Richard J. Trudeau's book The Non-Euclidean Revolution opposes his Story Theory of truth to the more traditional Diamond Theory of truth.  This site defends the Diamond Theory with various geometric examples, beginning with the diamond in Plato's Meno.

 

Foundations Study Guide:  Philosophy of Mathematics

by David S. Ross, Ph.D.

Excerpt:

The philosophy of mathematics is the philosophical study of the concepts and methods of mathematics. It is concerned with the nature of numbers, geometric objects, and other mathematical concepts; it is concerned with their cognitive origins and with their application to reality. It addresses the validation of methods of mathematical inference. In particular, it deals with the logical problems associated with mathematical infinitude...

 

History of Mathematics

Web resources; Regional mathematics; Subjects; Books and other resources; Chronology; Timelines. The Canadian Society for the History and Philosophy of Mathematics, founded in 1974 and more.

 

Indispensability Arguments in the Philosophy of Mathematics

Stanford Encyclopedia of Philosophy

Excerpt:

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It’s not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics...

  

Modern Development of the Foundations of Mathematics in the Light of Philosophy

by Kurt Gödel, 1961

Excerpt:

I would like to attempt here to describe, in terms of philosophical concepts, the development of foundational research in mathematics since around the turn of the century, and to fit it into a general schema of possible philosophical world-views [Weltanschauungen]. For this, it is necessary first of all to become clear about the schema itself. I believe that the most fruitful principle for gaining an overall view of the possible world-views will be to divide them up according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion). In this way we immediately obtain a division into two groups: skepticism, materialism and positivism stand on one side, spiritualism, idealism and theology on the other. We also at once see degrees of difference in this sequence, in that skepticism stands even farther away from theology than does materialism, while on the other hand idealism, e.g., in its pantheistic form, is a weakened form of theology in the proper sense...

 

The Paideia Archive:  Philosophy of Mathematics

The papers indexed below were given at the Twentieth World Congress of Philosophy, in Boston, Massachusetts from August 10-15, 1998. Additional papers may be added to this web site as electronic versions are acquired and formatted for the archive.

  

Philosophy of Experimental Mathematics and Visualization
The increasing levels of experimentalization in mathematics and other traditionally theoretical fields has brought focus to bear on the issues of human perception and thought, intuition, perception and the nature of knowledge. These issues are embodied in the problem of scientific visualization and its relationship to mathematics, philosophy and traditional scientific methodologies.

The projects described here represent ongoing work being done at the CECM with respect to the experimentalization of mathematics and related processes.

 

Philosophy of Mathematics

Paul Ernest

This site lists The Philosophy of Mathematics Education Journal and Philosophy of Mathematics Education Newsletters.  Site also includes essays by Paul Ernest on the Philosophy of Mathematics.

 

Philosophy of Mathematics

RBJ Mathematics Philosophy Foundation

Writings on the Philosophy of Mathematics mainly concerned with the foundations of mathematics, but also touching upon other problems in the Philosophy of Mathematics and providing some historical background.

  

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