PHILOSOPHY OF MATHEMATICS
1. Körner, Stephan: The Philosophy of Mathematics. An Introductory Essay (Dover Publications 2009).
This compact and comprehensive volume was published originally in 1960, but it is not obsolete even today. Sketches about Plato’s, Aristotle’s, Leibniz’s, Kant’s views; foundations of pure and applied mathematics; expository and critical chapters about the main modern schools of mathematics (formalists, logicists, intuitionists).
2. Hersh, Reuben: What is mathematics, really? (Oxford Univ. Press 1997)
More detailed than Körner with chapters about the “Criteria for a Philosophy of Mathematics”; the role of intuition and proof; misleading mathematical myths (unity, universality, certainty, objectivity of mathematics); “Humanists” and “Mavericks” in mathematics etc.
3. Barrow, John D.: Pi in the Sky. Counting, Thinking and Being (Little, Brown and Company 1992)
Interpreting the subject in a broader sense and mixed with history, general philosophy, even theology. Connections with other sciences (i.e. computability and compressibility); mathematical heroes to Gödel and “transhumanist” and “Marxist” mathematics… This book is not only about the philosophy of mathematics, but about how it is embedded into our culture and natural sciences.
4. Wolfram, Stephen: A New Kind of Science (Wolfram Media 2002)
online: http://www.wolframscience.com/nksonline/toc.html
One of the most exotic and controversial book about the rethinking the foundations of mathematics from a computational point of view. Wolfram uses “cellular automatas” (CA – it is a deterministic dynamical system) to model different processes offering an alternatives for the traditional mathematical/modelling approach. Although some of his generalizations are questionable in connection with biology or with the dynamics of society, the conclusion that “simple programs can produce great complexity” offers a new interpretation of both mathematics and natural sciences.
COSMOLOGY AND OTHERS
5. Ellis, George F. R.: Issues in the Philosophy of Cosmology (2008)
online: http://arxiv.org/pdf/astro-ph/0602280v2.pdf
A short and deep survey about the main themes of philosophy of cosmology from the uniqueness of universe to the question of its origins to multiverses and the nature of existence. If one reads this paper (which appendix contains a summary table about the main thesis and issues), then he/she meets every fundamental problem of cosmology.
6. Carr, Bernard (editor): Universe or Multiverse? (Cambridge Univ. Press 2007)
Reading into some frontiers in cosmology – this volume explains this subject from both philosophical and physical approach. Since the multiverse proposal is one of the most exciting questions of this field, almost every subject from Anthropic Principle to the concept of a mathematical universe based on Platonist ideas appears in this volume. The different chapters are written by different authors, so sometimes you find equations, but these studies are both understandable and readable.
7. Lem, Stanislaw: Summa technologiae (Univ. of Minnesota Press 2014)
One of most unique book about the possible future of science and technology. Written more than a half a century ago by Polish science fiction writer and thinker Stanislaw Lem, it begins with an interpretation of evolution as a kind of universal problem solving process, and continues with diverse and exotic subject from cosmic civilizations to “intelectonics” (intelligence+electronics); phantomology (virtual reality), etc. A critics of this book compared Summa technolgiae to Thomas Aquinas’ Summa theologica, and it gives a superb example for creative thinking inspired by the promises of technology and science.
8. Stapledon, Olaf: Star Maker (Dover Publications 2008)
Another SF author of this list – with a science-fiction book. Stapledon writes about the life in universe and its strange, almost unimaginable form and ideas and technological solutions which inspired the next generation of thinkers form Arthur C. Clarke to Dandridge Cole to Freeman Dyson. Beyond the fact that it is amusing science fiction work with philosophical insights, it is a gold mine for those hunt for new ideas.
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