Fall 2020 PHIL309A Science, Proof, and Paradox (DePauw University)
Course Overview
This course is an introduction to the philosophy of science and mathematics. It takes as its entry point the paradoxes and surprising results of late 19th and early 20th century science and mathematics from Albert Einstein, Bertrand Russell, Kurt Gödel, and Georg Cantor which challenged widely held ideas about the nature of scientific and mathematical knowledge. Einstein’s theories challenge the ideas that geometrical truths are unique, unable to be revised, and known by our intuitions about space. Bertrand Russell’s famous paradox in set theory challenge the idea that truths about sets, and the mathematical axioms definable in terms of them, are known by obvious logical principles. Kurt Gödel’s incompleteness theorems challenge the idea that arithmetical truths are exhausted by what we can prove. Georg Cantor’s set theory challenges the idea that there are no actually infinite sets. We will enter into the philosophical conversations generated by these results and ask questions such as:
How do we justify our mathematical beliefs? Can we appeal to the rules of a mathematical language or our intuitions to do so?
Do sets with infinitely many members really exist; if so, how could we know that?
How do our observations justify accepting or rejecting scientific theories, if at all?
Does observational evidence have any role in justifying mathematical truths?
Are parts of our scientific theories conventional, or arbitrary, or underdetermined by evidence? What does ‘conventional’, ‘arbitrary’, and ‘underdetermined by evidence’ mean here?
How do society, gender, and our ethical values interact with scientific knowledge and methods, if at all?
Some low-stakes, fairly informal exposition of some scientific and mathematical ideas and theories will be part of the reading and lecturing material, but the course will be taught so that no science or mathematics courses are prerequisites.
Course Calendar
‘T’ marks a reading for Tuesday, ‘R’ for Thursday.
Week 1: Introduction and Kant
T: Introduction. R: Immanuel Kant, Prolegomena to Any Future Metaphysics, Introduction, Preamble.
Week 2: Geometries and experience
T: Kant, Prolegomena, Part I. John Norton, Einstein for Everyone, Chapters 18-21 on geometry: https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/index.html R: Henri Poincaré, Science and Hypothesis, Chapters 3 and 4, “Non-Euclidean Geometries,” “Space and Geometry,” selections. Recommended: John Norton, “Philosophy of Space and Time: Part I.”
Week 3: Logical empiricism: a philosophy after Einstein
T: Hans Reichenbach: “The Philosophical Significance of the Theory of Relativity,” Sections I-III, V. Discussion of Einstein. [Optional: John Norton, Einstein for Everyone, Chapters 2-5 on special relativity, 24-25 on general relativity] R: Carl Hempel: “Problems and Changes in the Empiricist Criterion of Meaning.”
Week 4: Russell’s paradox, Gödel’s Theorem, and Rudolf Carnap’s reaction
T: W.V. Quine, “The Ways of Paradox.” Discussion of Frege, Russell’s paradox, Gödel’s Incompleteness Theorems. R: Tuesday’s reading continued, time for questions and discussion.
Week 5: Carnap’s program for scientific philosophy
T: Rudolf Carnap, The Logical Syntax of Language, Foreword, Sections 1, 2, and 17 "Formal and Factual Science,” "Intellectual Autobiography," Sections 6 and 8. R: Carnap continued.
Week 6: Quine against Carnap
T: W.V. Quine, “Two Dogmas of Empiricism,” Sections 1-3. R: “Two Dogmas,” Sections 4-6.
Week 7: Wrapping up Quine and Carnap, Science and Value
T: Quine, “Carnap and Logical Truth,” Sections 4-6. Richard Rudner, “The Scientist Qua Scientist Makes Value Judgments.” R: Rudner continued, Isaac Levi, “Must the Scientist Make Value Judgments?”
Week 8: Science and Value: Feminism, Society, and Underdetermination
T: Elizabeth Anderson, “The Use of Value Judgments in Science: A General Argument, with Lessons from a Case Study of Feminist Research on Divorce.” R: Helen Longino, Fate of Knowledge, Chapter 6: “Socializing Knowledge.”
Week 9: Thomas Kuhn: Scientific theories, worlds (or words?) apart
T: Godfrey-Smith, Theory and Reality, Chapter 5 and Chapter 6, Sections 1 and 2. [Optional reading: Thomas Kuhn, The Structure of Scientific Revolutions, Chapters 1-4]. R: Kuhn, The Structure of Scientific Revolutions, Chapters 9 and 10, Godfrey-Smith, Chapter 6, Sections 3, 4, and 5.
Week 10: Karl Popper: Falsification
T: Karl Popper, “Science: Conjectures and Refutations.” R: Hilary Putnam, “The ‘Corroboration’ of Theories.”
Week 11: Truth, models, and complexity in empirical science, with climate science as a case study.
T: Catherine Elgin, “True Enough.” R: Elisabeth Lloyd, “Confirmation and Robustness of Climate Models.”
Week 12: Philosophy and/of the infinite.
T: Basic ideas of set theory, including Cantor’s theorem. R: David Hilbert, “On the Infinite.” Set theory continued.
Thanksgiving Break
Week 13: Epistemology and the infinite.
T: Elijah Chudnoff, “Intuition in Mathematics.” R: T.M. Scanlon, Being Realistic about Reasons, Chapter 4, selections
Week 14: Epistemology and the infinite continued, and final paper discussion
T: Wrap-up discussion of course content. R: Individual meetings with students on their final paper drafts.