This class will investigate some of the fascinating philosophical issues presented by mathematics. Some of these come in the form of epistemological questions, such as "What is mathematical knowledge--is it absolutely certain and immune to rational doubt?" or "What is a mathematical proof (and what does one prove)?" Digging a bit deeper we might ask, "Is observation involved in mathematical knowledge, or is it a matter of pure reason?". Then there are semantic questions, such as "What are mathematical claims about, that is, what is the subject matter of mathematics?" or "What do mathematical statements mean?". These queries point to various metaphysical questions, such as "Do the putative entities of mathematics--numbers, lines, points, functions, sets, etc.--really exist?" or "What are these putative entities like--are they abstract objects, existing outside of space and time and causally inert?" and "If numbers, etc. are abstract entities, why is mathematics so useful for dealing with the concrete physical world within the spatiotemporal-causal nexus?" We will consider questions such as these, by considering both what philosophers and mathematicians have thought about mathematics throughout history, and by examining the views presented on these issues by contemporary theorists. Some of the thinkers we will read include Plato, Kant, Mill, Frege, Russell, Gödel, Ayer, Quine, Poincare, Lakatos, Benacerraf, Field, Shapiro, Kitcher, Maddy, and Yablo.
II. REQUIRED CLASS MATERIALS
Benacerraf, P. and Putnam, H. Philosophy of Mathematics: Selected Readings (Second Edition)
Cambridge: Cambridge University Press, 1984.
Shapiro, S. Thinking about Mathematics: The Philosophy of Mathematics.
Oxford: Oxford University Press, 2000.
The books for the course are available at The UNLV Bookstore. (And on reserve at Lied Library.)
There will also be several photocopied or online required readings.
III. CLASS REQUIREMENTS AND GRADING SCHEME
About the Requirements:
Class Participation--This requirement covers a couple of things. First, there is your contribution during class. Class attendance is thus necessary. However, to do well you must do more than just attend. You are expected to show up having read the assignment for the day and ready to talk about it. Second, everyone must make at least six contributions to the Electronic Discussion Board (accessible through WebCampus) during the term: three before October 20 and three after.
Reading Quizzes--These are like a Final Exam, spread out across the term. We will give 12 Reading Quizzes during the semester and count your best 10. They will be graded according to a ranking scale of check plus/check/check minus/zero, where a check plus = 10 points, check = 8.5 points, check minus = 6.5 points, and zero = 0 points. Your Reading Quiz total is thus a score out of a possible 100 points.
The First Paper--There will be a 6-8 page paper due in mid October. Paper topics will be distributed 12 days before the paper is due.
The Second Paper--There will be a second 6-8 page paper due in early December. Again, topics will be distributed 12 days before the paper is due.Note: All course requirements must be satisfactorily completed in order to pass the course. More than 3 unexcused absences reduces your final grade by 1/3 of a letter grade, more than 5 is a full letter grade deduction, more than 8 is automatic failure of the course.
IV. CLASS FORMAT
This is an upper-level philosophy course, so while we will present a lot of the material, our class meetings should also include a lot of student discussion, not just lectures. We hope that you will all have views about the theories we are going to examine, and we want you to express and explore those views whenever possible. It is typical of philosophical topics that people's views on them will differ. You are encouraged to question your classmates (and us) whenever anyone says something you disagree with, but on either side of this sort of exchange, everyone should always keep in mind that expressing disagreement is not a personal attack. Philosophical discussion thrives under this kind of interaction and often stems from disagreement. At the same time, philosophical discussion aims at reaching some sort of agreement. We probably won't reach agreement every time, but we should aspire towards it.V. CLASSROOM ETIQUETTE
In recent years it has become necessary to make a further comment about classroom etiquette. Engaging in activities like text messaging, surfing the web, checking Facebook, tweeting, IMing, etc. during class is entirely inappropriate. In fact, it is extremely rude and highly disrespectful of our joint enterprise of teaching and learning. Whether you are sitting in the back and presume you are not interfering with anyone else is irrelevant. It is not a question of what you are caught doing; it is a matter of what you do, noticed or not. We expect everyone to behave appropriately during class, engaging with our cooperative project and refraining from inappropriate activities at all times.
Readings from the Benacerraf and Putnam anthology (Henceforth, BP) are indicated by article author, title, and BP page numbers (in parentheses). Readings from the Shaprio book are indicated by author, Chapter, and page numbers (in parentheses). Reading from other philosophers are listed by author and title, and are labeled "online".
A note about the readings: As you well know, philosophical writing is often subtle and difficult. This is especially true of the reading assignments for this course, many of which are also somewhat technical; most of them should be read at least twice. It also helps to take notes on separate paper while reading.
The course units and readings for them (though not necessarily in the order they will be assigned) are as follows.
1. Overview and Orientation
Shapiro, Chapters 1-2 (pp. 3-45)2. History: Figures and Movements
Shapiro, Chapters 3-7 (pp. 49-63, 73-168, 172-189)
Plato, selections from Meno and Republic (online)
Kant, selections from Critique of Pure Reason (online)
Mill, selections from System of Logic, Ratiocinative and Inductive (online)
Frege, selections from Die Grundlagen der Arithmetik (online)
Weiner, "Understanding Frege's Project" and "What Was Frege Trying To Prove?" (online)
Russell, "Selections from Introduction to Mathematical Philosophy" (BP, pp. 160-182)
Carnap, "The Logicist Foundations of Mathematics" (BP, pp. 41-51)
Cook, "New Waves on an Old Beach: Fregean Philosophy of Mathematics Today" (from
New Waves in Philosophy of Mathematics) (online)
von Neumann, "The Formalist Foundations of Mathematics" (BP, pp. 61-65)
Brouwer, "Intuitionism and Formalism" (BP, pp. 77-89)
Shapiro, Chapters 8-10 (pp. 201-289)4. Epistemology: A Priori Knowledge, Proofs, Anti-Deductivism, and Falliblism
Linnebo, "Platonism in the Philosophy of Mathematics," SEP (online)
Benacerraf, "What Numbers Could Not Be" (BP, 272-294)
Resnik, "Mathematics as the Science of Patterns" (online)
Wigner, "On the Unreasonable Effectiveness of Mathematics for Science" (online)
Colyvan, "The Miracle of Applied Mathematics" (online)
Balaguer, "Fictionalism in the Philosophy of Mathematics," SEP (online)
Wagner, "Arithmetical Fiction" (online)
Field, "Introduction: Fictionalism, Epistemology and Modality" (from Realism,
Mathematics and Modality) (online)
Yablo, "The Myth of the Seven" (online)
Ayer, "The A Priori" (BP, pp. 315-328)*The instructors of this course reserve the right to change any aspect of the syllabus, with the understanding that any such changes will be announced in class.
Hempel, "On the Nature of Mathematical Truth" (BP, pp. 377-393)
Quine, "Two Dogmas of Empiricism", Sections 1-4 (online)
Poincare, "On the Nature of Mathematical Reasoning" (BP, pp. 394-402)
Kitcher, "The Apriorist Program" (from The Nature of Mathematical Knowledge) (online)
Lakatos, "A Renaissance of Empiricism in the Recent Philosophy of Mathematics" and
"What Does a Mathematical Proof Prove?" (online)
Quine, "Review of Imre Lakatos's Proofs and Refutations" (online)
Gödel, "What is Cantor's Continuum Problem?" (BP, pp. 470-485)
Maddy, "Believing the Axioms, I and II" (online)
Brown, "Proofs and Pictures" (online)
Folina "Discussion: Pictures, Proofs, and 'Mathematical Practice': Reply to James Robert Brown" (online)
Fallis, "The Epistemic Status of Probabilistic Proof" (online)
Easwaran, "Probabilistic Proofs and Transferability" (online)
Sherry, "On Mathematical Error" (online)