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 causally inert abstract objects, existing outside of space and time?" 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, Aristotle, Kant, Mill, Frege, Russell, Gödel, Ayer, Quine, Poincare, Lakatos, Benacerraf, Field, Shapiro, Kitcher, Maddy, and Yablo.
Learning Objectives:
To demonstrate knowledge about central problems in the philosophy of mathematics, as well as some
epistemology, metaphysics, and philosophy of language
Upon completion of the course, students should be able to:
Identify central issues or debates in philosophy of mathematics,
Articulate and, when appropriate, compare or contrast, different views that might be taken with
respect to these issues,
Summarize major motivations or arguments for these alternative positions,
Present significant objections that have or could be raised to these positions,
Assess the relative merits of these arguments and objections.
II. REQUIRED CLASS MATERIALS
Shapiro, S. Thinking about Mathematics: The Philosophy of Mathematics.
Oxford: Oxford University Press, 2000.
The book for the course is available at The UNLV Bookstore.
There will also be numerous online readings, available via the course website or WebCampus.
III. CLASS REQUIREMENTS AND GRADING SCHEME
Requirements.............................................Percent of
Final Grade
Class Participation......................................................10%
First Paper...................................................................30%
Second Paper...............................................................30%
Final Exam..................................................................30%
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.
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.
The Final Exam--There will be a timed (2 hour), in-class final exam given on Wednesday, Dec. 14, 2016 at 10:10am in our regular classroom. The final will consist of a choice of essay questions.
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.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 or Instagram, 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.
VI. TOPICS AND READINGS
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
Maddy, selections from Realism in Mathematics
Shapiro, Chapters 3-7 (pp. 49-63, 73-168, 172-189)
Plato, selections from Meno and Republic (online)
Aristotle, selections from Physics and Metaphysics (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, Ch. 4 and Ch. 7 of Frege (online)
Russell, selections from Introduction to Mathematical Philosophy (online)
Carnap, "The Logicist Foundations of Mathematics" (online)
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" (online)
Hilbert, correspondences with Frege (online)
Brouwer, "Intuitionism and Formalism" (online)
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" (online)
Resnik, "Mathematics as the Science of Patterns" (online)
Leng, "'Algebraic' Approaches to Mathematics" (from New Waves in Philosophy of
Mathematics) (online)
Wigner, "On the Unreasonable Effectiveness of Mathematics for Science" (online)
Colyvan, "The Miracle of Applied Mathematics" (online)
Pincock, "Towards a Philosophy of Applied Mathematics" (from New Waves in
Philosophy of 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" (online)*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" (online)
Quine, "Two Dogmas of Empiricism", Sections 1-4 (online)
Poincare, "On the Nature of Mathematical Reasoning" (online)
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?" (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)