Philosophy of Mathematics

Topics for Paper #1. Due Tuesday Feb 25.

Choose one of the following topics and respond to it in a paper of no less than 4 pages and no more than 6 pages. Your paper should be typed, 10- or 12-point, double-spaced and spell-checked. It should conform to the guidelines for writing philosophy papers handed out in class. Make Absolutely Certain that you have read and understood these guidelines before you attempt to begin writing your paper. Make sure to retain an extra copy of your paper (hard copy or on disk).

1. Describe in detail why Benacerraf thinks a single account of both mathematical truth and mathematical knowledge is hard to come by. In particular, what is the problem he thinks Platonism faces? How might a Platonist like Brown respond? Between Benacerraf and Brown, which do you think makes the better argument?

2. What are the core claims of Platonism? What are its fundamental problems? How does it differ from conventionalism, empiricism, and nominalism? Do you think Brown makes a good case for it?

3. How does Tait respond to Benacerraf’s critique of Platonism? In particular, how does Tait apply the "use" theory of meaning as a way to respond to the Problem of Access for the Platonist? Do you think he is successful?

Topics for Paper #2. Due Thursday March 27.

Choose one of the following topics and respond to it in a paper of no less than 4 pages and no more than 6 pages. Your paper should be typed, 10- or 12-point, double-spaced and spell-checked. It should conform to the guidelines for writing philosophy papers handed out in class. Make Absolutely Certain that you have read and understood these guidelines before you attempt to begin writing your paper. Make sure to retain an extra copy of your paper (hard copy or on disk).

1. What are the core claims of Formalism? What are its fundamental problems? How can Godel's Incompleteness Theorems be used in an argument against Formalism?

2. How can Godel's 1st Incompleteness Theorem be used to argue that the human mind cannot be a machine and that Platonism must be correct?
(See David Chalmer's list of references for inspiration.)

3. What are the core claims of Intuitionism? What are its fundamental problems?

Topics for Paper #3. Due Tues April 29.

Choose one of the following topics and respond to it in a paper of no less than 4 pages and no more than 6 pages. Your paper should be typed, 10- or 12-point, double-spaced and spell-checked. It should conform to the guidelines for writing philosophy papers handed out in class. Make Absolutely Certain that you have read and understood these guidelines before you attempt to begin writing your paper. Make sure to retain an extra copy of your paper (hard copy or on disk).

1. Explain in detail Zeno's paradox of the runner and Zeno's paradox of the arrow. In what sense are they paradoxes involving the concept of infinity? How does Aristotle attempt to avoid them? How can the concept of a limit introduced in the calculus be used to avoid them?

2. Explain in detail Cantor's diagonal proof that there are more real numbers than natural numbers. In what sense could it be said to replace the paradoxes of the infinitely big with the paradoxes of the One and the Many?

3. What is the Continuum Hypothesis? How might one argue that, if true, it demonstrates that the infinity of the infinitely small is larger than the infinity of the infinitely big? Do you think this is a good argument?

1.	Describe in detail why Benacerraf thinks a single account of both mathematical truth and mathematical knowledge is hard to come by. In particular, what is the problem he thinks Platonism faces? How might a Platonist like Brown respond? Between Benacerraf and Brown, which do you think makes the better argument?
2.	What are the core claims of Platonism? What are its fundamental problems? How does it differ from conventionalism, empiricism, and nominalism? Do you think Brown makes a good case for it?
3.	How does Tait respond to Benacerraf’s critique of Platonism? In particular, how does Tait apply the "use" theory of meaning as a way to respond to the Problem of Access for the Platonist? Do you think he is successful?

1.	What are the core claims of Formalism? What are its fundamental problems? How can Godel's Incompleteness Theorems be used in an argument against Formalism?
2.	How can Godel's 1st Incompleteness Theorem be used to argue that the human mind cannot be a machine and that Platonism must be correct? (See David Chalmer's list of references for inspiration.)
3.	What are the core claims of Intuitionism? What are its fundamental problems?

1.	Explain in detail Zeno's paradox of the runner and Zeno's paradox of the arrow. In what sense are they paradoxes involving the concept of infinity? How does Aristotle attempt to avoid them? How can the concept of a limit introduced in the calculus be used to avoid them?
2.	Explain in detail Cantor's diagonal proof that there are more real numbers than natural numbers. In what sense could it be said to replace the paradoxes of the infinitely big with the paradoxes of the One and the Many?
3.	What is the Continuum Hypothesis? How might one argue that, if true, it demonstrates that the infinity of the infinitely small is larger than the infinity of the infinitely big? Do you think this is a good argument?