80-110: The Nature of Mathematical Reasoning

Carnegie Mellon University

Spring 2015

Homework 9

Due Wednesday, April 29

  1. Read this paper, written in 1951 by Kurt Godel. (Focus on pp. 309-314, but feel free to read the rest if you're interested! Don't worry about the mathematical details; some of the language he uses is slightly different from what we've used in class.) Here, Godel describes what he sees to be the important applications of his incompleteness theorems, and what they mean for the philosophy of mind and the practice of mathematics.

    Explain as best you can, in a few paragraphs, the points that you think Godel is trying to make. We've discussed the tension between subjectivity and objectivity in mathematics; what is Godel's position in this debate? Why does incompleteness support this view?
  2. We discussed the Monty Hall problem in class. Suppose now that you are on a slightly different game show. As before, you are presented with three boxes, arranged on a circle: one contains a prize, the others contain nothing. You choose one box. Before you open it, the host flips a fair coin. If the coin lands on heads, she removes the box clockwise from the one you chose; if the coin lands on tails, she removes the box counterclockwise from yours. You are then given the choice to open either the box you originally chose, or the other remaining box.

    What do you choose to do? Explain your reasoning!