Study Questions on Benacerraf:  "Mathematical Truth", Hart pp. 14-30

1. What are Benacerraf’s initial two concerns with accounts of mathematical truth?

2. According to Benacerraf, under what conditions is the sentence "There are at least three large cities older than New York" true?

3. What is the problem of using similar truth conditions for evaluating the truth of the sentence "There are at least three perfect numbers greater than 17"?  (Aside:  A perfect number is a number equal to the sum of its positive divisors other than itself.  Example:  6 = 1 + 2 + 3.)

4. According to Benacerraf, did Hilbert think the truth conditions for the sentence in #3 above are the same as those for the sentence in #2?  What is it about #3 that Hilbert thought makes it different from #2?

5. Benacerraf calls Hilbert’s account of mathematical truth, and other similar accounts, "combinatorial" views.  Explain what he means by this.

6. Benacerraf describes two components that he thinks a coherant over-all philosophical account of truth and knowledge must have.  What is the first component, and which account of mathematical truth in Section I does Benacerraf think best satisfies it?

7. What is the second component?  Which of the two accounts of mathematical truth in Section I does Benacerraf think best satisfies it?

8. What is the main advantage Benacerraf sees with the "standard" (Platonist) view of mathematical truth?

9. According to Benacerraf, what does it mean to say "X knows that S is true"?

10. Explain why the standard (Platonist) view of mathematical truth is hard to reconcile with a causal theory of knowledge.

11. According to Benacerraf, Godel adopts a standard (Platonist) view of mathematical truth, but fails to make good on the second component that a coherant over-all philosophical account of truth and knowledge must have.  What is Godel’s explanation of how mathematical knowledge is acquired?  Why does Benacerraf think this is unsatisfactory?

12. According to Benacerraf, the "combinatorial" view of mathematical truth equates "truth" with "proof".  Explain.

13. What is the conventionalist view of mathematical truth?  What problems does it have, according to Benacerraf?

1. According to Tait, what is the truth/proof problem for Platonists?

2. According to Tait, on standard accounts what is the warrant for the assertion, "There is a prime number greater than 10"?  What is the warrant for the assertion, "There is a chair in the room"?  Does Tait think this distinction between types of warrant will hold up?

3. According to Tait, what is really involved in warranting the assertion, "There is a chair in the room"?

4. According to Tait, what is the analogue in the sensible world of the truth/proof problem in mathematics?  Ultimately for Tait, is verifying that there is a chair in the room fundamentally any different from proving that there is a prime number greater than 10?

5. According to Tait, under what circumstances can we be said to perceive a set?  Is this any different from the circumstances under which we can be said to perceive a chair?

6. What is the distinction between a "mythological" Platonist and an "ontological" Platonist?  What type of platonist is Tait?

7. According to Tait, Benacerraf thinks that our account of mathematical knowledge should be extendable to empirical knowledge, since they are interdependent.  How does Tait respond?

8. Tait agrees with intuitionists and constructivists that "truth equals proof".  In particular, he does not think that truth requires reference.  But he differs from intuitionists and constructivists over the distinction between a mathematical object A and a proof of A.  What is his view, and how is it different from the view of intuitionists and constructivists (like Dummett).

9. What is Tait’s final answer to the question "What does it mean to say that a mathematical statement A is true"?

10. Why does Tait think that the Platonist view that truth is independent of what we know or can know is correct?

1. In Putnam’s view, what is the chief characteristic of mathematical propositions?

2. What are the two "equivalent descriptions" of mathematics that Putnam thinks are particularly important?

3. According to Putnam, what are two ways of stating that there is a counter-example to Fermat’s Last Theorem?  (Note:  In modal logic, the box symbol means "it is necessarily the case that".)

4. What is the difference between the "object" picture of mathematics and the "modal" picture?  According to Putnam, which picture does Platonism subscribe to?

5. According to Putnam, what is the modal way of saying "Numbers exist"?  What is the modal way of saying "There is a set of integers that satisfies a given condition"?

6. Does Putnam think mathematics is unrevisable in principle?  Why does he think we are justified in accepting its truths?

7. What is the relation between science and mathematics, according to Putnam?

8. What is a mathematically undecidable proposition?  According to Putnam, are undecidable propositions peculiar to mathematics?  Describe the example he considers.

9. Does Putnam think the fact that a proposition is undecidable entails that it has no truth value?

10. Why do you think the claim that mathematically undecidable propositions (like the Continuum Hypothesis) have truth values is a claim usually associated with Platonism?  What is the modal view of mathematically undecidable propositions?

1. How does Parsons define the structuralist view of mathematical objects?

2. What is the set-theoretic definition of a structure?

3. What is the first difficulty in stating the structuralist view, according to Parsons?

4. In non-technical terms, what is a simply infinite system?  According to Parsons, how does Dedekind state claims about the natural numbers in terms of simply infinite systems?

5. What does Parsons mean by "eliminative structuralism"?

6. What is the multiple reduction problem?  How does eliminative structuralism avoid it?

7. What are external relations?  How do they pose a difficulty for a structuralist view of the natural numbers?

8. What is the difficulty associated with Dedekind’s claim that simply infinite systems exist?

9. What is if-thenism?  Does it face the same difficulty referred to in #8 above?  How does Putnam avoid this difficulty?

10. What is nominalism, according to Parsons?  How is it related to eliminative structuralism?

11. Why is the notion of infinity somewhat problematic for nominalism?

12. What is modalism, according to Parsons?  How is it related to eliminative structuralism?

13. According to Parsons, what is the problem for the eliminative structuralist if he adopts a second-order logic?  What is the problem if he does not?

14. Why does Parsons think that eliminative structuralism fails for set theory?

15. Parsons proposes a non-eliminative, context-dependent structuralism based on the notion of an "incomplete" mathematical object.  What are some characteristics of such objects?

16. Why does Parsons think that "quasi-concrete" objects cannot be given an eliminative structuralist description?