Ph103: Discussion Questions 10

Short Answer Questions

1. What is mathematical Platonism?
2. What is mathematical antirealism?
3. Explain the difference between a priori and a posteriori knowledge, and why the former poses a challenge for empiricism.

For Further Discussion

• A posteriori vs A priori knowledge. A priori and a posteriori knowledge get their justification in different ways. (a) Which of the following statements would typically be justified a priori?
  • The earth goes around the sun.
  • All bachelors are unmarried.
  • 2+2=4
  • The London School of Economics is in the UK, since the London School of Economics is in London and London is in the UK.
  (b) How might someone argue that the statement 2+2=4 is a posteriori? (c) What would it take to argue that all mathematical facts are a posteriori? (d) Do you find this (conventionalist) response plausible? (e) Do you find the platonist characterisation of the justification of 2+2=4 more plausible?
• Knowing without seeing. An interesting aspect of mathematical proof is that you can know a certain kind of object is possible even if you have no idea what it actually looks like. A famous example is that you can know the following Proposition: there are irrational numbers \(a\) and \(b\) such that \(a^b\) is rational. And you can know this even without knowing what \(a\) and \(b\) actually are!
  • Recall that a real number \(n\) is rational if it can be written as a fraction \(n=\frac{a}{b}\) of other integers \(a\) and \(b\). In lecture, we used this to show that \(\sqrt{2}\) is not rational.
  • Take an real number number \(n\), and suppose that it is rational. Does it follow that it is irrational? Is a given number, like say \(\sqrt{2}^{\sqrt{2}}\), guaranteed to be either rational or irrational?
  • Suppose \(\sqrt{2}^{\sqrt{2}}\) is rational. Then is the Proposition true?
  • Suppose \(\sqrt{2}^{\sqrt{2}}\) is irrational. Define \(a = \sqrt{2}^{\sqrt{2}}\) and \(b = \sqrt{2}\). Now consider the number, \[a^b = \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}.\] Then it is a rule of arithmetic that, \[a^b = \left(\sqrt{2}^{\sqrt{2}\times\sqrt{2}}\right) = \left(\sqrt{2}^{2}\right) = 2.\] So, is \(a^b\) rational or irrational? In this case is the Proposition true?
  • So, what can you conclude about the truth of the Proposition in general?
  • Does this argument tell us which numbers \(a,b\) in particular are the ones such that the Proposition is true?
  • This kind of argument is called non-constructive; it provides information about the existence of certain mathematical objects without actually constructing them. Should we be less convinced by non-constructive arguments than we are by constructive ones?