Background

• As part of our interest in the limits of reason, with our focus on mathematical reason, we have seen and will see a number of theories of what the objects of mathematics might be.
• The platonist (Godel) believes that they are real entities, independent of us; the intuitionist believes that they are either grasped or produced by a special human intuition; the formalist believes they are just the products of formal procedures.
• An alternative view, one which has not been popular, is that we refer to objects in mathematics as a convenient fiction, but that really mathematics is a series of conventions: this is how we do a thing called "addition," this is how we do a thing called "multiplication," and so on.
• Wittgenstein proposed and defended some form of conventionalism. He is also interpreted by Saul Kripke as having proposed a very radical form of conventionalism. We will consider Kripke's interpretation, which many consider not an accurate account of Wittgenstein's views, but which is nonetheless of philosophical interest. Philosophers sometimes thus refer to this view as "Kripkenstein's."
• Kripkenstein's paradoxical conventionalism is of interest to the philosophy of language, also. We might wonder whether the meanings of terms, for example, are things which we grasp, or rather whether meaning is just a convention use of the term. Wittgenstein expressed the latter view with his maxim, "meaning is use."
• Kripke refers to passages in Wittgenstein's Philosophical Investigations, particularly sections 201, 202, 243, 258, and 265. Wittgenstein (1889-1951) published only three works in his lifetime, and the Philosophical Investigations was the last of these (published posthumously but Wittgenstein intended to publish it and we think the published version is close to what he intended, and so let's call it an intended publication).

The Kripkenstein Skeptical Paradox

• Rules are supposed to apply to indefinitely many cases. You learn how to do addition from a few examples, and then are supposed to apply it to indefinitely may new cases.
• The paradox is not about what we talking about the paradox mean by "plus." To make the paradox clear, we will assume that we (the philosophers and students discussing this problem) mean the same thing by "plus" and understand what that is. We will assume that some hypothetical individual, on the other hand, may not know what "plus" means.
• Kripke's skeptical paradox is just this: suppose that I've never added any numbers larger than 56. Now I add 58 and 67. Note that though this example is a little silly because the numbers are small, the point still stands: I will need to add numbers at some point larger than any numbers I added when being taught to add.
• How do I know what 58+67 is? The skeptic says, perhaps the answer is 5, because the rule I learned is really (and we will call this rule in our language here in class, "quus") to do like addition up to 56, and then any numbers greater than 56 when added are to have the result 5.
• What, then, is it to learn a rule? It's not enough to say, keep doing the same thing you did in the past. We need to also know what it is to do the same thing when doing the rule for new cases.
• The challenge is to explain (1) what fact in the subject in questions distinguishes plus from quus, and (2) to explain how this fact justifies choosing to do plus over quus. Kripke's skeptic is claiming that there is nothing about the subject's past actions which can do these two things.
• Note that this applies to words or concepts just as it applies to addition. See pages 19ff.

Some Solutions That Kripke Rejects

• Our subject learned an algorithm when she learned to do something called "plus." This algorithm applies to 58+67 just as it does to 10+10. K's response: you can apply the same skeptical reasoning to the algorithm.
• A rule is a disposition, not some fact about me at the moment that I obey the rule. K's response: dispositions don't justify. (p. 23ff)
• Our dispositions or algorithm is like a machine. K's response: a machine is finite, so in the same problem; and a machine can break, and we would not be able to say why it was broken (as opposed to doing what it was supposed to do). That is, there is no justification in citing a machine (p. 33ff).
• Plus is simpler than quus. K's response: simplicity is in the eye of the beholder, and also simplicity does not provide justification.
• I have some inner experience of correct adding. K's response: how could this be accurate? How could it justify my doing what I do?

A Skeptical Solution

• A straight solution to a paradox solves it. A skeptical solution tells us that we have to live with the seemingly intolerable outcome.
• Hume offered a skeptical solution to his claim that we do not know that the future will be like the past, and that all expectations about the future are unfounded: he said there is no solution to this problem, there is no reason to expect the future to be like the past, but we simply must live as if we do so expect it. Furthermore, he redefines "cause" to mean something like having-seen-correlated, and not something stronger like a necessary relation. To say "fire causes heat" now means something like "in the past I have observed that fire causes heat" and perhaps also that I expect this to continue.
• Wittgenstein famously argues in the Philosophical Investigations that there cannot be a private language (a language that only you know and speak). Kripke claims that this is a natural outcome of his skeptical solution to his paradox.
• The skeptical conclusion is that there is no fact about you that determines that you should be doing plus instead of quus. Thus:

  There is no objective fact -- that we all mean addition by '+', or even that a given individual does -- that explains our agreement in particular cases. Rather our license to to say of each other that we mean addition by '+' is part of a 'language game' that sustains itself only becuas of the brute fact that we generally agree. (Kripke 1982: 92)

• This is extremely radical: it in essence means there is nothing like traditional meaning in math or language or any other kind of reasoning, but rather a set of traditions without any other norm (any other right or wrong) other than, that's how we tend to do it.

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Next: Godel's Incompleteness Theorems, Turing's Halting Problem.