Wittgenstein's Conventionalism and Its Kripke Interpretation
- 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
- 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"
- 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
- 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
- I have some inner experience of correct adding. K's response:
how could this be accurate? How could it justify my doing what I
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.
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
- 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.
Next: Godel's Incompleteness Theorems, Turing's Halting Problem.