Can someone explain this Wittgenstein joke?

Question

Schoolmaster: Suppose x is the number of sheep in this problem

Pupil: But, Sir, suppose x is not the number of sheep

(I asked professor Wittgenstein if this is not a profound philosophical joke, and he said it was.)

Answer 1

The quote comes from Littlewood's Miscellany.

Edward Bardeau, in More Fallacies, Flaws, and Flimflam, gives the following account:

The point is simply that, if we assume that an equation has a solution when it does not, we are introducing a false assumption into our discussion and we should not be surprised by the resulting reductio ad absurdum.

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Oddly, there is a more recent example of just this case - the Gettier problems of epistemology.

In a 1966 scenario known as "The sheep in the field", Roderick Chisholm asks us to imagine that someone is standing outside a field looking at something that looks like a sheep (although in fact, it is a dog disguised as a sheep). They believe there is a sheep in the field, and in fact, they are right because there is a sheep behind the hill in the middle of the field. Hence, they have a justified true belief that there is a sheep in the field. But is that belief knowledge?

Adding to the oddity, the Gettier problems were first identified by Littlewood's colleague, Bertrand Russell, using the stopped clock example.

Answer 2

I would rather not comment on jokes that need explaining. As I get it, it is a joke about an equivocation or double meaning.

"Any" in mathematics (and logic) means "all"; the teacher proposes x as "any number" and the pupils says it is not this, meaning some definite number called x. Actually a negation of something definite opens an indefinite field of possibilities which is seen to point that he has totally failed to grasp the point intended. There is a similar joke somewhere in Kant's Critique(s) about a man who is told to eat fruit but he refuses to eat apples, pears, cherries etc...

Answer 3

At one level this is just a play on words: had the schoolmaster said "let x be the number of sheep," there is no response from the pupil (that I can think of) that would still work as a joke.

But I think there is a deeper point here (perhaps the point Wittgenstein had in mind): that there is no formal distinction between axioms and definitions. No matter how the schoolmaster phrases it, formally they're introducing "x = number of sheep" to the system. This is fine as long as x isn't otherwise constrained, but you can't tell from this fragment of conversation that it isn't. If the schoolmaster previously introduced x = 5 as an axiom then the pupil's response is reasonable.

People sometimes ask whether F = ma is a hypothesis subject to empirical testing, or whether it's merely the definition of force, or of mass, or of acceleration. The answer is that you can treat it as any of those; it could also be a nontrivial theorem following from other axioms/definitions. As long as the system generates the same theorems, it is formally the same theory. Every experiment tests the theory as a whole, and if the test contradicts the theory, you can't pin the failure unambiguously on any part of the system.

(Newton actually said F = dp/dt; but people tend to ask about F = ma, and of course the argument is the same either way.)

Nick R's answer to this question quotes another interpretation of the joke by Edward Bardeau: that "if we assume that an equation has a solution when it does not, we are introducing a false assumption into our discussion and we should not be surprised by the resulting reductio ad absurdum." This is an interesting point too, but I think it probably isn't what Littlewood had in mind, because the pupil's response is phrased as though they believe that there is a number of sheep, and are only concerned that it might not be equal to x.

Answer 4

Mathematically, suppose x is the solution is an assumption, which can lead to difficulties, for example the infamous proof that 1+2+3+4+.... = -1/12 runs into trouble when you question the labelling of 1+2+3+4+..... = x and then apply operations which have only been proven for finite quantities.

I suppose there is a question of language. If language and words is defined by how they are used and their meaning to people, one might say that saying 'suppose x means so and so' could be a meaningless statement, if x is not in fact a description of that.

Answer 5

1 is the answer (without assumption)

Schoolmaster: Suppose 1 is the number of sheep (pupil) in this problem

Pupil: But, Sir, suppose 1 is not the number of sheep (pupils)

(I asked professor Wittgenstein if this is not a profound philosophical joke, and he said it was.)

The schoolmaster simply wanted to teach the sheep (pupil) something about life and the pupil said he isn't the problem.