Q&A: Induction and Linear Generalization
Induction and Linear Generalization
Question
In the past I read something you wrote about the problem with generalizing observations into a mathematical law.
According to your claim, there are infinitely many possible generalizations (polynomials, in the example you gave), and it is not necessarily the simple generalization (a linear line) that is correct.
I used to relate to this argument as another argument in praise of intuition. But recently I happened to think about it again, and it seems to me that it is simply a special case of using the principle of induction: until now, experience has shown that usually the simpler generalizations work, and therefore we continue to use them.
In light of this, I have two questions:
1. Do you agree with my description—that using linear generalization is simply a special case of using induction?
2. Besides induction and causality, are there additional principles that we use even though they do not arise from observation?
Answer
- I agree that this is a special case, but when you come to justify the use of induction, you cannot use induction in order to do so. David Hume already pointed this out.
- There are many. All the rules of logic, for example. Probability calculations. Occam’s razor.
Discussion on Answer
In my opinion, no. They preceded experience. To some extent, experience itself is based on them. Of course, in the case of anything, one can ground it in experience, like mathematics for example. We have experience that combining two objects with three more gives five. But my claim is that the insight that 2+3=5 precedes experience. Experience may perhaps help us notice it (as a didactic tool), but it is not its source. That, of course, is where our argument here began.
I think it is hard to say that the use of Occam’s razor is justified because of our intuition about it. Unlike the rules of logic and mathematics, where our thinking has no meaning if we deny them, one can imagine a world in which the razor does not work. If our experience did not justify using the razor, we would probably abandon its use.
Another way to look at the razor is that it assumes that every component in a theory that is not grounded in observation or logic is “suspect of error,” and because of that we want to minimize such components as much as possible. That is, the razor wants us to use as little intuition as possible.
Can’t the rules of probability and Occam’s razor both be grounded in the principle of induction? After all, the reason we use them is that we have had good experience with them.