A Causal Theory of Knowledge II

Continuing with our thoughts from yesterday....

What are we saying when we say we know that 45 + 303 = 348?

That depends on what we understand numbers, and arithmetic, to be.

I think the sensible course is to adopt an operational/pragmatic view of mathematical knowledge according to which it is a shorthand for facts concerning the way the world works. In G.E. Moore's spirit, I might hold up a hand, "here is one hand." Then I might hold up another, :"Here, also, is a hand." Finally, with both of them in my field of sight, I can say, "there are two hands here."

So we can take 1 + 1 = 2 as settled.

But 45 + 303 might be tedious to prove in that way, with some object other than hands (marbles?). Further, there would be the threat of false refutations. I might lose track of the first marble before I collected the 348th. Then I'd count them all and declare that the real answer to the question must be 347!

A naive empiricism about such matters becomes operationally difficult.

But my operational answer to the answer might just be that 45 + 303 equals that which a well-functioning calculator will show when that question is posed to it. In this case, when I come up with an answer by using a calculator, I have knowledge of the answer to the question in the same way as I know the answer to the question "what color are the leaves?"

The workings of the world caused human beings (smarter ones than I!) to design calculators in the way they have. The calculator provides the answer.

And the third element of the definition of knowledge should be causation, not justification. I know that 45 + 303 = 348 because it is true, I recognize it as true, and it is the way the world works which both constitutes that truth and causes me to have that belief.

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