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Thursday, December 20, 2012

That's an Empirical Question: Part IV

This is the 4th part in a series called "That's an Empirical Question". For the other parts of this series see here:

Part I
Part II
Part III

I think this one deserves an appropriate subtitle.

Is Induction Always Empirical?

In Part II, I suggested that induction is an important "empirical tool". Assuming that's true, that still raises the question whether all uses of the "induction tool" necessarily amounts to an "empirical investigation".

To explore this, as per usual, I will use an example: The Goldbach Conjecture.

The Goldbach Conjecture posits that all even numbers greater than 2 can be written as the sum of two primes. For illustration, I will list the first few:

4 2+2
6 3+3
8 3+5
10 5+5
12 5+7
14 7+7
16 3+13
18 5+13
20 7+13

Computers have verified that this holds true to at least 4x1018 (see here).

But isn't this approach to the Goldbach Conjecture kind of like an inductive investigation? After all we are attempting to establish that the statement is true by checking a sample of the even natural numbers (in this case, those less than 4x1018) and checking to see if the Goldbach relationship holds for that sample.

I'd like to suggest that it's an inductive approach to the problem. It's not an ideal inductive approach. Ideally we ought to be taken random samples from the population but that's not exactly feasible. This is, however, a reasonable alternative approach.

So if we accept that the above investigation of the Goldbach Conjecture is an inductive approach does that make it empirical?

I'd like to list a few observations:

First, since the sample of even numbers was verified by a computer program, I have a hard time believing that "sense impressions" are involved here.

As I noted in the last post of this series, I think one critical aspect to "empirical" method could be entailed by some sort of interaction (such as a measurement). This raises the question on whether or not we humans "interact" with a mathematical "reality". If we do, then perhaps there's something "empirical" about this kind of "induction".

If neither of those suffice, however, then it's possible that inductive methods are not strictly "empirical" but can be applied to other disciplines. My own "intuition" suggests that is a more appropriate conclusion.

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