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David Hume problematized

David Hume divides knowledge into matters of fact, which are about things outside of our minds abut admit of no necessity, and relations of ideas, which involve necessity but do not inform us about the outside world.

What if I had a hunch that all 3x4x5 triangles are right triangles?  That is, a hunch based upon a pattern I have observed in drawings of such triangles in the past.  Such a hunch or expectation would be a matter of fact, based upon experience, measurement, rather than a relation of ideas.  BUT what if I later proved tomyself (with the help of the Pythagorean formula) that such a triangle HAD to be a right triangle (with appropo qualifications re Euclidian vs. non-Euclidian geometry)?  The latter is, in Hume's schema, a relation of ideas; so it is not supposed (in Hume's opinion) to give one information about matters of fact.

But it does.  Ask yourself:  after learning the Pythagorean formula don't you know MORE about the drawing of the image of a 3x4x5 triangle than you did when you only had a measurement-based hunch?

As a matter of fact, you would.

In other words, relations of ideas can serve as models of how the things in the world behave; and the things in the world world can instantiate (sometimes imperfectly) what we understand about the relations of ideas.

A more straightforward example of the same point would be 2+3=5 in the mind and in nature (2 apples plus three..).  You really do know that the sum of these three apples and those two apples MUST BE five--if you have counted them correctly. 
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