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In R. W Hamming's response to the "Unreasonable effectiveness of mathematics" by Eugene Wigner, he gave four 'partial explanations' as to why it may be unreasonably effective. However, I am struggling to comprehend how the first one offers an explanation at all.

His first partial explanation, roughly speaking, is that a lot of physical phenomena arise from the mathematical tools we use to approach it. Or as he put it,

"We approach the situations with an intellectual apparatus so that we can only find what we do in many cases"

So using the mathematics we made, we found physical phenomena. If anything, doesn't this reinforce the notion of mathematics being unreasonably effective, rather than trying to explain it? If that is the case, how is this even an explanation then?

Am I misunderstanding something here?

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    This is a sort of Kantian argument: we "concetualize" the world according to the "conceptual tools" of our mind, and they are a sort of "firware" that we cannot escape. – Mauro ALLEGRANZA Jun 2 '16 at 18:07
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What we as humans can understand is automatically limited by our processing apparatus. If part of the world were inherently contradictory, we would find some way to pretend that it wasn't, simply because inherent contradiction is beyond the bounds of what we can reasonably abide. We might give up on explaining it at all, but we would never accept that it is simply two opposite things at once.

We have a hard enough time with something like the wave-particle dualism where the two things something is at once arise from sort of orthogonal or independent models and are not really contradictory. We can, after all, imagine particles that maintain some kind of rhythm by becoming more and then less real as they move, or we can imagine a transitory wave that is suddenly called upon to be solid and changes instantaneously into a particle. The idea of a wave and the idea of a particle are not opposed, they just fit together poorly in our minds as metaphors.

But our experience of true impossibilities are limited to the vagueness of words or the natural weakness of planning in rulesets. We cannot truly imagine a real and necessary contradiction that is not just an error, yet cannot be resolved, and is still meaningful and useful. We impose that expectation on nature as though it is not part of us, and that means that if we encountered it in reality, we simply could not decide to include it in our physics. It cannot be part of our physics, because it is alien to our mathematics.

Other parts of logic and mathematics may make other parts of reality world equally hard to accept. And we might, therefore never succeed at explaining them. We might stop trying, or never start. They might pass us right by, and not appear real to us.

If mathematics is so deeply a part of human understanding that it filters everything we perceive or interpret in this way, how is it mysterious that everything the filter lets through happens to agree very well with mathematics?

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