[esahione]

"Why" is a question that is aims to answer the cause of a property or effect.

Why do apples fall from trees? Gravity. Why does gravity exist? Because matter bends space. Why does matter bend space? We don't know yet.

Ultimately, any X is either caused or uncaused. Not "X=Y", but either "Y=>X" or "X exists" - the latter being uncaused.

This is pretty basic philosophy.

[turtledragonfly]

You bring a small constellation of points to mind; sorry for the multi-part reply:

1.

The original question was about the natural world, not math, and I'd say that the connection between the two is not guaranteed by anything. So any conclusion in math does not dictate reality. We simply have no "givens" in reality to work from. Though certainly mathematical models are useful.

2.

Going back to your Noether example: are you saying that conservation laws are caused by symmetries?

If the theorem says "X => Y", then does that mean Y is "caused by" X?

I don't think so. It just means that if you observe X, you can be sure Y is there too.

Suppose later we find out additional information: Y => X

Now, we have "X <=> Y", and certainly it would be unfair to say one of these "causes" the other, no?

That would fall in your "uncaused" category, I believe.

3.

The scientific method does not prove things to be true, ever. It only disproves wrong theories, by providing counterexamples.

So, if you have a theory about "why" something is a certain way, you will never fully confirm it with the scientific method. You will only discover when one of your proposals was wrong, never that it was right.

It may be the case that a lot of people try very hard to prove it wrong, and fail. And it may be the case that it is useful at predicting the future and other novelties. But it could still be wrong, and you would never know — maybe the counterexample will be found tomorrow.

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I really do think the bedrock answer to "why do apples fall from trees" is "we don't know." There's just a lot of interesting stuff to be discovered in the (failed) attempt at answering it, in the meantime.

[esahione]

Mathematics is constantly proving things true. Physics also has theorems, for instance, the Stone–von Neumann theorem.

The scientific method is a method for testing of hypotheses, yes, but that is simply one way of discovering what is true. Logic, testimony, written accounts and records, mathematical proofs, and so on are all other ways of discovering truth.

For instance, you cannot prove who was the Emperor of Rome at a particular date in the past with experiments. You must use historical record to do so. Unless you call your experiment opening a book - but that's not a controlled experiment. And even if all books said that Marcus Aurelius was the Emperor of Rome on March 16, 180 you still wouldn't be able to prove it mathematically or using logic. You're using inference to the best explanation in a form of Bayesian probabilities.

Even though it is a matter of fact that either he was or wasn't, there's no experiment you can run on that hypothesis that could tell you the truth.

On your point 2, "X=>Y" indeed means X implies Y, so given X is true, Y is true as well. It could mean causality, but it could also mean necessity. So saying "X is true means Y is true" can be applied to numerous different contexts, including one of causality. I let go of the apple, and the apple falls. This last statement being equivalent to every body that is not subject to forces follows the geodesic created by the spacetime manifold. Which can be put in a logical form X=>Y.

And on your first point: the natural world is clearly following mathematical truths, and that is the entire point of the conversation. The ask of "Why is nature consistent" implies something much much deeper about the nature of reality than what scientific experiments can show. It is a metaphysical question, not a scientific question.

[turtledragonfly]

> It is a metaphysical question, not a scientific question.

Well here I soundly agree with you. I believe that's what I was trying to communicate at the very start, when I said "those 'why' questions are exactly what math and science do not answer."

I suppose our difference relates to how we conceive of other kinds of "why" questions.

> Mathematics is constantly proving things true.

Also agreed with you there. That's one key difference between math and science, in my mind. For that reason, I suppose I'd put theoretical physics more in the "math" realm than the "science" realm, at least w/respect to the theorem you mention, and similar.

Small aside: thank you for the tenacious-but-respectful discussion — I always worry when a thread goes 3+ replies deep that it'll just become an angsty flamewar, so it's nice to have a counterexample in my training set (: