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I'm not sure I follow your question. You seem to be stating the 'trustworthiness' of mathematics has to be earned. I'm saying that this is false. We define the meaning of the symbols in mathematics. We also have methods of showing if something is internally consistent (which we cannot do with everything). It is always trustworthy because we define it to be as such. I'd be glad to see a counter example of this though.
It has been proven that any logical system robust enough to incorporate basic arithmetic cannot be proven to be consistent.
I do not understand how we can "define" math to be trustworthy. I agree that it is, but just because I make a series of claims involving symbols (like "1+1=2") does not inherently suggest that the overall scheme that is being used can be fully trusted. In fact, on its own, I could claim that there's no particular reason to believe that math should be trusted. The trustworthiness of math follows from the fact that it faithfully represents elements of the physical universe. (And *then* the ideas are extended into an abstract framework, and so forth...)
For example, if it were the case that when we put one brick into a bag, and then a second brick into the bag, that the bag contained five bricks, I don't think we would believe that "1+1=2" is a meaningful statement, and should therefore be rejected as being math (in the same way we reject "1+1=5").
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The application of mathematics to the natural world is quite different though. That would be an inductive reasoning (not mathematical induction), and that trustworthiness has to be earned.
Trustworthiness in a purely mathematical context relies upon the trustworthiness of a base logic. Why should logic be trusted?
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Oh, but I am arguing something different. If you could have an ether detector that you can know with certainty never has a false negative then you would know ether cannot exist (at least where you tried to detect it). But you don't have a detector that is certain to have no false positives, so you cannot rule out that the ether exists. It is certainly marginalized to the point of not being worth researching, but that is distinct from knowledge of its lack of existence.
There are two lines of logic in play.
1) If ether existed, then we would see effect X. We don't see effect X, therefore the ether does not exist. -- Your challenge at this point is that there could be a problem with the detector for effect X. That's fine.
2) The ether hypothesis is incompatible with observation Y. Since we see Y, we know that ether does not exist. -- It's this second line of logic where I don't think your argument works. Moving to the present tense, we know that the existence of a medium through which light must travel is inconsistent with the observations of special relativity.
Since special relativity and ether are inconsistent theories, we know that they can't both be true. Does the fact that we are able to confirm the effects of special relativity inform us that the ether does not actually exist?
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On a related note, can I claim that there are no elephants in my bedroom? (No, I'm not talking about stuffed animal elephants, or pictures of elephants, or anything like that. I mean the actual living creatures.)