And if unfalsifiable propositons are not useful, then the Halting Problem is not useful. Isn't that silly?
I strongly object to categorizations which discriminate against valid science. Is all.
The human process of trying to udnerstand reality is continuous, not discrete, so it's silly to reason about it in terms of discrete categories. It necessarily leads to confusion; and the sort of gatekeeping Carl Sagan is guilty of.
Of course, until the laws of thermodynamics are revised we can provisionally say that all programs actually running in nature will indeed stop at some point, no matter what is proven about idealized Turing machines.
And before I'm misunderstood. There are many ways the laws of thermodynamics can be tested. This prediction, unfortunately, cannot be tested. But it is a predicted consequence of the simplest known theory that explains of all sorts of observations about thermodynamics. Which is the limit of what the natural sciences aim to do here. Provisional truth based on observation vs. proven truth based on stated axioms.
I am explicitly not claiming that one truth is to be valued more than the other. I honestly don't think that. Merely noting, again, that the distinction is there to be made. I may be "discriminating between", but I'm certainly not "discriminating against".
It may or may not be a continuum. But even if only as an academic exercise can't we describe these two modes of knowledge? Curious researchers on both sides can certainly be informed and inspired by each others work, and can use the same techniques and tools.
Will the laws of nature stop running at some point?
The whole point is that the theoretical/abstract/mathematical constructs are what we use to compute consequences.
If you have no formalisms you can't compute any consequences - there is nothing to test. You have no science.
So treating Mathematics and science as "separate disciplines", even though they function as one symbiotic whole - that's the conceptual error.
Sure. And I suspect a subset of pure mathematicians would want terminology to make clear that they produce theorems out of intellectual curiosity rather than because they have any regard for whether those theorems can be applied by other fields. Fortunately we can categorize things in multiple ways. I'm open to suggestions on the semantics, but something more widely understood and less clunky than my own theorem/theory-producers would be good! Perhaps "Natural Sciences" or "Empirical Sciences" might be more specific terms for fields that produce theories, if you like.
I differentiate simply because seems possible to do so. And as I said, because it's worth considering whether different processes and cultures are useful. I'm intrigued as to why you object so strongly.
I am afraid my intellect isn't quite up to the application of scientific principles to the philosophy of science itself this morning. I'll have to think harder about whether that's even a valid thing to do.
I don't think you've shown that falsifiability makes no difference in practice. The fact that it's possible to come up with some borderline or problematic examples (which themselves aren't terribly practical) doesn't mean it's not a useful criterion for a scientific theory. Falsifiability is a valuable filter for ideas that the natural sciences are not able to speak to. String theory has been criticized as unfalsifiable. I think a good string theorist would accept that it's a serious accusation that requires an answer.
To be honest I'm quite happy to say "All humans are mortal" is not a well-stated scientific theory. "Human lifespan is limited to 180 years" is better, as it may one day be falsified.