We who? Don't "we" also need a grouping to make it even clearer that some fields can't produce any falsifiable theories if other fields don't produce at least some unfalsifiable theorems? A terra firma of sorts.

It's like a dependency graph. Or something.

Your insistence on "making a difference" seems to echo the sentiment of many pragmatists:

  It is astonishing to see how many philosophical disputes collapse into insignificance the moment you subject them to this simple test of tracing a concrete consequence. There can be no difference anywhere that doesn’t make a difference elsewhere – no difference in abstract truth that doesn’t express itself in a difference in concrete fact and in conduct consequent upon that fact, imposed on somebody, somehow, somewhere, and somewhen. The whole function of philosophy ought to be to find out what definite difference it will make to you and me, at definite instants of our life, if this world-formula or that world-formula be the true one. --William James

Does falsifiability make any difference? If something is only falsifiable in principle (e.g in theory), but not in practice then is it really falsifiable? On pragmatism - it's not a difference that makes any practical difference. And yet you insist on differentiating. Why?

Is "All humans are mortal." falsifiable or unfalsifiable? It sure is falsifiable in theory, but unfalsifiable in practice. Any living human is potentially immortal until they actually die.

Any running process is potentially non-halting, until it actually halts.

If falsifiability doesn't make a difference in practice (and it doesn't!) then I guess we can all get on with whatever scientific discipline we are busy practicing.

So, I'm going to carry on my life knowing at least one unfalsifiable scientific truth: the theorem known as The Halting Problem.

It's not even wrong, because it's right.

Anybody who insists the Halting Problem is falsifiable (even in principle) is welcome to solve it in principle.

> Don't "we" also need a grouping to make it even clearer that some fields can't produce any falsifiable theories if other fields don't produce any unfalsifiable theorems?

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.

It's pointless to speak of usefulness without specifying a utility function.

It is just as possible to differentiate as it is to integrate.

If it is determined a priori that unfalsifiable propositions are not useful, then knowing the result of the Halting Problem is not useful. Isn't that silly?

I strongly object to categorizations which discriminate against valid science (knowledge? truth? understanding? reasoning? Useful facts?). 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 and self-justification Carl Sagan is guilty of.

Science benefits much more from being defined too broadly; than being defined too narrowly.

I'd rather be too permissive then ignore the junk; than be too restrictive and never even encounter good ideas which were erroneously discarded as junk.

I don't think I said unfalsifiable propositions are not useful! A proven theorem is sacred!

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. Curious researchers on both sides can certainly be informed and inspired by each others work, and can use the same techniques and tools. But even if only as an academic exercise can't we describe these two modes of discovery. And isn't it worth being clear about their respective limits?

You seem to be missing the point. Ignoring for a second that the laws of thermodynamics themselves are based upon a handful of idealizations (the idealization of "thermal equilibrium", the idealization of "perfectly isolated system", the idealization of "perfect zero)...the laws of nature are encoded as formalisms/equations. Symbolic computations.

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.