Math is itself indeed not science. It is the language of science. It follows different rules than empirical sciences. But note that word "empirical" there; Popper was really only talking about empirical science, and according to him, that was the only real science. You could argue that there are non-empirical sciences.
Another problem with Popper is probably that outside of physics and chemistry, there are a lot of less exact sciences where predictions and refutations of a theory are never that clear cut. Like his issues with the theory of evolution.
Ultimately, I guess science is also simply "getting to stuff that works by trial and error".
How much engineering could we do without Mathematics? How much commerce?
I don't see it as exclusionary. You won't find many scientists in doubt about the fact that everything they do is built upon Logic and Mathematics, in addition to observation.
But don't we need a word to group fields that try to systematically describe, understand, and make predictions about the physical world? (Rather than seeking to explore and characterise idealised logical constructs?). What would you suggest?
You may not see it as exclusionary but many people do. Just look at the comments!
It's precisely the grouping I am talking about.
If you group science in such a way so that logic/mathematics/computer science falls outside the group then isn't that an erroneous grouping?
Isn't that a silly definition?
True and False are idealized logical constructs. It's the idea; and the idealization of the notion that there is a difference between Truth and Falsehood. Or if you want to get biblical - there is a difference between Right and Wrong.
We need a grouping to make it clear that some fields produce theories and others produce theorems.
We need theory-producers to be more humble and provisional in their statements. We need theory-producers to forever remain open for their theories to be falsified or refined (whilst not being paralysed by doubt about theories that have stood the test of time). In other words, we need a slightly different culture.
But we also need a way to rebut someone who says "OK, but can you prove we're not living in a perfect simulation of reality with a fabricated history that was created yesterday?". In science, the rebuttal is "No, I can't prove that, science depends falsification rather than proof. Can you suggest a way I could falsify it? If not, then I'm going to get on with my work because it doesn't make a difference to my field either way"
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.
Another problem with Popper is probably that outside of physics and chemistry, there are a lot of less exact sciences where predictions and refutations of a theory are never that clear cut. Like his issues with the theory of evolution.
Ultimately, I guess science is also simply "getting to stuff that works by trial and error".