I will confess up front that I am honestly confused why people still cling to Popper's work.
Even Popper himself eventually came to admit that his philosophy of science wasn't much more than an abstract aspiration, having little relation to the actual practice of science. Popper had some useful insights, sure, and an important place in the history of the philosophy of science, but seriously... Time to let that old dog rest.
Whinging aside, if we ask how many 'kinds' of science there are, we're going to get different answers depending on who we put the question to:
Popper would say there's one thing called 'Science' that is done within certain limits, and nothing else applies.
Kuhn would argue that every field creates a version of 'science' around its own central paradigm, and that even these paradigms can occasionally shift so that fields might have different understandings of 'science' at different points in their history.
- Feyerabend (whose name I always misspell, dang it) would argue that there is no 'science' in the general sense; there is only the singular behavior of individual scientists in their practice, with a wide multitude of 'sciences' that results from all their efforts.
Popperian falsification as non rigorous
The common element of most philosophies of science is the observation that science is an ongoing practice: an activity/verb, not a thing/noun.
In this sense, when we try to think about whether something is 'scientific,' we don't mean whether it is/isn't within the class of objects labeled 'science.' Instead, we are asking whether the topic is approached in a manner that conforms to certain analytical/methodological constraints. This was part of the problem with Popper's falsificationism: the act of 'falsification' is not itself methodologically rigorous.
Just to take your example, let's say I wanted to 'scientifically' test math. Simple test: I decide to see whether 1+1=2 in some concrete context.
- I take one apple and I put it on a table
- I repeat that process;
- I count the apples on the table.
- If there are more or less than two apples on the table, then I've
Of course, there are two apples on the table (unless I got hungry), so addition survives the first effort at falsification. So I try again:
- I take one water and pour it into a bowl
- I repeat the process
- I look in the bowl and count the waters I have
- If there are more or less than two waters, then I've falsified addition!
Oh, err... Everyone will object that I've made a mistake by using 'waters' (which are not properly 'countable') and so addition has not actually been falsified. Popper would be forced to say, here, that this kind of manipulation — where mathematicians redefine the subject matter with new concepts (like 'countability') in order to preserve the core theory — means that mathematics is unfalsifiable, and thus not scientific. But as a process this is a perfectly normal scientific activity, where researchers redefine and recategorize observable data in order to make theories more exacting and coherent. So don't place quite so much credit on Popperean accounts of 'science.'