First of all I would argue that science includes both empirical and analytical sciences, of where math and logic are analytical. Math and logic is the study of

First of all I would argue that science includes both empirical and analytical sciences, of where math and logic are analytical. Math and logic is the study of the consequences based on some very simple assumptions and have proven to be extremely valuable to the empirical fields of science. Second, I would argue that the assumptions are not arbitrary but what most people would hold as true (smartasses and a few more excluded) and even though we have different fields based on what assumptions we choose (fuzzy logic vs classic logic for example) they have all come up with a lot of very usable tools for us for use in the empirical sciences. Thirdly, "observed" can be argued to both apply to empirical (in the real world) and non-obvious analytical consequences. Sometimes, in analytical science you pull on a thread and see where it leads. The result might surprise you and it is not really wrong to say that you observe the consequence just like you would if you observe a experiment in the empirical sense.