Logic and the real world
Since the objects of mathematics are nothing but mental constructs, then really anything goes in talking about them as long as you're consistent; and so there isn't a great deal for us to say on the topic. When you get to the empirical sciences, however, where you're trying to find out facts about the world as it actually is, the problems in how you go about most efficiently achieving your goal become quite extensive.
The main point of what I have to say, however, about science will be found in the theory of effect and cause that I developed in Section 2 of the first part of this book. I think it forms the basic core of what scientists are doing. I put it there, of course, because I happen to think that philosophy's goal is the same as that of the empirical sciences--to find out what is really going on--and so it shouldn't be surprising if its method is basically the same one.
What I am going to do in this section is go through the traditional Five Steps of the scientific method and show how my theory of effect and cause makes sense out of what scientists do. I am not really going to try to refute the myriad other views there are on the topic, except on the basis of an established canon of scientific theory: if my theory explains all that they can explain and does it more simply and more logically, then my theory is to be preferred to theirs.
While I am at it, I will also find occasion to talk about a couple of topics that I have not been able to fit in as yet, such as the laws of probability (and the function of statistics), and the logic of induction. Both of these are heavily used in science, and so discussing them is appropriate here; but neither of them really belong anywhere else except back in Section 2 of the first part; but there, they would have been incidental and only cluttered up the basic theory of cause. The logic of induction, by the way, does not belong in the section on formal logic, because it doesn't proceed, as we will see, from the nature of statements.Next