School:Mathematics/Philosophy of Mathematics/Scope

This is an introductory meta chapter on the topic. It answers the basic questions about the scope of the course and gives information for deciding on taking the course or not.

Why Study The Philosophy of Mathematics?
The answer to this question is not that we are studying, but what we are studying.


 * PoM is better than taking another Maths course, since:
 * Just doing something without questioning is like a hamster in a running wheel; you never really understand why and how things are done.
 * The evaluation of mathematical results itself is not usually done purely mathematically, since every math problem should be taken contextually as well as mathematically.
 * Communications with non-mathematicians from time to time is unavoidable, so it helps to be able to explain things.


 * PoM is better than taking another Philosophy of xyz course, since:
 * Mathematics is THE apriorical science, so here one learns more about the a priori than anywhere else.
 * The concept of 'theory' is an elementary one in epistemology, mathematics is where it is defined.
 * When talking about mathematics many people lack confidence, here is clarified what one can and can't expect from it.

What does a Philosophy of Mathematics contain?
When doing mathematics 'the usual way' one can easily find that it's not a purely formal science at it's rock bottom. So it can be thought of a mathematical foundation of mathematics. This is usually referred to as metamathematics.

Also one can question the formal method of mathematics at all from a cognitive point of view: is it really better to write confusing lines of symbols than writing a poem about the stuff? Moreover, a mathematician is not alone. There is a whole mathematical community out there, and although the results of mathematical research are formally presented, doing mathematics is highly creative and evaluating these creative results as well as deciding about directions of research is primarily a social problem.

Similarly, when thinking about the application mathematics in science, engineering, and other such subjects, one can take two approaches: modelling the situation mathematically and examining the cognitive and community aspects. The application of mathematics can and can't be seen as part of the subject of philosophy of mathematics. Here it is included, but when reasons should appear to source it out to another course this could be done easily.

Exclusions: What it does NOT contain:
 * an introduction to logic or model theory
 * an introduction to epistemology
 * an introduction to cognitive science (should it?)
 * an introduction to social science or game theory (should it?)
 * any training of mathematical abilities or capabilities
 * any issues of non-formal linguistics
 * aesthetics of mathematics (as a standalone subject), since this is truly art and not only a subchapter of philosophy. Although it might appear as part of some philosophical considerations.

How is this course structured?
Problem: Philosophers often make statements in epistemology and theory of science, but Scientists ignore or are even annoyed at them (do you know any formal scientist who loves T.S.Kuhn?). Actually, Scientists are interested in the philosophical background of their job, and Philosophers are interested in the justification of their Theories.

Approach: Structuring the whole subject from the perspective of a questioning Mathematician. Such a course structure may be build up along the following questions, expanding the underlying Model of the 'Mathematician's job' step by step.

So the course will be beneficial for Scientists AND Philosophers as well as it enables both sides also to participate in EDITING the content - leveraging the cooperative spirit of Wikiversity!