From Pendulums to p-Adic Numbers - A Philosophy of Mathematics

For quite some time I have been working on an essay on the philosophy of mathematics. It's gone through many changes since I first started jotting notes on the commute and in various cafes around London, and bears no resemblance to anything I thought I would write when I started. It isn't complete and probably never will be, since there are always more insights and examples to add. It does have most of the philosophical points I want to make, and talks about most of the maths I feel even half-way competent to discuss. So I'm going to make it available for whoever needs some light entertainment. It will get updated from time to time.

The link is here (link)

It's an attempt to answer these questions:

How is it mathematical techniques and tools are so suited to describe physical processes? 

How do mathematical concepts work?


What kinds of knowledge does mathematics provide?


How do we know that a theory does not harbour fatal inconsistencies? 

How do mathematicians get and develop their ideas?


How do we judge the value of a technique, theorem or subject? 

What constitutes progress in maths?

It proceeds through discussions of these issues in the context of differential equations, functional analysis, infinity, functions, numerical analysis and recursive functions, and the various types of numbers, from the counting numbers to the p-adics. There's a discussion of axiomatics and model theory and a brief look at category theory; the way mathematical ideas are structured and what mathematical knowledge is (epistemology); how we might appraise different mathematical theories (methodology); and what constitutes progress and then a discussion of how to get ideas and solve problems (heuristics).

What there isn't is detailed presentations and rebuttals of existing philosophies of mathematics, what I've called the “where Smith mistakes Jones’ summary of Brown’s critique of Frege’s

misunderstanding of Kant” school of scholarly discussion.