[begriffs]

If you really want to go back to first principles, try "Foundations of Analysis" by Edmund Landau. It builds the integers, fractions, Dedekind cuts, and the real and complex numbers from scratch.

It's totally rigorous and starts from, "the ability to read English and to think logically -- no high-school mathematics, and certainly no advanced mathematics."

[nicklaf]

I would be very careful before unleashing a beginner on this book. It would be too easy, IMO, for the reader to end up with the wrong idea that mechanical proofs like the ones in this book are all that are needed in mathematics, since it's possible to get as far as the real numbers (or complex numbers) with so little geometric intuition. Furthermore, the real numbers are the most concrete, familiar setting to do analysis in, but it is not healthy to spend so much focus on the concrete details of the real line so early on: a student of mathematics needs variation to keep alive her or his curiosity. Pugh does an excellent job of explaining the simple geometric essence of Dedekind cuts. In principle, one might learn something about proof writing by reading the Landau book. However, it is much, much better, IMO, to defer detailed study of something so specific, until after first surveying the setting in which the results of Landau's book are used. In most real analysis books, the reader is asked to prove a few of the results covered in Landau.

Mathematics is foremost a conceptual subject rather than a mechanical one, and it is immaterial that the reader have firsthand experience that all the theorems are proven. As one learns mathematics, it soon becomes apparent that there will always be gaps in her or his knowledge, and that is therefore best to skip steps that s/he believes could be done in principle.

[DavidWanjiru]

Are you on twitter, nicklaf? You sound like someone I'd want to be following.

[nicklaf]

Negative, but I do plan to set up a sort of homepage for my research sometime in the future. When I do, I'll link to it in my HN profile.

[ericssmith]

Since I have painstakingly gone through the series of proofs in Landau's book, I feel I need to weigh in here. On the surface, this book does appear to start from sets and convincingly proceed up to real numbers etc. It's interesting that you include the quote from the beginning about "high-school mathematics", which I think is laughable.

I personally believe that Landau was caught up in the spirit of the times and optimistically believed that math could be built up from "first principles". The famous kickstart to this is Hilbert's 1900 presentation. And it certainly continued up through Nicolas Bourbaki.

In fact, Landau's mathematics is presented in a somewhat archaic style and his proofs are extremely hard to follow in spots, as if he is making unstated assumptions. Overall, it is an interesting, but ultimately thankless, task to go through that book. It is a mostly a historical curiosity. The same can be said of Hardy's "Course of Pure Mathematics", which was recommended elsewhere in this thread. I find it hard to believe that anyone who recommends these books have actually read them.

To the OP, while I can relate to the goal from personal experience, after decades of going down a similar path, I can tell you that the history of math is very messy. Our textbooks and notation reflect this messiness. My recommendation is to dive into whatever part strikes your fancy, although it may help to start from where you are. For instance, if you program, you might want to get a book on physics in game programming or learn Haskell.