Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

Question

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, consequences.

I would like to know if there are books:

• perhaps with a historical bent, characterizing the thoughts that went behind the design of a particular system of axioms for a catalogue of theories;

• perhaps with a non-historical bent, just exploring axiomatization in general;

• perhaps with a "fun" bent, attempting to recreate a well established mathematical theory from scratch;

• perhaps that are clearly a text on abstract algebra, but one that explores qualitatively the various consequences of choosing particular axioms?

Does such a book exist, or am I dreaming?

Answer 1

I suggest you :

Robin Hartshorne, Geometry: Euclid and Beyond (2005)

and

Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements (1st ed 1981).

Both deal with geometry and have an historical bent (no "fun").

You can see also :

Howard Eves, Foundations and Fundamental Concepts of Mathematics (1st ed 1990), with a nice Chapter 6 on Formal Axiomatics; it deals also with algebra and number systems (no "fun").

Regarding the development of modern algebra and axiomatics, see :

Leo Corry, Modern Algebra and the Rise of Mathematical Structures (2004).

With a different point of view, can be interesting also :

PhilipJ Davis & Reuben Hersh, The Mathematical Experience (1st ed 1981).

In conclusion, I think that it will be not easy to find references "helping" with the "design of a particular system of axioms for a catalogue of theories".

Answer 2

I'm not sure that there's a single book that fits your varied desiderata. But, just addressing the first of the bullet points, the history of the axiomatization of geometry (and then geometries, plural) is surely an obvious place to start. There is a rich but very readable article by Marvin Greenberg here. And then, for a lot more, there is Greenberg's book mentioned in his biblio.