As I have told you earlier I have this strange guest. He simply appeared. After some small talk in my dining room he realized that I will not be of much help if he wanted to return home. To be honest, I do not even have the slightest idea where he actually comes from.
What other options does he have? For a correct assessment he needed an overview about our scientific potential, he told me. What better place could there be than a library? Of course he was interested in the very foundation of our science and since mathematics is a solid basis he wanted volume one of mathematics! The librarian, a student, offered Analysis I and Linear Algebra I, but no book even remotely resembling mathematics volume I. Understandably, my guest was more than disappointed.
There must be some book describing how mathematics bootstraps itself out of nowhere into existence. This science is more than 2.000 years old and has accomplished so many things, but has no first book. There is no chapter one. How shaky is that? We teach people to use mathematical rigor and our own origins are more than frayed.
I told him that my understanding of the situation is that mathematics is based on three concepts. A formal language, needed to express assertions, a logic to attach truth values to these assertions and a notion of set to have something noteworthy to be talked about. I lent him Introduction to automata theory, languages and computation by J.E. Hopcraft and J.D. Ullman which I consider an excellent introduction into formal language and complexity theory. Furthermore I gave him Theory of Sets by N. Bourbaki a book he has to read anyway and Sets for Mathematics by F.W. Lawvere and R. Rosebrugh just to impress him. The logic part was covered by the rather inaccessible Logic of Mathematics by Z. Adamowicz and P. Zbirski. I told him that these three columns all start with a small number of given facts and then develop more complicated facts inductively by given rules.
‘Where from do you get this small number of given facts and rules’, my guest asked suspiciously. I told him, that we simply assume them. That seemed to be too much for him. His jaw literally dropped.
I fail to see his problem. Maybe you can help me. In case you have, or in case you know somebody who has Mathematics Volume I, please send me a copy. In the unlikely case that there is no well-defined starting point, please give me some sort of explanation. Thanks in advance.