This has directed me to Irreducible Complexity in Pure Mathematics and finally to a delightful, easy to read but very deep Meta Math! The Quest for Omega. The last one is especially worth recommending. Mathematics that is told with gusto and personal interest.
There are whole passages that are so close to my soul, that I can only quote them in extenso:
In my opinion, the view that math provides absolute certainty and is static and perfect while physics is tentative and constantly evolving is a false dichotomy. Math is actually not that different from physics. Both are attempts of the human mind to organize, to make sense, of human experience; in the case of physics, experience in the laboratory, in the physical world, and
in the case of math, experience in the computer, in the mental mindscape of pure mathematics.
And mathematics is far from static and perfect; it is constantly evolving, constantly changing, constantly morphing itself into new forms. New concepts are constantly transforming math and creating new ﬁelds, new viewpoints, new emphasis, and new questions to answer. And mathematicians do in fact utilize unproved new principles suggested by computational experience, just as a physicist would.
And in discovering and creating new mathematics, mathematicians do base themselves on intuition and inspiration, on unconscious motivations and impulses, and on their aesthetic sense, just like any creative artist would. And mathematicians do not lead logical mechanical “rational” lives. Like any creative artist, they are passionate emotional people who deeply care about their art, they are unconventional eccentrics motivated by mysterious forces, not by money nor by a concern for the “practical applications” of their work.
I know, because I’m one of these crazy people myself! I’ve been obsessed by these questions for my whole life, starting at an early age. And I’ll give you an insider’s view of all of this, a ﬁrst-hand report from the front, where there is still a lot of ﬁghting, a lot of pushing and shoving, between different viewpoints. In fact basic questions like this are never settled, never deﬁnitively put aside, they have a way of resurfacing, of popping up again in transformed form, every few generations. . .
So that’s what this book is about: It’s about reasoning questioning itself, and its limits and the role of creativity and intuition, and the sources of new ideas and of new knowledge. That’s a big subject, and I only understand a little bit of it, the areas that I’ve worked in or experienced myself. Some of this nobody understands very well, it’s a task for the future. How about you?! Maybe you can do some work in this area. Maybe you can push the darkness back a millimeter or two! Maybe you can come up with an important new idea, maybe you can imagine a new kind of question to ask, maybe you can transform the landscape by seeing it from a diﬀerent point of view! That’s all it takes, just one little new idea, and lots and lots of hard work to develop it and to convince other people! Maybe you can put a scratch on the rock of eternity!