This book is too sprawling to wait and review all at once at the end, so I've decided to do it little by little as I go along.
I thought the prologue sucked, but immediately after that it became deeply fascinating, so don't get discouraged. I guess I should say why I hated it, though. It seemed as though he was judging former times and societies through a "presentist" lens, as though all people have always and only been scientists since the start of time, only they were really bad at it back then. It's kind of a scientist's way of ignoring everything else about reality besides science, and made me a bit nauseated, thinking "oh no I hope he's not going to be this dumb all the way through." Luckily, he quickly transitioned to extreme brilliance, in which he's jaw-droppingly continued since then. Even though he's talking so far about seemingly simple stuff, he keeps knocking me for a loop with his deep insights which I've never considered before.
Only in chapter 3 so far, and discussing integers, irrational numbers, and the real numbers. I keep having to stop and think hard about the things he's saying. He asks the question if we lived in a universe where things were an amorphous soup would the integers exist there. He also points out that calculus (and stuff like momentum, velocity, and many of our physical concepts depend on calculus) is defined on the real numbers. If it turns out that the universe is discrete at the tiniest level, this math won't apply anymore (except as an approximation). However, he also observes that the real numbers first invented in Euclid's day when we had physical evidence spanning only some 15 orders of magnitudes (the smallest to largest distances known) are still going strong now when our knowledge spans something like 150 orders of magnitude, so they aren't doing too badly! These are the notions of someone who has thought deeply about how math and physics are intertwined. I keep being dumbstruck with things he casually asks about things that are ostensibly simple which I've known forever but never thought to ask that. Really important stuff. He is breathtakingly brilliant! I'm so glad I'm reading this book!
Aside: The more I read the more sure I am that Platonic essences exist independent of the nature of physical reality, and independent of their instantiation in some physical reality.
Spent some time going over familiar ground in the complex plane. It's been long enough since I studied or used this stuff that it's quite enjoyable and satisfying to do that. I think I've settled on the slow savoring method of reading this book rather than the quick devour. This review's going to be very long, but I hope it'll admit of savoring a bit as well. =)
In Chapter 5 now, and talking about e and logarithms, I wondered why it is again that e is a more natural base for logarithms than any other number. So I spent some time adding it up from the formula e = (1/0!)+(1/1!)+(1/2!)+(1/3!)+(1/4!)+(1/5!)+... and watched the digits slowly materializing 2.7182.... so I believe that much. =) Next I'm reading again how e originally came up in playing around with logs and powers. This book has that effect that it makes me think again about stuff that I haven't thought about since I was young. I would really like to feel I understood what we know of reality inside and out when I'm done. I want to see the whole chain starting from one cow, two cows on up to the standard model and beyond. It's always been an obsession of mine just to understand how things freaking work, what the universe is like, what nature is based on, and I have this feeling I could get much closer by going through this volume carefully. The title keeps reminding me of the Royal Road to Geometry, which Aristotle reportedly told Alexander the Great did not exist, so that's some kind of warning, hah!
So far I've resisted the urge to jump ahead, except for reading the section called "beauty and miracles" near the very end. You have to admit that's an attractive section name! Alas, I understood it only in the broadest way, that beauty (mathematical elegance) and miracles (seemingly crazy mathematical coincidences such as all the complicated terms happening to drop out or whatever) act as a powerful but not unfailing guide so far to finding theories that fit how nature behaves. At that moment, my dear kitten Alai jumped up and sat right on the book, as if to say, "you want beauty and miracles? Just look at me!" As I pet him, I kept saying "beauty and miracles" affectionately.