God Created the Integers

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1. javajoshJul 2020
   God Created the Integers: The Mathematical Breakthroughs That Changed History. Stephen Hawking. I bought mine for cheap on alibris (https://www.alibris.com/God-Created-the-Integers-The-Mathema...)

   From the blurb:

   "...includes landmark discoveries spanning 2500 years and representing the work of mathematicians such as Euclid, Georg Cantor, Kurt Godel, Augustin Cauchy, Bernard Riemann and Alan Turing. Each chapter begins with a biography of the featured mathematician, clearly explaining the significance of the result, followed by the full proof of the work, reproduced from the original publication, many in new translations."

   What's great about this book for a teenager is that they get to read original sources for the stuff they've already learned! And indeed, as they learn more they can keep coming back for more original sources. Personally, reading Descartes original words in Geometry was awe-inspiring, not because every word was so perfect, but because he comes across as just so damn human, the ideas he presents are subtle and profound, and yet presented with an interesting combination of humility and pride that is instantly recognizable. I truly wish I'd had something like that book before embarking on my own journey through math - we stand on the shoulders of giants, but we so rarely look down to see their faces.

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2. javajoshJul 2020
   Not at all. My own recommendation, God created the Integers, isn't a math textbook. I doubt Hofstadter himself would claim GEB had a point - it was more of an intellectual fugue put to paper. If GEB was a novel it would be more along the lines of Finnegan's Wake than Les Misérables, and I would never ever give the former to a teenager.
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3. javajoshJul 2020
   I would give God Created The Integers as a reference to read up to where you are in math, not as something to get through. So, you could get a feel for what Euclid wrote, or what Descartes wrote, either after or during learning those lessons. As you move through your education, you can keep moving through the book, Cauchy, Galois, Riemann, etc. Anyway, that's the context in which I would give it. BTW Cantor's original diagonal proof is in GCTI. :)
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4. javajoshMay 2020
   God Created the Integers: The Mathematical Breakthroughs That Changed History. Stephen Hawking. I bought mine for cheap on alibris (https://www.alibris.com/God-Created-the-Integers-The-Mathema...)

   From the blurb:

   "...includes landmark discoveries spanning 2500 years and representing the work of mathematicians such as Euclid, Georg Cantor, Kurt Godel, Augustin Cauchy, Bernard Riemann and Alan Turing. Each chapter begins with a biography of the featured mathematician, clearly explaining the significance of the result, followed by the full proof of the work, reproduced from the original publication, many in new translations."

   My absolute favorite is Descartes Geometrie. Reading the original humanizes him. He is at once humble and yet confident that he'd discovered something important. Today, we take Descartes ideas completely for granted, and even find them trivial, and yet to realize the immense shift in thinking it represented in his time, makes it all the more impressive. I would even argue that Descartes' ideas were more important than any save Euclid in forming the foundation of modern math and physics - he was the one that divorced the notion of "quantity" from "length of a line segment"! It's just so great to read the original.

   Interestingly, Riemann got almost all of his papers included in this book, the most pages by far of any author. Clearly Hawking loved Riemann - and no wonder. Riemann was the one who truly generalized geometry, and its interesting that his work in that area was mostly ignored until 50 years after it was published, when Einstein used it for General Relativity. He made enormous contributions to lots of other fields, too.

   Oh, and it was really cool reading Boole's words, and recognizing that he was explicitly talking in terms of creating a calculus of thought!

   So, yeah, I love this book because we often talk about "the shoulders of giants" but we rarely actually read them directly. And it's amazing, inspiring, and wonderful to relate to them!

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5. amaiFeb 2019
   I can recommend https://en.wikipedia.org/wiki/God_Created_the_Integers

   "God Created the Integers: The Mathematical Breakthroughs That Changed History is an anthology, edited by Stephen Hawking, of "excerpts from thirty-one of the most important works in the history of mathematics.""

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6. vinaymsMay 2018
   I studied Mechanical Engineering but gradually ended up as a full fledged software engineer from one who wrote ME related programs. So I have sort of always been in the midst of mid level maths and physics, but, as it happens, I lost it all except for some fundamental concepts. I had a similar epiphany as yours in my early thirties and this is what I did and it helped me greatly.

   First of all, you have to realize that you are learning these stuff only for the sake of learning, as an intellectual challenge, rather than making a career out of it. So, you need not follow a pattern that is made for late teen students attending university. That includes not needing to religiously solve problems in textbooks, especially the numerical ones. If you have to, solve the conceptual ones. Further, you wouldn't even need textbooks.

   What you do need, however, is a perspective. Unlike university students whose primary aim is to pass the exams that matter, graduate and get a job, for which they learn by rote, what you need is to understand the reason why learning certain topics is essential. You need to learn about the topics first in order to appreciate whatever you will encounter later. For this purpose, well regarded popular science books should be your first choice. Any worthy textbook will also give a bit of perspective in the preface, early chapters or the appendix, but it can never compete with popsci books which are designed for the purpose of elucidating academically complex stuff to an otherwise educated public. Good popsci books provide enough stimulation to your mind. They prime you for further, formal exploration of the topic.

   Often times what happens is that highly technical topics such as those you mentioned look attractive from the outside but get painfully dry, boring and difficult once you pick up a text book and start studying from it. What I realized is that for many topics what I really crave is a deep understanding as an educated person as opposed to deep academic knowledge. For such topics popsci books regarded as 'hard' are better than proper textbooks. These days, with the availability of opencoursewares, you can simply watch a few lectures before deciding if its worth your time delving deep into the topics with a textbook. The textbooks are often listed in the course content, and we all know where we can find the appropriate pdf versions.

   These books worked for me. They are in my home library and I try to flip through some of them once a year.

   For maths, books like Prime Obsession by John Derbyshire, e The story of a Number by Eli Moar, Number by Tobias Dantzig, The Unknown Quantity by John Derbyshire provide good orientation for number theory, analysis & calculus and algebra. The second book might be tough to finish as its really really deep for a thin book. You might also look at books like God Created the Integers by Hawking, The Calculus Gallery by William Dunham that curate interesting and historically important results, where original works are often reproduced.

   For physics, my goto book (which I have never finished) would be The Road To Reality by Roger Penrose. This self contained 1000+ page monster builds up to advanced physics in a methodical fashion from scratch. The first half deals with all the necessary maths from elementary to advanced and the second half covers physics. It has chapters with names such as calculus on manifolds, hypercomplex numbers, the entangled quantum world, gravity's role in quantum state reduction etc which sound pretty deep and complex for a recreational learner. Beyond that, you can also read books such as The Evolution of Physics by Einstein himself, Thirty Years that Shook Physics by George Gamow to get a historical perspective on the development of advanced physics. For a midway academic treatment, you can read Feynman's lecture on Physics volumes.

   For discrete maths, books like The Algorithm Design by Skiena and Algorithm Design by Kleinberg and Tardos should serve you well. They don't push you into a maze of mathematics like Knuth books do, or a maze of source code like CLRS or Sedgewick books do, but show you the practical side of things with limited code and pseudocode. That said, Knuth's The Art of Computer Programming 4A Combinatorial Algorithms Part 1 will just blow your mind.

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