Forum favorite Good and Real looks reasonably accessible to me, and covers a lot of ground. Also seconding Gödel, Escher Bach.
The Mathematical Experience has essays about doing mathematics, written by actual mathematicians. It seems like very good reading for someone who might be considering studying math.
The Road to Reality has Roger Penrose trying to explain all of modern physics and the required mathematics without pulling any punches and starting from grade school math in a single book. Will probably cause a brain meltdown at some point on anyone who doesn’t already know the stuff, but just having a popular science style book that nevertheless goes on to explain the general theory of relativity without handwaving is pretty impressive. Doesn’t include any of Penrose’s less fortunate forays into cognitive science and AI.
Darwin’s Dangerous Idea by Daniel Dennett explains how evolution isn’t just something that happens in biology, but how it turns up in all sorts of systems.
Armchair Universe and old book about “computer recreations”, probably most famous is the introduction of the Core War game. The other topics are similar, setting up an environment with a simple program that has elaborate emergent behavior coming out of it. Assumes the reader might actually program the recreations themselves, and provides appropriate detail.
Surely You’re Joking, Mr. Feynman is pretty much entertainment, but still very good. Feynman is still the requisite trickster-god patron saint of math and science.
Code: The Hidden Language of Computer Hardware and Software explains how computers are put together, starting from really concrete first principles (flashing Morse code with flashlights, mechanical relay circuits) and getting up to microprocessors, RAM and executable program code.
Good and Real is superb, but really too dry for a 13-year-old. I’d wait on that one.
Surely You’re Joking is also fantastic, but get it read and approved by your nephew’s parents first; there’s a few sexual stories with a hint of a PUA worldview.
In the 3,000 categories of mathematical writing, new mathematics is being created at a constantly increasing rate. The ocean is expanding, both in depth and in breadth.
By multiplying the number of papers per issue and the average number of theorems per paper, their estimate came to nearly two hundred thousand theorems a year. If the number of theorems is larger than one can possibly survey, who can be trusted to judge what is ‘important’? One cannot have survival of the fittest if there is no interaction. It is actually impossible to keep abreast of even the more outstanding and exciting results. How can one reconcile this with the view that mathematics will survive as a single science? In mathematics one becomes married to one’s own little field. [...] The variety of objects worked on by young scientists is growing exponentially. [...] Only within the narrow perspective of a particular speciality can one see a coherent pattern of development.
Peer-review is the predator. But if the prey population is higher than can be sheltered by selection of promising ideas from nonsense, nonsense will prevail. That is, those people producing valuable results won’t be favored over those that come up with marginal or wrong results.
Yes, that’s exactly the kind of stuff I recommended The Mathematical Experience for. It takes a bird’s eye view instead of going for the usual textbook minutiae, but still feels like it’s talking about the actual practice of mathematics instead of something simplified to death for the benefit of popular audiences.
Forum favorite Good and Real looks reasonably accessible to me, and covers a lot of ground. Also seconding Gödel, Escher Bach.
The Mathematical Experience has essays about doing mathematics, written by actual mathematicians. It seems like very good reading for someone who might be considering studying math.
The Road to Reality has Roger Penrose trying to explain all of modern physics and the required mathematics without pulling any punches and starting from grade school math in a single book. Will probably cause a brain meltdown at some point on anyone who doesn’t already know the stuff, but just having a popular science style book that nevertheless goes on to explain the general theory of relativity without handwaving is pretty impressive. Doesn’t include any of Penrose’s less fortunate forays into cognitive science and AI.
Darwin’s Dangerous Idea by Daniel Dennett explains how evolution isn’t just something that happens in biology, but how it turns up in all sorts of systems.
Armchair Universe and old book about “computer recreations”, probably most famous is the introduction of the Core War game. The other topics are similar, setting up an environment with a simple program that has elaborate emergent behavior coming out of it. Assumes the reader might actually program the recreations themselves, and provides appropriate detail.
Surely You’re Joking, Mr. Feynman is pretty much entertainment, but still very good. Feynman is still the requisite trickster-god patron saint of math and science.
Code: The Hidden Language of Computer Hardware and Software explains how computers are put together, starting from really concrete first principles (flashing Morse code with flashlights, mechanical relay circuits) and getting up to microprocessors, RAM and executable program code.
Good and Real is superb, but really too dry for a 13-year-old. I’d wait on that one.
Surely You’re Joking is also fantastic, but get it read and approved by your nephew’s parents first; there’s a few sexual stories with a hint of a PUA worldview.
I loved “The Mathematical Experience” when I was 13-ish, and I re-read it recently; still good! I strongly second this recommendation.
Thanks, I just ordered ‘Darwin’s Dangerous Idea’ and ‘Code: The Hidden Language of Computer Hardware and Software’. I’ve already got the others.
Here a tidbit from ‘The Mathematical Experience’
I’ve ordered a copy, but on a second look, I’m not sure that the argument is sound, or even interesting.
Biological evolution runs on the local non-survival of the least fit (and sometimes the unlucky), not on an overview-based evaluation of the fittest.
Peer-review is the predator. But if the prey population is higher than can be sheltered by selection of promising ideas from nonsense, nonsense will prevail. That is, those people producing valuable results won’t be favored over those that come up with marginal or wrong results.
Yes, that’s exactly the kind of stuff I recommended The Mathematical Experience for. It takes a bird’s eye view instead of going for the usual textbook minutiae, but still feels like it’s talking about the actual practice of mathematics instead of something simplified to death for the benefit of popular audiences.
Wow, great list. Thanks!
Oh, oops— I intended my review of the above selections to show up on your replies, not Risto’s.