I've recently read a book called Visions of Infinity by Ian Stewart. Stewart provides a nice overview of all the most formidable and important problems mathematicians have solved throughout time and problems that have yet to be solved yet. The flow of Visions of Infinity was really interesting because each chapter, excluding the first and last chapter, looks at a different famous problem and is independent from each other chapter. So each chapter looks at a famous problem that someone proposed to true or unsure if it was true or false, then Stewart gives us a glimpse into how from once the problem was first brought up the different conjectures made for it all the way to either how the problem was solved or how far the problem was been taken and what is potentially needed to solve them or if they may ever be solved such as the Riemann Hypothesis and P/NP problem. One of the main points that I felt Stewart was trying to show in each chapter was that these famous problems aren't famous so much because of how useful the knowledge we get from solving them is but instead all the great things we learned while trying to solve these famous problems that may have never come up without people trying to solve these famous problems such as the Taniyama-Shimura conjecture about elliptic curves that was proven while trying to solve Fermat's Last Theorem.
One of my favorite chapters in the book was the chapter dealing with the four color theorem. The reason I really enjoyed this chapter was because it gave a lot of background and insight about the four color theorem along with why it was so important. I found it very interesting that the problem was first proposed outside a mathematical setting by Francis Guthrie while he was coloring the counties in a map of England. When I first learned about the four color theorem in my Discrete Math class I thought that the importance of the proof was just proving the theorem itself. I learned from Visions of Infinity that the importance of the four color theorem was that it changed what would be accepted as proofs. Before the four color theorem proofs use to rely solely on human brainpower so when the theorem was proven with computer assistance many mathematicians were hesitate to accept the proof. I agree with the author when he said the proof was not radically altered if some of the steps were done by a computer. My favorite quote from the author is one regarding computer assisted proofs is, "a proof is a story; a computer-assisted proof is a story that's too long to be told in full, so you have to settle for the executive summary and a huge automated appendix."
Overall I really enjoyed Visions of Infinity, one thing I really liked about the book is that it showed me how math theorems and proofs in real life are much more of a team effort. Through my undergraduate career it seems like all the proofs I write for my math classes have to be done on my own and no collaboration with anyone else, so I found it very interesting when the author wrote about other mathematicians getting stuck on theorems and reaching out to other mathematicians to help them. Another thing I really liked about the book was the author pushing the idea that some of the most important proofs of all time were not because of the things they proved but because of the questions that arose from trying to solve them and the important things we proved. A good example being the Taniyama-Shimura conjecture because the author mentioned no one ever really thought about elliptic curves until mathematicians tried to prove Fermat's last theorem. One downside to the book is that to really understand a lot of these proofs they require a lot of knowledge about all different areas of math but it is understandable that the author expected the reader to possess quite a bit of math knowledge before reading the book because if he tried to teach the reader all the topology, number theory, and other math areas needed to understand the thinking and proofs the book would be exponentially longer. That being said I would still recommend this book to anyone interested in mathematics because there is still lots of information about logic, proofs, and mathematics to take away from this book.