How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

Category: Mathematics
This Year Hacker News 3
This Month Hacker News 2


by todd8   2018-01-07
First, keep solving problems. Mathematical brainteasers got me started problem solving when I was in grade school. I really like Martin Gardner's books[1] and those by Robert Smullyan[2]. These kinds of problems develop the flexibility of thought that helps find creative solutions.

Actual, specific approaches to tackling tough problems are taught by the famous Hungarian mathematician George Polya in his classic book How to Solve It [3].

Discrete Mathematics is a field that covers a number of areas, but especially in counting problems (from combinatorics) and graph theory there are a number of results that are not hard to grasp but lead to beautiful solutions for real problems. There are many powerful theorems and principles in discrete math that will unlock seemingly impossible problems. Unfortunately, the books for this subject are mostly written for math majors that are interested not only the application of these results but how to prove them and consequently the books may not appeal to everyone. Perhaps something like Schaum's Outline of Discrete Mathematics would suitably cover the way to apply some of the important theorems without bogging down in the proofs.

Finally, I think there is value in doing small interesting programming projects. Two older books, available used, with interesting projects are Etudes for Programmers by Wetherell [5] and Software Tools in Pascal by Kernighan and Plauger [6]. Etudes has my favorite exercise for trying out new programming languages, building an interpreter for the simple TRAC programming language; Software Tools has a number of programs in Pascal that do interesting things, try implementing the programs in your programming language--they cover a range of difficulties and the book has a nice discussion for each that explains why the programs are structured the way they are.

A more advanced book, Structure and Interpretation of Programming Languages, available as a downloadable pdf [7] is a classic book for those wanting to become better programmers.






by pmoriarty   2017-08-19
There's certainly some truth to this, especially in certain anti-intellectual societies and time periods. But many highly intelligent people have been truly gifted not only with intelligence but with the respect and even adoration of others.

You should read the biography of John von Neumann.[1] He's deserved the term "genius" if anyone ever did. George Polya[2], famous author of the math classic "How to Solve It"[3] wrote of him "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."

These were unsolved math problems -- unsolved to the entire field of mathematics that he was able to solve right after hearing them for the first time in class. The ability to do that is simply staggering.

Von Neumann went on to make so many contributions to so many fields that this would turn in to a huge post if I was to try to briefly mention them all. Some of the most notable was coming up with the von Neumann architecture on which virtually all modern computers are based, the central role he played in the development of the atomic bomb and the development and use of computers, the invention of cellular automata, and many, many others.

He was extremely highly regarded during his life for his intellect, and was enormously influential.

That's just one really obvious example, but you'll find many, many others. Einstein springs to mind as the quintessential intellectual superstar, as do Richard Feynman and Stephen Hawking. Socrates and Plato had a gigantic influence on virtually all of Western philosophy and through that on much of the modern and ancient world. Aristotle, a student of Plato, had an incredible influence on more fields of study than can easily be counted, and could arguably be one of the most influential people in history. He also tutored Alexander the Great, one of the greatest of all military conquerors. Diogenes got away with telling Alexander to get out of his light.

Many many people "3 standard deviations above the mean" (or more) have been eagerly sought out and highly rewarded. Michelangelo got to paint the Sistine Fucking Chapel. Newton and Leibniz created fucking calculus, and were both highly regarded and influential in their time and after. Voltaire influenced all of France and was hugely popular even in his life, as was Benjamin Franklin.

It's actually getting to be a little exhausting to do an adequate summary of the hugely influential brilliant people throughout history, and I think this post could go on for quite some time and not be nearly complete.

Yes, plenty of "geniuses" do get overlooked during their lifetimes, and many more will probably never be "discovered" or acknowledged even after they are dead. Van Gogh only sold one painting in his life, and that was to his brother. Many anti-intellectual regimes have deliberately committed mass murder of their intellectual classes, staged mass book burnings, etc. Many intelligent people are bullied as children, and as adults are persecuted for being too far ahead of their time, as Gallileo was. But many others are recognized and rewarded -- much more so than most "average" people will ever be.

As for the "Curse of Smart People", I'd rather live with my eyes open, as painful as that might be, than be lulled in pleasant slumber.

[1] -

by allemagne   2017-08-19
This isn't directly what you're asking for, but the book "How To Solve It" by George Polya is a great tour of how to be a teacher of mathematical concepts and how to "simulate" that teacher in your mind if you're encountering a problem no one else has.