Question was definitely too general – my apologies! I am definitely in the situation you described – the many ‘paths’ of math are quite confusing from the basic center. I’ve found some openlib Linear Algebra courses, and so I’ll tackle those over the next couple months, spending extra times in gaps. Then perhaps some combinatorics. Thanks so much!

]]>I have no idea if that’s true, but maybe they should!

]]>This might be obvious, but mistakes, serendipitous discoveries and the winding road of scientific progress are key ingredients in “The Structure of Scientific Revolutions” by Kuhn. A worthwhile well written book. Also “Against method” by Feyerabend, but here the whole process is hailed as even more chaotic. That one left me a bit confused.

Any thoughts on Lakatos’ more science-general “The Methodology of Scientific Research Programmes”? It’s on my reading list.

]]>Question is too general, I think you need to be more specific about the topics, but you may be in a situation where you don’t know what the topics are. in which case, I think a really useful thing to learn is Linear Algebra, both from an abstract and from a very concrete matrix basis. Can’t help you with suggestions for material offhand though.

Another thing to consider is perhaps combinatorics, which is very useful for understanding the underlying ideas behind probability (in a discrete setting, but it’s a good intuition pump), as well as ideas surrounding graphs, and decision trees, and soforth.

]]>Very interesting.

]]>If I want to read an interesting reflection on proofs and progress in mathematics, I read Thurston or Gromov or Langlands, or someone similar.

]]>For several years, I taught a course on Problem Solving for prospective secondary math teachers. I drew heavily on ideas from Polya’s How to Solve it, and also from the book Mathematical Problem Solving, by Alan Schoenfeld. But I did a lot of tailoring for the intended audience, many of whom had experience only with “routine” (i.e., learn the algorithm and apply it) problems. For example, one of Polya’s maxims is “draw a picture”. For this audience, I gave some examples, but also some “exercises” that stated a problem and asked the students to draw a suitable picture. (For example, one such problem that I recall is on figuring out how to design a window overhang that will shade from the sun certain times and let sun in at other times.)Then students presented their picture in class, and the class discussed them. This helped them understand how a suitable picture needed to be tailored to the problem being asked.

I’ve got some of the materials posted at http://www.ma.utexas.edu/users/mks/360M05/360M05home.html.

]]>Is it everyone now? I thought it was people using browsers other than the latest google chrome (always released last month) or something.

]]>I’m sure you’re right, but I suffer from Polya-phobia because way back when I did math team, the coaches were always pushing Polya and How to Solve It, and I associated this with other irritating attitudes they had, such as not wanting to use calculus or analytic geometry, or being opposed to applications.

]]>Hate to keep writing the same reply, but Lakatos was explicitly building on the work of a very well-known mathematician.

]]>As mentioned below, HTSI and the other plausible reasoning books were a primary and acknowledged source for P and R.

]]>As mentioned below, check out Mathematics and Plausible Reasoning

]]>Proofs and Refutations hews especially close to the principles of How To Solve It. Not coincidentally, Lakatos did the Hungarian translation of HTSI.

]]>Andrew,

Congratulations on your NYTimes piece

]]>Are there books on more empirical branches of science that do something similar by focusing on the productivity of error if it’s incorporated in a larger process of reflection?

]]>1. I can get to new articles from the blog list of (for example)

http://observationalepidemiology.blogspot.com/ long before they appear on your home page

2. Comments in the Recent Comments list cannot be accessed for posts that do appear for a day or so. ]]>

Victor Miller

]]>Thanatos:

Yes, this is annoying me too. I’ll talk with the sysadmin on Monday.

]]>Yes. Automatic theorem proving is not yet at a point where it has any practical effect on the work of the large majority of mathematicians.

]]>Roger:

I disagree. I don’t think the book is about “sloppy definitions” at all. Mathematical definitions are, by their nature, as precise as they need to be. As the theory is expanded, the definitions become more specific. And I think the general point of the interplay between proofs and refutations is very relevant to modern mathematics.

]]>By the time I picked it up, I’d already done abstract algebra and several graduate level math classes in logic and set theory. So maybe I was already too familiar with what it meant to present a proof. Or maybe I’d have enough dialectic after reading *Gödel, Escher, Bach* in high school (as much as that book influenced me to specialize in logic and computation, I wouldn’t recommend rereading favorite childhood pop books after obtaining a graduate-level understanding of the material).

@Joshua Pritikin: Computers are helpful for some proofs (finite simple groups, four color problem, etc.), but they’ve hardly replaced mathematicians.

]]>