I really love math. It can be an art or a science, depending on your motivations. I enjoy it as an artist; I appreciate it for its beauty. It is the single most independent form of truth. Its pulchritude is that of pure thought.

Product Integration
No, it's not a business buzzphrase. Rather, it's the continuous analog of multiplication (as the usual integral is to addition). Currently this is the focus of my PhD research. If your browser can handle MathML in an XHTML page (Mozilla can do it!), you should check out my extraordinarily brief introduction to product integration.

One fun thing about math that most people can relate to is a good brianteaser. Problems that seem simple, but either are deceptively tricky to solve, or require a cunningly clever insight to master. My advisor, Dennis Shasha, collects his own share of computationally interesting teasers. He's written a fun book about it, and ran a graduate course based on some of his puzzles. My friend Christof Konig also has his own page full of brainteasers. Finally, I also occasionally (meaning once so far) put a puzzle in my blog.

What is math?
This is sort of a tricky question. I feel that most of the traditional answers are overly simplistic. To paraphrase a certain online dictionary, math is the study of numbers and geometry along with the generalizations thereof. That's what most people would logically conclude having taken the typical requirements for any non-math major. But this really misses a big piece of the point. One of the main purposes of math is to prove things. Yes, there is still the question of what to prove, but in actuality the best theorems and their elegant proofs go hand-in-hand.

In other fields, you cannot prove things the way you can in math. Many people don't really comprehend this because they never appreciably studied math from the proof perspective. An ignorant lawyer may think that he or she proves things in a courtroom the same way a mathematician does in a paper. Closer to the real confusion, a physicist may wantonly use the word "proof" in reference to some defense of his or her theory. Yes, mathematical physicists can prove math theorems which relate to physics, but no law of the physical universe can ever be fundamentally "proven" the way a mathematical theorem can. (Although, I could imagine a math-minded physicist using certain laws of physics as axioms and using those to logically and rigorously demonstrate that some other law must follow, but this is part of what I meant by "mathematical physics." The physicist who does this has not actually proven a theory, but rather has shown that if certain assumptions about the universe are true, then this other fact must also be true. Mathematicians use a similar trick by assuming their axioms, but they are constantly aware of the dependence on these axioms -- they never prove Pythagoras's Theorem is always true, but rather they prove that if you have these geometric axioms, then Pythagoras is correct.) You can never prove beyond all doubt that gravity works a certain way -- not the way you can prove the Prime Number Theorem.

So let me offer my own definition: Math is the study of precise and logical reasoning. It is looking at how to solve problems in a manner that is at once exact and rigorous. In order to gain precision, we must use abstract notions. In order to be as reasonable and correct as possible, we use the strict rules of logic in the form of proofs.

The two primary branches within mathematics -- applied or pure, are really overlapping with a lot of grey area between. The pure end of the spectrum is occupied by the reasoning which is as nicely abstract and widely general as possible. The applied is inhabited by those issues which more directly affect our everyday lives.

Probability Theory by R. Durrett
Most math books which actually say something are pretty difficult to read, but worth the effort. Unfortunately, this trend can be very discouraging to students (such as myself). I think it is a good idea to give something like Cliff's® notes for traditional math textbooks. So as I was recently studying a particularly useful book - Probability: Theory and Examples, 2nd edition by Richard Durrett - I took careful notes for each part of the text. I've compiled these notes as an HTML commentary which can be used by anyone learning from the book. Note that there is also an official typ-o list available from the author's webpage.

www. Tyler Neylon .com