This is an experiment in trying to make sense of my thoughts on mathematics. The first problem is to answer the question "What is Mathematics?". I think that the way to go is to focus on mathematical language, behavior and its tangible results. That is, look at what people are saying and doing when they "do mathematics", look at the resulting activity and its tangible resulting output. An important activity is asking questions and answering them while trying to make sense of both. This is not what is normally thought of as mathematics in the classical sense. In the Western world Since the time of Plato, mathematics has been thought of as existing in some ethereal world of "forms". This is referred to in the title of a 1993 book "Pi in the sky" by John D. Barrow. I wrote a brief essay"What is a Mathematician?".
In a 1999 book What is Mathematics Really?, Reuben Hersh refutes the Platonic view. Hersh talks about his ideas in What Kind Of Thing is A Number? at Edge. In Notices of the AMS is a 1997 review by Kenneth C. Millett of The Mathematical Experience, Study Edition co-authored by Hersh. At Cut The Knot is an article by Hersh titled Math Lingo vs. Plain English originally published in the American Mathematical Monthly. I like a published in 2003 book A Handbook of Mathematical Discourse by Charles Wells. He details the language used to talk "in the mathematical register". A quick overview of the problem of math lingo is in Spud's Dog House which is worth reading.
In the area of experimental mathematics
and visualization the Centre for Experimental and Constructive
Mathematics at Simon Fraser University is doing great work. A brief
quote is "
...issues of human perception and thought, intuition,
preception and the nature of knowledge. These issues are embodied in
the problem of scientific visualization...".
The articles Mathematics and Language by Tony Brown, and An Exploration into the Teaching and Learning of Geometry and the Possible Effects of Dynamic Geometry Software by Dave Wilson echo similar thoughts.
My current thinking focuses on the great importance of graphics, in the most general sense, for the understanding and applications of mathematics and for making connections with people. The visual point of view is a very ancient one and has a direct connection with the physical universe. Just how direct is a matter of debate. Sara Smollett has some ideas about mathematics and writing. Allen Klinger has a list of math books that are good reading. To have a real understanding of mathematics you have to know its history and MacTutor History of Mathematics is a good place to find it.
Meanwhile, many places on the web have mathematical content. To find them you can begin at Yahoo or Penn State. Another list is EDU2. Another list is by Phil Hughes. A particularly good place to find mathematical reference information is the MathWorld by Eric Weisstein. For sequences of integers, the place to go for information is the On-Line Encyclopedia of Integer Sequences (OEIS) originated by Neil Sloane where I have authored over a thousand sequences.
I am interested in some particular topics in mathematics. For my own use I have built a few pages with links to helpful information about these topics of possible general interest. The list so far is:
I am interested in algebraic identities and for my own use I have collected over 500 of them which have a simple, special form. They are used in proving theorems and importantly some have associated funcional equations which characterise elliptic functions. My list of Special Algebraic Identities (229kB) are here embedded in PARI/GP code.
I am interested in graph theory and for my own use I transcribed an almost exact copy of a short 1881 article by Arthur Cayley titled "On the Analytical Forms Called Trees". I have links to both PDF form and LaTeX form.
A small part of my work resulted in an essay "A Remarkable eta-product Identity" (23Mar2010) (PDF,DVI,text,LaTeX) on one of thousands of eta-product identities collected at my Dedekind eta function product identities website. Another small part of my work is an essay "Rational Function Multiplicative Coefficients" (21Feb2011) (PDF,DVI,text,LaTeX) on two related conjectures on rational functions with multiplicative coefficients.
Circa 2000 Jim Propp looked at closely related topics. He was teaching at Harvard and his home page there links to Math 192r video lectures and some of his work. I have translated his list of "A Dozen Laurent Recurrences" into my own version in PARI-GP code.
Oh, and by the way, there is now a combinatorial interpretation of Somos-4 and Somos-5 sequences via perfect matchings of planar graphs. There is even a T shirt which depicts this matching. For more details goto this page at Harvard. This is closely related to the work by Eric Kuo on Graphical Condensation which is an example of bilinear mathematics.Back to my home page