There is no doubt that the vast majority of people find mathematics dry, dull, boring, painfully difficult, and irrelevant. Unfortunately, there seem to be few people who are really interested in convincing them otherwise. Most mathematicians fail to relate to the difficulties that others have trying to learn the subject, and most school teachers know terribly little about the essence of mathematics to instill a passion for it in their students. I personally believe that mathematics is both aesthetically beautiful and tremendously powerful. It has the power to unravel the nature of the physical world and simultaneously fill you with the same sense of awe that comes from listening to great music or appreciating great art. My hope is that this small library helps some people see these aspects of math that the experts know well but often fail to convey to others.
My main goal is to teach the curious layperson about the kinds of things that mathematicians think about, in a way consistent with how mathematicians think about them. There are plenty of books, magazine articles, and web pages available that discuss small mathematical puzzles and curiosities. Although these provide valuable and tangible ways to learn mathematical ideas, they often fail to penetrate into the heart of mathematics by not emphasizing abstractness and rigor. Of course, in trying to reach people with little experience, one has to avoid excessive technical details and straightforward rigorous presentations. Many of the pages here are slightly whimsical and long-winded, and I hope that such a presentation keeps the subject matter interesting without sacrificing the certainty of the arguments.
One of the themes that I hope to emphasize in this library is that although mathematics thrives on abstract and intangible ideas, investigations along these lines can lead to a more solid understanding of basic mathematical objects (like the natural numbers 0,1,2,... and the real numbers) along with practical benefits to science and technology. Most people are probably quite familiar with the success of some branches of mathematics, such as calculus, in fields like physics. More surprisingly, the cryptographic protocols used to protect your credit card number when you transmit it over the Internet, the codes on your CDs and DVDs that correct errors on the disks without you realizing it, and the logic used inside your computer are all made possible only through the investigation of more abstract branches of mathematics like Number Theory, Linear Algebra, Abstract Algebra, and Mathematical Logic. There are many other examples throughout the following pages, and I hope they help to dispel the myth that mathematics is abstract nonsense reserved only for people that like to walk with their head in the clouds. However, I do not think that mathematics should be judged based only on its applications. The artistic nature of mathematics, the richness and beauty of the structures that it studies, provide enough reason to spend time working to unravel its mysteries. I hope that by spending time discussing applications, this library does not lose sight of the importance of studying mathematics simply for enjoyment and to satisfy the thirst for knowledge.
Unfortunately, mathematics is such a vast field that I am unable to treat all of its many branches fairly. Readers will find that I give little attention to both geometry and topology even though these subjects are of central importance in modern mathematics. By spending less time on this material, I do not mean to imply that it is less important, but only that I personally am not a suitable guide to the terrain. On the other hand, I treat mathematical logic quite thoroughly even though many mathematicians would consider it an isolated branch of mathematics with little to say about true mathematical practice. I spend more time on this subject since I disagree with this sentiment, I work in the field, and because many people seem to be fascinated by results like Godel's Incompleteness Theorem, while much of the information readily available to general public about these subjects is nonsense (at least from a mathematician's point of view).
I chose to call this place a "Mathematical Museum" because I hope that it has much in common with a typical museum. There are different sections to explore, and each section has a variety of exhibits that highlight the main points without spending too much time on technicalities. I try to present ideas in a playful manner without sacrificing their integrity. If browsing through these pages gives you both the feeling of exhilaration that comes from exploring an art museum, and the satisfaction of learning about the world that comes from a spending time in a natural history museum, then I will have achieved my goal.