Math Awareness Month
For Math Awareness Month we have a guest post from Noson Yanofsky, author of The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us.
April is Math Awareness Month. This is a special time when organizations make events that increase the public understanding and appreciation for mathematics. This year’s theme is “Mathematics, Magic, and Mystery.” We all know the amazing magic that mathematics does. It is simply thrilling when you understand the way a math trick works or when you comprehend how a short equation can summarize a whole phenomenon. We are all amazed at the many different areas that use mathematics and the myriad things one can do with math. Most science classes demand extensive knowledge of math. Many other areas also have mathematics playing a major role: whether it is the stock market, statistics in politics, or measuring how our favorite team plays, there is a lot of deep math all around.
But mathematics cannot answer all questions. There are many areas where the tools of mathematics do not give the answers. For these areas the results remain a mystery. Recently, MIT Press published a book that I wrote titled The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us. The book deals with unanswerable questions in many different areas of science as well as mathematics, computers, logic, language and philosophy.
In the book, I elaborate on different limitations of mathematics. There are equations that cannot be solved; math statements that are true but cannot be proved; shapes that cannot be constructed with a straightedge and a compass; simple problems in mathematics that can be solved by a computer but the computer would need trillions of centuries to actually come up with the solution; and there are easily stated math problems that no computer can ever solve. While most of physics is stated in the language of mathematics, there are simple physical phenomena that mathematics offers no help in describing or solving. These limitations are not failures of our present knowledge nor because we are not clever enough to solve these problems. Rather mathematicians have proven that no amount of cleverness can help. These problems are inherently unsolvable and their results must remain mysterious.
The book also deals with some deep philosophical issues at the heart of mathematics.
—Why does mathematics work so well?
—How come the truths of mathematics are so universal? When scientists attempt to send messages to space aliens they use mathematical ideas. After all, every alien must understand and know about pi and prime numbers. Why should this be?
—What is the nature of mathematical proof and its relationship with truth? Mathematicians have shown that it is impossible to prove that arithmetic is consistent. Should this bother us?
—Where does mathematics come from?
These and many other philosophical quandaries are discussed in the book.
While we are all aware of what mathematics can do in many areas, it is fun and instructive to think about those phenomena that are beyond the powerful tools of mathematics. In a sense, the phenomena that even mathematics cannot deal with are even more mysterious… and more interesting.