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Saturday, September 24, 2005

The unreasonable effectiveness of mathematics

If you look at almost any set of modern standards for mathematics teaching — such as the NCTM’s or the Massachusetts curriculum frameworks or Weston’s own standards — you will see a prominent role for applications of mathematics. This is a welcome change from the ’60s, when I attended high school (together with George W. Bush, but that’s another story and will wait for another day).

But it’s important for us to us to understand why it’s a welcome change. The reason is almost entirely motivational. Some students love math for its own sake and never need to worry about applications. At the other extreme, some students hate math and ask “When will I ever need to use this?” without ever waiting to listen to an answer, because the question is really a complaint rather than a request for information. But the vast majority of students, who are in between these two groups, are more motivated to learn math when they can see that it has authentic applications than when it’s totally abstract (though even they can still be hooked by intriguing puzzles).

The trouble comes when students reverse cause and effect in trying to understand mathematicians’ motivations. Mathematicians practically never start with an application and then develop the theory to support it. Instead, they develop the theory and then eventually it finds an application. This has even happened to prime numbers, long considered one of the most abstract and least applicable ideas in math, being part of number theory, which for millennia was safe from applications. Now number theory in general and prime numbers in particular have become the cornerstone of encrypting messages over the Internet, a practical application if there ever was one.

Eugene Wigner’s famous 1960 article, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” discusses fundamental philosophical justifications for the surprising applicability of math, at least in the area of physics. If you haven’t read the article, do so.

One side-comment on this idea: although science teachers and math teachers have similar sensibilities and overlapping interests, a source of continual dispute between the two groups is the use of units. Science teachers and science textbooks rightly insist on the careful use of units, as their principles and their answers can’t make sense without the correct units. But math teachers and math textbooks usually refer to things like “3-4-5” triangles, with no units attached. Kids are often confused by the mixed messages that this conflict sends, especially since math teachers sometimes do insist on units. Students want to know who’s right, the math teacher or the science teacher. After I say that of course the math teacher is right (at least if the student has a sense of humor), I point out that both of the teachers are right, becasuse they’re engaged in different endeavors. Science almost always requires correct units in order for statements to make sense, but math is more abstract, and part of its unreasonable effectiveness is that mathematical theorems are general and do not depend on particular units. When we’re doing an applied problem, we do use units and make sure that they are correct.

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