![]() The role that words play in mathematics was on the mind of the French mathematician Jacques Hadamard when, during the 1940s, he asked colleagues around the world how they thought about their subject. Words come to us automatically when we view images. Its examples are intended to leave the reader speechless.ĭespite the many proofs without words, mathematical thought ohne Worte might be impossible. Since 1975 the Mathematical Association of America has published a column devoted to them in its Mathematics Magazine. Just as the 19th-century musician Felix Mendelssohn enjoyed writing Lieder ohne Worte (Songs Without Words), so mathematicians like to craft Proofs Without Words. Similarly we can find the sum of the first N integers, for any N whatsoever. + 100 must be equal to 100 × 101 divided by 2, which is 5,050. Moreover, there is nothing special about 5 columns of dots. The novelty of the picture is that we can grasp the idea at a glance. In order to find our original sum, we need only divide by 2, thereby correcting for our double counting. By arranging all in rectangular fashion as we have, it is an easy matter to count the total, which is 5 × 6 = 30. ![]() ![]() The figure at right displays 1 + 2 + 3 + 4 + 5 dots twice, once in black and again in red. But what about adding the first 100 integers? Let’s see how that might be done.Ĭonsider the task of adding the first few integers, say 1, 2, 3, 4, 5. If mathematics is a language, then we should be able to understand its ideas without the use of words. The idea does not seem so far-fetched when we consider musical notation, which is readable by trained musicians everywhere. ![]() If mathematics is a language, then just as any ordinary language, such as French or Russian, does not rely on another one to be understood, so mathematics should be independent of ordinary languages. Gibbs, who is responsible for much of the vector calculus that we use, would have added a modern symbol or two of his own. For Galileo, the letters of mathematics were triangles, circles, and other geometric figures. But just as students at Plato’s Academy were reputedly greeted with the warning “Let no one ignorant of geometry enter here,” so Galileo’s readers were being cautioned that the book before them had some language prerequisites. ![]() Galileo wrote in Italian rather than scholarly Latin, hoping to reach readers who were literate but not necessarily scientific. In Il Saggiator ( The Assayer), published in Rome in 1623, the Italian astronomer wrote: “ cannot be read until we have learned the language and become familiar with the characters in which it is written. Galileo Galilei beat him to it by more than 200 years. Gibbs wasn’t the first notable scientist to call mathematics a language. ![]()
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