Q1 Writing Assignment

Chapter 8 of Letters to a Young Mathematician explains the fundamental properties of a proof and explains why proofs matter in mathematics. Ian Stewart says that proofs are essential to mathematics because they make the study “honest,” meaning that they give authority to the study, since it proves that a mathematical contention cannot be questioned. He gives an example of a very simple SHIP-DOCK proof in a verbal context, without mathematical symbols, to illustrate the fundamental qualities of proofs— intuition is not evidence, simplification is useful and sometimes necessary, and proofs cannot allow any exceptions. He also explains that proofs are not always the most ideal way to assert the authority of theory, citing experimentation as another means of proving, especially in the sciences. The SHIP-DOCK proof is a perfect example of how to apply mathematics to a situation where it appears impossible to do so. In order to prove the theory that in order to get the word DOCK from the word SHIP by changing only one letter each time to form a new word, and in that sequence of words there is a word with exactly two vowels, you have to analyze the pattern of the solutions. Ian Stewart used given information— information given to us by the restraints of the puzzle—to make the key observation that, in order for the placement of the vowel to change, the word must eventually change to a word with two vowels, since every word requires at least one vowel, and you can’t simply switch the placement of the vowels in the word since that would mean altering more than one letter in the word in one turn. The proof is intuitively convincing, but for me it lacks a certain authority because it seems to lack a raw mathematical feel to it. That doesn’t mean that the proof CAN’T be represented in a more traditional way, I’m saying that Ian Stewart chose a non-traditional way of proving it simply because his focus was on explaining the purpose and contents of a good proof. I understand very well the textbook proof that A(x+y) = Ax + Ay, since the proof is extremely simple and algebraic in nature. The proof simply expands all of the components and columns out into a sort of series Σ(k = 1, n = n) (xkAk + ykAk) and then just paired each x value with a y value so that you could factor out the A matrix an have (x1 + y1)A1 + … + (xn + yn)An. Then you have to recognize that this is the long-hand version of A(x+y), and your proof is complete. The proof works logically too, since if you add the x and y components and then multiply them by a matrix, it is the same thing as carrying out the multiplication of the matrix and each vector separately and then summing the components. The order doesn’t matter. The proof that to me seems unconvincing is the invertibility theorem, specifically clause (a) The only solution to the homogeneous linear system Ax = 0 is the trivial solution x = 0.