Expectations

Hard thinkin' being done today.

It is useful (for me!) to think about the importance of math as teaching us how to think about problems, rather than providing us with useful factoids (I’m looking at you, history class).  There are a lot of problems/puzzles/patterns in the world, and the chance of seeing the same problem twice is very low (and really, I’ve never seen Batman use the Pythagorean theorem even once, so what’s the point?), so we focus on solving problems in as broad of a context as possible.  In this way, I’d argue, mathematicians become very good problem solvers (“toot! toot!” <– my own horn)

One method of problem solving I would like to focus on today is to name and describe your answer before you have found it.  As a simple example, in order to answer the question “what number squared is equal to itself?”, we would:
1. Name the answer: Suppose x squared is equal to x.
2. Describe the answer: This is where the explicitly developed machinery comes in: We know that x^2 = x, so we deduct that x also has the property x(x-1) = 0, and conclude that either x = 0 or x = 1.

A geometric way of looking at the word problem. NOT TO SCALE.

As a second example, much of linear algebra is naming objects, describing them, and then realizing you accidentally completely described them.  For example, suppose we wanted to identify every matrix with a number, and make sure that every singular matrix has determinant 0:
1. Name the answer: Let’s call the answer the determinant, or det() for short.
2. Describe the answer: det() should be a function from matrices to numbers, and at least satisfy the following properties: (i) det(I) = 1, so that the identity matrix is associated with the number 1 (so at least some nonsingular matrices will not have determinant zero), (ii) if the matrix A has a row of zeros, then det(A) = 0 (so that at least some singular matrices will have determinant zero, and (iii) the determinant is multilinear, which takes some motivation, will definitely respect identifying singular matrices.

Well, it turns out that these three properties have already completely determined the object we are looking for!  If I had been greedy and asked (iv) each nonsingular matrix is associated with a unique number, then I would have deduced that no such map exists.  If I had not included property (iii), then I would have found there are many such maps.  It is a fairly enjoyable exercise to deduce the other properties of determinants starting from just these three rules.

More filler photos! This is from Cinqueterre in Italy, between some two of the towns.

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