Duck Theorems

Another difference between Math and the rest of STEM: the strong use of if and only if relations which lets us determine something exclusively by its properties.

A follow-up to Does Math Really Fit in STEM?

An example: Suppose you wanted to show that for an arbitrary value a, a\cdot 0 = 0, assuming you have a proper context for all the other symbols, etc. The first problem is that 0 is only really known as an (or the) additive identity, i.e. b + 0 = b. And we don’t know anything about how it works with multiplication. But there’s a way:

a + a \cdot 0 = a \cdot 1 + a \cdot 0 = a \cdot (1 + 0) = a \cdot 1 = a,

using the rest of the field axioms.  But why is this good enough? We rely on what I call a/the Duck Theorem, which in this case is the fact that a + x = a has a unique solution (0), and since a \cdot 0 also satisfies this equation, we must have a \cdot 0 = 0.

Aside: Why do I call it a Duck Theorem?  From the old saying: If it looks like a duck, walks like a duck, and sounds like a duck, it must be a duck. In this case a \cdot 0 acts like 0, and since 0 is unique, a \cdot 0 = 0.

There are many things in the background that make this work, all of which rely upon the exceptionality gained from using if and only if and the corresponding deductive arguments. The definitions involved exclude any other options. The Duck Theorem provides a distinctive uniqueness (which could be rewritten, in our case, as x + a = a if and only if x = 0.) I’m not an expert on the other STEM fields, but I’m not aware of many (or any) cases where some physical result can be used to 100% identify some particular object or property. Most definitions are descriptive and, unless they’ve been abstracted into mathematics, are not given in an if an only if form. (A mammal is warm-blooded (as well as other things), but being warm-blooded doesn’t make you a mammal). Of course I’m only talking about how math relates to math, this doesn’t always extend to the results when math is applied (e.g. statistics is as precise within its own context, but it doesn’t extend the if and only if nature to the results of a statistical test as understood in the application area, i.e. statistical results are not if and only if/causal).

This all is not to say math is better (I’d argue that this feature makes math worthy of the many jokes made about math); it is just different, and maybe different enough that method proven to be good for teaching physics, biology, engineering, programming, etc., might not translate over to being good methods for teaching math.

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