Wednesday, September 4, 2013

Formal reasoning and the real world

This week we started on the course material proper. At the beginning of the book, the authors get a little silly with their "problems," for instance how a knight could cross a perfectly square 20ft moat onto a perfectly square landmass within, while for some reason only having 19ft 8in logs available.

Realistic? No. So what's the point? Maybe nothing. Could be the authors are weird or a little crazy. But then Ed Burger is a professor and consultant, and Mike Starbird is a distinguished professor at UT and both are professional mathematicians, so maybe they're ivory-tower types giving students flip little problems meant to "teach" something condescending. It could be that's too harsh; maybe the problems were just meant to be readable: math-y enough to communicate something, but easy enough that people don't get turned off to the book.

There is another purpose though. Just like skills in writing and communication, mathematical skills have a point, which they are trying to bring in. Yes the problems are silly, but they do resemble things in the real-world. Of course the real world does not come with its own interpretations or its own structures: it's there, doing its own thing, and we humans have to deal with it. Reality doesn't have problems, humans have problems, and most of us want to deal effectively with them, and of the many ways of dealing with what's out there, the process of formal reasoning, when used properly, is extremely effective. It works like this. Something in reality is encountered and a mental model is formed to reflect it, at least parts which can be understood as a formal structure. Formal or mathematical reasoning can be used to solve the problem in the mental model, and one attempts to translate this into a real-world solution.

The moat-crossing knight is an example. He finds a moat, which he perceives to be a problem (for whatever reason he wants to be on the moat's interior). He ascertains the situation, and creates a mental picture---in this case, a useful interpretation is as a geometry problem. The items around him are also given geometric interpretations: nearby logs can be seen as line segments for instance. His task is to use these line segments, and a few concrete rules, to draw a path across the geometrically-abstracted moat. If this strictly formal problem can be solved, he then attempts to translate the mental solution into the real world.

This is quite a philosophical approach to what is ultimately a collection of very simple story problems, and possibly my observations are pedestrian or ridiculously obvious. Maybe you see some value in this way of thinking. But I'm going through this because I want to encourage you to do likewise: don't just ask how to solve a particular problem from the book (yes, actual problem solving is important too), but why you are solving it. Don't be led by the nose, doing just what the book asks, trying to get through. Be active. Try to see why you're being asked to do it, and ask what of value is in it. You're not in this class to solve ridiculous little problems after all!

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