The Gift of Clear Thinking

Celebrating the 12 days of Christmas

with short reflections on 12 gifts

Did you hate math class?  OK, you econometrics, engineering, IT, math and physics majors reading this (all five of you!) are excused from answering this one.  You arts, education and business types—how did you feel about math class?  Endless hours of tedious, incomprehensible and purposeless stuff, right?

What a shame.  Worse, what a failure of imagination—if not in individual math teachers then in curriculum designers.  Mathematical concepts can clarify our day-to-day thinking in areas as diverse as public policy discussions and proposal writing.  Join me in celebrating just two such…

The necessary but not sufficient’ construct, in which we distinguish between conditions that are surely required, yet not by themselves enough, to bring about a desired outcome.  How much argumentation about social problems and solutions could be avoided by adopting this simple phrase and, by extension, admitting that complex problems often do not have single solutions?

The disjoint set’, in which we carve up the universe of things into mutually exclusive categories that cover all instances.  Think of the ‘animal, vegetable, mineral’ of the Twenty Questions game we played as kids (And who knew those categories originated with Linnaeus?  Thanks again to Wikipedia!).  Simple categorizations help us cover all the bases, while keeping ‘like’ with ‘like’ and ‘different’ somewhere else!  To see the quintessential example of a non-disjoint set (albeit likely an artificial one), check out this supposed ancient Chinese taxonomy of animals, and contrast it with what Linnaeus did.

No matter where we find its tools—even in math class—may we treasure the gift of clear thinking.

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6 Responses to The Gift of Clear Thinking

  1. Dave

    So let’s see.. going back to the gift of motivation .. sufficient is like good enough and perfection is more than good enough.
    There are conditions which are not necessary, that is not required for sufficiency, but desirable none the less, therefore beyond sufficient which when included move the outcome closer to perfection.
    So let’s see we could have two disjoint sets of conditions : unacceptable and acceptable. A proper subset of the acceptable conditions would be a sufficient subset. Would the perfect set be equivalent to the acceptable set?
    What is perfect? We need a metric here. Or maybe we could define super sufficient to mean more than sufficient and in the limit super sufficient converges to perfect as we include more and more of the set of acceptable conditions. Thus some of the acceptable conditions could have infinitesimally small value. Then of course we might ask whether the presence or absence of any one acceptable condition enhances the value of any other. So how about a value function which is attached to each possible subset of acceptable conditions? Also a probability function for each of these outcomes?
    Oh woe is me I am losing it! No I have lost it. That’s why I am retired.

    • Isabel Gibson

      Warning: Readers who are not econometricians or statisticians many find some language in this comment to be disturbing.
      Dave – If only there were a metric for ‘good enough’!

  2. Jim taylor

    Okay, I’m an arts and education type, and I’m very good at math. I even did one of my two majors for my BA in math. What turns me off is not math, but the way it’s taught. “First we must assume a quantity ‘x’ which is greater than ‘y’ but less than ‘z’ for which we wish to perform the function……..” Argh! Now, if they had told me that integration added a dimension to, say, the area of a circle, so that you now had the volume of sphere with the same radius, I might have been interested. Or that set theory is a tool of logic. Or that the ancient Mayans used trigonometry instead of math to calculate star positions over thousand year intervals. But no, “Given an integer >1 such that….”


    • Isabel Gibson

      Jim – Yeah, for sure – What is this thing for? Always a good question to anticipate/answer. Working with tech experts in my adult day-job, I find that they often don’t explain the ‘why’ very well. I suspect it’s obvious to them…

  3. Dave

    So you folks enjoy reading insurance contracts? My blood starts to boil as soon as I read the first clause. And yes we speak the same language. Now communicate with individuals whose language is not English.
    There would be a lot less miscommunication if we could employ mathematical concepts. Of course people would have to be taught more and in a different way in school at an early age. Just think lawyers would be replaced by mathematicians. The documents would be much shorter.
    Oh well some people like brussels sprouts and some even like spinach!

    • Isabel Gibson

      Dave – You must love the Tax Act! Every few years, there is a splashy but short-lived and ineffectual movement to promote plain language in government. It never takes. First, plain language requires clear thinking. Second, plain language doesn’t provide any (often helpful) ambiguity about intentions. But I’m all for teaching folks how to communicate better…so is anyone who edits or teaches writing or any field that depends on writing to communicate results.