On Mathematics: Approximation and Exactitude

The first chapter of Here’s Looking at Euclid (which is numbered at Chapter Zero) deals with the cultural differences between dealing with numbers and instinctive human reactions to dealing with numbers, which leads to a lot of discussion of approximating numbers versus exact counting.

The author ends the chapter with the conclusion that numbers are a human construct imposed on the outside world as a way to try to make sense of it. I find this conclusion baffling as it follows on the heels of this question, “If our brains can represent numbers only approximately, then how were we able to ‘invent’ numbers in the first place?” Perhaps the whole thing would make more sense if one assumed that God created numbers and mathematics, and that even our attempts at exactness are derivative from His truly exact calculations. (Pi, anyone?)

To me, the most interesting part of this exploration of the human brain and numbers was the idea that we innately tend to think logarithmically rather than linearly. That is, we tend to think in terms of comparisons and ratios rather than exact numbers as laid out on a number line. Don’t believe me? Which sounds more drastic, the difference between one and a million or the difference between one million and two million?

See what I mean?

Even those of us who lean toward logic and precision of calculation still have a human inclination to view numbers in an approximate and comparative way. (Possibly because this is more useful in everyday life, as people who tend to get caught up on precise calculations are often reminded. Counting how many items are in the carts of each person in each line of the grocery store isn’t going to save you any time, even if you do manage to calculate which line is mathematically shortest, but a quick estimate and comparison of heaping full carts vs one nearly empty cart might save you quite a bit of time.)

Now, here’s one of the interesting bits: Teenagers who were tested on their ability to rapidly compare groups of dots and accurately estimate the differences in sizes of the groups varied greatly in their ability to make these estimates. The ones who scored highest on these tests correlated to those who tended to score highly  on their school test in the precise calculations of formal mathematics. In other words, the better you are at estimating and comparing, the better you likely are at precise calculations.

This brings to mind teaching approaches that focus on the natural developmental stages of children. Perhaps rushing children past the early, colorful, comparative stages of learning math into ‘proper’ academics actually slows down their progress in the long run. I have no idea off the top of my head what that means about teaching math as specific ages, but it does seem to lend general support in the direction of allowing younger children time to focus on creative play instead of formal academics.

This chapter of the book sparked one last ponderable thought for me: If most of our formal mathematics are based on a logical, linear scale, are there similar levels of advanced mathematics yet to be discovered along the path of more intuitive, logarithmic scale?

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