Two kinds of mathematics

Mathematics, at almost any level, can be either practical or ideal.  Problems arise when the two kinds are confused.

Our tutoring consultant, because he teaches mathematics, confronts the ubiquitous electronic calculator.  He has developed techniques and advice about the machine; we’ll go into detail about that another time.  In doing so he realized that there were two kinds of mathematics that the students were trying to learn, and sometimes their troubles lay in confusing the two.

To explain: the civilizations of ancient Mesopotamia and Egypt developed a very sophisticated mathematics.  The aim was to solve practical problems: surveying the land, building pyramids and temples, predicting (to a certain accuracy) where the planets were going to be.  There were rules-of-thumb and procedures that could come up with answers to difficult and complicated situations, things we’d turn to our calculators for now, or even more powerful computers.  Call this kind of work (somewhat inaccurately) Babylonian mathematics.

Then came the classical age of Greece.  The Greeks learned from the older civilizations, but also came up with new ideas, like the rigorous proof and the idea of an exact answer.  It led them to such things as irrational numbers: quantities that could not be expressed as a ratio of integers, however large.  They could come up with approximations of these, like the Babylonians, but were acutely aware that they were approximations and not the real answer.

Most mathematics schooling in the past couple of centuries has been Greek, with an emphasis on proofs and manipulating symbols.  An answer might be the square root of 3 divided by two, left in symbols most of the time.  There was a section devoted to actual calculation, extracting roots by hand or looking things up in a table of sines, but it was pretty clearly set off from the basic work that led up to the calculation.

Numerical electronic calculators are Babylonian.  [There exist symbol-manipulating calculators, which are a different kettle of fish.]  They use a finite number of digits to come up with an approximate answer.  The approximation is very good in most cases, far better than you’ll need in almost all practical work, but it’s still an approximation.  They cannot give you an exact answer, or a rigorous Greek-style proof.  The answer may look exact; one divided by two comes out to 0.5, to any number of decimals.  But the calculator itself can only say it’s 0.5 to perhaps ten decimal places, which is not the same thing.  You have to pull in other knowledge to say it’s exactly 0.5.

After their initial horror at the idea, mathematics teachers eventually took up the calculator as a teaching tool.  It’s especially useful for graphing functions, quickly showing what they look like and suggesting some of their important characteristics.  They’re certainly much quicker and more accurate than the old method of plotting points by hand.  But they work by choosing a finite number of points and connecting the dots, or better, by allowing the human eye to connect the dots into an illusion of a smooth curve.  If you don’t have some idea of where and how close to look, you can miss important things.

Digital Babylonian mathematics assumes that the universe between its plotted points (which is most of the universe) is well-behaved and holds no surprises.  Sometimes it does.  You have to turn Greek to find out.