Writing down the equation

We notice how mathematical notation affects scientific progress.

We have observed things about progress in science before, particularly in the replacement of one theory by another.  A better theory (we talked about Newtonian mechanics replacing the geocentric astronomy of Ptolemy) could actually be harder to calculate than an earlier one.  And it’s not obvious that Newton is actually simpler: to start the calculation for each planet, one needs six numbers (its position and velocity in three dimensions, or the equivalent), hardly an improvement on the Ptolemaic tables.  But scientists talk a lot about how an improved theory is a simplification of previous ideas.  What do they mean?

One part is the replacement of several different ideas by a unification.  The motions of each planet, separate problems for Ptolemy, were shown to be differing results of the same law of gravity.  In place of numerous geometric diagrams and tables, there was one equation.  Granted, actually applying the equation to calculate motions was more difficult, but it was one equation.

However, we can write a single simple equation now because of the vector notation developed only in the nineteenth century.  Previously there were more words and terms to collect and organize; the advance in simplicity became apparent mostly through an advance in mathematical notation.

Let’s take another example.  In the 1860s James Clerk Maxwell managed to put together all that was known about electricity and magnetism, and connect them through four equations (known as Maxwell’s equations, in a rare instance of appropriate nomenclature).  They’re short; you can put them on a t-shirt.  But they’re only short because of an advance in notation, one that did away with a plethora of partial derivatives and the necessity to choose a coordinate system in advance.  In their original form any simplicity is much more difficult to see.

Finally, consider General Relativity.  You can write down two equations that look very simple, and mean (in the words of a popular textbook) “matter tells space how to curve; space tells matter how to move.”  But you can only do it by interpreting one symbol as representing a four-by-four arrangement of quantities, details of which depend on what coordinate system you’re using: a tensor; and similarly for the rest.  If you had to write out the ten different equations that this notation represents (and our astronomer has had to do it), you could quickly become lost or confused.

Actual calculations in a more advanced theory are often more difficult than in the one that was replaced; the importance is that they’re more accurate.  The gain in simplicity, it appears, comes from a better mathematical notation.  Maybe the next step in science will come from a different way to write down a certain equation.