As a kid I was always confused by the requirement to “show your work” on math tests, which is to say, demonstrate how you derived your answer. Why did it matter how I got the answer? What mattered was whether the answer was right or wrong, wasn’t it?
This comes out of the street-fighting approach to math commonly taught in elementary schools, in which math is treated as a skill used to come up with answers to potentially difficult puzzles --- or, worse, merely a means to pass math tests. (And surely one motivation for demanding the “showing of work” is to reduce cheating on tests.) One alternative approach is to see math as a medium of creative expression, as explained in Lockhart’s Lament, in which the tools and materials are abstract ideas rather than clay or paintbrushes. But another alternative is to see math as a form of argument, whose quality is to be judged by its convincingness and fallibility.
That is, although I could tell you that 48303 / 27 = 1789, even if you trust me, it may not be immediately obvious to you whether I am mistaken or not. If you are going to rely on this fact for some purpose, such that you will put yourself in a position to be harmed if it turns out to be false, you might want some sort of stronger assurance than merely my fallible assertion. And this is the objective of “showing your work” if you write down the partial sums:
1789
× 27
-----
12523
+ 3578
-------
48303
This is an abbreviated notation for a syllogistic argument for the truth of my original assertion, which we could partly unpack as follows: 1789 × 7 = 12523; 1789 × 20 = 35780; 12523 + 35780 = 48303; therefore 48303 / 27 = 1789. (There are several other implicit premises as well, such as the distributive law of multiplication.)
Although it happens to be correct, this is not a very good argument, because each of the three premises I stated explicitly above is less than obvious. If I had written 1789 × 20 = 35870, for example, it might take you a while to spot the error. I claim that a principal objective of math is to state arguments in such a way as to make any errors obvious. Such an argument can be far more convincing: if it contains no obvious errors, then it contains no errors at all. Then, if its premises are correct, so is its conclusion, even if its author is untrustworthy.
I think this is a better argument for the same proposition: 1789 + 1789 = 3578; 3578 + 1789 = 5367; 53670 - 5367 = 48303; therefore 48303 / 27 = 1789. These calculations are simpler and so if there is an error in them it should be easier to spot, although perhaps the reasoning requires a little more explaining (30 - 3 = 27, so 30 × 1789 - 3 × 1789 = 1789 × 27).
In practice, though, I checked these calculations mostly by doing them with computer programs that I believe are unlikely to produce wrong answers, and it’s common nowadays for people to use spreadsheets, cash registers, or pocket calculators for this purpose. Arguably, repeating a calculation a few times with different calculators is more trustworthy than mental checking. But there’s still a great deal of potential for error in the process of invoking the calculator, as well as from hardware and software bugs.
This mathematical form of argument is the central ratchet that has allowed human knowledge to advance rather than falling backward over the last few centuries, because, just as money permits us to gain safety and sustenance by the efforts of not only honest hardworking folk but even treasonous cutpurses and greedy misers, math allows us to gain true and trustworthy knowledge of the universe from the reasoning of half-mad alchemists and deluded fools, because we can sift the occasional flake of gold from the mountains of superstitious dross they produced; by mathematical argument we can recognize a truth even when beset on all sides by nonsense, and often we can perfect a near-truth into a truth, and just as easily we can spot a flaw even in the sweetest honey of theory, dripping from the mouth of the finest of philosophers.
Unfortunately, at present we cannot do the analogous operation for results produced from a computer program rather than a mathematical formula. Often not enough information is published to even allow us to reproduce the published results by re-executing the program used by the original researcher; when such reproduction is possible, often the results diverge, and the error is quite frequently a different environment on the computer of the researcher attempting the reproduction, a situation more closely resembling chemistry than mathematics. Even if the results reproduce the original results correctly, they may well be due to a bug present in software installed on both computers.