# Parable of the Adding Machine

A reader comments, in regard to an ongoing conversation: “We can create physical systems that when operating normally behave exactly like the purely mathematical systems we’re trying to capture.”

Aha! Now this is a good point, but it cuts both ways.

Suppose I had a clockwork, like an adding machine, with three gears of ten teeth each, and had the top teeth visible to a clerk through 3 small windows in the machine.

The gears are cleverly connected one to the next so that when the clerk pressed a lever ten clicks to a small gear, a larger gear advances by one click.

An elf takes chalk and writes Arabic numerals zero through nine on the teeth.

The clerk, when he presses a lever, finds that the machine will now, when nine is showing in the righthand window, upon clicking the lever once, advance the gears so that a one appears in the middle window, and a zero in the right.

Experimenting, he depresses the lever ninety or so more times, and discovers, to his delight, that when nine and nine are shown in the middle and the right window, clicking the lever once more changes the displayed digits, so that a one appears in the left window, zero in the middle, and zero in the right.

The clerk thinks, “Why, if I multiply the value seen in the rightmost window by one hundred, and the value seen in the middle window by ten, the number read in the three windows is the same as the sum of the times I have pressed the lever, expressed in base ten notation! Eurika! THE MACHINE KNOWS HOW TO ADD! — or, to be precise, the engineer has created a physical system that when operating normally behaves exactly like the purely mathematical systems of adding numbers one by one!”

An elf then takes chalk and writes the Roman numerals for numbers one through nine on the teeth of the gears, and puts an omicron on the remaining gear. He does this for all three gears in the machine.

Now, when the clerk goes to back to the machine, and he sees a IX in the rightmost window, and presses the lever once, he does NOT see the next number, X. Instead he sees an I in the middle window and an omicron in the left. This is not the number X. It is not a number at all.

Frantic, he clicks the lever one more time. He expects to see the number XI appear.

Instead, the number I appears in the middle window and I appears in the right.

One more click. Instead of XII, he see I and II. This is three rather than thirteen.

The clerk then thinks, “Why, if I multiply the value seen in the rightmost window by one hundred, and the value seen in the middle window by ten, the number read in the three windows is the same as the sum of the times I have pressed the lever, expressed in base ten notation! But what the heck do I do with the omnicron?”

The next day, the elf erases the chalk and replaces the digits with the Japanese signs for one through ten, or perhaps the Greek or Hebrew letters representing these numbers.

The clerk cannot read them, and so he concludes that the machine is broken.

Is the machine broken? What is wrong with the clerk’s logic?

Hint: the clerk is attributing to the machine something he himself is doing in his head.

Second hint: the elf is not a physical being, and not a part of the machine at all.

Your answers, gentlemen? Ladies?