From: Talen Rabe
Da Vinci, Turing, and Wittgenstein are enjoying lunch together when Da Vinci brings up a recent invention.
Da Vinci: I’ve invented a combination lock which unlocks anytime the first two numbers sum to the third. The mechanism is neat on its own, but it’s really got me thinking about the nature of mathematics. If I have a machine that adds, an adder, it puzzles me to say which came first, the adder or addition itself.
Turing: That does sound neat. I’ve done much theorizing on similar computational machines myself and would propose to say that the adder comes before the addition. The nature of mathematics is manifest through language. In your case that language is in the positions of cogs and wheels. A more robust machine could, hypothetically, follow any mathematical process, and any two such machines would be equivalent to each other, even if their mechanisms were quite different. For example, the function of the DNA in our very bodies constitutes such a machine, as well as the process of neurons firing in our brains. It seems the world is filled with these computational machines just popping out of each other.
The very fact that our most fundamental laws of physics are mathematical at all suggests that the universe itself is operating as one of these complete computing machines, thereby implying that the adder, indeed, comes before the addition.
Wittgenstein: Not so fast, Turing, your argument brought me to just the opposite conclusion. Let’s say we have two very different machines, a mechanical machine like Da Vinci’s and a chemical machine like DNA, both of which are carrying out a process of addition. Here I’ll remind you; these machines have nothing to do with numbers. They are just cogs spinning about, or chemicals floating around. So I ask you, what do these machines have in common? They are equivalent in that you can map their functions onto one another, but what are you mapping? The fact that machines are adding is only apparent through the interpretation of the positions of their parts. The essence of addition must lie outside the machines and inside the interpretation. Therefore, addition comes before adder.
Da Vinci (twisting mustache): So we agree that language, whether a language of words, symbols, or mechanisms, is the essence of math and logic, and that a machine following instructions in a language can add two numbers.
Wittgenstein then claims that addition exists not in the machine, but in the language used to interpret the machine, thus addition comes first. While Turing claims that one such language must be fundamentally built into the universe, thus the adder comes first.
Just then the three notice a person nearby listening to their conversation intently.
Da Vinci (to listener): What do you think of all this?
Person: Addition, adder, that’s all fine and dandy, but I can’t help but wonder … Where do you stand on politics?