How do we get the system to do it?
First, let’s think about the whole notion of “puzzles” in an abstract theoretical sense. For instance, in the film Labyrinth, the “puzzle” of the book is “how do spiders climb on the ceiling?” The solution may not be obvious; after all no one has ever tried it. This is the most abstract possible way of doing mathematics; and that’s why it’s called “puzzles”; they have no practical meaning. But it does mean it is possible. There’s a world of mathematics that can be used to solve the “puzzle.”
The next step in the analysis is to see what different puzzle systems really do: “puzzles” are not so abstract after all; they are all systems with their own set of specific rules for how to do the same “workings”. The different systems we see in our everyday lives will not be so different from each other, once we understand the rules that make it work. If we were to go through every single puzzle in a modern language, we would quickly find out that the first-order rules are what distinguish different languages. If a different language had a First-order logic system, it would be different from another and be completely different. So the whole idea of “puzzles” is silly and useless. Instead, we should use different systems with a First-order set of rules, and try to find the solution that works for the problem we face: how to do math.
When we talk about “puzzles” in general, we are talking about the whole picture of what works and what doesn’t work. Some system does what we want it to, while others don’t. We must try to work out which way the systems are “worse” than any other type. But this will only work by using a language that has a very complex set of First-order rules.
Next, we have to think about what sort of problem solving is going on behind the scenes. The actual problem to solve isn’t really a problem of mathematics at all: sometimes we have to find out how to add fractions, or how to write a computer program in a language that can be understood by the person writing it; and sometimes we have to find out how a computer is supposed to behave in some special case of our problem in general. The “programming” and “reading” part of the metaphor is just so they aren’t considered “work” in
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