I saw a lady yesterday who was looking for a five dollar bill in her purse. She was certain that the five dollar bill was in there, but the issue caused me to think. I guess it was obvious to me that if the five dollar bill was in her purse, then either it is directly in her purse, or it is directly inside something else that is directly in her purse. Obviously, to prove that the five dollar bill was in her purse, she would have to inspect each item in her purse, and then recursively inspect the contents of each item.
But given what you’ve said above:
Now, citing axioms and theorems to justify a step in a proof is not a mere
social convention to make mathematicians happy. It is a useful constraint on
your cognition, allowing you to make only inferences that are actually valid.
But is the proposition I cited above really an axiom? Is it true? I’m sure it’s possible to generalize this spatial relation of containment to the universe as a whole, would the proposition be true then?
I think it’s interesting that you site a programming example. I’m not sure that you’ve left the laboratory. More importantly, I’m not so sure how useful it is to prove my proposition to determine if the five-dollar bill is in her purse.
But I’ve been thinking about heuristics lately. Surely you know that, at least outside the laboratory, that we all use heuristics for problem solving and making decisions. And often we engage in heuristics when it really isn’t practical to stop and write a formal proof of our answer.
This doesn’t mean, however, that we can’t analyze our heuristic when we get home. So I guess I’m imagining rationalists as people who regularly refine their heuristics, and has more heuristics, more rational tools, than other people have. You can prove the effectiveness of a heuristic using logic.
I’m not a programmer, but I think programmers probably have a leg up on this over other people. Heuristics for searching and sorting aren’t just algorithms for machines, but apply to real life. There are probably other possibilities as well. Just something I’ve been thinking about.
I saw a lady yesterday who was looking for a five dollar bill in her purse. She was certain that the five dollar bill was in there, but the issue caused me to think. I guess it was obvious to me that if the five dollar bill was in her purse, then either it is directly in her purse, or it is directly inside something else that is directly in her purse. Obviously, to prove that the five dollar bill was in her purse, she would have to inspect each item in her purse, and then recursively inspect the contents of each item.
But given what you’ve said above:
But is the proposition I cited above really an axiom? Is it true? I’m sure it’s possible to generalize this spatial relation of containment to the universe as a whole, would the proposition be true then?
I think it’s interesting that you site a programming example. I’m not sure that you’ve left the laboratory. More importantly, I’m not so sure how useful it is to prove my proposition to determine if the five-dollar bill is in her purse.
But I’ve been thinking about heuristics lately. Surely you know that, at least outside the laboratory, that we all use heuristics for problem solving and making decisions. And often we engage in heuristics when it really isn’t practical to stop and write a formal proof of our answer.
This doesn’t mean, however, that we can’t analyze our heuristic when we get home. So I guess I’m imagining rationalists as people who regularly refine their heuristics, and has more heuristics, more rational tools, than other people have. You can prove the effectiveness of a heuristic using logic.
I’m not a programmer, but I think programmers probably have a leg up on this over other people. Heuristics for searching and sorting aren’t just algorithms for machines, but apply to real life. There are probably other possibilities as well. Just something I’ve been thinking about.