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That’s called Mathematics. For example, most NP-complete problems we know of were proven to be NP-complete by showing them to be no easier than some other probl

That’s called Mathematics. For example, most NP-complete problems we know of were proven to be NP-complete by showing them to be no easier than some other problems we know to be NP-complete. https://en.wikipedia.org/wiki/NP-completeness#NP-complete_problems https://en.wikipedia.org/wiki/NP-completeness#NP-complete_pr...: “The easiest way to prove that some new problem is NP-complete is first to prove that it is in NP, and then to reduce some known NP-complete problem to it” As a second example, in combinatorial game theory, the Sprague–Grundy theorem states “every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization” (https://en.wikipedia.org/wiki/Sprague–Grundy_theorem https://en.wikipedia.org/wiki/Sprague–Grundy_theorem) That means that, presented with an impartial game (both players can make the same moves, so chess is ruled out because white can’t move black pieces and vice versa) under the normal play convention (last player who can make a move wins), mathematicians look for a way to translate game positions to nimbers (https://en.wikipedia.org/wiki/Nimber https://en.wikipedia.org/wiki/Nimber) in order to learn how to play them. Doing such mappings, if not trivial, requires creativity, so I think research on the subject would be in the psychology department. Polya, in “How to Solve it” has some discussion on this (https://en.wikipedia.org/wiki/How_to_Solve_It#Heuristics https://en.wikipedia.org/wiki/How_to_Solve_It#Heuristics)

Msurrow 5y ago
Agree with this. And I guess a more “applied” branch of science that covers this would be Computer Science in the area of Computational Complexity (https://en.m.wikipedia.org/wiki/Computational_complexity_theory https://en.m.wikipedia.org/wiki/Computational_complexity_the...) See for example the section on Complexity classes wrt the P/NP/Pspace class problems mentioned above
Someone 5y ago
Reply to self: of course, the prime example is counting. If you have five breads, you can sell three, and have two breads left after that. Similarly, If you have five apples, you can sell three, and have two apples left after that. You didn’t have to learn separate addition tables for breads, apples, etc. Instead you learned that all such problems essentially are the same, and you learned how to manipulate the equivalence classes of those (which are called 5, 3, and 2)
- belter 5y ago
  I don't think your examples are examples of solving a different problem.
lee 5y ago
I love this answer, because it is such a "duh" answer but also so non-obvious (at least for me). We're taught so much about the mechanics and details about mathematics in school, that I've never taken a step back to understand it from a fundamental/philosophical level for what it is. Beautiful answer.
kevinwang 5y ago
Good answer. Seems like a relevant time to post an old math joke: There were two men trying to decide what to do for a living. They went to see a counselor, and he decided that they had good problem solving skills. He tried a test to narrow the area of specialty. He put each man in a room with a stove, a bucket of water, and an empty pot on the stove. He said, “Boil some water in the pot.” Both men filled the pot with water from the bucket and turned on the burner to boil the water. Next, he put them into a room with a stove, a bucket of water, and a pot full of water on the stove. Again, he said, “Boil the water in the pot.” The first man immediately turned on the burner. The counselor told him to be an Engineer. The second man emptied the pot and proudly said that now the problem is reduced to the previously solved problem. The counselor told him to be a mathematician. This version from: https://www.davidmarcus.com/Humor/MathJokes.htm https://www.davidmarcus.com/Humor/MathJokes.htm and other versions at https://www.math.utah.edu/~cherk/mathjokes.html https://www.math.utah.edu/~cherk/mathjokes.html
time_to_smile 5y ago
To add to this, another mathematical term to be familiar with is "isomorphism". It quite literally means showing that two structures are the same, and perserve the same binary relations (coming from modern Algebra). If you can show that domain A is isomorphic to domain B, then all of the things you know how to do for domain A can be trivially applied to domain B. Thinking about solving problems in terms of isomorphisms has been tremendously helpful for my day-to-day problem solving.

Msurrow 5y ago

Agree with this. And I guess a more “applied” branch of science that covers this would be Computer Science in the area of Computational Complexity (https://en.m.wikipedia.org/wiki/Computational_complexity_theory https://en.m.wikipedia.org/wiki/Computational_complexity_the...) See for example the section on Complexity classes wrt the P/NP/Pspace class problems mentioned above

Someone 5y ago

Reply to self: of course, the prime example is counting. If you have five breads, you can sell three, and have two breads left after that. Similarly, If you have five apples, you can sell three, and have two apples left after that. You didn’t have to learn separate addition tables for breads, apples, etc. Instead you learned that all such problems essentially are the same, and you learned how to manipulate the equivalence classes of those (which are called 5, 3, and 2)

belter 5y ago
I don't think your examples are examples of solving a different problem.

lee 5y ago

I love this answer, because it is such a "duh" answer but also so non-obvious (at least for me). We're taught so much about the mechanics and details about mathematics in school, that I've never taken a step back to understand it from a fundamental/philosophical level for what it is. Beautiful answer.

kevinwang 5y ago

Good answer. Seems like a relevant time to post an old math joke: There were two men trying to decide what to do for a living. They went to see a counselor, and he decided that they had good problem solving skills. He tried a test to narrow the area of specialty. He put each man in a room with a stove, a bucket of water, and an empty pot on the stove. He said, “Boil some water in the pot.” Both men filled the pot with water from the bucket and turned on the burner to boil the water. Next, he put them into a room with a stove, a bucket of water, and a pot full of water on the stove. Again, he said, “Boil the water in the pot.” The first man immediately turned on the burner. The counselor told him to be an Engineer. The second man emptied the pot and proudly said that now the problem is reduced to the previously solved problem. The counselor told him to be a mathematician. This version from: https://www.davidmarcus.com/Humor/MathJokes.htm https://www.davidmarcus.com/Humor/MathJokes.htm and other versions at https://www.math.utah.edu/~cherk/mathjokes.html https://www.math.utah.edu/~cherk/mathjokes.html

time_to_smile 5y ago

To add to this, another mathematical term to be familiar with is "isomorphism". It quite literally means showing that two structures are the same, and perserve the same binary relations (coming from modern Algebra). If you can show that domain A is isomorphic to domain B, then all of the things you know how to do for domain A can be trivially applied to domain B. Thinking about solving problems in terms of isomorphisms has been tremendously helpful for my day-to-day problem solving.