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I took a lot of math in my schooling, and have continued using math on a daily basis in my engineering career. I even subscribe to many math channels on YouTube

I took a lot of math in my schooling, and have continued using math on a daily basis in my engineering career. I even subscribe to many math channels on YouTube, but this is the first time in my life that I've even heard of this function. I know there aren't real rules to this puzzle, but this function doesn't seem well-known at all.

tantalor 2y ago
It has a rich history going back to the formalization of arithmetic in the 1800s. https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms It's probably something that only folks who study the foundations of mathematics would know. You may have also heard about it if you learned about Gödel's incompleteness theorems, or read Gödel, Escher, Bach
empath75 2y ago
The successor function is how natural numbers are defined in most axiomatic arithmetic systems. A natural number is either zero or a successor of a natural number. Addition is defined as a recursive application of successor functions. m + 0 = m, m + S(n) = S(m + n). as an example -- 3 + 2: S(S(S(0))) + S(S(0)) = S( S(S(S(0))) + S(0) ) = S( S( S(S(S(0))) + 0 ) ) = S(S(S(S(S(0))))) There's nothing really in the definition of the successor function that necessarily requires that it's interpreted as n+1, though. It's just an interpretation from the context in which it's used. It could represent any operation as long as it is isomorphic to adding one -- but there's nothing special about "adding one". You could have it represent multiplying by a constant, and interpret "zero" as the number one. So a number in this interpretation is either 1; or 2 times a natural number. 1, 2, 4, 8, 16, 32 -- you're working only with powers of 2 now. The above "addition" rule above still works, but now it represents "multiplication" instead of addition. I'll replace all the S's in the above example with x2 2x2x2x1 "+" 2x2x1 = 2x(2x2x2x1 + 2x1) = 2x(2x(2x2x2x1 + 1) = 2x2x2x2x2x1 = 32 So now, instead of addition, we've recursively defined multiplication where the successor function is interpreted as multiplying by 2. There's an infinite number of ways that you can interpret the successor function. So basically, I do think it's cheating, and if you do want to define it as n+1, it would be even simpler to just define a function that takes any number to the desired output.
kccqzy 2y ago
I'm not sure what kind of math schooling you did, but did you learn to construct the natural numbers from scratch? This may have been the fault of math education. In my college people learn real analysis (constructing the real numbers) before they learn to construct the natural numbers, which is backwards to me. I recommend learning it: constructing the natural numbers from just sets in the tradition of Zermelo–Fraenkel is mind blowing the first time you see it. Of course you could just use Peano axioms without touching set theory too.

tantalor 2y ago

It has a rich history going back to the formalization of arithmetic in the 1800s. https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms It's probably something that only folks who study the foundations of mathematics would know. You may have also heard about it if you learned about Gödel's incompleteness theorems, or read Gödel, Escher, Bach

empath75 2y ago

The successor function is how natural numbers are defined in most axiomatic arithmetic systems. A natural number is either zero or a successor of a natural number. Addition is defined as a recursive application of successor functions. m + 0 = m, m + S(n) = S(m + n). as an example -- 3 + 2: S(S(S(0))) + S(S(0)) = S( S(S(S(0))) + S(0) ) = S( S( S(S(S(0))) + 0 ) ) = S(S(S(S(S(0))))) There's nothing really in the definition of the successor function that necessarily requires that it's interpreted as n+1, though. It's just an interpretation from the context in which it's used. It could represent any operation as long as it is isomorphic to adding one -- but there's nothing special about "adding one". You could have it represent multiplying by a constant, and interpret "zero" as the number one. So a number in this interpretation is either 1; or 2 times a natural number. 1, 2, 4, 8, 16, 32 -- you're working only with powers of 2 now. The above "addition" rule above still works, but now it represents "multiplication" instead of addition. I'll replace all the S's in the above example with x2 2x2x2x1 "+" 2x2x1 = 2x(2x2x2x1 + 2x1) = 2x(2x(2x2x2x1 + 1) = 2x2x2x2x2x1 = 32 So now, instead of addition, we've recursively defined multiplication where the successor function is interpreted as multiplying by 2. There's an infinite number of ways that you can interpret the successor function. So basically, I do think it's cheating, and if you do want to define it as n+1, it would be even simpler to just define a function that takes any number to the desired output.

kccqzy 2y ago

I'm not sure what kind of math schooling you did, but did you learn to construct the natural numbers from scratch? This may have been the fault of math education. In my college people learn real analysis (constructing the real numbers) before they learn to construct the natural numbers, which is backwards to me. I recommend learning it: constructing the natural numbers from just sets in the tradition of Zermelo–Fraenkel is mind blowing the first time you see it. Of course you could just use Peano axioms without touching set theory too.