4

The first operation, addition, is abelian and so is multiplication. However, the next operation, exponentiation is not! Why is this? I understand that $2^3$ and $3^2$ are not equal but why do we suddenly lose this property after adding one other operation? Is there a property that we lose for tetration? Or Pentration (the next operation)? If so what property is lost at the nth operation?

Edit: I there anything lost when going from addition to multiplication?

Ethan Bolker
  • 95,224
  • 7
  • 108
  • 199
mtheorylord
  • 4,274
  • 1
    We loose associativity when $x^{(y^z)}\ne (x^y)^z$ – Doug M Dec 30 '16 at 18:35
  • Well firstly just because one thing has a certain property then we change something we certainly don't expect it to follow. Rather you need to deductively prove it. I don't just expect there to be infinitely many primes, it very well could just stop at some large number but we can see that if there was a last prime number we could form another one from it. – marshal craft Dec 30 '16 at 18:44
  • I think the only way for you to answer that question is for you or someone to set out to prove that n-tration operation is commutative for all $n \in \Bbb N$. You can't prove it because it's false. You can step through it and fine a spot where it breaks down maybe, but as with prime numbers having no end, it's fundamental to logic. I don't think it would be any different than if you found a quadratic equation which yielded primes for say first 5 integers then didn't for a while. – marshal craft Dec 30 '16 at 18:50
  • From addition to multiplication, you lose the inverse of $0$, and it also becomes much harder in general to obtain multiplicative inverses (since there are many groups which don't have a natural ring structure). – Patrick Stevens Dec 30 '16 at 19:01
  • Why would we not lose commutativity? There is absolutely no reason for operations to be commutative, really. – Mariano Suárez-Álvarez Dec 30 '16 at 19:25
  • What do you mean inverse of 0. Identity, right? – mtheorylord Dec 31 '16 at 20:22
  • Ok. But then what do we lose when going from addition to multiplication? – mtheorylord Dec 31 '16 at 20:22

1 Answers1

3

This is my take on the question, namely that we loose an important bit of symmetry as we go from multiplication to exponentiation.

When adding two numbers, you might think of both numbers as translations, and addition as doing first one translation, then the other. When multiplying two numbers, you can think of both numbers as scalings, and the multiplication itself as doing first one scaling, then the other. In other words, the two numbers on either side of the operation symbol are entities of the same kind.

However, exponentiation is repeated multiplication. In the expression $2^3$, $2$ signifies the scaling in question, and the $3$ tells you how many times to do it. In other words, the $2$ and the $3$ signifies qualitatively different entities (and I have seen no interpretation of exponentiation where the two numbers signify the same kind of thing). Therefore you break the symmetry, and thus there is no reason to expect it to be commutative.

Arthur
  • 199,419