Discussion
You know how the identity to addition is 0, like 2 0 + is 2. Then, on top of that, you have the identity of multiplication that is 1, 2 1 * is 2.
Okay, so, add 0, mul 1, now. Is there an identity somewhere that is 2? Serious question, I've been scratching my head on this for a good 15 minutes and I'm drawing a blank. It's not pow, and it's not square.. It might be a stupid question, if it is, please do tell me it's stupid.
@neauoire Hmm, for a "obvious" extension of the 0 & 1 identities, I'm reminded of the idea of log probabilities https://en.wikipedia.org/wiki/Log_probability (which I had used, like, a lifetime ago)
So if we have a mapping
f (x) = log₂ x = lg x,
where f : R⁺ → R⁺; this will give
f (x ⋅ y) = f (x) + f (y) = lg x + lg y.
We can then consider a binary operator ⁑ such that
f (x ⁑ y) = f (x) ⋅ f (y) = lg x ⋅ lg y
which yields
x ⁑ 2 = 2 ⁑ x = x.
any operation that flips between two states sequentially will work with 2 (or any even positive integer). I think for this reason we don't over have a second parameter for these operations. *But* if you would allow a 'number of repeats' for the 'bitwise not'
3 2 ~ == ~(~3) == 3
@neauoire since I’m a humanities/social science I’m going to just blab and say: for identity, addition (a linear-ish process) = 0, multiplication (a geometric-ish process) = 1. So I’d dig around for something that is scaling at the “next level” maybe exponentials? maybe something four dimensions? Mostly I’m going to just enjoy thinking about and try not to rabbithole the entirety of math history.
@neauoire addition identity is to change something by 0. multiplication identity is to keep 1 set of.
What do you do twice to end up where you started? Reflection, inversion, half-rotations.
@neauoire The integers overall are defined in terms of 0 and 1, so I don't think you will find any other identities. You could of course define a some other type of number where 2 is an identity. For instance, 2 is an identity for the min operation on ternary digits.
@neauoire an idea that just came to mind: How about an operator that rotates vectors around origin by multiples of pi, in radian. 2 by would be a full circle, therefore mapping the input vector into itself
@neauoire there are an infinite number of monoids (sets + operations) where 2 is the identity element (also known as the neutral element).
one example would be the MAX operation defined on the set {2, 3, 4, ...}.
another example would be the set of rational numbers and area of the triangle with sides x and y, i.e. (x*y)/2.
EDIT: originally i said groups but one of my examples (MAX) is a monoid (there is no inverse). the general point still stands.
@neauoire
Dragging @johncarlosbaez into the conversation using a vaudeville hook
@neauoire Often times, it's actually defined the other way around: Rather than "0 is the neutral element of addition, 1 the neutral element for multiplication", it's "for a set we want addition with a neutral element, and we want multiplication with a neutral element. Oh look, natural / real / complex numbers with 0 and 1 work for this". But you can define sets were that isn't the case. Especially for finite sets it can get a bit funky: I faintly remember seeing 2 as a neutral element once
@lhp do you remember which weird set had it? I'd happily pull on that thread to put this idea to rest..
@neauoire ok, I think I recalled it: Since a group is just a set with any internal operation, define addition on a finite set of integers such that it "wraps". Now you can just define the set small enough so that adding 2 wraps around to the same element again.
Not sure how useful an answer this is, but sets and operations on sets can be defined more or less arbitrarily, so you can define effectively whatever you want (even the "is element of set" operation is arbitrary and re-definable)
@neauoire it was a finite set small enough for the professor to show all possible operations on the blackboard as an example, but that was years ago and my notes did not survive, so not right now.
I'll try to look it up, but no promises :)
@neauoire multiply-and-divide-the-product-by-two, I think.
@neauoire I mean, we're talking about groups right.
@screwlisp I'm not sure! Maybe?
@neauoire oh wow thats a interesting question.....
maybe something involving operations of ternery numbers that are mapped onto {0,1,2} instead of the usual {-1,0,1}??
@angelwood Oh! That's a good hint, and then I suppose it could be all bases, and infinity is identity to a high order arithmetic.
Lemme try your hypothesis.
@neauoire @angelwood my math foo is not that strong, but I think your question is coming at it from the wrong direction. Identity is a property of the operator not the value. So to restate your question, what operation maps 2 to 2? Stated like that it’s clear there are many such operations.
@Jmj @angelwood yes! that's the operator I'm looking for, sorry not the value. Do you mean compound operations like.. +2-2, that sort of things?
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@neauoire maybe e^i2PI, a full rotation in imaginary space? The sequence could be the hierarchy of operators +, *, ^ relating to 0, 1, 2? Just an idea…