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I was thinking more about my post from the other day (Translate THIS), and the fact (or is it a fact?) that if everything were a function of everything else, how would you write … well, everything in mathematical form? With apologies to Douglas Adams, the answer can’t be as simple as “42”.
I wrote to my brother, Jonathan Collier (Director of Finance at Kimpton Hotels by day; Mathematician, Philosopher, and Daddy by night) with the question. His answer is below. (Who knew that my annoying little squirt of a brother would end up being so smart!)
I’m somewhat philosophically opposed to the implication of a singularity. Functions are at best a transformation from a chaotic (or less discernable) set to a less chaotic (or more discernable) set. There is nothing to implicate that the product set of a function is smaller or somehow simpler.
I think you’re spot-on in your treatment of the function, though: the goal for the “translator” (as opposed to the translation) is to find the function, not the variables.
So sure: Everything is the product of a function. Absolutely. It's the identity function: F (x) = x.
What I think you’re getting at, though, is “does everything exist in the range of some function?” (Range in mathematical terms is the set into which a function’s output is contained). So, what you’re really talking about the space of functions, not necessarily the sets of inputs and outputs.
Before going further, let’s talk about a function being “well-defined”. For a function to be defined, it must satisfy two criteria:
1. The first is to take function F that maps elements of a set X to elements of a set Y. For all elements x in set X, if F is well-defined, then F (x) = y implies that y is an element of the set Y.
So, for the language translation example, F translates phrases of English to Spanish. If F ("You're pulling my leg") = "me tome el pelo", the function is not well-defined. You need to define it further.
2. The second requirement is not so easy. With the function F, if x = y, then F (x) = F (y).
This is particularly hard for the translation function. In fact, this is particularly hard for anything outside of math. Using the California vernacular for translating "yes" and "no", does F (“yeah, no”) = F (“no way”)? What about F (“no, yeah”) = F (“obviously”)? It’s this kind of leap that Facebook’s algorithm made to approach being a true transformation in the mathematical sense (as opposed to being a simple algorithm).
So now, take the function that translates perception from one person to another. F (Car accident) is most certainly undefined when considering bystander perception.
Sliding to the other end of complexity is the axiom of choice. For the clear majority of modern logic (of which mathematics is merely a subset), it must be taken as an axiom — an accepted fact without question or available proof — that for any set, there exists a function that allows you to organize that set in an intelligible way. No translation, no mapping — simply a way to discern one element as different than every other element.
So essentially, what I think you’re getting at is the heartfelt but unprovable existence of a function for every set we can dream up that will provide order to that set.
Consider this: I drop a pile with an infinite number of tube socks in your lap. [How is this theoretical? In my house, it’s called “Saturday”. But I digress. —Ed.] Can you create a function that allows you to match them in pairs of two without leaving any unmatched socks? If you accept the axiom of choice, then yes. Without the axiom of choice, you will be folding socks for many, many years to come to find the answer.
I already have enough laundry in my life as it is, thankyouverymuch. (And thanks, Jon, for the information!)
Regarding the axiom of choice: I think I should clarify something: the analogy should be not just pairing the socks but organizing them in to "left" and "right" socks without a rule for determining right from left. The point is about the ability to always create an arbitrary function between infinite sets!