behold the mathematical field...

[ursula]

yhlee asked for a math-check of this Tumblr post. Here are some thoughts!

I've seen a few variants on this string of jokes; in particular, I'm pretty sure that the claim that "behold the field in which my fucks are grown" implies that fucks are elements of a field is not original to my comment on that post. But let's take this Tumblr iteration as a starting point.

The first commenter provides evidence that nonzero fucks admit both additive and multiplicative inverses, and that there are imaginary fucks. They claim that fucks must therefore be isomorphic to the field of complex numbers.

The second commenter objects, "Technically, we can only conclude that 'a fuck' is an element of some norm space over a field containing fractional values." The part about "containing fractional values" is unclear--any field contains fractional values--but the "norm space" part is an interesting extension of the joke. The first commenter used the phrase "what the absolute fuck" to argue for the existence of additive inverses, and this commenter is pointing out that the mathematical symbol |x| can be used both for the absolute value and to represent a "norm", which is a mathematical generalization of the idea of length. The second commenter goes on to say that isomorphism to the complex numbers is a much stronger claim! This is very true. The quickest example of a field that has an imaginary element but isn't isomorphic to the complex numbers is Q(i), the rational numbers together with the square root of -1. You can extend on that example by taking solutions to polynomials more complicated than x²+1 = 0. That will net you some sort of purely imaginary number field. If you prefer a more exotic example, you could try the algebraic closure of the p-adic numbers.

The third commenter is clearly thinking about the Q(i) example: they object that if anybody ever said "what the rational fuck" one could assume that there were irrational fucks (and thus that there were fucks that live in the real numbers but not in the rationals). They go on to say, "We could then generously assume completeness and thus a Banach space that is at least a subset of" the complex numbers. This is an actual mathematical error, because there is more than one way to complete the rational numbers: one can choose between the real numbers, where you fill in the gaps between numbers that can be expressed as fractions using our usual notion of distance, or the p-adic numbers I mentioned earlier, which instead use a notion of distance that depends on divisibility by a prime number. (It's not unusual to get through an undergraduate degree in math without hearing about the p-adics. I missed them as an undergrad, got grumpy at a grad school algebraic number theory professor who skipped that section on the grounds we had all heard about them already, and then forgot about them until I started doing research in a related area. The basic idea is really cute, though, and I recommend reading up on them if you like weird number systems!)

The Banach space part of the third comment is interesting, though. It's picking up on the earlier reference to a "norm space", and thus on the many meanings of |x|. Intuitively, a Banach space is a vector space where you can measure length and where sequences that look like they should have limits actually have them. If you assume that you're filling in sequences of fractions in the usual way, to get the real numbers, and that you have some way to make sense of "an imaginary element", it makes sense to jump to the complex numbers at this point.

I need to catch a plane, so I'm not going to talk about why people care about fields or Banach spaces right this second, but you should ask me those questions in comments!

Comments

[davidgillon]

Expanding on a theme, 'get fucked' implies 'fucked' is a function or command or value in various computer languages ;)