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

[yhlee]

...i had not heard of the p-adic numbers but i am very definitely intrigued.

i know undergraduately about fields. but why do people care about banach spaces, if you are up for explaining when you have time?

i hope your flight goes well!

Banach spaces part 1: different norms

[ursula]

Thanks! I am in Maine for the weekend, doing math.

Rⁿ is a Banach space, where you measure the length of a vector in the usual way, by taking the square root of the dot product of a vector with itself. If that's the only kind of vector space you ever think about, then the concept of a Banach space seems kind of boring. The advantage of the definition is that you can come up with either weirder vectors or weirder concepts of length, and then use the Banach space axioms to transfer all of your intuition about the way length in vector spaces work to this new, stranger context.

Two fundamental examples of different norms are the taxicab norm, |(x₁,...,x_n)| = |x₁|+...+|x_n|, and the max norm, |(x₁,...,x_n)| = max{|x_1|, ..., |x_n|}. The source of the name of the taxicab norm is that if you're trying to get from place to place in a city, you have to follow the rectangular street grid, rather than taking diagonal shortcuts. One way I think about the max norm is trying to find a way to encapsulate people's tastes mathematically. If Alisaundra and Bailey rate a bunch of movies and Alisaundra picks all 7s but Bailey picks all 8s, they probably have a lot in common, despite the slight difference in scores. Then if Cassandra comes along and rates everything between 7 and 8 except Phantom Menace, which she scores at 0, she is much more likely to fight with Alisaundra and Bailey about movies, because the greatest difference is the one that predicts disagreement.

Re: Banach spaces part 1: different norms

[yhlee]

Oh, cool, thank you!

I remember coming across a book in a high school called Taxicab Geometry or something, which I don't think had the taxicab norm above but was literally a high school accessible exploration of what happens if you literally take as your distance norm "how long does it take to get from point A to point B in city X during regular traffic." I didn't buy or read the book, however, so I may be mangling this in memory.

Banach spaces part 2: different vectors

[ursula]

Banach spaces get more exciting when you pick vector spaces where the objects don't look like lists of real numbers. One fundamental example is continuous functions f(x) on the interval [0,1], with norm given by the maximum of |f(x)| on that interval. You can define more complicated norms if you start doing things like taking integrals of functions.

My personal reason to be interested in Banach spaces is quantum physics. In that setting, the vectors are wave functions which you can use to measure the probability of observing a particle at a particular point. Quantum physics is complicated and weird, so being able to think about it in vector space metaphors is really helpful.

[glasseye]

You are my favorite math nerd! ❤️

[ursula]

♥

[davidgillon]

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

[landofnowhere]

Hi, this is your soon-to-be-coworker Alison, and I approve this message :-D (Have added you to my circle. Back when I was in high school and college I was very active on LJ, and my profile reflects my interests from back then, but all that stuff is now locked. Nowadays I only post publicly, on a very occasional basis.)

[ursula]

Hi!

I'm kind of surprised we haven't encountered each other before now--I've been around lj & now dw since my own college years.

[landofnowhere]

Hi! (I actually found your account back in the fall, but felt self-conscious at the time).

The old LJ had a lot more math people :-/ I think I occasionally commented in the mathematics community, but don't remember seeing you there.

[ursula]

I guess I have always thought of this as more a book space than a math space, but we seem to have common interests there, too!

[landofnowhere]

I think there was just more of everything on the old LJ. During my highschool years my LJ circle (and also my AIM friendslist) was mostly people I knew from math competitions/programs. Then in college it expanded to induce people from the college SF club. Then I started lurking around the the edges of LJ booklogging and eventually got drawn in.

We do seem to have common book interests! The ones listed in my profile are representative of what I was reading when younger, being mostly YA fantasy + classic literature, but since then I've gotten more into adult speculative fiction. (Aaron is also a big genre reader.)

[ursula]

I was definitely into Diana Wynne Jones when I was younger!

I got into lj in college (but I'm a bit older than you); part of that was SCA friends & my (future) spouse glasseye, and part of that was my roommate, who was one of the people involved in the founding of AO3.

[landofnowhere]

I still find Diana Wynne Jones to be excellent comfort reading :-) Though I haven't read /Fire and Hemlock/, which was my favorite book as a teenager, for quite a while, because at some point I reached the point where I pretty much had the whole thing memorized.