Origin/Composer: M. Salomome
Year: 2004

I first heard about this from Dustin Foley on the SUMaC group. I emailed and IM’d a bunch of people later that day about it. Then, a day later, a similar message about it came through the mn-arml list from Mitcho. What makes this song great is that its audience is pretty much only math and related majors who have taken quite a few “serious” college math classes. Most people I showed this to did not laugh much, simply because phrases like “open but too dense” doesn’t have any mathematical meaning to them. So to make this funny for everyone, I made a list of the math references in the song below:

Please visit Klein Four’s Official Site for more information about the a cappella group.

Lyrics

The path1 of love is never smooth2
But mine’s continuous3 for youa
You’re the upper bound4 in the chains5 of my heart
You’re my Axiom of Choice6, you know it’s true

But lately our relation’s7 not so well-defined8
And I just can’t function9 without you
I’ll prove10 my proposition11 and I’m sure you’ll find
We’re a finite simple group12 of order two13

I’m losing my identity14
I’m getting tensor15 every day
And without loss of generality16
I will assume that you feel the same way17

Since every time I see you, you just quotient out18
The faithful image19 that I map20 into
But when we’re one-to-one21 you’ll see what I’m about
‘Cause we’re a finite simple group of order two

Our equivalence22 was stable23,
A principal love bundle24 sitting deepb inside
But then you drove a wedgec between our two-forms25
Now everything is so complexified26

When we first met, we simply connected27
My heart was open28 but too dense29
Our system30 was already directed31
To have a finite limit32, in some sense

I’m living in the kernel33 of a rank-one map34
From my domain35, its image36 looks so blue,
‘Cause all I see are zeroes37, it’s a cruel trapd
But we’re a finite simple group of order two

I’m not the smoothest38 operator39 in my class40,
But we’re a mirror pair41, me and you,
So let’s apply forgetful functors42 to the past
And be a finite simple group, a finite simple group,
Let’s be a finite simple group of order two
(Oughter: “Why not three?”)e

I’ve proved my proposition now, as you can see,
So let’s both be associative43 and free44
And by corollary45, this shows you and I to be
Purely inseparable46. Q. E. D.47

### Math References

1 Path: Basically the set of points that can be described by a function, or the curve along which a path integral is taken.
2 Smooth: A function that can be differentiated infinitely many times.
3 Continuous: A function without “breaks.”
4 Upper Bound: The upper bound is a value that can sometimes be calculated to determine the maximum possible value of a certain solution.
5 Chain: A totally ordered set in set theory.
6 Axiom of Choice: An axiom of set theory stating, in the vernacular, that one can choose an element from each set in a set of sets.
7 Relation: A definition relating two or more mathematical objects.
8 Well-defined: Mathematically defined in a logical way from base axioms, and without ambiguity.
9 Function: Everyone should be able to get this joke 😀
10 Prove: Logically establishing a statement to be true from previous axioms and proven statements.
11 Proposition: A conjecture.
12 Finite Simple Group: Umm…these three words take a lot to explain. Well, a group is a mathematical concept that combines a set of objects together with an operation, along with three axioms. Then, you can look at symmetries and other cool stuff in that object called a group. A finite group is simply one that has a finite set of objects defined within it. A simple group, in plain English, is the most basic type of group, similar to the prime numbers in the natural numbers.
13 Order Two: The order is the number of elements in a group. A simple finite group of order two is very interesting in that it is the smallest prime cyclic group, with some interesting properties.
14 Identity: A value e, such that axa-1=e
15 Tensor: A geometric entity describing a certain space.
16 Without Loss of Generality: Often abbreviated “WLOG,” this is often used in proofs to (correctly) substitute a simpler, equivalent special case for the general case to be proven.
17 An example of a bad use of WLOG. 😀
18 Quotient Out: In a quotient group, a specific type of simple finite group, certain values are said to be “quotient out” to create the group.
19 Image: The resultant figure after a mathematical tranformation.
20 Map: A function that takes each value in an object and corresponds it with another value.
21 One-to-One: For each x, there is one f(x), and for each f(x), there is one x.
22 Equivalence: A binary relation that is transitive, reflexive, and symmetric.
23 Stable: A polynomial in which all the roots lie in the left half-plane, or in the open unit disc.
24 Principal bundle: I know this is math, but I have no idea what it means. Check Mathworld or Wikipedia or something.
25 Two-forms: Another one I don’t know.
26 Complexified: A vector space that has been extended to the complex numbers.
27 Simply connected: In plain terms, an object that is in one piece, and without any “holes” or “loops.”
28 Open: An interval that does not include the end-point.
29 Dense: Basically, a set that there are infinitely many values between any two values in the set.
30 System: A group of equations to be solved simultaneously.
31 Directed: In graph theory, a graph where the edges have specific directions.
32 Finite limit: A limit is something in calculus…and a finite limit is a limit that is finite.
33 Kernel: There’re a lot of definitions. Check Wikipedia.
34 Rank-one map: In linear algebra, the number of rows or columns that are linearly independent.
35 Domain: The “input” of a function.
36 Image: I think I’ve done this already…yep. Check number 19.
37 Zeroes: The roots of a function, or just the number zero.
38 Smoothest: Check number 2.
39 Operator: A symbol that defines a specific relationship between mathematical objects.
40 Class: Any group of set with a certain distinguishable trait.
41 Mirror Pair: The paired point in a reflection. Or in physics, the anti-particle or supersymmetric particle of a particle.
42 Functors: Don’t know. Check Mathworld or Wikipedia.
43 Associative. The property that, with three values a, b, and c, and operator ^, (a^b)^c=a^(b^c). Associativity is one of the axioms of groups.
44 Free: A variable that is not dependent.
45 Corollary: A result or statement that comes obviously from a statement that has just been proven.
46 Purely inseperable: ???
47 Q.E.D.: Quod Erat Demonstrandum, or “Which was to be demonstrated” in Latin. It is often used at the end of proofs as a completion.

### Perhaps a Math Reference

a You: Maybe a play on the letter u, which is often used in calculus?
b Deep: A theorem that has many implications and goes to a core of mathematics.
c Wedge: A mathematical shape–a triagular prism.
d Trap: Just sounds like it could be mathy.
e “Why not three?” Obviously, many of the propositions in this song would be false if the finite simple group became of order three. Also, this hints at a three-some. I won’t go into what that means, if you’re a poor confused soul