**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 path^{1} of love is never smooth^{2}

But mine’s continuous^{3} for you^{a}

You’re the upper bound^{4} in the chains^{5} of my heart

You’re my Axiom of Choice^{6}, you know it’s true

But lately our relation’s^{7} not so well-defined^{8}

And I just can’t function^{9} without you

I’ll prove^{10} my proposition^{11} and I’m sure you’ll find

We’re a finite simple group^{12} of order two^{13}

I’m losing my identity^{14}

I’m getting tensor^{15} every day

And without loss of generality^{16}

I will assume that you feel the same way^{17}

Since every time I see you, you just quotient out^{18}

The faithful image^{19} that I map^{20} into

But when we’re one-to-one^{21} you’ll see what I’m about

‘Cause we’re a finite simple group of order two

Our equivalence^{22} was stable^{23},

A principal love bundle^{24} sitting deep^{b} inside

But then you drove a wedge^{c} between our two-forms^{25}

Now everything is so complexified^{26}

When we first met, we simply connected^{27}

My heart was open^{28} but too dense^{29}

Our system^{30} was already directed^{31}

To have a finite limit^{32}, in some sense

I’m living in the kernel^{33} of a rank-one map^{34}

From my domain^{35}, its image^{36} looks so blue,

‘Cause all I see are zeroes^{37}, it’s a cruel trap^{d}

But we’re a finite simple group of order two

I’m not the smoothest^{38} operator^{39} in my class^{40},

But we’re a mirror pair^{41}, me and you,

So let’s apply forgetful functors^{42} 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 associative^{43} and free^{44}

And by corollary^{45}, this shows you and I to be

Purely inseparable^{46}. 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 *a*x*a*^{-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