1856: Existence Proof
Title text: Real analysis is way realer than I expected.
In mathematics, an existence proof is a proof that only shows that an object with a specific property exists, but does not tell what this object is. For instance, if f is a continuous function such that f(0) = 0 and f(100) = 2, it is easy to prove that there exists an x between 0 and 100 such that f(x) = 1 (as in the comic). However, this proof gives no way to find such an x.
In many situations, a proof of existence is enough to satisfy a mathematician, but in others, it is desirable to actually identify the object whose existence has been proven.
The full statement itself seems like a solution to some kind of function composition problem. Seems like what the class has proven is that if you apply certain function G(x) to a starting point of function f(0), then what it will do is just give you a value of f(x) at some other value of x, existence of which is stated to be proven. The sentence "There exists some number x such that f(x)=G(f(0))=1." boils down to "There is an x such that f(x)=1". The part with G(f(0)) is only a way to arrive at 1. For some reason there is an x that satisfies f(x)=G(f(0)), and since G(f(0))=1, it is equivalent to f(x)=1.
In the comic, Miss Lenhart (and students) take this one step further, by taking up arms to destroy the function value, which they have proven to exist. In the last panel, some students off screen begin to wonder if they are in the right class, as normal math classes do not take up swords to fight abstract concepts. Another student remarks that they are finally in the right math class, implying that this is the kind of thing they wanted from their math curriculum all along.
The phrase "We ride" is commonly used in rallying battle cries, particularly in fantasy or medieval dramas where characters are preparing to enter combat on horseback. Variations of the phrase are used several times in The Lord of the Rings, for example.
The title text refers to Real Analysis, a branch of mathematics dealing with real numbers and real-valued functions (as opposed to studies dealing with integers, rational numbers, imaginary numbers in the complex plane, etc.). As the speaker implies, Real Analysis is supposed to remain confined to the theoretical realm of mathematics; certainly nobody signing up for such a class would ever expect to be embroiled in a crusade against intangible constructs! Taken out of its mathematical context, "analysis" literally means "breaking down", referring to the teacher's intention to cut things up with a sword. The use of the uncommon word "realer" conveys that the situation has suddenly developed unusually high stakes. This nuance would be lost if the word "realer" were replaced with the technically correct phrasing of "more real".
This may be a continuation of 982: Set Theory, where numbers were "executed" to prove a point.
- [Miss Lenhart stands in front of a whiteboard and points at calculations written on it.]
- Miss Lenhart: There exists some number x such that f(x)=G(f(0))=1.
- [Miss Lenhart moves her arm in a frameless panel.]
- Miss Lenhart: Oh yes.
- Miss Lenhart: Somewhere out there, it exists.
- [Zoom-in on Miss Lenhart raising a fist.]
- Miss Lenhart: And we must find it... and destroy it.
- [Miss Lenhart raises a sword.]
- Miss Lenhart: Grab your swords, students! We ride!
- Student #1 (off-screen): I think I'm in the wrong math class?
- Student #2 (off-screen): I'm finally in the right one.