xkcd.WTF!?

Explanation ➲

Miss Lenhart is teaching a math class. She begins a proof when one of her students (Cueball) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She claims no, but in a matter of seconds Cueball is calling out that he was right.

The proof she starts setting up resembles a proof by contradiction. However, after Cueball's interruption Miss Lenhart's proof takes a turn for the absurd: rather than assuming there will be a point in the function that correlates to co-ordinates (x, y), Miss Lenhart assumes that the act of writing numbers on the board will correlate to co-ordinates (x, y).

A normal proof by contradiction begins by assuming that a particular condition is true; by demonstrating the implications of this assumption, a logical contradiction is reached, thus disproving the initial assumption. One example of a proof by contradiction is the proof that √2 is an irrational number:

1. Assume that √2 is a rational number, meaning that there exists a pair of integers whose ratio is √2.
2. If the two integers have a common factor, it can be eliminated using the Euclidean algorithm.
3. Then √2 can be written as an irreducible fraction a/b such that a and b are coprime integers (having no common factors other than 1).
4. The equation a/b = √2, when multiplied by itself, gives a²/b² = 2, which can be rearranged as a² = 2b².
5. Therefore, a² is even because it is equal to 2b². (2b² is necessarily even because it is 2 times another whole number, and multiples of 2 are even.)
6. It follows that a must be even (as squares of odd integers are never even).
7. Because a is even, there exists an integer k that fulfills: a = 2k.
8. Substituting 2k from step 7 for a in the second equation of step 4: 2b² = (2k)² is equivalent to 2b² = 4k², which is equivalent to b² = 2k².
9. Because 2k² is divisible by two and therefore even, and because 2k² = b², it follows that b² is also even, which means that b is even.
10. By steps 6 and 9, a and b are both even, which contradicts that a/b is irreducible as stated in step 3.

Q.E.D.

Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.

The way that Ms Lenhart's proof refers to the act of doing math itself, is characteristic of metamathematical proofs, for example Gödel's incompleteness theorems, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While typical mathematical theorems and their proofs deal with such mathematical objects as numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 Kurt Gödel was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in set theory, where many statements become impossible to prove or disprove if the axiom of choice is not taken as a part of the axiomatic system.

Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to Cantor's diagonal argument where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same cardinality (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.

The axiom of choice itself states that for every collection of disjoint nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. Determinacy of infinite games is used as a tool in the set theory, however the deterministic process is rather a term of the stochastic processes theory, and the dynamical systems theory, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in 804: Pumpkin Carving and later in 982: Set Theory, another comic about a math class with a similar theme on how teachers teach their student mathematical proofs.

Although Miss Lenhart did retire a year ago after 1519: Venus, she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class. A very similar Miss Lenhart comic was later released with 2028: Complex Numbers.