1724: Proofs

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Proofs

Title text: Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...

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.

Transcript

[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]

Miss Lenhart: ... Let's assume there exists some function F(a,b,c...) which produces the correct answer-

Cueball (off-panel): Hang on.

[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]

Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.

[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]

Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.

Cueball (off-panel): All right...

[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]

Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (x, y). If we—

Cueball (off-panel): I knew it!

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Discussion

Judging from my experience when I first encountered proofs in math classes (or my general experience from math classes), the teacher is going to write down a "proof" which makes absolutely no sense to students and is also never explained in a way that actually makes them understand. Instead, they are just going to use "dark magic" and write what seems to be completely senseless to students. 141.101.91.223 04:24, 24 August 2016 (UTC)

'Dark magic' might also refer to the supernatural, so when the teacher said that an answer 'will be written' in a specific location, Cueball took this to mean that a spirit would be summoned to write it, like a ouija chalk board. 141.101.70.67 09:27, 25 August 2016 (UTC)

Transcript generated by the BOT was murdering me, had to change it. Proposing miss Lenhart is party 1. EppOch (talk) 04:45, 24 August 2016 (UTC)

I support that. 141.101.91.223 06:13, 24 August 2016 (UTC)

Me to, but I am on mobile, so editing is a pain 162.158.86.71 06:51, 24 August 2016 (UTC)

Done Elektrizikekswerk (talk) 08:26, 24 August 2016 (UTC)

Note that the BOT doesn't create any text - see here. The transcript was made by several people. Agree completely that this is Miss Lenhart, but even if it was not "party 1 and party 2" is not the way to describe a woman with long blonde hair and Cueball ;-) There is at the moment a discussion what to call other women looking like this (i.e. those that are not clearly Miss Lenhart, Mrs. Roberts or her daughter Elaine Roberts). Chip in there if you have any opinions on that regard... --Kynde (talk) 11:01, 24 August 2016 (UTC)

Irrationality proof isn't really a proof by contradiction (it doesn't use double negation elimination). You're showing (exists a,b. ...) -> False by assuming (exists a, b. ...) and showing False, which is implication introduction --162.158.85.105 07:33, 24 August 2016 (UTC)

I'm thinking she's doing one of those proof that write down a formula or function out of nowhere, and proceeds to proof everything with it. 108.162.222.125 08:43, 24 August 2016 (UTC)

This comic reminds me of "divination" rituals, where a magical spirit is summoned to write out an answer. Usually not something as complex as here, but hey, XKCD! --Henke37 (talk) 10:04, 24 August 2016 (UTC)

Man, Reductio ad absurdum never made any logic. If we could assume any thing, why use logic? Oh wait, it has already been covered in XKCD 162.158.49.12 (talk) (please sign your comments with ~~~~)

"Dark magic" proofs are centered around properties of functions, and abstract concepts, rather than manipulating the functions themselves?? 108.162.246.113 11:26, 24 August 2016 (UTC)

My assumptions is that the "Dark Magic" being referred to here is more "A technique that works, though nobody really understands why." [see http://catb.org/jargon/html/B/black-magic.html] In this case, the teacher is setting up a proof in an manner which will lead to the desired goal, but to the student it is exceedingly unobvious as to why one would do it this way, other than "it works" 108.162.219.52 15:30, 24 August 2016 (UTC)

I was thinking that a "dark magic proof" referred to those ridiculous "party trick" proofs like 'proving' that 1 = 0 via some confusing train of logic, and mathematical sleight of hand. 108.162.237.213 (talk) (please sign your comments with ~~~~)

Maybe he meant "dark patterns"? 162.158.126.139 (talk) (please sign your comments with ~~~~)

It seems pretty obvious to me that by "weird, dark magic proofs", the student is talking about proofs that drag in far-flung reaches of mathematics so distant that they no longer appear to be mathematics, especially ones that involve meta-reasoning. Gödel's proof of the incompleteness of Peano arithmetic is the archetypical example, but others include Lob's theorem and any proof by contradiction involving the halting problem. Ms Lenhart's proof starts out by setting up a proof-by-contradiction, already a warning sign, and she then escalates it at the end by implying that this proof will somehow involve the actual physics of where the solution can and cannot be written. 108.162.241.123 17:27, 24 August 2016 (UTC)

Agreed, although I think starting out with a proof by contradiction setup is by itself not that much of a warning sign. However it heads straight into meta-space by making the assumption of the existence of a function that produces a solution of something. Zmatt (talk) 18:52, 24 August 2016 (UTC)

The fact that the proof mentions the actual blackboard on which it is written is of course problematic in numerous ways, as is predicating on whether something "will eventually" happen. This is well outside the scope of the usual mathematical foundations. Since careless use of meta-recursion is a trap, such a proof would have to very very carefully consider foundational issues and cannot handwave over them. Zmatt (talk) 19:13, 24 August 2016 (UTC)

---

"In the title text the decision of whether to take the axiom of choice is made by a deterministic process. The axiom of determinacy is incompatible with the axiom of choice..." The axiom of determinacy is not really relevant to deterministic processes - it is about (certain types of two-players-) games and says that any such game is determined (that is, some player has a winning strategy). So this axiom is not relevant to the title text --162.158.83.66 17:39, 24 August 2016 (UTC)

I agree. I read the title text in almost exactly the opposite way - that the proof relies on the existence of a deterministic process for selecting objects, and therefore the invocation of the axiom of choice as a part of the process is superfluous (but not a contradiction). Anyhow, the axiom of determinacy isn't ever mentioned, so it probably shouldn't be shoehorned in here. 162.158.74.53 20:36, 24 August 2016 (UTC)

I feel like it is a stretch to assert Lenhart is setting up a proof by contradiction. It sounded to me more like an prior knowledge proof (not sure it's technical name). For example, "calculate the space between two concentric circles of differing diameter when the longest straight line you can draw is length d." If you assume there is a function F(r1, r2) which has been previously proven to calculate this space, then it is easy to show that the space is in fact .5*pi*(.5*d)^2 (as you have a degenerative case where r1=0, and you have an ordinary circle). I also think this type of proof is more "dark magic"-feeling than a simple proof by contradiction. 108.162.216.87 (talk) (please sign your comments with ~~~~)

While technically the same pattern, I would assume something more like NP-complete proofs: Assume we have function F which solves this problem in polynomial time ... then we can solve that problem in polynomial time as well. Just, instead of "polynomial time", the existence of function is the question here, so it will likely be something around recursively enumerable/countable stuff. -- Hkmaly (talk) 13:02, 25 August 2016 (UTC)

I don't like how this explanation uses the word "standard". Non-standard mathematical objects are subjects of non-standard analysis, not metamathematics. --108.162.218.185 02:25, 27 August 2016 (UTC)

Simplest explanation would be Cueball suspect Ms Lenhart already made-up an answer for a made-up function (hence magic), which is confirmed at the last panel. Laymen like myself wouldn't grasp any of those methamathematical stuff explanation. :) 162.158.167.35 07:20, 29 August 2016 (UTC)

There is no such thing, like "answer for a function", so you can't be right. And this interpretation is completely ignoring the mathematical similarities, yet it was introduced as a summary of the mathematical explanation. If you don't grasp the idea, don't try to summarize it, please. 162.158.133.138 14:35, 29 August 2016 (UTC)

I think the explanation of Godel's incompleteness theorem is not quite right. I've always heard the precise formulation of it as "Any logical system powerful enough to include basic arithmetic has statements that are true but cannot be proven or disproven within the system." I would edit the page to reflect this, but to be honest I'm not that confident in my understanding of it.172.69.42.134 01:40, 16 April 2021 (UTC)