1724: Proofs

Explain xkcd: It's 'cause you're dumb.
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Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...
Title text: Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...


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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. 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. 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. 06:13, 24 August 2016 (UTC)
Me to, but I am on mobile, so editing is a pain 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 -- 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. 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 (talk) (please sign your comments with ~~~~)

"Dark magic" proofs are centered around properties of functions, and abstract concepts, rather than manipulating the functions themselves?? 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" 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. (talk) (please sign your comments with ~~~~)

Maybe he meant "dark patterns"? (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. 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 -- 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. 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. (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. -- 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. :) 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. 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. 01:40, 16 April 2021 (UTC)