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Wed, 09 Aug 2023
No plan survives contact with the enemy
According to legend, champion boxer Mike Tyson was once asked in an interview if he was worried about his opponent's plans. He said:
It's often claimed that he said this before one of his famous fights with Evander Holyfield, but Quote Investigator claims that it was actually in reference to his fight with Tyrell Biggs. Here's how Wikipedia says the fight actually went down:
Tyson's prediction was 100% correct! He went on to knock out Biggs seventh round. NoteThe actual quote appears to have been more like “Everybody has plans until they get hit for the first time”. The “punched in the mouth” version only seems to date back to 2004. Further research by Barry Popik. [Other articles in category /misc] permanent link Fri, 04 Aug 2023The standard individual U.S. Army ration since 1981, the so-called “Meal, Ready-to-Eat”, is often called “three lies for the price of one”, because it is not a meal, not ready, and not edible. [Other articles in category /misc] permanent link
Worst waterfall in the U.S. and Doug Burgum pays me $19
North Dakota is not a place I think about much, but it crossed paths with me twice in July. WaterfallsLast month I suddenly developed a burning need to know: if we were to rank the U.S. states by height of highest waterfall, which state would rank last? Thanks to the Wonders of the Internet I was able to satisfy this craving in short order. Delaware is the ⸢winner⸣, being both very small and very flat. In looking into this, I also encountered the highest waterfall in North Dakota, Mineral Springs Waterfall. (North Dakota is also noted for being rather flat. It is in the Great Plains region of North America.) The official North Dakota tourism web site (did you know there was one? I didn't.) has a page titled “North Dakota has a Waterfall?” which claims an 8-foot (2.4m) drop. The thing I want you to know, though, is that they include this ridiculous picture on the web site: Wow, pathetic. As Lorrie said, “it looks like a pipe burst.” The World Waterfall Database claims that the drop is 15 feet (5m). The WWD is the source cited in the official USGS waterfall data although the USGS does not repeat WWD's height claim. I am not sure I trust the WWD. It seems to have been abandoned. I wrote to all their advertised contact addresses to try to get them to add Wadhams Falls, but received no response. Doug BurgumDoug Burgum is some rich asshole, also the current governor of North Dakota, who wants to be the Republican candidate for president in the upcoming election. To qualify for the TV debate next month, one of the bars he had to clear was to have received donations from 40,000 individuals, including at least 200 from each of 20 states. But how to get people to donate? Who outside of North Dakota has heard of Doug Burgum? Certainly I had not. If you're a rich asshole, the solution is obvious: just buy them. For a while (and possibly still) Burgum was promising new donors to his campaign a $20 debit card in return for a donation of any size. Upside: Get lots of free media coverage, some from channels like NPR that would normally ignore you. Fifty thousand new people on your mailing list. Get onstage in the debate. And it costs only a million dollars. Money well spent! Downside: Reimbursing people for campaign donations is illegal, normally because it would allow a single donor to evade the limits on individual political contributions. Which is what this is, although not for that reason; here it is the campaign itself reimbursing the contributions. Anyway, I was happy to take Doug Burgum's money. (A middle-class lesson I tried to instill into the kids: when someone offers you free money, say yes.) I donated $1, received the promised gift card timely, and immediately transferred the money to my transit card. I was not able to think of a convincing argument against this:
Taking Doug Burgum's $19 was time well-spent, I would do it again. Addendum: North Dakota tourismOut of curiosity about the attractions of North Dakota tourism, I spent a little while browsing the North Dakota tourism web site, wondering if the rest of it was as pitiful and apologetic as the waterfall page. No! They did a great job of selling me on North Dakota tourism. The top three items on the “Things to Do” page are plausible and attractive:
Good stuff. I had hoped to visit anyway, and the web site has gotten me excited to do it. [Other articles in category /geo] permanent link Tue, 01 Aug 2023
Computational content of Gantō's axe
Lately I have been thinking about the formula $$((P\to Q)\land (\lnot P \to Q)) \to Q \tag{$\color{darkgreen}{\heartsuit}$}$$ which is a theorem of classical logic, but not of intuitionistic logic. This shouldn't be surprising. In CL you know that one of !!P!! and !!\lnot P!! is true (although perhaps not which), and whichever it is, it implies !!Q!!. In IL you don't know that one of !!P!! and !!\lnot P!! is provable, so you can't conclude anything. Except you almost can. There is a family of transformations !!T!! where, if !!C!! is classically valid !!T(C)!! is intuitionistically valid even if !!C!! itself isn't. For example, if !!C!! is classically valid, then !!\lnot\lnot C!! is intuitionistically valid whether or not !!C!! is. IL won't prove that !!(\color{darkgreen}{\heartsuit})!! is true, but it will prove that it isn't false. I woke up in the middle of the night last month with the idea that even though I can't prove !!(\color{darkgreen}{\heartsuit})!!, I should be able to prove !!(\color{darkred}{\heartsuit})!!: $$((P\to Q)\land (\lnot P \to Q)) \to \color{darkred}{\lnot\lnot Q} \tag{$\color{darkred}{\heartsuit}$}$$ This is correct; !!(\color{darkred}{\heartsuit})!! is intuitionistically valid. Understanding !!\lnot X!! as an abbreviation for !!X\to\bot!! (as is usual in IL), and assuming $$ \begin{array}{rlc} P\to Q & & (1) \\ \lnot P\to Q & & (2) \\ \lnot Q & (≡ Q\to\bot) & (3) \end{array} $$ we can combine !!P\to Q!! and !!Q\to\bot!! to get !!P\to\bot!! which is the definition of !!\lnot P!!. Then detach !!Q!! from !!(2)!!. Then from !!Q!! and !!(3)!! we get !!\bot!!, and discharging the three assumptions we conclude: $$ \begin{align} \color{darkblue}{(P\to Q)\to (\lnot P \to Q)} & \to \color{darkgreen}{\lnot Q \to \bot} \\ ≡ \color{darkblue}{((P\to Q)\land (\lnot P \to Q))} & \to \color{darkgreen}{\lnot\lnot Q} \tag{$\color{darkred}{\heartsuit}$} \end{align}$$ But what is going on here? It makes sense to me that !!(P\to Q)\land (\lnot P \to Q)!! doesn't prove !!Q!!. What I couldn't understand was why it could prove anything at all. The part that puzzled me wasn't that !!P\to Q!! and !!\lnot P\to Q!! wouldn't prove !!Q!!. It's that they would prove anything more than zero. And if !!(P\to Q)\land (\lnot P\to Q)!! can prove !!\lnot\lnot Q!!, then why can't it prove anything else? This isn't a question about the formal logical system. It's a question about the deeper meaning: how are we to understand this? Does it make sense? I think the answer is that !!Q\to\bot!! is an extremely strong assumption, in fact the strongest possible statement you can make about !!Q!!. So it's easist possible thing you can disprove about !!Q!!. Even though !!(P\to Q)\land(\lnot P\to Q)!! is not enough to prove anything positive, it is enough, just barely, to disprove the strongest possible statement about !!Q!!. When you assume !!Q\to \bot!!, you are restricting your attention to a possible world where !!Q!! is actually false. When you find yourself in such a world, you discover that both !!P\to Q!! and !!\lnot P\to Q!! are much stronger than you suspected. My high school friends and I used to joke about “very strong theorems”: “I'm trying to prove that a product of Lindelöf spaces is also a Lindelöf space” one of us would say, and someone would reply “I think that is a very strong theorem,” meaning, facetiously or perhaps sarcastically, that it was false. But facetious or sarcastic, it's funny because it's correct. False theorems are really strong, that's why they are so hard to prove! We've been trying for thousands of years to prove a false theorem, but every time we think we have done it, there turns out to be a mistake in the proof. My puzzlement about why !!(P\to Q)\land (\lnot P\to Q)!! can prove anything, translated into computational language, looks like this: I have a function !!P\to Q!! (but I don't have any !!P!!) and a function !!\lnot P\to Q!! (but I don't have any !!\lnot P!!). The intutionistic logic says that I can't use these functions to to actually get any !!Q!!, which is not at all surprising, because I don't have anything to use as arguments. But IL says that I can get !!\lnot\lnot Q!!. The question is, how can I get anything from these functions when I don't have anything to use as arguments? Translating the proof of the theorem into computations, the answer one gets is quite unsatisfying. The proof observes that if I also had a !!Q\to\bot!! function, I could compose it with the first function to make a !!P\to\bot\equiv \lnot P!! which I could then feed to the second function and get !!Q!! from nowhere. Which is very strange, since operationally, where does that !!Q!! actually come from? It's manufactured by the !!\lnot P\to Q!! function, which was rather suspicious to begin with. What does such a function actually look like? What functions of this type can actually be implemented? It all seems rather unlikely: how on earth would you turn a !!P \to \bot!! value into a !!Q!! value? One reasonable answer is that if !!Q = \lnot P!!, then it's easy to write that suspicious !!\lnot P\to Q!! function. But if !!Q=\lnot P!! then the claim that I also have a !!P\to Q!! function looks extremely dubious. An answer that looks good at first but flops is that if !!Q=\mathtt{int}!! or something, then it's quite easy to produce the required functions, both !!P\to Q!! and !!\lnot P\to Q!!. The constant function that always returns !!23!! will do for either or both. But this approach does not answer the question, because in such a case we can deduce not only !!\lnot\lnot Q!! but !!Q!! itself (the !!23!! again), so we didn't need the functions at in the first place. Is the whole thing just trivial because there is no interesting way to instantiate data objects with the right types? Or is there some real computational content here? And if there is, what is it, and how does that translate into the logic? Does this argument ever allow us to conclude something actually interesting? Or is it always just reasoning about vacuities? NoteAs far as I know the formula !!(\color{darkgreen}{\heartsuit})!! was first referred to as “Gantō's Axe” by Douglas Hofstadter. This is a facetious reference to a certain Zen koan, which says, in part:
(See Kubose, Gyomay M. Zen Koans, p.178.) Addendum 20230904Simon Tatham observed that this is a special case of the theorem: $$((P\to Q)\land ((P\to R) \to Q)) \to {(Q\to R)\to R}$$ which definitely has nontrivial computational content, and that the vacuity or !!\color{darkred}{\heartsuit}!! arises because !!R!! has been replaced by the empty set. Full details are here. [Other articles in category /math/logic] permanent link |