I'm watching Pardon the Interruption after work, and they're talking about the Belmont Stakes. They show a clip of horses running, and Emmy pipes up: "I like horses!" She does this when she feels I'm not paying her enough attention.
"Horses are okay," I say.
"Okay? Horses are really neat!" She thumps her tail on the floor, to emphasize the point.
"I guess." A really bad idea comes to me. "Say, did you know that all horses have an infinite number of legs?"
"Yeah," I say, pausing the DVR. "All horses have an infinite number of legs, and I can prove it with logic."
"Well, we know that horses have an even number of legs, right?"
"And we also know that horses have both forelegs and back legs, right?"
"Now, four legs plus two back legs is six legs, which is an odd number of legs for a horse..."
"I guess, but..."
"...And we already said that horses have an even number of legs. And the only number that is both even and odd is infinity, therefore all horses have an infinite number of legs."
She's silent. I look over, and she's staring at me with huge eyes. "Whoa...," she says. "I think you just blew my mind."
"Pretty good, huh?"
"Dude, I had no idea..." She continues to look amazed. "But why do they look like they only have four legs on tv?"
"No, no-- it's a joke. They really do only have four legs."
"But... you used logic.... What about the back legs?"
"It's not really logic. It's a kind of Groucho Marx fast-talking fake logic thing. Punning off 'forelegs' meaning the two front legs, and acting as if it were 'four legs," giving the number of legs."
"Oh." She still looks confused. "Why would you do that, though?"
"It's an old math joke. There's a whole big set of joke math proofs out there if you look for them. There are a bunch of different ways to 'prove' that 1=2, as well." I do the wiggly fingers for the quote marks, just to make sure she gets the point. Being a dog, she sometimes misses those little details. It's very trying.
"Why would anybody want to do that?"
"Well, for one thing, it's funny." She cocks her head sideways. "Okay, fine it's funny if you have the right sort of personality. It's also a useful reminder about the importance of checking all your assumptions."
"Well, the 'proofs' that 1=2 rely on slipping some illegitimate operation in in a way that looks plausible. You always end up dividing by zero, or something like that, but if you're not paying close attention, you can miss the incorrect step."
"I guess that makes sense. But how many reminders to be careful about math do humans need, anyway?"
"More than you might think. Whole fields of math and science have been created by people realizing that they were assuming something that wasn't true."
"Well, relativity, for example. One of the key realizations in Einstein's special theory of relativity is that time is different for different observers. Prior to 1905, people kept running into contradictions when they thought about light emitted by moving objects, because they were making the assumption that there was a single, universal time for every observer in the universe."
"There isn't, though, because moving clocks run at different speeds, right?" She thumps her tail on the floor again. She's always happy when she sees the connection to other areas of physics.
"Exactly. Special relativity really flows from the realization that different observers will disagree about the timing of events. Once you have that idea, you can work out the rest of it relatively easily."
"Pardon the pun..."
"Hmmm? Oh, yeah, pardon the pun. General relativity also makes use of an area of mathematics that comes from challenging an assumption that seems really obvious. The mathematics describing curved spaces was developed by people who decided to see what you could do with geometry if you lifted the requirement that parallel lines never intersect."
"Why would you do that?"
"I think it started as an attempt to prove that the parallel line postulate had to be true, by showing that geometry wouldn't work without it. It turned out to work just fine, though, and gave rise to a bunch of interesting mathematics. And then Einstein worked out that acceleration is like a curvature of space, and all of that stuff turned out to be useful."
"Yep. So, you see, joke 'proofs' are useful. In addition to being funny."
"I still don't see the humor, but whatever."
"Right, whatever." I sit back again, and hit "Play." Mike and Tony resume talking about racing.
"Anyway, as I was saying, horses are neat!"
"Even if they don't have an infinite number of legs."
"Yeah. Their crap smells fantastic!" She thumps her tail enthusiastically. I really don't know what to say to that, so I just don't say anything.
Ah, but what COLOR are they? Granted that's not a joke, just a bad proof, but still.
Also, I'm surprised that Emmy didn't realize that SHE ALSO has an infinite number of legs!
And Alexander the Great also had an infinite number of limbs....
And Alexander the Great also had an infinite number of limbs....
I've always wondered if the geometry of curved spaces developed partly because people doing the great land surveys (like the one in India that resulted in the first measurement of Mt Everest) of the early 19th century knew that the angles of a triangle added up to more than 180 deg if you drew a big enough one on the earth.
CCPhysicist #4: Non-Euclidean geometry was discovered in the early 19th century, before the surveys you referred to. I'm not sure exactly when spherical trigonometry was developed, and that's really all you need for this kind of survey. Negatively curved geometric space would have been a purely mathematical curiosity before general relativity was developed, although positively curved geometry does have application to navigation on a sphere (of course, flat means Euclidean).
I would argue that joke proofs are useful, but not funny.
The survey of India was started at the beginning of the 19th century (1802), well before Riemann was even born. Although it took decades to complete this extremely precise measurement, the geometry of positively curved spaces was well known by surveyors in the 1700s, a century before Riemann's work on that subject.
Hence my question whether this knowledge played any role in motivating Riemann to unify flat and spherical geometry and extend this to include negative curvature. According to Wiki, Riemann's first lectures on non-Euclidean geometry were just two years after Mt. Everest was measured. He would certainly have known about that, and might also have known that the value for the oblateness of the earth had been revised decades before based on the survey of India.
CCPhysicist: I think the connection you are looking for involves Gauss. Riemann drew heavily upon Gauss' monograph (IIRC, Gauss was one of the examiners who approved Riemann's Habilitationsschrift). In addition to basically inventing differential geometry, Gauss was also involved in land surveys.
Reminds me of the old logic proof that cat have three tails--(a)no cat has two tails, and (b)one cat has one more tail than no cat, therefore (c)a cat has three tails.
I don't think that one will impress Emmy, though.
Yeah, but if you change cats for squirrels in the above preposition, then she'll get it first go.
And then probably mention that she knew all along that those squirrels were up to something and has now found out it's tail smuggling.
Some comments of CCPhysicist's speculation:
1)Riemann's Habilitationvortrag (1854) was two years before the public announcement of the height of Everest (1856).
2)I don't see why Riemann must have known about the survey of India!
3)For Eric Lund: To do surveying you only need plane and not spherical trigonometry .
4)Also for Eric Lund: Spherical trigonometry first saw the light of day in the Sphaerica of Menelaus in about 100 C.E.
I've heard a story that Gauss actually climbed three mountains repeatedly to measure the angle between their peaks. He wanted to see if the sum of the angles really was pi. If it's true, it combines two big ideas - the first that the real world doesn't necessarily have to be Euclidean, just because we think it should be, the second that you should take lots of measurements and use statistics to judge the significance of the result. I've got no idea if there's any truth to the story, though.