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Conterintuitive facts in mathematics, CS, and physics

• kbenson 5y ago
  So, are there any popular or commercial games that make use of intransitive/non-transitive dice? That seems like it would be fun to break out with friends over.
• toolslive 5y ago
  the planet closest to earth is Mercury.
• MatteoFrigo 5y ago
  The list is missing one of the most astonishing discoveries of all time: if you reflect the universe in a mirror, you can tell whether you are in our universe or in the mirror because the laws of physics are different in the mirror. See https://en.wikipedia.org/wiki/Wu_experiment https://en.wikipedia.org/wiki/Wu_experiment
  • kadoban 5y ago
    Really like that one, especially because it's just so ... random. All the other forces just don't work that way, but the Weak force does, wtf.
  • cycomanic 5y ago
    It is a great discover I agree, it might not quite fit in the list because it's not so counterintuitive for someone not into particle physics though. There are actually many examples of things working different for the mirror image, e.g. many medical/chemical compounds work differently based on chirality. The topic of chirality is fascinating, one of the big puzzles in nature is for example why it so strongly prefers right-handedness.
    • goohle 5y ago
      Maybe nature prefer right-handedness just because it started in North hemisphere.
    • Ginden 5y ago
      > There are actually many examples of things working different for the mirror image, e.g. many medical/chemical compounds work differently based on chirality. It "works different" because you consider only change in one part of system. You change compound, but rest of system stays the same. If you mirror entire system, it works almost the same way (differences due to parity violation by weak force are millions time smaller than required to even detect differences).
  • amelius 5y ago
    How can someone rigorously prove that? E.g. perhaps there is something inside quarks that has chirality which we haven't discovered yet.
    • oehtXRwMkIs 5y ago
      One can't. It's science. One can't rigorously prove anything via induction.
• 1f60c 5y ago
  > 15. Two 12-inch pizzas have less pizza than one 18-inch pizza. That is surprising, because it's not true. Any calculator will confirm that 2 (2 π×6) > 2 π×9. Or am I missing something?
  • detaro 5y ago
    That's the formula for circumfence, not area.
    • 1f60c 5y ago
      Ahhhh, of course. facepalms
      • erdewit 5y ago
        It's still a good proof that the smaller pizza's have more crust.
  • _Microft 5y ago
    Area is calculated using pi*r^2: 2 * pi * 6^2 < pi * 9^2
    • 1f60c 5y ago
      Thank you. I feel kind of stupid now.
  • bryanrasmussen 5y ago
    it's true but really what they're missing is the number and variety of toppings on these pizzas.
    • rawling 5y ago
      Variety maybe, but surely number is also proportional to area (and presumably it's worse because "crust width" doesn't scale with diameter?)
• jcims 5y ago
  I needed a half cup of something for a recipe and only found the 1/3 cup measure. Then it occurred to me that a third and a half (of a third) is equal to a half. So simple but somehow doesn't feel right.
  • raldi 5y ago
    3 measures = 1 cup Now divide both sides of the equation by 2.
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• whiterock 5y ago
  very cool collection - truly something to pick from at a party. should have much more upvotes imho.
  • anonymousDan 5y ago
    Agreed, some real mindbenders there!
  • reidjs 5y ago
    I have actually told this one at a party (well, not exactly a rager) "Two 12 Inch Pizzas have less Pizza than one 18 inch pizza." and we did the math to prove it to ourselves. Blew our minds... an actual valid use for middle school math.
    • deleted 5y ago
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• tunesmith 5y ago
  Remedial question about zero-knowledge proof. Isn't "proof" a misnomer since the concept is about just making it incredibly likely to be true?
  • eruleman 5y ago
    No — the word 'proof' is accurate, it establishes with certainty that you know the value of x.
    • Kranar 5y ago
      I thought you were right until I decided to look into it myself, turns out we were both wrong. Zero-knowledge proofs are probabilistic and contain a soundness error which is the probability of guessing the correct answer. This can, in some but not all cases, be brought down arbitrarily close to 0, but it can never be 0 exactly. https://en.wikipedia.org/wiki/Zero-knowledge_proof#Definition https://en.wikipedia.org/wiki/Zero-knowledge_proof#Definitio...
  • pcmonk 5y ago
    That's why it's sometimes called an "argument" or specifically a "cryptographic proof". You can construct the statement such that it can be "proven" in a traditional sense by adding qualifiers such as "with probability more than 1-1/(2^256)". You'll generally need an assumption like knowledge-of-exponent or at least hash soundness.
  • db_admin 5y ago
    In cryptography you prove security through showing that an adversary has a negligible [1] chance of winning a game of guess the secret. [1] https://crypto.stackexchange.com/questions/5832/what-exactly-is-a-negligible-and-non-negligible-function#5833 https://crypto.stackexchange.com/questions/5832/what-exactly...
  • dlubarov 5y ago
    In the literature [*], the "proof" is an actual proof, but that's not what the prover sends to the verifier. Rather, the verifier queries random pieces of the proof, eventually convincing themselves that with high probability, the prover knows a valid proof. [*] At least in most of the older literature descending from the PCP literature. Modern papers sometimes abandon that distinction and just use "proof" and "argument" interchangeably.
• jancsika 5y ago
  > Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it: https://en.wikipedia.org/wiki/The_Market_for_Lemons https://en.wikipedia.org/wiki/The_Market_for_Lemons From that Wikipedia page: > This means that the owner of a carefully maintained, never-abused, good used car will be unable to get a high enough price to make selling that car worthwhile. This is bullshit-- the word "enough" was sneaked in there without any rationale provided. The most we can say is that a seller-- if they decide to sell-- won't get as high a price as they would if the information assymmetry didn't exist. But that's just a truism. I'd be willing to agree for the sake of argument that we are representing humans here as some commonly known set of JSON values. But before we go anywhere else from that argument, I'd need to know that the speaker will at some point halt the simulation and come back to Earth with insights into the real world. Does that happen in this paper? If not, then how does the paper have relevance for the economic transactions among the set of bona fide human beings?
  • rumpelstilz18 5y ago
    > Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it > This means that the owner of a carefully maintained, never-abused, good used car will be unable to get a high enough price to make selling that car worthwhile. Both statements are a wrong understanding of the phenomenon. The market collapses because the transactions happen outside of the market. Something similar has in my opinion happened on the job market. Most good jobs are outside the regular job boards. Most good applicants could not be bothered to apply for a job opening but are asked by recruiters or friends.
    • pishpash 5y ago
      Under this model, if the transaction of quality goods happened anywhere, some buyer reached information symmetry. But among the assumptions in the article is that sellers have no alternate market (they just "leave", whatever that means) and there is no credible way to provide information on the quality of goods. But then how do buyers know to lower their prices with what's left in the market? There're a lot of unreasonable assumptions and maybe that's why the paper was rejected 3 times. You can also run the thought experiment backwards: lemons should be removed from the market first since, as stated, those are the ones that sell, but then you should be left with a market full of peaches.
• tunesmith 5y ago
  Someone asked about #4 then deleted after I typed the response, so here it is.. :) It's queuing theory in general, related to "utilization". The utilization curve is always shaped the same; 50% utilization always doubles waiting time, and the curve is practically vertical when you get to 99% utilization. That's what explains the difference - 5.8 customers per hour, for a throughput of 6 per hour, shoves the efficiency to the almost-vertical part of the curve, which impacts waiting time.
  • hinkley 5y ago
    What really throws people for a loop is the wait time after queuing starts. People expect the wait time to drop once arrivals return to normal, but there just isn't capacity to catch up. In the real world, except at the DMV, people give up and shorten the queue (or don't join it to begin with), causing the arrival rate to go below nominal allowing the workers to catch up. In must-have or automated situations they see the full consequences of under-provisioning.
    • stevetodd 5y ago
      This also explains the supply chain issues we are experiencing now because of the move to just-in-time manufacturing across the board.
      • galaxyLogic 5y ago
        Cool. Ships waiting to dock on California cost. This explains it!
• antognini 5y ago
  Here is one of my favorites: The specific heat of a star is negative. (This also applies to a galaxy or any other gravitationally bound object.) As a star loses energy, it heats up. If you inject it with energy, it cools down. It's a trivial corollary from the virial theorem, but it leads to counterintuitive behavior (like the gravothermal catastrophe).
  • joombaga 5y ago
    So what happens to a star inside a "perfect" Dyson sphere?
    • antognini 5y ago
      The same thing that happens to a star not in a Dyson sphere. It slowly radiates energy away and in the process contracts and heats up. In fact when the Earth was initially forming the Sun was only about 70% as bright as it is today. (How liquid water could form under those conditions is still something of a mystery: https://en.wikipedia.org/wiki/Faint_young_Sun_paradox https://en.wikipedia.org/wiki/Faint_young_Sun_paradox)
      • raxxorrax 5y ago
        I really wish we would have a class K or M star as they would have a far higher life expectancy. The theorized instability in form of solar flares would probably be an acceptable compromise. Perhaps we could just use a straw to give our sun a liposuction.
• bo1024 5y ago
  Great list! > 33. "...if you flip fair coins to generate n-dimensional vectors (heads => 1, tails => -1) then the probability they're linearly independent is at least 1-(1/2 + o(n))^n. I.e., they're very very likely independent! Counterintuitive facts about high dimensional geometry could get their own list. A side-1 cube in n dimensions has volume 1 of course, but a diameter-1 sphere inside it has volume approaching zero! The sphere is tangent to every one of the 2n faces, yet takes up almost none of the space inside the cube. Note that the distance from the middle of any face of the cube to the opposite face is 1, yet the length of a diameter of the cube (corner to opposite corner) is sqrt(n).
  • spott 5y ago
    as an aside: what the hell is the "o(n)" here? Some term that is proportional to n?
  • tgb 5y ago
    Only sort of true. It doesn't make sense to compare n dimensional volume to n+1 dimensional volumes, so the limit of the volume of an n-sphere isn't meaningful. The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising. In particular, it's equally valid and frankly nicer to define the unit n-sphere to be volume 1 rather than the unit cube. Do that and we see that this statement is just saying that the n-cube grows in volume to infinity, which makes sense given the fact you point out that it contains points increasingly far from the origin. I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.
    • bo1024 5y ago
      > The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising. This is why I start by recalling that the volume of the n-cube is always one, as the frame of reference. But I think people still find it surprising. Hard to tell, because... > I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy. Hard to tell, because I don't find any of these facts surprising anymore -- would guess you're in the same boat! Another good one is how you can fit exp(n) "almost-orthognal vectors" on the n-sphere.
• 6gvONxR4sf7o 5y ago
  > 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox https://en.wikipedia.org/wiki/Potato_paradox I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another. > 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood https://gwern.net/Littlewood This one, on the other hand, I don't like. Depending on a whole bunch of subjective definitions, a one-in-billion event can happen a million times a second or practically never or whatever else you choose.
  • lotsofpulp 5y ago
    >https://en.wikipedia.org/wiki/Potato_paradox https://en.wikipedia.org/wiki/Potato_paradox The Wikipedia link above says: > Fred brings home 100 kg of potatoes, which (being purely mathematical potatoes) consist of 99% water. He then leaves them outside overnight so that they consist of 98% water. What is their new weight? The surprising answer is 50 kg. It annoys me when mass is used interchangeably with force (weight), so I went to the Wikipedia source, and the source is accurate in using units of force throughout. https://web.archive.org/web/20140202214723/http://www.daviddarling.info/encyclopedia/P/potato_paradox.html https://web.archive.org/web/20140202214723/http://www.davidd... Wonder why the person that wrote the Wikipedia article changed it up when it is supposed to be a direct quote.
    • deleted 5y ago
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    • jgtrosh 5y ago
      In this situation, the mass and weight are proportional and irrelevant to the problem. Other than proper respect of units, why would it really matter? I would agree that using mass+kg would remain correct and be less unusual, but it doesn't matter a lot.
      • lotsofpulp 5y ago
        It does not matter, it is just a pet peeve of mine. Might be due psychological trauma from when I was a kid and arguing with an older cousin about how pounds and kilograms are not units of the same thing, and the older cousin "winning" the argument in the eyes of the elders because the cousin was quite a few years older than me and considered to be smart in school.
• hodgesrm 5y ago
  > 0% selected the right answer on this SAT question: Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will circle A revolve in total? That's fun. I of course immediately selected 3 which means I could have a bright career in test preparation ahead of me.
  • clon 5y ago
    It was implied that the correct answer is 4.
    • caf 5y ago
      Similarly, if you set out in a boat and circumnavigate the world in an easterly direction, ticking off a day on your calender every sunset, when you arrive back at your setting-off point your calendar will be one day ahead of everyone else.
      • bongoman37 5y ago
        Wasn't this a major plot point in Around the World in 80 Days?
  • kristjansson 5y ago
    That one got me good, so my future at College Board is as bright as yours. However, I don't think the argument by demonstration in that video is particularly convincing. Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - it traces a circle whose radius is the sum of the two radii. So, if 3 * r_small = r_big, the center of the small circle has to travel 2 * pi * (r_s + r_b) = 4 * 2 * pi * r_s, then divide by 2 * pi * r_s per rotation to get 4 rotations.
    • hinkley 5y ago
      Without the math: Radius is always proportional to circumference, so a circle twice the size is twice as big around. Take the case of two identical circles. To move a point on the first circle from 12 o'clock back to 12 o'clock, it only goes halfway around the other circle, which you can prove to yourself by imagining you've wrapped a string around the circle and marked it at 12 and 6 o'clock. If you unwrap half of the string and wrap it around the other circle, then the end of the rope is at 6 o'clock. To roll the string back up by moving the circle, the top of the circle will be pointing upward again when it reaches the bottom. 1 full revolution. Now wrap the other half of the string around the other side, 6 has to go back to the bottom again to roll the string back up. 2 revolutions.
• dwohnitmok 5y ago
  29 is not correct as stated and falls prey to logical errors. Hamkins presents a formal take on it here: https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb https://mathoverflow.net/questions/44102/is-the-analysis-as-... His conclusion (which I agree with) is > The claims made in both in your question and the Wikipedia page on the existence of non-definable numbers and objects, are simply unwarranted. For all you know, our set-theoretic universe is pointwise definable, and every object is uniquely specified by a property. Despite arguments about countability, which ignore how difficult it is to pin down "what is definable," it is possible (although not necessary) for all real numbers to be describable/definable in ZFC.
  • wodenokoto 5y ago
    It’s also terribly phrased. > The vast majority of real numbers can't be described. But it is impossible to give a single example. If we accept the first sentence and assume the second refers to indescribable numbers, then isn’t it obvious that we have no examples of things we cannot describe? If the second sentence refers to real numbers in general I can give one example or two.
    • tshaddox 5y ago
      How could you give an example? The example you give would be a description of a number.
      • Kranar 5y ago
        He just means the phrasing is kind of poor. A general description of an undefinable number could be something like: given a sequence of Turing Machines, T_1, T_2, ..., T_n, where T_1 is the smallest representation of Turing machine over a grammar G, and T_2 is the second smallest representation of a Turing machine over G, and T_3 is the third smallest etc... take the limit of the ratio of said Turing machines that halt to Turing machines that don't halt as n goes to infinite. I mean that's a description of some number, you could even write it out mathematically or write an algorithm to express that number. Of course neither the algorithm or the formula will ever converge and yet it will also always be bounded between 0 and 1 (hence it doesn't diverge to infinity). So is that a description of a number? Well sure in one sense I just described it, there is only one single real number that can satisfy that description, and as I said I could in principle write it out formally and rigorously... and yet in another sense it also doesn't describe anything since no matter how hard you try, there will always be at least two real numbers that could potentially satisfy the definition and no way to eliminate one of them.
• brrrrrm 5y ago
  a couple more: - at any time while stirring a cup of coffee, there will be a point on the top that is right where it started. (if we pretend coffee stirring is 2-dimensional, Brouwer's Fixed Point theorem) - a drunk man will eventually make it home, unless he can fly. in which case he only has a 34% chance. (if we assume the man is walking/flying on a grid, Pólya's recurrence theorem)
  • robocat 5y ago
    - a chair with four even legs placed on any undulating continuous surface can always be rotated such that all four legs touch the ground at once.
    • caf 5y ago
      Do the legs have to be even?
      • istjohn 5y ago
        Yes. Imagine a chair with one pair of diagonally opposite legs very long and the other pair of legs very short. On even a flat surface it is impossible to touch all four legs to the floor simultaneously.
      • twic 5y ago
        Not even, but the feet have to be coplanar.
        • caf 5y ago
          And since the spacing between the feet is obviously irrelevant, that further implies that it's always possible to find 4 coplanar points on the surface arbitrarily close together. Or arbitrarily far apart.
  • nicolas-siplis 5y ago
    Huh, I wonder if there's any relationship to the Hairy Ball theorem. They appear to describe similar situations, but on different dimensions.
• Causality1 5y ago
  Considering spacetime, matter, and energy are all quantized, why is something like Gabriel's Horn significant? I don't see how it has any more relation to reality than phrases like "negative surface area" would. Also, it's patently absurd someone would include Fitch's Paradox, a piece of philosophy, on a list of "counterintuitive facts."
  • School-Cotton 5y ago
    Of course Gabriel’s horn doesn’t exist in physical reality, but it’s still interesting that such a thing exists in a mathematical theory that is normally a pretty good model of physical reality.
  • mrestko 5y ago
    I don't think we know that spacetime is quantized.
  • pdonis 5y ago
    > Considering spacetime, matter, and energy are all quantized First, we don't know that spacetime is quantized; that's a plausible speculation but we have no theory of quantum gravity. Second, "quantized" is not the same as "discrete". A free particle in quantum theory is "quantized" but the spectrum of all of its observables is continuous.
    • wnoise 5y ago
      It's not even a plausible speculation; all of the best models we have, quantum mechanics, special relativity, quantum field theory, general relativity, and string theory have a fully continuous space-time. The one notable exception is loop quantum gravity.
  • guerrilla 5y ago
    I think you're taking things too seriously (and in one case not seriously enough): These are all conclusions that are true if their premises are true. Some of their premises can obviously be satisfied, others obviously can't and many others are in between (unknown or debatable.) They're all counterintuitive results though.
• pishpash 5y ago
  The most counterintuitive fact, er... fib is this nugget: "0% selected the right answer on this SAT question: Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will circle A revolve in total?" You know how hard it is to get 100% of the people to do something? Don't insult our intelligence like that, or of SAT test takers in general.
  • IainIreland 5y ago
    The correct answer was not provided as a multiple choice option, which means by definition that 0% of test takers selected it.
    • pishpash 5y ago
      All right, that's certainly counterintuitive...
• niccl 5y ago
  Missing one that still makes no sense to me: The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12 https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF
  • t8e56vd4ih 5y ago
    what? that's incredible ...
    • kadoban 5y ago
      Literally incredible, because it's not. It's just really awful notation. It's really more like F(1+2+3+...) = -1/12 for a certain F (pedantically it's not a function I don't think, but whatever).
  • bpodgursky 5y ago
    > where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning
  • charcircuit 5y ago
    If you don't use the regular definitions of how things are defined, you can get surprising results. If I define + as -, it might be surprising if I said 1 + 1 = 0.
  • barbs 5y ago
    This is a good video explaining it: https://www.youtube.com/watch?v=w-I6XTVZXww https://www.youtube.com/watch?v=w-I6XTVZXww Although it relies on a particular interpretation of the left side of the equation, it's used in some areas of physics.
    • crvdgc 5y ago
      I find a response video to it more enlightening. It's often found as the number one related video. https://youtube.com/watch?v=YuIIjLr6vUA https://youtube.com/watch?v=YuIIjLr6vUA
• lapetitejort 5y ago
  My favorite paradox in physics: what happens when you spin a disk at relativistic speed? The circumference should contract, since it's parallel to the direction of motion, but the radius is perpendicular, and thus should not contract. https://en.wikipedia.org/wiki/Ehrenfest_paradox https://en.wikipedia.org/wiki/Ehrenfest_paradox
  • UnFleshedOne 5y ago
    I think if you rotate the disk at maximum speed ignoring all material material properties, all points on disk will approach speed of light and you will end up with "spin through" because layers won't be able to maintain same angular speed (needed to consider the thing a disk) with fixed linear speed...
  • reidjs 5y ago
    Relativistic paradoxes are the most insane. Twin paradox, ladder paradox, grandfather paradox, things get so mindbendingly weird at very large, very fast, and very small scales.
  • hinkley 5y ago
    I believe Veritasium covered this: spinning at relativistic speeds generates centripetal forces that overcome the nuclear force. There is no material you can use to test this paradox, and there can be no such material because your device would atomize - or worse - in the attempt. Effectively there are g-forces so high that you end up with subatomic particles.
    • caf 5y ago
      Yes, this is similar to transmitting a message faster than the speed of light by moving a very long perfectly rigid rod; a perfectly rigid rod is not realisable and neither is a perfectly rigid disk.
    • Jensson 5y ago
      Relativistic speed rotations means that you are close to trapping light, meaning your force is similar to being very close to the event horizon of a black hole. And what is one of the properties of gravity wells like black holes? Well, their circumference isn't equal to 2 pi times radius, since the space has enough curvature to not be flat, similar to how circumference isn't 2 pi times radius if your space is a 2d sphere. So the paradox is correct, you would get that effect, at least if you used a black hole to cause the fast rotation. Although I'm not sure if the maths adds up to be the same.
• doublepg23 5y ago
  I work in customer service and the queuing theory is something I noticed while working. A single cashier is usually unsustainable for very long but two will get you through quite a rush. Very cool to see it formally expressed.
• GPerson 5y ago
  I didn’t see this one listed, and thought it was pretty cool when I studied it in a course a few years ago: https://en.m.wikipedia.org/wiki/Skolem's_paradox https://en.m.wikipedia.org/wiki/Skolem's_paradox “ Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says (or that precisely says in the standard model of the theory) that there exist sets that are not countable.” “ Skolem went on to explain why there was no contradiction. In the context of a specific model of set theory, the term "set" does not refer to an arbitrary set, but only to a set that is actually included in the model. The definition of countability requires that a certain one-to-one correspondence, which is itself a set, must exist. Thus it is possible to recognise that a particular set u is countable, but not countable in a particular model of set theory, because there is no set in the model that gives a one-to-one correspondence between u and the natural numbers in that model.”
  • H8crilA 5y ago
    Just to make this more concrete: 1. There is a countable model of real numbers. 2. There even is a countable model of the entire set theory.
    • Viliam1234 5y ago
      > There is a countable model of real numbers. What exactly happens when you try to apply Cantor's diagonal argument to this model? I guess that at some step, you get an answer like "outside of the model, yes this exists, but inside the model the answer is no", but I would like to see it precisely, how exactly the in-model reasoning diverges from the outside-model reasoning.
      • matt-noonan 5y ago
        Inside the model, “the reals are uncountable” means you have two sets R and N, and there is no surjective function from N onto R. That function would be a set as well; a certain subset F of NxR, say. But even if we can externally enumerate R, there is no reason to expect that our external enumeration corresponds to a set F that exists in the model.
• smoldesu 5y ago
  They list homomorphic encryption as the first fact, but has anyone created a truly complete HE implementation yet? My impression is that there are a lot of great theories and experiments in the field, but nobody has really created a practical standard for it. I'd love to be proven wrong though, it's a fascinating field.
• charcircuit 5y ago
  >It is possible to compute over encrypted data without access to the secret key I don't think this is counterintuitive for most people. The most basic encryption scheme that everyone knows is the Caesar cipher. It's easy to see that shifts of the cipher text will cause shifts in the plain text.
  • randomswede 5y ago
    But in a Caesar cipher only some types of computation is possible (to wit, addition and subtraction). Now, "multiplication" of letters is, well, dodgy as a concept. But, the thing is that you can build encryption systems where D(E(k) op V) is equal to k op V, and op contains both addition and multiplication.
  • dfdz 5y ago
    I agree, I really don’t like this one either. There are many things in math that are counterintuitive, but the idea of a homomorphism is not one of them in my opinion. Once someone explains the idea, and provides a few examples it is very natural. I also don’t like the text explaining zero knowledge proof. It needs the phrase “practically speaking” somewhere or “for practical purposes” since it’s not true in a strict sense But overall there were some fun ideas on the list!
    • Ar-Curunir 5y ago
      What do you mean by the line about zkps? We have perfectly-hiding proofs that reveal no information about the secret information, no matter how powerful the adversary is.
      • chriswarbo 5y ago
        Yes, but they're not proofs in the mathematical sense, since there's always an (exponentially-shrinking) chance that the answers were only correct due to coincidence.
        • dfdz 5y ago
• gfody 5y ago
  John Conway's free will theorem could go here
• api 5y ago
  I love “causation does not imply correlation.” We still reason so much from the implicit premise that correlations are meaningful. It’s so intuitive, but wrong. The thing that finally got me to wrap my mind around it was this: https://tylervigen.com/spurious-correlations https://tylervigen.com/spurious-correlations
• Ansil849 5y ago
  For some reason, the one I have the most trouble intuitively grasping is "Two 12 Inch Pizzas have less Pizza than one 18 inch pizza."
  • guerrilla 5y ago
    The area enclosed by a circle is πr^2 while 12 and 18 are the diameters, right? The radius of a disc is half the diameter, so 6 and 9, respectively. 2π6^2 < 1π9^2 ~~ 226.2 < 254.5. In other words the area is proportional to the square of the radius, not linearly proportional to the diameter in any way.
    • Ansil849 5y ago
      > In other words the area is proportional to the square of the radius, not linearly proportional to the diameter in any way. Awesome, thanks for that geometry refresher, that makes sense now of course :).
      • caf 5y ago
        Similarly, if the toppings are distributed evenly across the pizza, the average distance of a piece of topping from the edge is one third of the radius: most of the pizza is near the edge.
  • 77pt77 5y ago
    18/12 = 1.5 > sqrt(2) = 1.4..
  • Viliam1234 5y ago
    To make it simpler, let's assume that the shape of pizza is a square, i.e. we are talking about a 12×12 inch pizza and a 18×18 inch pizza. This increases the size of both pizzas by the same ratio, so it shouldn't change the answer to "are two small pizzas smaller than one large pizza?". Now let's measure the size in a new unit which is 6 inch long. So the small pizza is 2×2 units, and the large pizza is 3×3 units. twice 2×2 = twice 4 = 8 3×3 = 9
• londgine 5y ago
  > There are as many whole positive numbers as all fractions According to a specific non-colloquial definition of "as many".
  • adenadel 5y ago
    Hm, I'm not so sure that just because the definition of a bijection is technical that it is not intuitive. I'll start an enumeration of the rationals 1 1/3 2 1/4 3 1/5 ... If you can prove that you can do this, is that really so non-colloquial? It is certainly what we mean by "as many" for all finite sets, so what is wrong with doing this for infinitely many sets?
    • londgine 5y ago
      If every whole positive number is a fraction, but not every fraction is a whole positive number, then colloquially, I wouldn't define them as having "as many" elements as each other. Now, if you want to say they have the same cardinality (and you define cardinality as existing a bijection), then I would agree fully.
      • istjohn 5y ago
        What's worse, there are an infinite number of fractions that equal each whole number.
      • iovoid 5y ago
        Wouldn't there be exactly twice (or twice + 1, if you allow negative fractions) as much fractions, since fractions are represented as two positive numbers (plus a bit if you consider the sign). (The encoding could be "represent both numbers in binary, put the denominator in the odd bits (LSB = first bit), and the numerator in the even bits" so 2/3 => 10/11 => 1110 => 14)
      • foldr 5y ago
        The thing is that you could also use this kind of logic to show that there are more natural numbers than there are natural numbers. For example, you could associate 1 with all of the numbers from 1 to one million, and still have enough numnbers 'left' to associate each of 2,3,... with distinct natural numbers above one million.
• campital 5y ago
  How is 27 counterintuitive? > Let alpha = 0.110001000000000000000001000..., where the 1's occur in the n! place, for each n. Then alpha is transcendental. (Calculus, 4th edition by Michael Spivak) Nearly all infinite sums involving factorial are transcendental.
  • bloak 5y ago
    I agree. Probably most people who know what "transcendental" means would guess that the number described is transcendental. However, only a small proportion of people who know what "transcendental" means are capable of proving that any number is transcendental.
  • adenadel 5y ago
    In fact, almost all numbers are transcendental (algebraic numbers have measure zero).
• enthdegree 5y ago
  Great list. The statement “causation does not imply correlation”, while true in some contrived settings, is obstructive as a heuristic in my opinion. It is an anti-productive and unimportant diversion in any setting I can imagine it coming up. Causation does indeed imply correlation, by tautology, using a very reasonable definition causation in the context of linear correlations.
• quickthrower2 5y ago
  Monty Hall is there, albeit inside the answer to a stack overflow question linked in Misc #33
• geogra4 5y ago
  >Two 12 Inch Pizzas have less Pizza than one 18 inch pizza Love this one. Always go for the biggest pizza you can
  • TMWNN 5y ago
    No. While two 12" pizzas have more pizza than one 18" pizza, it's possible for the two 12" pizzas to be a better value. If the total price of the two 12" pizzas is $7.20, for example, they are a better value than the 18" pizza once the price of the 18" pizza is greater than $8.10. More generally, the two 12" pizzas are a better value if the 18" pizza's price is higher than 112.5% of the two 12" pizzas' total price.
    • dartharva 5y ago
      But that's almost never the case.
      • TMWNN 5y ago
        How do you know, unless you do the calculation?
        • dartharva 5y ago
          Don't know about you, but I have never seen any place that charges more for one large pizza than for two medium pizzas.
          • geogra4 5y ago
            For example the place by me charges 11 for a 12 inch pizza and 25 for a 24 inch. That's 4x the pizza for a little over twice the price!
• analog31 5y ago
  >>> 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood https://gwern.net/Littlewood This is certainly counterintuitive, given that one-in-a-billion and 8 per month have different units of measure.
  • elcomet 5y ago
    The implicit information is that there is one-in-billion chance that this event happens to someone in a given month.
• d_burfoot 5y ago
  Here is one of my favorites. Take a light source and shine it at a filter that is polarized up/down that is in front of a filter that is polarized left/right. None of the light will get through: the first filter removes all of the L/R components of the light, and the second filter removes all the remaining light. Now add a third filter, between the two, which is polarized at 45 degrees. Now some of the light goes through! If this doesn't surprise you, imagine there was a man firing a machine gun at a pair of walls. The two walls are thick enough that they absorb all the bullets. But when you add a third wall in between them, some of the bullets go through.
  • marcosdumay 5y ago
    Not surprisingly, this is related to the observer effect of Quantum Mechanics.
  • ffhhj 5y ago
    For me this is as surprising as the double slit experiment, but much easier to reproduce.
  • jiggawatts 5y ago
    It makes more sense if you think of the polariser sheets as fences with vertical bars, and light like a string going through between two bars and being wiggled. If the direction of the waves in the string are at right-angles to the slit, the wave can't propagate through the bars. If the wave is at an angle, then some of the wave gets through, with a maximum of 100% when the wave wiggles up and down in the same direction as the slit. In this model you can visualise how three 45-degree slits will allow some light to go through even though two 90 degree slits don't -- the light is transformed by its passage through the polarisers. Where QM people get themselves confused is that they assume that the light is not transformed, even though it clearly is... I.e.: If the polarisers didn't rotate the angle of the light waves, then successive aligned polarisers would result in very close to 0 light making it through, but this is not what happens. (Excluding losses due to non-polariser-related effects) All of this goes back to a fundamental confusion between two possibilities: 1) Is it the EM field in free space that is quantised? OR 2) Is it just the interactions of the EM field with matter that are quantised? If you actually run the experiments to check which one, it turns out that (2) is true, but most QM people think (1) is true because most of the time they're indistinguishable. See this video for such an experiment: https://www.youtube.com/watch?v=SDtAh9IwG-I https://www.youtube.com/watch?v=SDtAh9IwG-I
• d_burfoot 5y ago
  Surprising fact about the sun: it actually produces less heat per unit volume than a compost pile - this makes sense when you consider that fusion events are very rare. The reason the sun is so hot is that it has an enormous ratio of volume to surface area. Heat emission due to radiative cooling is proportional to the surface area, but heat creation due to fusion is proportional to the volume.
  • behnamoh 5y ago
    It's so humbling to think about how huge the Sun is, and then, how small it is compared to the really big giants in the universe.
  • GolDDranks 5y ago
    I learned about this many years ago, and even now, I am occasionally amused by this thought.
  • Synaesthesia 5y ago
    The density of the sun is very low, like much lower than the atmosphere (throughout most of it, anyway,) But yes it has an enormous volume.
    • hamilyon2 5y ago
      Given vastness of sun, it's age and existence of extremophiles, I would be surprised if there is no life there.
      • amelius 5y ago
        Perhaps a Boltzmann-brain every now and then.
      • simonh 5y ago
        Life requires a stable environment in which persistent structures can be maintained over long periods of time. High temperatures are inimical to that. This is why we live in a part of the universe where temperatures rarely rise much above a few hundred degrees Kelvin. Above that most complex chemical structures break down.
• IgorPartola 5y ago
  Closing roads to improve commute times makes obvious sense if you think of the TCP back pressure mechanism.
• wobsta 5y ago
  Nice list, like it. Let me add my favorite: Wet air is actually lighter than dry air. Consider an ideal gas. H₂O is lighter than both N₂ and O₂. Now water replaces some of the oxygen and nitrogen ...
• ncmncm 5y ago
  Number 19 is a clinker. Banach-Tarski applies only to objects in real-number space, but there are no such objects. For that to work, objects have to be infinitely divisible, but all of our objects are made out of atoms. Real-numbered space is a good enough approximation to our experience that we hardly ever encounter a model failure like this one.
  • 0xmarcin 5y ago
    Also this only works when the highly contested Axiom of Choice is used.
    • BalinKing 5y ago
      But isn't Banach-Tarski one of the main reasons why AoC is highly contested in the first place? (I don't know much about higher math, so this only my amateur impression.)
      • kcolford 5y ago
        It's also contested because while intuitive to a countable set of countable sets, generalizing to higher cardinals is not intuitive. Although the same could be said of the Power Law as well...
    • intuitionist 5y ago
      The paradoxical decomposition of the free group doesn’t rely on choice though, and that’s arguably the “heart” of BT.
  • mjburgess 5y ago
    I'd deny that our objects do not live in a real space. Spacetime is real-valued. The issue isn't the reals, but that "solid object" isn't defined properly, ie., the sets under question don't have well-defined volumes. As soon as you fix that problem, via measure theory, the paradox resolves. You dont need to ditch real numbers.
    • 5y ago
• mlang23 5y ago
  There is a typo in the headline: Conterintutive -> Counterintuitive
• nwallin 5y ago
  > 17. At any given moment on the earth's surface, there exist 2 antipodal points (on exactly opposite sides of the earth) with the same temperature and barometric pressure: youtube.com/watch?v=cchIr1OXc8E This is not necessarily true. They say a picture is worth 1000 words: https://media.deseretdigital.com/file/5894488349 https://media.deseretdigital.com/file/5894488349 Regardomg Gabriel's Horn and Banach-Tarski, the paradox is described as a trumpet, or a ball, made out of molecules, atoms, electrons, protons, neutrons, quarks-- but the mathematical proof is... not that. It's pretty common that intuition about objects made out of a finite number of parts breaks down when describing a construct with infinitely many parts.
  • raldi 5y ago
    I don't get what thought that picture is supposed to provoke.
    • ruuda 5y ago
      Borsuk-Ulam applies to surfaces homeomorphic to a sphere, but the picture shows that the earth's surface is a sphere with at least one handle, a donut with at least one hole.
    • Kwantuum 5y ago
      The earth has "holes" and as such the proof of that statement for a sphere does not apply to the earth.
      • lasc4r 5y ago
        Thank you, something about this didn't make total sense to me.
• rsj_hn 5y ago
  Picard's Great Theorem would be one of my favorite counterintuitive facts: As a holomorphic function approaches an essential singularity, it takes on every possible value (except at most one) infinitely often. An astonishing result. https://en.wikipedia.org/wiki/Picard_theorem https://en.wikipedia.org/wiki/Picard_theorem And the nice thing is you can prove this with fairly elementary techniques (e.g. first year complex analysis is all you need).
• implements 5y ago
  Number 7: “The Earth makes 366.25 rotations around its axis per year.” Errr, the Earth doesn’t ‘roll around the sun’ - hence the author’s (+1) is wrong, I think (hope!). It’s 365.256 according to Wikipedia.
  • rawling 5y ago
    > Both the stellar day and the sidereal day are shorter than the mean solar day by about 3 minutes 56 seconds. This is a result of the Earth turning 1 additional rotation, relative to the celestial reference frame, as it orbits the Sun (so 366.25 rotations/y). https://en.wikipedia.org/wiki/Earth's_rotation https://en.wikipedia.org/wiki/Earth's_rotation
    • implements 5y ago
      So we would need 367 unique date identifiers … but we’ve only got 366 (Feb 29th being the non-annual one). I get I may be being unintelligent, but isn’t the author confusing the rolling coin paradox with an obscure astronomical reference system and coming up with a ‘mistakenly technically correct’ result that doesn’t match experienced reality?
      • kgwgk 5y ago
        Why 367 dates? Dates are not used to count Earth revolutions. They are counting days/nights and there are only a bit more that 365 of those in one year.
      • rawling 5y ago
        It matches experienced reality if you look at the stars rather than the sun, or if you use something like a Foucault pendulum to measure the rotation speed.
      • mmmmmbop 5y ago
        Imagine you're standing on a set point on the surface on the outer coin, e.g. the one touching the inner coin. As the outer coin rotates around the inner coin, your experienced reality will be that you see n-1 rotations. In the example of the outer coin having 1/3 the radius of the inner coin, as the outer coin rolls around the inner coin 4 times, you would actually only touch the inner coin 3 times.
• simonebrunozzi 5y ago
  > The Earth makes 366.25 rotations around its axis per year Isn't it 365.25?
  • folli 5y ago
    Here's a nice image that explains the difference between a solar day (meaning the time it takes until the sun is at the same azimuth again) and the sidereal day (meaning the time it takes for earth to rotate around it's axis once): https://qph.fs.quoracdn.net/main-qimg-ab0d69361311b4f15b00641ae1fcf727 https://qph.fs.quoracdn.net/main-qimg-ab0d69361311b4f15b0064...
    • deleted 5y ago
      [deleted]
  • kgwgk 5y ago
    Imagine that the Earth was showing always the same face towards the Sun (like the Moon/Earth situation). At the end of the year the Earth would have rotated once (not zero times). Edit: Imagine now that you knew only that the Earth was rotating around its axis (not parallel to the orbital plane) once per year. Then either we are in the previous case (no day/night cycle) or there are two day/night cycles.
• _0ffh 5y ago
  17 reminds me of this one: If you have a three-legged stool on an uneven (but continuous) surface you can always find a stable position for the stool just by rotating it.
• vic-traill 5y ago
  Is there a joke or double entendre to the misspelling of 'counterintuitive' in the HN title (the article title is correct) that I'm missing?
  • mkl 5y ago
    I'm pretty sure it's just a typo. The actual title is "The most counterintuitive facts in all of mathematics, computer science, and physics", which is slightly too long for HN and a bit click-baity. No need to retype or abbreviate CS though.
• gokhan 5y ago
  77+33 is not 100 and that's quite saddening.
• vadim_lebedev 5y ago
  My favorite is a variation of #7: Earth rotation period is actually 23h 56m, not 24 hours https://en.wikipedia.org/wiki/Earth%27s_rotation https://en.wikipedia.org/wiki/Earth%27s_rotation
• mkl 5y ago
  > The Earth makes 366.25 rotations around its axis per year. This one isn't quite correct, unless you're talking about artificial Julian years. It is closer to 366.24, and even closer to 366.24219 = 1+365.24219 (https://pumas.nasa.gov/sites/default/files/examples/04_21_97_1.pdf https://pumas.nasa.gov/sites/default/files/examples/04_21_97...).
• enimodas 5y ago
  Do lemon markets actually occur in real life? It seems a combination of: people may need the money, inventory is not free, and most goods depreciate in value will inhibit the forming of a lemon market.
• st_goliath 5y ago
  > 19. Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. While you're at it, you can completely turn that sphere inside out without creating any holes or creases[1]. [1] https://www.youtube.com/watch?v=wO61D9x6lNY&t=92s https://www.youtube.com/watch?v=wO61D9x6lNY&t=92s
• bogosmith 5y ago
  There is a typo in the title.
• echopurity 5y ago
  Epistemological conflation is par for HN.
• deleted 5y ago
  [deleted]
• svachalek 5y ago
  > 11. Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it: https://en.wikipedia.org/wiki/The_Market_for_Lemons https://en.wikipedia.org/wiki/The_Market_for_Lemons This is new and interesting to me, although I think the phrasing of 11 is untrue as it's more about a cumulative effect in a market than an individual sale. Still I think this explains a lot of things in a way I've never really thought about it before. For example, dating apps.
  • curiousgal 5y ago
    I remember I struggled to wrap my head around this in my microeconomics class when we first explored information asymmetry.
• btbuildem 5y ago
  The SAT question one is brutal.. they did not include the correct answer on a multiple-choice question!
• trevortheblack 5y ago
  > 21. In two dimensions, there are infinitely many regular polygons. In three dimensions, there are five Platonic solids. In four dimensions, there are six platonic polychora. In all higher dimensions than four, there are only ever three regular polytopes. (Maths 1001, Richard Elwes) This implies that Platonic Solids are the 3D analogue of the 2D regular polygon. This is not the case. The Platonic Solids are merely all of the convex solid regular polyhedra. When the caveats are removed there are (at least) 48 regular polyhedra: https://www.youtube.com/watch?v=_hjRvZYkAgA https://www.youtube.com/watch?v=_hjRvZYkAgA
• GoblinSlayer 5y ago
  Fitch's paradox has an incorrect assumption about conjunction: if I know that all digits of pi are between 0 and 9, that assumption then suggests that I know all digits of pi, because they are a part of known truth. Or it's used incorrectly.
• justusthane 5y ago
  > Adding 3 feet to a tightly tied rope around the earth would allow you to raise it uniformly by almost 6 inches. There is no way my brain can accept that this is true.
  • dragonwriter 5y ago
    > > Adding 3 feet to a tightly tied rope around the earth would allow you to raise it uniformly by almost 6 inches. > There is no way my brain can accept that this is true. Yeah, my brain wants the size of the Earth to matter even though I know c = πd => c1 - c0 = π(d1-d0) means only the delta matters. The Earth, a soccer ball, the Milky Way galaxy...the amount of additional diameter you get from the same added circumference is constant.
• crvdgc 5y ago
  > 6. Causation does not imply correlation: https://arxiv.org/abs/1505.03118 https://arxiv.org/abs/1505.03118 And in the paper, from the abstract, > We demonstrate that the Faithfulness property that is assumed in much causal analysis is robustly violated for a large class of systems... In preliminaries, > The Faithfulness assumption is that no conditional correlation among the variables is zero unless it is necessarily so given the Markov property. So probably a more accurate laymam description would be "not all causations are observable as a statistical correlation relationship".
• abandonliberty 5y ago
  I was mistaken. Here's the video! https://www.youtube.com/watch?v=yCsgoLc_fzI https://www.youtube.com/watch?v=yCsgoLc_fzI
  • rinnan 5y ago
    The professor lost in the video -- he did not win. The one who _didn't believe_ that one could go faster downwind than the wind gave the $10,000 to charity. Or did I miss something?
    • abandonliberty 5y ago
      Thanks!