[HN Gopher] Conterintuitive facts in mathematics, CS, and physics
___________________________________________________________________
Conterintuitive facts in mathematics, CS, and physics
Author : raviparikh
Score : 850 points
Date : 2021-10-05 19:28 UTC (1 days ago)
(HTM) web link (axisofordinary.substack.com)
(TXT) w3m dump (axisofordinary.substack.com)
| charcircuit wrote:
| >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.
| geoduck14 wrote:
| >It is possible to compute over encrypted data without access
| to the secret key
|
| This is counter intuitive to me. For one, I don't consider the
| Caesar Cipher to be an encryption scheme that I would actually
| use for data.
|
| In addition, when I want to "compute" data, I want to do things
| like identify sentiment analysis in free form text or identify
| key themes in a paragraph - and I'm not sure this actually IS
| possible with data that is encrypted
| charcircuit wrote:
| Maybe a simpler example would be doing a not operation on a
| ciphertext for a one time pad? When you decrypt it you will
| get the plain text with all of the bits flipped.
|
| Doing computations on cipher text is very limited which is
| why it's not very efficient to do complicated operations.
| dfdz wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Exactly, practically it makes no different that the
| method could be fooled with a very tiny probability, but
| when making these counterintuitive statements I think it
| is important to be precise.
|
| Ideally the reader should fully understand the statement
| and still feel amazed, rather than doubting the statement
| for a valid reason: perfect zero knowledge proof systems
| (which do not fail sometimes) are impossible and a reader
| would be right to think so
| Ar-Curunir wrote:
| The interesting part is performing _arbitrary_ computations
| over encrypted data.
| tunesmith wrote:
| 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 wrote:
| 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 wrote:
| This also explains the supply chain issues we are
| experiencing now because of the move to just-in-time
| manufacturing across the board.
| galaxyLogic wrote:
| Cool. Ships waiting to dock on California cost. This
| explains it!
| enthdegree wrote:
| 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.
| d_burfoot wrote:
| 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 wrote:
| 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.
| Synaesthesia wrote:
| 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 wrote:
| Given vastness of sun, it's age and existence of
| extremophiles, I would be surprised if there is no life
| there.
| simonh wrote:
| 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.
| jerf wrote:
| Even more fundamentally, a life form needs to be able to
| pump entropy out more quickly than it comes in, no matter
| what its substrate is. With all that heat and light and
| magnetic fields running around and pressing in on any
| conceivable life form living in the sun, there's no way
| it could possibly pump it out fast enough.
|
| With that analysis you don't have to get into the weeds
| of what exactly plasma and magnetic fields might
| theoretically be able to cohere into and whether it may
| be able to be life someday... it doesn't matter. There's
| no way sun life can pump out the entropy fast enough no
| matter what.
|
| (On the flip side, one can imagine some form of nebula,
| gas-cloud life, but they would have to be so slow that
| there's no chance any of it could evolve into anything
| terribly complicated in the life time of the universe. If
| we ever did find some it would double as proof that there
| must have been some _other_ life form that created it.)
| amelius wrote:
| Perhaps a Boltzmann-brain every now and then.
| FabHK wrote:
| Conclusion: to solve all our energy problems, we just need a
| sufficiently large compost pile.
| Loughla wrote:
| I mean, yeah. But at some point the pile will become large
| enough to collapse in on itself from gravity and start
| fusion, I would imagine.
|
| So, according to Cornell[don't know how to do a citation],
| "1,000 BTU per hour per ton" is a good heat capture rate from
| manure compost. Then, according to something called
| 'alternative energy geek' "the earth receives 82 million
| quads of Btu energy from the sun each year. A "quad" is one
| quadrillion British Thermal Units (BTUs) of energy."
|
| So, to recreate that, we would need. Um. One massive shitload
| of compost to recreate that.
|
| Citations: https://smallfarms.cornell.edu/2012/10/compost-
| power/ http://www.alternative-energy-geek.com/solar-energy-
| per-squa...
| GolDDranks wrote:
| I learned about this many years ago, and even now, I am
| occasionally amused by this thought.
| trevortheblack wrote:
| > 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
| whiterock wrote:
| very cool collection - truly something to pick from at a party.
| should have much more upvotes imho.
| reidjs wrote:
| 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]
| anonymousDan wrote:
| Agreed, some real mindbenders there!
| quickthrower2 wrote:
| Monty Hall is there, albeit inside the answer to a stack overflow
| question linked in Misc #33
| toolslive wrote:
| the planet closest to earth is Mercury.
| wobsta wrote:
| 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 ...
| Ansil849 wrote:
| 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 wrote:
| The area enclosed by a circle is pr^2 while 12 and 18 are the
| diameters, right? The radius of a disc is half the diameter, so
| 6 and 9, respectively. 2p6^2 < 1p9^2 ~~ 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 wrote:
| > 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 wrote:
| 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 wrote:
| 18/12 = 1.5 > sqrt(2) = 1.4..
| Viliam1234 wrote:
| To make it simpler, let's assume that the shape of pizza is a
| square, i.e. we are talking about a 12x12 inch pizza and a
| 18x18 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 2x2 units, and the large pizza is 3x3
| units.
|
| twice 2x2 = twice 4 = 8
|
| 3x3 = 9
| antognini wrote:
| 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 wrote:
| So what happens to a star inside a "perfect" Dyson sphere?
| antognini wrote:
| 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)
| raxxorrax wrote:
| 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 wrote:
| 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).
| tgb wrote:
| 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 wrote:
| > _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.
| svachalek wrote:
| > 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
|
| 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 wrote:
| I remember I struggled to wrap my head around this in my
| microeconomics class when we first explored information
| asymmetry.
| 6gvONxR4sf7o wrote:
| > 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
|
| 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
|
| 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.
| pessimizer wrote:
| > > 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
|
| >I really like this one. It's a perfect combo of intuitive from
| one perspective and mind bending from another.
|
| Comes up a lot lately because of vaccine effectiveness e.g. 95%
| is twice as effective as 90%.
| sib wrote:
| It's probably more intuitive if you say that it's "half as
| ineffective"
| Gibbon1 wrote:
| I first saw this in a discussion of power supply
| efficiency. A 95% efficient supply generates ~half the heat
| that a 90% one does.
| umvi wrote:
| Wait, how is 95% effective twice as effective as 90%?
| anandoza wrote:
| It halves your risk.
| fgonzag wrote:
| just as going from 98% effectiveness to 99% effectiveness
| halves your risk (you go from 2% chance of falling ill to
| 1%)
|
| its a common concept in games in which armor follows a
| linear formula (each point of armor is more effective than
| the last when calculating effective health)
| anthk wrote:
| Then there's the polynomial wizard.
| zbaxrl wrote:
| 90% is 1 in 10, 95% is 1 in 20.
| xarope wrote:
| and sunscreen (SPF) calculations...
| [deleted]
| niklasbuschmann wrote:
| The strawberries remind me of the pricing of long-term bonds.
|
| Let's say newly issued 100yr Treasuries pay a 1% coupon today,
| but tomorrow the coupon will be 2%, how much does the price of
| the older bond change?
|
| The answer is a near 50% loss, simply because this is required
| to bump the yield of the older bond to 2%.
|
| (If we include the discounted principal payment the exact
| answer becomes a 41.3% loss)
| lotsofpulp wrote:
| >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.davidd...
|
| Wonder why the person that wrote the Wikipedia article changed
| it up when it is supposed to be a direct quote.
| mint2 wrote:
| I really don't like that example because it makes no sense.
| In no logical circumstance could the potatoes dehydrate so
| quickly when left out over a single night.
| tharkun__ wrote:
| And they also don't consist of 99% water. That is why they
| called them "purely mathematical potatoes" and they
| could've chosen any type of fruit or vegetable. Heck, I'm
| just waiting for a car analogy now!
|
| Brake fluid anyone?
| mint2 wrote:
| But that 99% water simplification is needed for the
| purpose of the exercise.
|
| The left out overnight is basically an absurd statement
| that is intended to confuse and not really related to the
| actual question.
|
| The statement could be "left in a dry environment until"
| or simply "left to dry a few weeks"
| tharkun__ wrote:
| You definitely could write that but it wouldn't change
| anything and you could make the same "argument" you are
| making now. "The left to dry a few weeks is basically an
| absurd statement that is intended to confuse" and it
| still wouldn't be true. It's not intended to confuse at
| all. It's intended to get the point across that you let
| this imaginary thing dry from 99% to 98%. They could've
| said "sponge" and let it out to dry any number of
| minutes. The point isn't to make a 100% accurate example
| of the drying properties of any actual 'thing'. They just
| needed something that people intuitively know "has water
| content" and that "can dry".
| mint2 wrote:
| Wow by the downvotes I learned people react extremely
| negatively to any critique of math word problems.
| Spherical cows and 99% water potatoes are fine, those
| over simplifications are required for the analysis.
|
| Saying Leaving the potatoes out overnight to imply they
| halved in mass, sounds as reasonable as "the potatoes
| were in the ground for 12 hours and then doubled in
| amount, what percent of water are they now?" It's so
| gratuitous and requires ignoring all other laws of
| physics, while the goal of the spherical cow type
| simplification is to only ignore a few key challenging
| ones.
| sbakzbsmx wrote:
| Honest question to know how others think.
|
| It doesn't really matter, does it? The rate of evaporation
| is irrelevant to the problem. Mr. Potato could have waited
| a year, or dried them on the Uyuni salt plains.
|
| Why do you care? Would this distraction affect your ability
| to solve the problem?
| User23 wrote:
| You probably won't be thrilled by the spherical cows in the
| nearby pasture then.
| jean_tta wrote:
| Do spherical cows dream of mathematical potatoes?
| inkyoto wrote:
| Spherical cows, naturally, dream of spherical potatoes in
| a vacuum whilst grazing next to spherical chickens.
| [deleted]
| sbakzbsmx wrote:
| The difference between weight and mass is domain specific to
| physics.
|
| I actually get annoyed at people who are pedantic about these
| things. Precision is important in some conversations, but
| just elitist in other.
|
| Anyway, the term "weight" to refer to mass outdates its use
| as a force - its only since Newton that we distinguish the
| two, after all.
| menotyou wrote:
| When I buy potatoes I am interested to buy 10 kg mass of
| potatoes, not the amount or potatoes of which has a gravity
| force of 98,1N. On mars you need to eat 10 kg of potatoes a
| week, not the amount of which has 98,1N gravity force.
|
| The scales in the supermarket on earth automatically convert
| the weight into mass for my convenience by applying a
| constant factor of 1/9,81.
|
| I am hardly far away enough from earth that the constant
| changed, so I did not need to distinguish between the two
| measures so far. When carrying the potatoes home I just use
| the mass of 10 kg as a proxy for the force I need.
|
| And to determine the increased breaking distance of my car, I
| need to know the mass again.
| syncsynchalt wrote:
| > Wonder why the person that wrote the Wikipedia article
| changed it up when it is supposed to be a direct quote.
|
| They likely changed it from lb to kg because that would be
| more friendly to an international audience, without realizing
| that lb is a measure of weight and kg is a measure of mass.
| Therefore they didn't know to change "weight" to "mass".
| lotsofpulp wrote:
| Oh yes, that would make sense.
| kergonath wrote:
| I have been to the US quite regularly, and been living in
| the UK for a number of years, and I have _never_ seen
| someone using pounds as a unit of force instead of a unit
| of mass meaning roughly 500g, give or take. Second meaning
| was about 1.20EUR.
|
| FWIW, the pound is a proper unit of mass:
| https://en.m.wikipedia.org/wiki/Pound_(mass) .
| syncsynchalt wrote:
| I'm a civil engineer. In statute terms pound is the unit
| of force and slug is the unit of mass. That might have
| colored my thinking.
| p1necone wrote:
| You're being overly pedantic (and I would argue actually
| incorrect). Kilograms and pounds are both referred to as
| "weight" in general conversation and _nobody_ is going to be
| confused by this.
|
| Go to any supermarket in a country that uses the metric
| system, potatoes will be sold by the kilogram - it's the
| natural way to phrase this outside of America.
|
| In a physics context the definition of kilogram might be
| _specifically_ mass, with newtons referring to weight /force.
| But words can have different meanings outside of technical
| contexts.
|
| If you go to a metric country, and ask someone how much they
| "weigh", approximately zero people will say "x newtons", they
| will say "x kilograms" (or "x pounds" still in a lot of
| commonwealth countries if we're being pedantic).
| p1necone wrote:
| Although the more I think about this the more I think the
| difference between technical and colloquial is actually
| that "weight" in colloquial use refers to mass, because
| force is not commonly relevant.
| eru wrote:
| Yes, in colloquial use weight refers to mass _most_ of
| the time. But can also refer to inertia or mass. Or be
| used metaphorically.
| jacobolus wrote:
| "Weight" historically referred to mass, in common speech
| dating back forever. It's the Germanic word which has
| been used throughout the history of English, whereas
| "mass" comes from Latin via French, like 5-6 centuries
| ago. The two words are almost exact synonyms, in
| historical/colloquial use.
|
| Both historically and today, a "pound" (Roman libra) is a
| unit of mass. People use a pound-force as a unit of force
| only in somewhat specialized contexts.
|
| At some point in the relatively recent past, someone (not
| sure who) decided that we needed to have 2 separate words
| for mass vs. force, and we should keep the Latin word for
| mass and use the Germanic word to mean force.
|
| Now pedantic people are constantly insisting that using
| the standard English word weight to mean mass is "wrong".
| adrian_b wrote:
| Actually in the past "weight" or the Latin "pondus" (=>
| pound) always referred correctly to what is now named
| "mass".
|
| When someone mentioned "weight" just in a qualitative
| way, as a burden, they might have thought at the force
| that presses someone down, but whenever they referred to
| weight in a quantitative way, they referred to the weight
| as measured with a weighing scale, which gives the ratio
| between the mass of the weighed object and the mass of a
| standard weight, independently of the local acceleration
| of gravity.
|
| Methods that measure the force of gravity and then the
| mass is computed from the measured force, i.e. with the
| force measured either mechanically with springs or
| electrically, have appeared only very recently.
|
| The distinction between force of weight and mass became
| important only since Newton, who used "quantity of
| matter" for what was renamed later to the more convenient
| shorter word "mass".
|
| Perhaps it would have been better to retain the
| traditional words like weight and its correspondents in
| all other languages with the meaning of "mass", because
| this meaning has been used during more than 5 millennia
| and use a new word, e.g. gravitational force, for the
| force of weight, because we need to speak about this
| force much more seldom than about the mass of something.
| jacobolus wrote:
| > _Actually in the past "weight" [...] always referred
| correctly to what is now named "mass"._
|
| That's the same thing I just said. Why add "actually" in
| front? Yes, weight was historically measured with balance
| scales.
|
| I guess I should have been clearer that the term "mass"
| as used in physics only dates from 3 centuries ago (from
| Newton), and did not historically mean weight in Latin.
| (Mass comes from Latin via French for lump of dough.)
| adrian_b wrote:
| You are right, I have misunderstood what you have said,
| because it indeed looked like if "mass" would have been
| some traditional word having anything to do in any
| language with what are now called "weight" and "mass"
| instead of a recent post-Newton word choice for naming
| one of the 2 quantities, while keeping the old names for
| the other.
|
| I still think that the choice of which of the 2 should
| get a new name was bad, because the traditional
| quantitative meaning almost always referred to what is
| now called "mass"(with extremely few exceptions such when
| somebody would be described as so strong as to be able to
| lift a certain weight).
| jgtrosh wrote:
| 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 wrote:
| 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.
| [deleted]
| Scarblac wrote:
| To me (Netherlands) a pound is simply 0.5kg. Force is
| expressed in Newton.
| version_five wrote:
| From what I remember from intro physics, we distinguished
| between pounds and pounds force, the latter having the
| 32ft/s^2 multiplied in.
|
| And wikipedia seems to agree with me, see pound (mass) vs
| pound (force).
| lotsofpulp wrote:
| Oh wow, learning a lot today. I was taught in the US that
| pounds are a measure of force, and that is how it was
| always used in physics problems.
| tharkun__ wrote:
| As a European I learned in metric. When I first learned
| pounds, it was as the imperial system's equivalent of
| grams and a conversion factor was given. Force in physics
| class was taught in Newtons (kg*m/s^2).
| kergonath wrote:
| Pounds as a mass unit are perverse enough. Things like
| pound force and psi (pounds per square inch) were used
| only to make fun of old mechanics papers and textbooks.
| Also, btu. It is quite amazing actually that someone
| would see the SI and think "no, too simple; I'll keep my
| pounds, ounces, inches, and feet".
|
| Anyway, yes, the proper unit of force is the Newton.
| treebog wrote:
| In engineering school (in the US), we used pounds mass
| (lbm) as the unit of mass, and pounds force (lbf) as the
| unit of force.
| version_five wrote:
| I think there is some weird dual usage that makes them
| either mass or weight depending on the context. For
| example, torque is in ft*lbs or N*m so the pounds there
| are lbs force.
|
| Though checking wikipedia again, it actually specifies
| that torque is measured in as lbf*ft. I take that to mean
| that 1 ft*lb is the torque of 1/2 oz (1/32 of an lb) at 1
| foot. I expect that's a test question almost everyone
| would get wrong, myself included.
|
| https://en.m.wikipedia.org/wiki/Pound-foot_(torque)
| stan_rogers wrote:
| Think like an ordinary person. You know, an ordinary
| person who would say that they _weigh_ about 80
| kilograms. Only science nerds would say that they _mass_
| about 80 kilograms, or that they _weigh_ about 785
| Newtons. Similarly, anyone who 's used to living with US
| customary (or Imperial) units understands that a pound of
| force is what a pound of mass weighs and would see no
| reason, under any circumstances, why anyone would want to
| divide the gravitational acceleration out of a pound-foot
| to arrive at a "real" torque value. When the pound value
| is expressing a weight-equivalent force, that force _is_
| the force of a pound under normal gravitational
| acceleration at or near the surface of the Earth.
| duxup wrote:
| 16 is like the money hall problem. I understand the answer, the
| answer makes sense to me. And yet but when I think of it how I
| think of it initially ... it still makes no sense.
| lifeplusplus wrote:
| For me the mistake was 1:1 relation between percentage and
| weight which didn't remain true after weight loss but I
| thought it did
| [deleted]
| silisili wrote:
| Easiest for me if thinking only in fractions and percentages,
| and realizing that the dry mass is a constant.
|
| The obvious example is 99g water, 1g dry mass. Knowing that
| 1g cannot change, what do we need water to be to equal 98%?
| 49g.
| machinelearning wrote:
| The way to make sense of this is not to think about the water
| weight but the solid matter. By changing the proportion of
| water from 99% to 98%, you're also doubling the proportion of
| solid mass from 1% to 2%.
| maweki wrote:
| > you're also doubling the proportion of solid mass from 1%
| to 2%.
|
| And the final step is then, that the solid mass didn't
| change and therefore the liquid must be halved, instead of
| the solid doubled.
| pvg wrote:
| The limit case can be helpful here, a strawberry made of 100%
| water can be dehydrated to practically nothing and is still
| made of 100% water.
| nighthawk454 wrote:
| Yes, it doesn't explicitly state the rate or distribution of
| events. But it is a good reminder of what happens when your
| whatevers/second are pretty high - see the famous "One in a
| million is next Tuesday" [1]. "Rare" is soon if you roll the
| dice fast enough.
|
| Any time your service has a high TPS, your API gets a lot of
| calls, a button in your app gets pressed a lot, ... this
| applies.
|
| Critically, "a lot" is defined relative to your failure
| tolerance. It may actually be very fast or a lot, or not
| particularly fast but it really really needs to work.
|
| It highlights the fallacy of equating "low probability" and
| "won't happen".
|
| [1] https://docs.microsoft.com/en-
| us/archive/blogs/larryosterman...
| magicalhippo wrote:
| > "Rare" is soon if you roll the dice fast enough.
|
| Indeed. One fun example is LHC[1], where the probability of a
| proton in a single bunch hitting a proton in the bunch going
| the opposite direction is on the order of 10^-21, yet due to
| huge number of protons per bunch and large number of bunches
| per second, it still results in ~10^9 collisions per second.
|
| [1]: https://www.lhc-
| closer.es/taking_a_closer_look_at_lhc/0.lhc_...
| leeoniya wrote:
| "Given the scale that Twitter is at, a one-in-a-million
| chance happens 500 times a day."
|
| https://www.ted.com/talks/del_harvey_protecting_twitter_user.
| ..
| curiousgal wrote:
| > _And I've seen some absolute doozies in my time - race
| conditions on MP machines where a non interlocked increment
| occurred (one variant of Michael Grier's "i = i + 1" bug)_
|
| I could not find any info about that bug, anyone got a link
| or a source?
| jerf wrote:
| I assume that bug is referring to the fact that while i = i
| + 1 may look atomic to you as a human, in the computer it
| turns into Read i into register.
| Add one to that register. Write i back to the
| memory location.
|
| And there's a window during that "add one to the register"
| where you can obviously have something jump in and write
| something else to that memory location.
|
| What happens on your real processor is more complicated
| since this is going to relate to cache coherency between
| the processors, not directly writing RAM at that point, and
| that's a deep rabbit hole. I couldn't describe it all in
| detail anyhow. But I can observe it doesn't take much at
| all to turn that one cycle vulnerability into something
| with a larger target.
| IIAOPSW wrote:
| Or everyone's newest favourite, the virus randomly mutating
| is "rare".
| kbenson wrote:
| >> 18. A one-in-billion event will happen 8 times a month:
| 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.
|
| I think this is about events happening to people, the number of
| people alive (and assuming they all communicate "miracle"
| occurrences"), and how many things they experience.
|
| That is, if I understand if correctly it's not that you can
| choose a random number between one and a billion and run a CPU
| to randomly check numbers in that range as fast as possible and
| get lots of results in seconds, it's that based one how we have
| roughly 8 billion people all communicating events that things
| we consider "one in a billion" occurrences will be experienced
| about 8 times a month across the populate, and we'll all pretty
| much hear about it, which may not match with our expectations
| of how often we should see a "one in a billion" event reported.
|
| Edit: Here's some relevant info from the paper "Methods for
| Studying Coincidences"[1]:
|
| _The Law of Truly Large Numbers. Succinctly put, the law of
| truly large numbers states: With a large enough sample, any
| outrageous thing is likely to happen. The point is that truly
| rare events, say events that occur only once in a million [as
| the mathematician Littlewood (1953) re- quired for an event to
| be surprising] are bound to be plentiful in a population of 250
| million people. If a coin- cidence occurs to one person in a
| million each day, then we expect 250 occurrences a day and
| close to 100,000 such occurrences a year.
|
| Going from a year to a lifetime and from the population of the
| United States to that of the world (5 billion at this writing),
| we can be absolutely sure that we will see incred- ibly
| remarkable events. When such events occur, they are often noted
| and recorded. If they happen to us or someone we know, it is
| hard to escape that spooky feeling.
|
| A Double Lottery Winner. To illustrate the point, we review a
| front-page story in the New York Times on a "1 in 17 trillion"
| long shot, speaking of a woman who won the New Jersey lottery
| twice. The 1 in 17 trillion number is the correct answer to a
| not-very-relevant question. If you buy one ticket for exactly
| two New Jersey state lot- teries, this is the chance both would
| be winners. (The woman actually purchased multiple tickets
| repeatedly.)
|
| We have already explored one facet of this problem in
| discussing the birthday problem. The important question is What
| is the chance that some person, out of all of the millions and
| millions of people who buy lottery tickets in the United
| States, hits a lottery twice in a lifetime? We must remember
| that many people buy multiple tickets on each of many
| lotteries.
|
| Stephen Samuels and George McCabe of the Depart- ment of
| Statistics at Purdue University arrived at some relevant
| calculations. They called the event "practically a sure thing,"
| calculating that it is better than even odds to have a double
| winner in seven years someplace in the United States. It is
| better than 1in 30 that there is a double winner in a four-
| month period-the time between win- nings of the New Jersey
| woman._
|
| 1: https://www.gwern.net/docs/statistics/bias/1989-diaconis.pdf
| teorema wrote:
| I agree with the parent that the cited "fact" is sort of
| questionable (along with some other things on the site even
| though I really enjoy it overall) because of ambiguity in
| definitions, assumptions, and so forth.
|
| However, the law of truly large numbers, as you frame it, is
| something you experience firsthand working in high level
| severity hospital settings in large metro areas. There's a
| large enough hospital catchment area that you start to see,
| on a fairly regular basis, the medical outcomes of all the
| bizarre and unbelievable things that happen rarely to any
| given person. It gets to a point it's difficult to know how
| to explain because the details of each case would be
| potentially identifying given how strange they are. And yet
| something happens all the time. Maybe not that one thing, but
| something of similar impact. It gives you a distorted sense
| of risk.
| I_complete_me wrote:
| Here's my attempt to understand what's going on:
|
| 100g strawberry total weight where the 100% is made up of 99%
| water and 1% solid matter. 100g strawberry total weight where
| the 100g is made up of 99g water and 1g solid matter. 99/100 =
| 0.99
|
| 50g strawberry total weight where the 100% is made up of 98%
| water and 2% solid matter. 50g strawberry total weight where
| the 50g is made up of 49g water and 1g solid matter. 49/50 =
| 0.98
| Aerroon wrote:
| #16 is something video games taught me, particularly Path of
| Exile. In PoE resistance are a flat multiplier to incoming
| damage. Eg monster does 100 damage per attack and you have 60%
| resistance then you take 40 damage.
|
| The interesting thing is that the higher your resistances the
| more effective each additional percentage point of resistance
| is.
|
| Let's say a monster does 100 damage per attack.
|
| If you have 0% resistance and increase it to 5%, then your
| incoming damage went from 100 to 95. You take 5% less damage
| than before.
|
| If you have 75% resistance and increase it to 80%, then your
| incoming damage went from 25 to 20. You take 20% less damage
| than before.
|
| It is pretty unintuitive until you realize that you need to
| focus on the remainder rather than the other part.
| FabHK wrote:
| Interestingly, the same logic applies to vaccination rates.
| Going from 0% to 5% vaccination has no impact on the course
| of the pandemic (except for those few vaccinated people, of
| course). Going from 75% to 80% has a much larger impact, and
| could stop the pandemic in its track (depending on R_0, and
| many other real-world complications of course).
|
| (And the reason is just the same: what matters is the
| remainder.)
| yodelshady wrote:
| A similar but not quite the same mechanic is fuel economy.
|
| Let's say you have two vehicles, both doing 10,000 miles per
| year. One gets 10 mpg and the other 50.
|
| Would you rather upgrade the 10 mpg vehicle to 13 mpg, or the
| 50 mpg to 100 mpg?
|
| Not only should you pick the former - you should pick the
| former even if you could upgrade the 50 mpg vehicle to run on
| _nothing_.
| Scharkenberg wrote:
| Yes, the fuel savings in the former case are larger than
| the initial fuel consumption in the latter case, but is it
| _really_ unintuitive in practice? What I mean is that we
| generally pay for fuel per volume, not per mileage. Now, I
| am not sure about others, but I would always base my
| decision based on the money I 'd save over a period of
| time, which in this case would require considering each
| vehicle's actual fuel consumption over that period of time.
| 123anonanonanon wrote:
| It is the assumption that the two cars always make the same
| number od miles indepedently od the cost that is unusual
| and unintuitive.
| skrtskrt wrote:
| If you think of it as a family that keeps obstinately
| driving both cars the same amount despite massive cost
| differences its weird but you could think of it as a
| mixed fleet of delivery or work trucks that are all
| needed regardless, the question is how you manage or
| prioritize upgrading them.
| aloer wrote:
| A similar example with league of legends:
|
| One point of resistance gives 1% effective extra health no
| matter how much you already have.
|
| 100 armor give 50% reduction (100/100+100) while 200 armor
| give 66% (100/100+200)
|
| The percentage 50 -> 66% is shown ingame and players often
| think the value per point of armor drops.
|
| What does actually happen is your effective bonus health
| changes from +100% to +200% and every additional point will
| be worth the same
| throwaway8451 wrote:
| OK, let's assume you have 75% elemental resistance. You also
| take 15% reduced damage and 10% less damage. You have 5%
| chance to avoid elemental damage and 10% to dodge it and are
| under the effect of Elemental Equilibrium and Gluttony of
| Elements. How much better is it to just kill everything
| before it can kill you?
| BigJono wrote:
| You joke, but PoE's use of stuff like 'increased/decreased'
| and 'more/less' to distinguish between additive and
| multiplicative calculations is one of the smartest game
| design decisions I've seen.
|
| The game has a lot of seemingly arbitrary distinctions and
| concepts that you just have to learn over time, but once
| you actually learn them, the consistency of it all makes it
| very easy to handle the large amounts of complexity in the
| game.
|
| It's completely unbearable going back to other games that
| just say crap like "+30% to x" without actually
| distinguishing between the different ways calculations can
| be done, forcing you to either experiment endlessly, look
| up every tiny thing on a community wiki, or just wing it.
|
| It's a nice contrast to something like WoW where every time
| you get a new item you just chuck the item code into some
| ten million LOC simulator and fuck around with limiting
| permutations until it doesn't take 15 minutes to run, just
| to find out through some totally opaque process that you
| have 189 more dps. And then 2 months later you find out
| there was a bug in the simulation and the item you deleted
| 1.9 months ago was actually better.
| londgine wrote:
| > There are as many whole positive numbers as all fractions
|
| According to a specific non-colloquial definition of "as many".
| adenadel wrote:
| 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 wrote:
| 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 wrote:
| What's worse, there are an infinite number of fractions
| that equal each whole number.
| iovoid wrote:
| 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)
| kristjansson wrote:
| There's an infinite number of each, so we're already stretching
| colloquial definitions by comparing them
| unholiness wrote:
| My sister and I used to figure out who had more candy at
| halloween by lining up the pieces next to each other. The
| concept of bijection might be more intuitive than counting
| itself.
| cedilla wrote:
| For all we know it's significantly older than counting.
| Pebbles representing bijections to wares like sheep (called
| calculi like in calculus) occur earlier than counting marks
| and much earlier than anything resembling numbers.
|
| There are still today human tribes that don't count at all.
| tzs wrote:
| > Pebbles representing bijections to wares like sheep
| (called calculi like in calculus) occur earlier than
| counting marks and much earlier than anything resembling
| numbers.
|
| How was this determined? I wouldn't expect that using
| pebbles this way would leave any distinctive marks or
| damage or residue on the pebbles that would allow an
| archaeologist several tens of thousands of years later to
| tell that was what the pebbles were used for.
| gfody wrote:
| John Conway's free will theorem could go here
| dwohnitmok wrote:
| 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-...
|
| 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.
| Kranar wrote:
| This is very interesting but I think it relies heavily on
| interpretation.
|
| For example there exists models of ZFC where all "real numbers"
| are definable, but said model does not include all the actual
| real numbers, it excludes any number that is not definable in
| ZFC. The issue is that the term "real number" is overloaded. In
| the formal sense it may refer only to numbers that are members
| of a model in which undefinable numbers are excluded. In
| another sense the term "real number" refers to actual real
| numbers as we humans intend for them to exist but do not have a
| precise formal definition.
|
| This actual set of real numbers does indeed contain members
| that are not definable in ZFC or any formal system, the issue
| is that there is no way to formalize this actual definition.
|
| This is similar to what another poster mentioned about Skolem's
| paradox:
|
| https://news.ycombinator.com/item?id=28767108
| dwohnitmok wrote:
| > In another sense the term "real number" refers to actual
| real numbers as we humans intend for them to exist but do not
| have a precise formal definition.
|
| Ah a Platonist in the flesh. Don't see many of you on HN. I
| don't think real numbers truly, objectively exist and think
| of them more as artifacts of human thought, but that's a deep
| deep rabbit hole.
|
| I'm kind of curious then, what do you believe the cardinality
| of the "real" real numbers is?
| Kranar wrote:
| I think I'm with you on that. Real numbers don't exist in
| an objective sense, I mean they exist in the same sense
| that an Escher painting of a hand drawing a hand exists,
| but they don't exist in the sense that a hand drawing a
| hand actually exists.
|
| When I was in high school I remember thinking that
| computers use the discrete to approximate the continuous
| and that it is the continuum that is real and the discrete
| that is an imperfect representation of the continuous. Then
| a high school teacher blew my mind when he told me to
| consider the opposite, that in fact it's the continuous
| that is used to approximate the discrete. The discrete is
| what's real and we humans invented the continuous to
| approximate the discrete.
|
| That simple twist in thinking had a profound effect on me
| that influences me to this day 30 years later.
|
| If anything I may have some extreme opinions that frankly
| no one takes seriously and I'm okay with that. For example
| I think the finitists had it right and infinity does not
| exist. There really is such a thing as a largest finite
| number, a number so large that it's impossible even in
| principle to add 1 to it. I can't fathom how large that
| number is, but there's physical justification to believe in
| it based on something like the Bekenstein bound:
|
| https://en.wikipedia.org/wiki/Bekenstein_bound
|
| At any rate, I like thinking about this stuff, I do
| appreciate it, but I don't take it literally. It's poetic,
| it can inspire new ways of thinking, but I also remind
| myself to compartmentalize it to some degree and not take
| these ideas too literally.
| dwohnitmok wrote:
| If you're sympathetic to the finitist cause, the idea
| that all mathematical objects are in fact definable is
| right up that alley. It's nice that this happens to line
| up acceptance of infinity, but finitism is basically
| entirely predicated on definability.
| wodenokoto wrote:
| 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 wrote:
| How could you give an example? The example you give would be
| a description of a number.
| Kranar wrote:
| 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.
| wodenokoto wrote:
| It's only the vast majority that can't be described.
|
| So either it is claimed that it is counter intuitive that
| you can't give an example of something you can't describe.
| That is not counter intuitive- that is basically the
| definition of indescribable.
|
| The other way the sentence can be read is that you can't
| give an example of a real number. Of course you can. It's
| only the vast majority of real numbers that can't be
| described. There's still infinitely many we can describe. 1
| is a real number.
| campital wrote:
| 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.
| adenadel wrote:
| In fact, almost all numbers are transcendental (algebraic
| numbers have measure zero).
| bloak wrote:
| 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.
| IgorPartola wrote:
| Closing roads to improve commute times makes obvious sense if you
| think of the TCP back pressure mechanism.
| enimodas wrote:
| 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.
| _0ffh wrote:
| 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.
| GoblinSlayer wrote:
| 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.
| implements wrote:
| 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 wrote:
| > 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
| implements wrote:
| 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?
| mmmmmbop wrote:
| 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.
| kgwgk wrote:
| 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 wrote:
| 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.
| lapetitejort wrote:
| 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
| hinkley wrote:
| 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.
| Jensson wrote:
| 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.
| caf wrote:
| 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.
| moralestapia wrote:
| It's mean to be a thought experiment but some people (like
| Veritasium) can't get around this simple realization.
|
| Like, "I wonder what could be happening inside of a black
| hole?"
|
| "Oh we would never know, because if we send a camera it would
| break. Also suppose we MaaaAaAKKeee aa cCAaaMeerra with ThE
| STROooonngGEEstt Material In the UUunniveersssee, so it
| wouldn't break, you would find that there's no wifi in there
| so how would you transmit your findings ;)"
|
| I hate this kind of "smart" people.
| yongjik wrote:
| (Disclaimer: I am not a physicist.)
|
| I think it's actually a pretty reasonable description?
| Sounds like it's a paradox because we assume a rigid body,
| but nothing in real life is a perfect rigid body. So the
| situation simply becomes a bunch of particles following
| circular orbits. If you measure distance along one
| direction (while constantly accelerating in relativistic
| speed) you get one number; if you measure distance in
| another direction you get a different number. But that's
| what relativity does.
|
| In other words, it's similar to the simpler question: "If I
| have a perfectly rigid rod that can reach the moon, and I
| push it, then the other end pushes the moon immediately.
| But speed of information cannot exceed speed of light. How
| come?" Answer: There's no perfectly rigid rod.
| Joker_vD wrote:
| It's one of the reason I dislike physical terms that has
| words like "ideal" or "perfect" as their part. They
| subliminally suggest that their properties and behaviours
| are the "true" ones while the real material things are
| their imperfect counterparts whose imperfections you can
| sometime disregard.
|
| Of course, it's exactly the opposite: it's those "ideal"
| concepts are imperfect approximations of the real things,
| omitting lots of details which sometimes are not that
| important but sometimes are absolutely crucial.
|
| My personal favourite example is attaching a perfect
| source of voltage (zero internal resistance) to a perfect
| wire (zero resistance). You can't arrive to this scenario
| starting with the real world entities: both the battery
| and the wire will have non-zero resistances and depending
| on their proportions, you end up with approximating
| either one of those as zero, or none, but never both.
| hinkley wrote:
| Hence the "perfectly spherical cow" joke.
| moralestapia wrote:
| The example you give and its answer follows a line of
| reasoning that leads you to an interesting conclusion,
| that's the point of thought experiments.
|
| What I was saying is more akin to answering your example
| with:
|
| "Oh no you can't! There's not enough steel on Earth to
| build such thing and even if you had it, it would require
| an EEeEeeenOOOOrrrMMMoooUUUusss amount of energy to put
| in place ;)."
|
| That would be quite a moronic interpretation of the
| problem that completely misses the goal of said thought
| experiment, which is, well, to make you think.
| cedilla wrote:
| The difference there is, I think, that the pragmatic
| argument of "not enough steel" doesn't prove the non-
| existence; while the argument about "there are no truly
| rigid bodies" does.
|
| I haven't seen that Veritasium video, but it sounds like
| it makes the exact same point. It's not that we haven't
| found a material that's rigid enough; it's that a rigid
| disk is counter-factual to begin with, even in non-
| relativistic conditions.
| robocat wrote:
| Spin the observer at the centre of the disc, and keep the disc
| stationary... now what happens?
| gpderetta wrote:
| not a physicist, but remember that accelerated frames break
| the symmetry.
| UnFleshedOne wrote:
| 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 wrote:
| 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.
| ncmncm wrote:
| 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.
| CJefferson wrote:
| I agree. Also, until you get super technical, it isn't really
| any different to "if you take the natural numbers, and split
| them into odd and even, you get two copies of the natural
| numbers".
| Cheezemansam wrote:
| One of the big things though, is that you _can 't_ know
| exactly what this division looks like. Like you said, at a
| high, non-technical level that is kind of fundamentally what
| is going on, with the caveat that the actual "cut" you are
| making isn't clean like splitting numbers down the middle.
| ColinWright wrote:
| I disagree ... the "two copies of the natural numbers" is
| sorta fine, except that they're more "spread out" so it's not
| at all surprising.
|
| The surprising thing about BT is that the "pieces" are "moved
| around" ... there's no expansion or contraction.
|
| Yes, the natural number thing helps to understand that simply
| counting things doesn't help, but the "rigid motion" aspect
| of BT takes it further.
| CJefferson wrote:
| True, that is the subtle bit -- but I think most people
| misunderstand (I'm not saying you are!), and don't realise
| you can always split an infinity into two -- this is just
| about splitting a sphere into some 'point clouds' such that
| you can cleverly stitch them back together into the same
| original space. In particular, the 'cutting' really makes
| no real-world sense (at least, as far as I can understand).
| ColinWright wrote:
| I agree that the "cutting" makes no real world sense. In
| a way, that's one of the points of the exercise.
|
| And I agree that most people don't (initially) understand
| that an infinite set can be divided into two infinite
| sets that kinda "look the same", such as dividing Z (or
| N) into the evens and odds.
|
| But BT is more than that. What follows isn't really for
| you, but is for anyone following the conversation.
|
| Let's take a set _A_. It 's a subset of the unit sphere,
| and it's a carefully chosen, special set, not just any
| random set. It's complicated to define, and requires the
| Axiom of Choice to do so, but that's what the BT theorem
| does ... it shows us how to define the set _A_.
|
| One of the properties of _A_ is that we can rotate it
| into a new position, _r(A)_ , where none of the points of
| _r(A)_ are in the same position as any points of the
| original position, _A_. So the sets _r(A)_ and _A_ have a
| zero intersection. For the set _A_ there are lots of
| possible choices of _r_ ... we pick a specific one that
| has some special properties. Again, the BT theorem is all
| about showing us how to do this.
|
| Now we take the union: _B = A u r(A)_
|
| The bizarre thing is this. If we've chosen _A_ and _r_
| (and therefore by implication, _B_ ) carefully enough, it
| ends up that there's another rotation, call it _s_ , such
| that _s(B)=A_ , the set we started with.
|
| So whatever the volume of _A_ , the volume of _A u r(A)_
| must be twice that, but that 's _B_ , and _B_ can be
| rotated to give _A_ back to us. So _B_ must have the same
| volume as _A_. So 2 times V(A) must equal V(A), so _A_
| must have zero volume.
|
| Well, we can kinda cope with that.
|
| But if we've chosen _A_ carefully enough, we find that a
| small, finite number of them, carefully chosen and
| rotated appropriately, together make up effectively the
| entire sphere (we miss out countably many points, but
| they have zero total volume, and we can fix that up
| later). So if finitely many copies of _A_ make up a solid
| sphere, they can 't have zero volume.
|
| And that's the "paradox".
|
| The conclusion is that we can't assign a concept of
| "volume" to the set _A_ , and this is explained a little
| more in a blog post I've submitted here before:
|
| https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiT
| heo...
|
| There's a lot more going on than just the "I can split
| infinite sets into multiple pieces that kinda look the
| same as the original", although that is certainly part of
| it, and lots of people already find that hard to take.
|
| To any who has got this far, I hope that's useful.
| ParanoidMarvin wrote:
| What do you mean "spread out"? Aren't there the same amount
| of even numbers as natural numbers?, because they both are
| countable sets. https://en.wikipedia.org/wiki/Countable_set
| ColinWright wrote:
| >>> _... if you take the natural numbers, and split them
| into odd and even, you get two copies of the natural
| numbers ..._
|
| >> _... the "two copies of the natural numbers" is sorta
| fine, except that they're more "spread out" ..._
|
| > _What do you mean "spread out"? Aren't there the same
| amount of even numbers as natural numbers?_
|
| Yes, there are the same number, but when you look at just
| the even numbers, they are each distance 2 from their
| neighbours, whereas the natural numbers are all distance
| 1 from their neighbours. So people are less surprised,
| because the even numbers are "spread out", they are less
| dense in any given area. To map the even numbers back
| onto the natural numbers you have to "compress" them.
|
| But this is not the case with the Banach-Tarski Theorem.
| There is a set, _A_ , and another set _B_ , which is just
| _A_ rotated around, and they are disjoint. So they have a
| union, _C=AuB_. But when you rotate _C_ , you can get an
| exact copy of _A_. There 's no squashing or spreading
| needed.
|
| So we have _A_ and _B_ , with _B=r(A)_ , and _A_
| intersect _B_ is empty. Then we have _C=AuB_. No problem
| here.
|
| The challenge comes that there is a rotation, _s_ , such
| that _s(C)=A_.
|
| So even though _C_ is made up of two copies of _A_ , it's
| actually identical to _A_. So start with _C_ , divide it
| into _A_ and _B_ , then rotate _B_ back to become a copy
| of _A_ , and then rotate each of those to become copies
| of _C_. So you start with _C_ , do some "cutting" and
| rotations, and you get two copies of _C_.
|
| Finally, when you take a few of these and put them
| together, you get a full sphere, so you can't say they
| have zero volume.
|
| Does that make sense? Does that answer your question?
|
| Does that help?
| 0xmarcin wrote:
| Also this only works when the highly contested Axiom of Choice
| is used.
| BalinKing wrote:
| 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 wrote:
| 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...
| hikarudo wrote:
| Feynman talks about exactly this in his book "Surely you're
| joking, Mr. Feynman":
|
| "Then I got an idea. I challenged them: "I bet there isn't a
| single theorem that you can tell me - what the assumptions are
| and what the theorem is in terms I can understand - where I
| can't tell you right away whether it's true or false."
|
| It often went like this: They would explain to me, "You've got
| an orange, OK? Now you cut the orange into a finite number of
| pieces, put it back together, and it's as big as the sun. True
| or false?"
|
| "No holes."
|
| "Impossible!
|
| "Ha! Everybody gather around! It's So-and-so's theorem of
| immeasurable measure!"
|
| Just when they think they've got me, I remind them, "But you
| said an orange! You can't cut the orange peel any thinner than
| the atoms."
|
| "But we have the condition of continuity: We can keep on
| cutting!"
|
| "No, you said an orange, so I assumed that you meant a real
| orange."
|
| So I always won. If I guessed it right, great. If I guessed it
| wrong, there was always something I could find in their
| simplification that they left out.
| mjburgess wrote:
| 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.
| anthk wrote:
| Sqrt(2) is a real number yet I bet you to properly measure up
| to the last point the diagonal of a square.
| zarzavat wrote:
| > Spacetime is real-valued
|
| There's really no evidence for this, as far as we _know_ the
| real numbers are a pure mathematical invention and don 't
| have any physicality.
|
| Even if you want to say that spacetime is _dense_ (i.e.
| infinitely divisible), there 's an infinite number of fields
| like that, the real numbers are just a convenient superset.
|
| There's no evidence that spacetime is dense either, and many
| practical ways in which it is not, as an obvious upper bound
| if you took all the energy in the observable universe to make
| one photon, it would still have a finite wavelength.
| jankovicsandras wrote:
| I have very little understanding of physics and math and
| might be wrong.
|
| But if we accept that the Planck length is the smallest
| possible length and the Planck time is the smallest
| possible time, then it seems logical that the universe is
| an integer lattice of these. ("Spacetime is not real-
| valued")
|
| https://en.wikipedia.org/wiki/Planck_length
| https://en.wikipedia.org/wiki/Planck_units#Planck_time
| czzr wrote:
| It's not known if those are fundamental limits.
|
| However the idea that space time is discrete is a
| reasonable hypothesis to test, we don't currently have
| any ways to probe at those resolutions, though.
| Pyramus wrote:
| That is correct and should be highlighted more.
|
| In a way the real numbers are a model (or maybe a
| 'language') to describe physical phenomena. They work
| exceptionally well at that, but they are not backed by
| evidence and do come with (theoretical) limitations.
|
| This bachelor's thesis is a good starting point [1], search
| for 'finite precision physics' or 'intuitionistic
| math/physics'.
|
| [1] https://www.math.ru.nl/~landsman/Tein.pdf
| mjburgess wrote:
| Well this is a popular idea imported from comptuer
| science, but there's absolutely no evidence for it -- and
| plenty against.
|
| Eg., QM is only linear in infinitely-dimensional real-
| spaces, etc.
|
| Essentially of a physics uses real spaces indispensably.
| There is no evidence _whatsoever_ that this is
| dispensible; other than the fever dreams of discrete
| mathematicians.
| jerf wrote:
| Remember, though: QM is wrong. Relativity also depends on
| continuous spaces, but it is also wrong. All the theories
| in physics that depend on continuous space are also
| wrong.
|
| By "wrong", I mean, we know they can't predict everything
| correctly. QM itself can't derive relativity. Relativity
| doesn't have QM in it, and break down at extremes like
| black holes. They're both very, very, very accurate in
| their domains, but physics _knows_ that neither theory
| has the domain of "the entire universe". This is not a
| wild claim by an HN commenter, this is consensus in the
| physics world, just perhaps not phrased in the way you're
| used to.
|
| It's _possible_ the eventual Grand Unified Theory will
| still have continuous space at its bottom, but it 's also
| entirely possible it won't. Loop quantum gravity doesn't.
| And personally I expect some sort of new hybrid between
| continuous and discrete based on physics history;
| whenever in the past we've had a similar situation where
| it couldn't be X for this reason, but it couldn't be the
| obvious Not-X for some other reason, it has turned out to
| be something that had a bit of both in them, but wasn't
| either of them.
| mjburgess wrote:
| They're not "wrong" in tests of their real-valuedness
| though.
|
| I'm somewhat confident there is an empirical test of
| real-valuedness in areas of physics which require
| infinite-valued spaces.
|
| However, either way -- the positions of the other
| commenters was that *geometry* is somehow a dispensable
| approximation in physics!
|
| This is an extremely radical claim with no evidence
| whatsoever. Rather some discrete mathematicians simply
| wish it were the case.
|
| It is true that *maybe* (!) spacetime will turn out
| discrete, and likewise, Hilbert spaces, etc. -- and all
| continuous and infinite dimensional things will be
| discretised.
|
| This however is a project without a single textbook.
| There is no such physics. There are no empirical
| predictions. There are no theories. This is a project
| within discrete mathematics.
| zarzavat wrote:
| The real numbers are a man-made axiomatic system. They
| were developed to make analysis mathematically rigorous
| to the high standards of pure mathematicians.
|
| The real numbers are popular outside of mathematical
| analysis because they provide a "kitchen sink" of every
| number you could possibly need.
|
| The downside is that the reals include many numbers that
| you don't need. The number 0.12345678910111213... is a
| transcendental real number, but it is not very useful for
| anything. It is notoriously difficult to prove that a
| given number is transcendental, i.e. part of the
| uncountable part of the reals and not the countable
| algebraic subset. Which is ironic because the uncountable
| part is infinitely larger!
|
| I'm not suggesting that physicists should drop their
| Hilbert spaces. Rather that a distinction should be drawn
| between mathematical model and physical reality.
|
| -
|
| As for whether spacetime is countably infinitely
| divisible:
|
| Infinity is big. Infinitely small implies that if you
| used all the atoms in the universe to write in scientific
| notation to write 10^-999..., that space would be more
| divisible than that. In fact for whatever absurdly tiny
| number you could think of, perhaps 1/(TREE iterated
| TREE(3) times) spacetime would be finer than that.
|
| I'll admit it's possible, but I have trouble believing
| it.
| mjburgess wrote:
| Well functions have properties in virtue of being defined
| over the reals, eg., sin(x) --
|
| I don't see that these properties are incidental.
|
| Yes they obtain in virtue of /any possible "dividing"
| discrete sequential process/ _never terminating_ , eg.,
| space being "infinitely divisible".
|
| However I dont think this is as bizarre as it appears.
| The issue is congition is discrete, but the world
| continuous.
|
| So we are always trying to project discrete sequential
| processes out onto the world in order to reason about it.
| Iterated zooming-in will, indeed, never terminate.
|
| I dont see that as saying anything more than continuity
| produces infinities when approached discretely. So, don't
| approach it that way, if that bothers you.
| jerf wrote:
| "They're not "wrong" in tests of their real-valuedness
| though."
|
| Yes, they are, or more accurate, they're not _right_
| enough for you to confidently assert the structure of
| space time at scales below the Planck scale. You are
| doing so on the basis of theories _known_ to be broken at
| that scale. You are not entitled to use the theories that
| way.
|
| Even the Planck scale being the limit is a mathematical
| number; I'm not sure we have _concrete_ evidence of that
| size being the limit. I 've seen a few proposed
| experiments that would measure at that resolution (such
| as certain predictions made by LQG about light traveling
| very long distances and different wavelengths traveling
| at very slightly different speeds) but I'm not aware of
| any that have panned out enough to have a solid result of
| any kind.
| geogra4 wrote:
| >Two 12 Inch Pizzas have less Pizza than one 18 inch pizza
|
| Love this one. Always go for the biggest pizza you can
| TMWNN wrote:
| 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 wrote:
| But that's almost never the case.
| TMWNN wrote:
| How do you know, unless you do the calculation?
| dartharva wrote:
| Don't know about you, but I have never seen any place
| that charges more for one large pizza than for two medium
| pizzas.
| niccl wrote:
| 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
| barbs wrote:
| This is a good video explaining it:
| 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.
| Kranar wrote:
| I love that one, what this taught me is that infinite sums are
| fundamentally different from finite sums, and that when
| operations performed on the finite are generalized to work with
| the infinite, there are often ambiguities and subtleties about
| how to perform that generalization.
|
| We wish to use the same familiar notation with the infinite
| that we do with the finite, but we must keep in mind that while
| they do share similarities, they are not the same operation.
| The way certain ambiguities are resolved in order to extend the
| finite to the infinite can lead to counterintuitive results and
| absurdities that may not even be apparent at first.
|
| For me, seeing how one approach to generalizing infinite sums
| yields the -1/12 result, and how that result actually has some
| relevance in physics is quite profound and insightful and I am
| happy that you reminded me of it.
| t8e56vd4ih wrote:
| what? that's incredible ...
| kadoban wrote:
| 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 wrote:
| > 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 wrote:
| 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.
| hodgesrm wrote:
| > 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 wrote:
| It was implied that the correct answer is 4.
| caf wrote:
| 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 wrote:
| Wasn't this a major plot point in Around the World in 80
| Days?
| kristjansson wrote:
| 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.
| db_admin wrote:
| You could also argue that it is a matter of perspective. From
| the perspective of either circle, A will only revolve 3
| times.
|
| Only by introducing a larger frame of reference, a grid or in
| the video a table, you gain an outside perspective. From this
| outside perspective you redefine a revolution according to
| some new orientation and end up with n+1 revolutions.
|
| Or maybe the argument is backwards and I just try to justify
| answering 3.
| rmu09 wrote:
| The demonstration would be clearer if the "point of contact"
| at the start of the rotation would be marked on the smaller
| circle and the larger circle would be divided into 3 segments
| of different color. That would make it obvious that 1
| rotation of the small circle doesn't trace a whole
| circumference of the small circle on the large one.
| hinkley wrote:
| 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.
| kristjansson wrote:
| It's interesting to read others reasoning on this. I find
| it hard to follow and much harder to generalize without
| some notation (and a diagram which this comment box
| struggles to reproduce)
| Linosaurus wrote:
| I like to decompose it.
|
| Hold center of small circle still, rotate big circle once
| counter clock wise, small one rotates 3x clockwise.
|
| Glue them together, rotate big circle once clockwise,
| small circle also rotates once clockwise.
|
| Sum them together, 0 rotations for big circle, 4 for
| small. I'm not at all sure how to rigorously generalize.
| unholiness wrote:
| I don't think that argument holds water. Imagine rolling it
| on the inside of the circle - the center of the circle traces
| a path with only twice its radius, yet it still rolls 4 times
| around.
| kristjansson wrote:
| I don't think that's right. It if the radius of the inner
| circle is one third the outer, it only rotates twice
| rolling around the inside, which makes sense as it's center
| traces a circle that's the the difference of the radii.
|
| Imagine the limiting case, as the inner circle approaches
| the size of the outer circle - the inner circle completes
| much less than one rotation per lap around the inside edge
| of the outer circle, and 'seizes' (if we're imagining these
| as gears), completing zero rotations per lap when the
| circles are the same size. However, rolling around the
| outside, a circle of the same size completes two rotations.
|
| In general the problem is like the old Spirograph toy
| (which I had to break out to convince myself)
| dartharva wrote:
| I'm confused, I also immediately came to 3 when I read this
| question. Is that wrong? What's the correct answer?
| ryathal wrote:
| the answer is 4. The reason it's 4 is because distance
| traveled is relative to the center of the circle. A circle
| will travel it's circumference in a rotation, but the
| distance traveled isn't actually the circumference of the
| inner circle, because that isn't where the center of the
| circle is. It actually travels the sum of the two circles
| radii.
| raldi wrote:
| Imagine Circle B is reduced to infinitesimal size, like rolling
| a quarter around a needle. It still makes one full revolution,
| even though the ratio of the circumferences is effectively
| infinite.
| cousin_it wrote:
| I just thought of a simple argument: unroll the bigger circle
| into a line. Then as the smaller circle rolls from one end of
| the line to the other, it makes 3 revolutions. After that, roll
| up the line back into a circle (with the smaller circle still
| attached to the end). That adds one more revolution.
| markc wrote:
| >A rolls one trip around circle B
|
| Without the diagram (which I didn't see until watching the
| video) this is ambiguous. In my visualization, the plane of the
| small circle (A) was perpendicular to the plane of the larger
| circle (B). (Think of circle B drawn on paper, while circle A
| is a coin on its edge)
|
| With that interpretation of "rolls one trip around", 3 is
| indeed is the correct answer.
| 1f60c wrote:
| > 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 px6) > 2 px9. Or am I missing something?
| detaro wrote:
| That's the formula for circumfence, not area.
| 1f60c wrote:
| Ahhhh, of course. _facepalms_
| erdewit wrote:
| It's still a good proof that the smaller pizza's have more
| crust.
| _Microft wrote:
| Area is calculated using pi*r^2:
|
| 2 * pi * 6^2 < pi * 9^2
| 1f60c wrote:
| Thank you. I feel kind of stupid now.
| bryanrasmussen wrote:
| it's true but really what they're missing is the number and
| variety of toppings on these pizzas.
| rawling wrote:
| Variety maybe, but surely number is also proportional to area
| (and presumably it's worse because "crust width" doesn't
| scale with diameter?)
| bryanrasmussen wrote:
| well not if the big one is a margherita pizza.
| jcims wrote:
| 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 wrote:
| 3 measures = 1 cup
|
| Now divide both sides of the equation by 2.
| GPerson wrote:
| 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
|
| " 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 wrote:
| 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 wrote:
| > 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.
| Kranar wrote:
| Consider that there are "real numbers" as in some kind of
| construct that we as humans are interested in
| understanding, and then there's ZFC-real numbers, which is
| an attempt to formalize "real numbers" rigorously. What we
| know is that we can never rigorously take "real numbers"
| and uniquely formalize them and so any formal investigation
| of "real numbers" will be prone to multiple
| interpretations.
|
| Given this, consider all models M that contain some set
| R(M) that satisfies ZFC's definition of real numbers and
| where R(M) is actually countable. Furthermore M also
| contains some set N(M) that satisfies ZFC's definition of
| natural numbers.
|
| Within this model M, since N(M) satisfies ZFC's definition
| of the naturals, it is ZFC-countable (that is it satisfies
| ZFC's definition of a countable set). Furthermore applying
| Cantor's diagonal argument to M, one can show that M does
| not contain a set that represents a surjection from N(M) to
| R(M), hence R(M) is ZFC-uncountable (it satisfies ZFC's
| definition of being uncountable).
|
| That said, all this means is that ZFC-countable, and ZFC-
| uncountable do not fully capture what it actually means to
| be countable or uncountable. ZFC-countable means a set has
| the same cardinality as whatever set satisfies ZFC's
| definition of natural numbers, which is not the same as
| what we as humans consider to be actual natural numbers.
|
| Similarly being ZFC-uncountable just means a set has a
| greater cardinality than the set that satisfies ZFC's
| definition of natural numbers, but that does not mean that
| such a set is actually uncountable.
|
| There is no way to extend ZFC so that what we consider to
| be actually countable or uncountable has one single unique
| interpretation. If there were then we could claim that said
| unique interpretation captured precisely our notion of
| countable and uncountable.
|
| What we can do is jump up a level to second order logic,
| and in that logic it actually is possible to have one
| unique interpretation of countable and uncountable sets so
| that there is a unique and countable set of naturals and a
| unique and uncountable set of reals, but second order logic
| comes with its own set of ambiguities and issues that for
| the most part mathematicians reject outright.
| matt-noonan wrote:
| 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.
| chriswarbo wrote:
| I like to think of this as a game, with one player choosing the
| axioms and the other choosing a model. If the first player
| picks a (countable) set of axioms, the second player can always
| respond with a countable model. Likewise, if the second player
| picks a countable model, the first player can always extend the
| axioms in a consistent way, to rule out that model. This can
| alternate back-and-forth forever.
|
| Uncountability is hence a 'leaky abstraction': something we
| want to investigate and study in general terms, even though
| _particular_ occurances might have some loophole /edge-case.
|
| I think about infinity and infinitesimals in a similar way,
| like iterative processes (e.g. the natural numbers arise from a
| process that increments; calculus arises from iteratively
| shrinking 'dx', e.g. by halving; etc.). Combining/interleaving
| such processes is tricky, so it's often more convenient to take
| their limits individually and manipulate those as objects;
| that's justified if those manipulations could potentially be
| implemented by _some_ interleaving, but can otherwise result in
| paradoxes (e.g. Thomson 's lamp)
| api wrote:
| 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
| [deleted]
| mlang23 wrote:
| There is a typo in the headline: Conterintutive ->
| Counterintuitive
| vic-traill wrote:
| 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 wrote:
| 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.
| btbuildem wrote:
| The SAT question one is brutal.. they did not include the correct
| answer on a multiple-choice question!
| brrrrrm wrote:
| 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, Polya's recurrence theorem)
| nicolas-siplis wrote:
| Huh, I wonder if there's any relationship to the Hairy Ball
| theorem. They appear to describe similar situations, but on
| different dimensions.
| gjm11 wrote:
| There's at least the following relationship: both the Brouwer
| fixed-point theorem and the hairy ball theorem are easy
| consequences of a more-highbrow thing called the Lefschetz
| fixed-point theorem.
|
| Unfortunately even the statement of the Lefschetz fixed-point
| theorem is a bit complicated, but let's see what I can do.
| I'll have to miss out most of the details. Depending on how
| much mathematics you know, it may not make much sense. But
| here goes.
|
| If you have a topological space X, there are a bunch of
| things called its "homology groups": H_0(X), H_1(X), H_2(X),
| and so on. I will not try to define them here. If you have a
| continuous map f from the space X to the space Y, then it
| gives rise to corresponding maps from H_k(X) to H_k(Y).
|
| The machinery that manufactures homology groups can be
| parameterized in a certain way so that you can get, instead
| of the ordinary homology groups, "the homology groups over
| the rational numbers", "... over the real numbers", and so
| on. (These can actually be obtained fairly straightforwardly
| from the ordinary homology groups "over the integers".) If
| you do it "over the rational numbers" or "over the real
| numbers" then the resulting things are actually _vector
| spaces_, and if your space is reasonably nice they're
| _finite-dimensional vector spaces_.
|
| (What's a vector space? Well, there's a formal definition
| which is great if you're a mathematician. If not: let n be a
| positive integer; consider _lists of n numbers_ ; for any
| given n, all these lists collectively form a "vector space of
| dimension n". You can do things like adding two lists
| (element by element) or scaling the values in a list by any
| number (just multiply them all by the number). A finite-
| dimensional vector space is a thing where you can do those
| operations, that behaves exactly like the lists of n numbers,
| for some choice of n.)
|
| And then the maps between these vector spaces, that arise
| (magically; I haven't told you how) out of continuous
| functions between topological spaces, are _linear maps_. You
| can represent them by matrices, with composition of maps (do
| this, then do that) turning into multiplication of matrices.
|
| OK. Now I can kinda-sorta state the Lefschetz fixed-point
| theorem.
|
| Suppose X is a compact topological space, and f is a
| continuous mapping from X to itself. Then you get
| corresponding maps from H_k(X) to itself, for each k. For
| each of these maps, look at the corresponding matrix, and
| compute its _trace_ : the sum of its diagonal elements. Call
| this t_k. And now compute t_0 - t_1 + t_2 - t_3 + ... . (It
| turns out that only finitely many of these terms can be
| nonzero, so the sum does make sense.) Then: _If this is not
| zero, then f must have a fixed point._
|
| So, whatever does this have to do with the Brouwer fixed-
| point theorem or the hairy ball theorem?
|
| The Brouwer fixed-point theorem is about maps from the
| n-dimensional ball to itself. It turns out that all the
| homology groups of the n-dimensional ball are _trivial_ (have
| only one element) apart from H_0, and that whatever f is the
| map from H_0 to itself that arises from f is the identity.
| And this turns out to mean that the alternating sum above is
| 1 - 0 + 0 - 0 + ... = 1. Which is not zero. So the map has a
| fixed point.
|
| The hairy ball theorem says that a continuous vector field on
| the 2-dimensional sphere has to be zero somewhere. Suppose
| you have a counterexample to this. Then you can make a whole
| family of maps from the 2-dimensional sphere to itself, each
| of which looks like "start at x and move a distance epsilon
| in the direction of the vector at x". If epsilon=0 then this
| is the identity map. If epsilon is positive and sufficiently
| small, then the fact that the vector field is never 0
| guarantees that the map does actually move every point; in
| other words, that it has no fixed points.
|
| But all the terms in that infinite sum that appears in the
| Lefschetz fixed-point theorem are (so to speak) continuous
| functions of f. And it's not hard to show that the value of
| the sum for f = identity is exactly 2. So for very small
| epsilon, the value of the sum must be close to 2, and in
| particular must be nonzero. So, for small enough epsilon, we
| have a map with no fixed points and a nonzero value of the
| sum, which is exactly what Lefschetz says can't happen.
| robocat wrote:
| - 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 wrote:
| Do the legs have to be even?
| istjohn wrote:
| 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 wrote:
| Not even, but the feet have to be coplanar.
| echopurity wrote:
| Epistemological conflation is par for HN.
| [deleted]
| gokhan wrote:
| 77+33 is not 100 and that's quite saddening.
| analog31 wrote:
| >>> 18. A one-in-billion event will happen 8 times a month:
| https://gwern.net/Littlewood
|
| This is certainly counterintuitive, given that one-in-a-billion
| and 8 per month have different units of measure.
| elcomet wrote:
| The implicit information is that there is one-in-billion chance
| that this event happens to someone in a given month.
| [deleted]
| Causality1 wrote:
| 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."
| BeetleB wrote:
| Gabriel's Horn was cool till someone pointed out to me that you
| can have a line of infinite length within a square (trivially).
|
| When comparing something of a certain dimension with something
| of a higher dimension, it's not at all surprising that the
| lower one can be infinite and the higher one finite.
|
| Usually it's phrased as "a finite amount of paint can paint an
| infinite area." But why do I need the Horn to realize this? It
| works in the Horn only if there is no lower limit to the
| thickness of paint. If you accept that, then I can take a drop
| and paint an infinite plane with it. Why do I need the Horn to
| demonstrate this?
| caf wrote:
| The way I'd heard the paint comment was along the lines that
| _" Gabriel's Horn can hold only a finite quantity of paint,
| but requires an infinite quantity of paint to cover the
| surface"_.
|
| So if you think of it as a bucket that can't hold enough
| paint to cover itself, that _is_ at least a little
| surprising.
| BeetleB wrote:
| But that's exactly my paint. If you allow for infinitely
| thin paint, then a finite volume of paint can always cover
| an infinite surface - you don't need Gabriel's Horn to show
| that.
|
| If you don't allow for infinitely thin paint, then no -
| Gabriel's Horn surface cannot be painted even with an
| infinite amount of paint.
| mrestko wrote:
| I don't think we know that spacetime is quantized.
| guerrilla wrote:
| 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.
| pdonis wrote:
| _> 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 wrote:
| 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.
| umanwizard wrote:
| 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.
| smoldesu wrote:
| 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.
| st_goliath wrote:
| > 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
| nwallin wrote:
| > 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
|
| 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 wrote:
| I don't get what thought that picture is supposed to provoke.
| ruuda wrote:
| 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 wrote:
| The earth has "holes" and as such the proof of that statement
| for a sphere does not apply to the earth.
| lasc4r wrote:
| Thank you, something about this didn't make total sense to
| me.
| bogosmith wrote:
| There is a typo in the title.
| vadim_lebedev wrote:
| 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
| simonebrunozzi wrote:
| > The Earth makes 366.25 rotations around its axis per year
|
| Isn't it 365.25?
| folli wrote:
| 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-ab0d69361311b4f15b0064...
| [deleted]
| kgwgk wrote:
| 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.
| tunesmith wrote:
| Remedial question about zero-knowledge proof. Isn't "proof" a
| misnomer since the concept is about just making it incredibly
| _likely_ to be true?
| pcmonk wrote:
| 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 wrote:
| 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...
| eruleman wrote:
| No -- the word 'proof' is accurate, it establishes with
| certainty that you know the value of x.
| Kranar wrote:
| 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#Definitio...
| dlubarov wrote:
| 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 wrote:
| > 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
|
| 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 wrote:
| > 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 wrote:
| 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.
| ameetgaitonde wrote:
| I really don't like how the OP phrased this one, because
| it's about more than knowing the value of your car.
|
| This paper was published in 1970, but it demonstrated how
| markets can fail, as well as a method of correcting that
| failure. Imagine the following scenario:
|
| There's a used car market of private sellers that is
| comprised of a mixture of peaches (good) and lemons (bad).
| To keep it simple, let's assume we're just talking about
| one model of car.
|
| Additionally, there's no way to identify which car is a
| lemon, but it's known that they're worth much less than a
| peach because of the much higher cost of maintenance.
|
| If you have a market where the above conditions exist (only
| seller knows if car is lemon/peach, and a mixture of
| lemons/peaches), you'd potentially end up with a market
| failure.
|
| This is because sellers of peaches can't get the price they
| want for their car, whereas sellers of lemons can profit
| over the expected value of theirs.
|
| The problem with assuming that lemons would be removed from
| the market is that any buyer of a lemon would want to sell
| it once they've realized what they purchased, putting it
| back on the market. This effect compounds to where a
| greater percentage of cars being sold on the market are
| lemons, further depressing the price and removing peaches
| from the market.
|
| Akerlof's solution to fix the market was to introduce
| warranties. Owners of peaches would be willing to offer
| warranties, because they trusted the quality of the cars
| they were selling. Eventually, buyers would see the lack of
| a warranty as the indication of a car being a lemon,
| forcing the sellers of lemons to either offer a warranty or
| lower their asking price below the market price of the
| vehicle.
|
| His work applied to the function of other markets, like
| insurance (older people are the costliest for health
| insurance companies) and employment markets (Certain
| classes of people have difficulty finding a job despite
| similar skills), as well as the institutions that have
| formed (Medicare, professional licensing) to improve the
| functioning of these markets.
| Ekaros wrote:
| Can make it, but I believe there is likely a point where
| sellers and buyers utility value considerations cross. That is
| seller considers price of car to be same as buyer considers
| this. As these two values are not necessarily tied together.
| Maybe needs of both parties are different.
| rsj_hn wrote:
| 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
|
| And the nice thing is you can prove this with fairly elementary
| techniques (e.g. first year complex analysis is all you need).
| MatteoFrigo wrote:
| 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
| cycomanic wrote:
| 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 wrote:
| Maybe nature prefer right-handedness just because it started
| in North hemisphere.
| kadoban wrote:
| 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.
| amelius wrote:
| How can someone rigorously prove that? E.g. perhaps there is
| something inside quarks that has chirality which we haven't
| discovered yet.
| mkl wrote:
| > 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...).
| doublepg23 wrote:
| 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.
| kbenson wrote:
| 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.
| d_burfoot wrote:
| 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 wrote:
| Not surprisingly, this is related to the observer effect of
| Quantum Mechanics.
| jiggawatts wrote:
| 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
| amelius wrote:
| Can you explain the double slit experiment with this
| approach?
| jiggawatts wrote:
| Not really, the waves in the double slit experiment travel
| through vacuum (or air), not through a solid like with
| polariser sheets.
| amelius wrote:
| But they interact with the matter at the boundary of the
| slit.
| joshuamorton wrote:
| IIRC, you can grok this with linear transforms:
|
| given X = [[1,0],[0,0]] and Y = [[0,0],[0,1]], aXY = 0 for
| all a (because XY == [[0,0],[0,0]]).
|
| But, given V = [[1,1],[1,1]] (or, in fact, a lot of other
| matrices/linear transforms), XVT = [[0,1],[0,0]], which
| blocks all vertical components and rotates the horizontal
| component 90 degrees. So if you imagine light as a wave with
| a horizontal and vertical component, and the polarizing
| filter applies a linear transform to the light.
| ffhhj wrote:
| For me this is as surprising as the double slit experiment, but
| much easier to reproduce.
| _Microft wrote:
| You can do that 'continuously' by inserting many layers of
| slightly rotated polarizers to rotate the polarization plane by
| an arbitrary angle.
|
| An applied example of this are liquid crystal screens.
| Molecules take the role of the polarizers there. The rotation
| angle depends on an applied electric field and when you
| sandwich a layer of crystals between polarizers, you almost
| have a screen:
|
| https://www.britannica.com/technology/twisted-nematic-cell
| akomtu wrote:
| My hunch is that light does get thru the first two filters, but
| it gets "squeezed" so it's hard to see it. The third filter
| "unpacks" the light.
| pishpash wrote:
| 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 wrote:
| The correct answer was not provided as a multiple choice
| option, which means by definition that 0% of test takers
| selected it.
| pishpash wrote:
| All right, that's certainly counterintuitive...
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