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.
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.
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.
> 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?
> 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.
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.
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.
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.
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.
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.
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).
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.
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.
> 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).
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.
> 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.
> 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.
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.
> 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.
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.
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.
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).
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.
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.
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.
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.
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!
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".
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.
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?
> 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".
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 1.20€.
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).
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.
"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".
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.
> 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.)
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).
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.
> > 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%.
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)
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".
> 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?
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.
> "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.
> 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.
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.
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.
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%.
#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.
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?
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.
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.)
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.
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.
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
> 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.
> 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.
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.
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.
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:
> 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?
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:
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.
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.
> 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.
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.
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.
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.
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)
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.
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.
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)
If circle A is rolling around the edge of circle B from within, it is actually revolving around a new, smaller circle C which has the Radius Circle B - Circle A.
Is this correct? wouldn't the radius of the circle it's revolving around be significantly smaller if you had it follow the edge of the larger circle but from the inside? I don't know much about math or physics, so I could be wrong, but I think it would be significantly less, closer to two, right?
The reason that this problem is tricky, and has a counterintuitive solution is that Circle A is rolling around Circle B, and so the 'radius; of the circular path it is following is the radius of Circle B + the radius of Circle A.
I'm no expert, but some quick math:
Circle B has a radius of 9, circumference then is 56.55
Circle A (1.3) has a radius of 3, circumference is 18.84
56.55 / 18.84 = 3. This suggests that you could "unwind" circle A (say it was made of pipecleaner), and you would need 3 circle As to fully ensconce Circle B
BUT that wasn't the question. The question states that Circle A is rolling AROUND circle B.
So the circumference of the circle we are now trying to 'ensconce' is Circle A Radius + Circle B Radius = 12, so the new circumference is 75.39, and divided by Circle A Circumference, we end up with 4, which makes sense, and matches the demonstration from the video.
HOWEVER, if we are 'rolling' around the inside of circle B, then I think you're right, the radius of the circular path Circle A will take is 9 -3 = 6, and the circumference of said circle is 38, so we it only will take two rolls. I think this is correct, i do not think it will be 4 rolls.
Think of it this way: when Circle A is rolling around the inside of Circle B, the way you're picturing it is circle A following the outside of Circle B, which is why I think intuitively it feels like the answer will still be 4 revolutions. BUT a better way to think of it is that Circle A is actually revolving around a new Circle, Circle C, which has a circumference of 38. Circle A is not revolving around Circle B, it's revolving around this new, smaller circle within circle b. Does that make sense?
I made a quick diagram to illustrate my point. I did it in a wire-framing tool that has snap to grid, so it's definitely not perfect:
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.
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.
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.
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.
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.
- at any time while stirring a cup of coffee, there will be a point on the top that is right where it started. (if we pretend coffee stirring is 2-dimensional, Brouwer's Fixed Point theorem)
- a drunk man will eventually make it home, unless he can fly. in which case he only has a 34% chance. (if we assume the man is walking/flying on a grid, Pólya's recurrence theorem)
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) continuou...
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.
And since the spacing between the feet is obviously irrelevant, that further implies that it's always possible to find 4 coplanar points on the surface arbitrarily close together. Or arbitrarily far apart.
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."
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.
> 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.
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.
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.
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?
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.
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.
It's not supposed to be especially tricky. Yes, as soon as you realise that the horn is a bucket with a very deep section that is narrower than the assumed thickness of a coat of paint, the surprise dissipates.
I believe the point of the exercise is an introduction to curves with infinite length but finite area under them, in order to expand one's intuition about such objects, which is then transferable to other examples like space-filling curves.
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
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.
> There are actually many examples of things working different for the mirror image, e.g. many medical/chemical compounds work differently based on chirality.
It "works different" because you consider only change in one part of system. You change compound, but rest of system stays the same. If you mirror entire system, it works almost the same way (differences due to parity violation by weak force are millions time smaller than required to even detect differences).
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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.
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...
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).
> 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
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.
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.
where I've written this estimate as the difference of a divergent term and a constant term.
Here's the point: the divergent term is really just left over (integrated up or 'induced') from our asymptotic description of the degree 0 sum. In a sense, it's degenerate and doesn't provide any sort of 'new' information. So in the same sense, the interesting semantic content of 1 + 2 + 3 + ... is just the constant term: -1/12.
Ps. This way of thinking is related to trace formula truncation.
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.
“ 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.”
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.
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.
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.
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)
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.
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.
>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.
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!
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.
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.
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
>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
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.
But in a Caesar cipher only some types of computation is possible (to wit, addition and subtraction).
Now, "multiplication" of letters is, well, dodgy as a concept. But, the thing is that you can build encryption systems where D(E(k) op V) is equal to k op V, and op contains both addition and multiplication.
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:
The area enclosed by a circle is πr^2 while 12 and 18 are the diameters, right? The radius of a disc is half the diameter, so 6 and 9, respectively. 2π6^2 < 1π9^2 ~~ 226.2 < 254.5. In other words the area is proportional to the square of the radius, not linearly proportional to the diameter in any way.
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.
To make it simpler, let's assume that the shape of pizza is a square, i.e. we are talking about a 12×12 inch pizza and a 18×18 inch pizza. This increases the size of both pizzas by the same ratio, so it shouldn't change the answer to "are two small pizzas smaller than one large pizza?".
Now let's measure the size in a new unit which is 6 inch long. So the small pizza is 2×2 units, and the large pizza is 3×3 units.
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?
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.
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)
The thing is that you could also use this kind of logic to show that there are more natural numbers than there are natural numbers. For example, you could associate 1 with all of the numbers from 1 to one million, and still have enough numnbers 'left' to associate each of 2,3,... with distinct natural numbers above one million.
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.
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.
> 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.
334 comments
[ 3.7 ms ] story [ 426 ms ] threadand we did the math to prove it to ourselves. Blew our minds... an actual valid use for middle school math.
https://en.wikipedia.org/wiki/Zero-knowledge_proof#Definitio...
[1] https://crypto.stackexchange.com/questions/5832/what-exactly...
[*] 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.
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?
> 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.
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.
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.
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.
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.
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).
(How liquid water could form under those conditions is still something of a mystery: https://en.wikipedia.org/wiki/Faint_young_Sun_paradox)
Perhaps we could just use a straw to give our sun a liposuction.
> 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).
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.
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.
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.
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.
And wikipedia seems to agree with me, see pound (mass) vs pound (force).
Anyway, yes, the proper unit of force is the Newton.
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)
Brake fluid anyone?
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”
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.
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?
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".
FWIW, the pound is a proper unit of mass: https://en.m.wikipedia.org/wiki/Pound_(mass) .
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.
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).
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".
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.
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.)
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).
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.
>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%.
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)
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...
https://www.ted.com/talks/del_harvey_protecting_twitter_user...
I could not find any info about that bug, anyone got a link or a source?
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.
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_...
> 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
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.
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.
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.
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.
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.
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
(And the reason is just the same: what matters is the remainder.)
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.
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)
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
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.
> 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.
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.
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.
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
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?
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.
That's fun. I of course immediately selected 3 which means I could have a bright career in test preparation ahead of me.
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.
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.
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.
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.
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)
I made a diagram that helped me think through it visually:
https://i.imgur.com/dospt2w.png
If circle A is rolling around the edge of circle B from within, it is actually revolving around a new, smaller circle C which has the Radius Circle B - Circle A.
The reason that this problem is tricky, and has a counterintuitive solution is that Circle A is rolling around Circle B, and so the 'radius; of the circular path it is following is the radius of Circle B + the radius of Circle A.
I'm no expert, but some quick math:
Circle B has a radius of 9, circumference then is 56.55 Circle A (1.3) has a radius of 3, circumference is 18.84
56.55 / 18.84 = 3. This suggests that you could "unwind" circle A (say it was made of pipecleaner), and you would need 3 circle As to fully ensconce Circle B
BUT that wasn't the question. The question states that Circle A is rolling AROUND circle B.
So the circumference of the circle we are now trying to 'ensconce' is Circle A Radius + Circle B Radius = 12, so the new circumference is 75.39, and divided by Circle A Circumference, we end up with 4, which makes sense, and matches the demonstration from the video.
HOWEVER, if we are 'rolling' around the inside of circle B, then I think you're right, the radius of the circular path Circle A will take is 9 -3 = 6, and the circumference of said circle is 38, so we it only will take two rolls. I think this is correct, i do not think it will be 4 rolls.
Think of it this way: when Circle A is rolling around the inside of Circle B, the way you're picturing it is circle A following the outside of Circle B, which is why I think intuitively it feels like the answer will still be 4 revolutions. BUT a better way to think of it is that Circle A is actually revolving around a new Circle, Circle C, which has a circumference of 38. Circle A is not revolving around Circle B, it's revolving around this new, smaller circle within circle b. Does that make sense?
I made a quick diagram to illustrate my point. I did it in a wire-framing tool that has snap to grid, so it's definitely not perfect:
https://i.imgur.com/dospt2w.png
You can play around with what I made here:
https://whimsical.com/r-DoXHwe2kUdgvSiAu2yuGpE
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.
- at any time while stirring a cup of coffee, there will be a point on the top that is right where it started. (if we pretend coffee stirring is 2-dimensional, Brouwer's Fixed Point theorem)
- a drunk man will eventually make it home, unless he can fly. in which case he only has a 34% chance. (if we assume the man is walking/flying on a grid, Pólya's recurrence theorem)
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) continuou...
Also, it's patently absurd someone would include Fitch's Paradox, a piece of philosophy, on a list of "counterintuitive facts."
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.
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?
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.
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.
I believe the point of the exercise is an introduction to curves with infinite length but finite area under them, in order to expand one's intuition about such objects, which is then transferable to other examples like space-filling curves.
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.
It "works different" because you consider only change in one part of system. You change compound, but rest of system stays the same. If you mirror entire system, it works almost the same way (differences due to parity violation by weak force are millions time smaller than required to even detect differences).
"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.
https://en.wikipedia.org/wiki/Ehrenfest_paradox
Effectively there are g-forces so high that you end up with subatomic particles.
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.
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.
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.
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.
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.
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.
https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF
Although it relies on a particular interpretation of the left side of the equation, it's used in some areas of physics.
https://youtube.com/watch?v=YuIIjLr6vUA
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.
Next, let's do something like integration, and increase those zeroth powers to one, ie. consider the function
In this case, the estimator ends up being where I've written this estimate as the difference of a divergent term and a constant term.Here's the point: the divergent term is really just left over (integrated up or 'induced') from our asymptotic description of the degree 0 sum. In a sense, it's degenerate and doesn't provide any sort of 'new' information. So in the same sense, the interesting semantic content of 1 + 2 + 3 + ... is just the constant term: -1/12.
Ps. This way of thinking is related to trace formula truncation.
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.”
1. There is a countable model of real numbers.
2. There even is a countable model of the entire set theory.
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.
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.
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)
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.
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!
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
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
Doing computations on cipher text is very limited which is why it's not very efficient to do complicated operations.
Now, "multiplication" of letters is, well, dodgy as a concept. But, the thing is that you can build encryption systems where D(E(k) op V) is equal to k op V, and op contains both addition and multiplication.
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
Awesome, thanks for that geometry refresher, that makes sense now of course :).
Now let's measure the size in a new unit which is 6 inch long. So the small pizza is 2×2 units, and the large pizza is 3×3 units.
twice 2×2 = twice 4 = 8
3×3 = 9
According to a specific non-colloquial definition of "as many".
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?
(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)
There are still today human tribes that don't count at all.
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.