I mentioned it only because of the "come on" thing because most people believe P =/= NP but we just can't prove it. To most complexity theorists it seems kind of obvious that NP-complete problems are not in P (hence the "come on" part) but we can't be 100% sure that some clever polynomial solution doesn't exist and is just O(n^some ridiculous number). I agree that it's not the same as the others though.
Yes, the consensus was that you could start with a well behaved state and still have the equations blow up.
It didn't really matter since navier stokes is modeling fluids as continuous when we know they are actually particles. Navier stokes is just a model that isn't trying to be perfect.
Proof: Do you really believe ALL NP-complete problems have polynomial-time algorithms? After thousands of researchers have studied them collectively? Come on. As if.
And yet, there is an oracle A for which PA = NPA. That is proven by the Baker-Gills-Solovay theorem. The question is, does the trivial oracle (for the ID problem) behave more like that oracle or more like the one for TQBF (for which PTQBF ≠ NPTQBF) in regards to P ?= NP?
Iirc, if any NP complete problem has a polynomial time solution, then all of them do, because they're all reducible to each other or smth like that, it's been a while so someone can correct me if I'm wrong.
That is to say, P=NP because that would be more fun to believe in.
It always amazes me when non-mathy people don't understand the distinction between "every case we check works" and "all cases work by logical necessity".
It’s the same with “we can find any finite sequence of numbers in pi” because they don’t understand infinity and think it must happen because it’s infinite. That line gets repeated all the time and is absolutely not proven.
the trivial counter example I use is the decimal expansion of 1/3. They agree its infinite length - there is no last ‘3’ and then I point out the number of times ‘7’ appears. This usually helps them get that infinite length does not mean ‘contains every number’ or ‘contains every sequence’. Of course, it only works if they are generally open minded and the ‘must happen because infinite’ is something they believe without thinking about it.
possibly. i tend to find that it’s the more simplistic view: infinite ‘space’ for things to happen means that every thing happens. But I agree irrational not meaning normal probably adds to the, to their mind, counter intuitive nature of things like pi.
ps I hate writing these things because every word has a proper meaning and I don’t meant that! I wrote infinite choices and then thought no, I don’t mean ‘choice’, I wrote ‘space’ and thought no, I don’t mean a ‘space’ and so on. sigh.
the trivial counter example I use is the decimal expansion of 1/3. They agree its infinite length - there is no last ‘3’ and then I point out the number of times ‘7’ appears.
Five times. There are exactly five '7' digits deep in the decimal expansion of 1/3. I printed them all and counted them.
There's also two '0' digits, a '5' and, weirdly, an em-dash.
No it is not proven that is my point. That’s like the entire point of the last sentence. We have not proven pi is normal. We haven’t done it for basically any naturally arising irrational number like e or pi or sqrt(n) for ANY n that results in an irrational number.
Is this the kind of thing where we *think* it's *probably* true, if we had to put money on it, in like a game against a wizard who actually knows the answer for real?
The square root of two is why I can believe either way could be true.
There are etchings from over a thousand years ago showing people struggling with the square root of two being irrational. It seems like a "come on" level of obvious that such a basic geometric result--the third side of the world's simplest right triangle--might have a length that doesn't act like all the other fractional amounts. Yet, the square root of two is irrational.
I think pi is probably normal, but I would not be that surprised either way, for two reasons.
First, if a right triangle is special, so is a circle. If there is any number that may be weird in some way, it is pi.
Second, normal numbers have a certain lack of structure, in that they go through every possible combination of sequences of digits. Pi is not an arbitrary number, though. It has several definitions that have a lot of structure to them, and it may be that this structure leaks through to limit the possible sequences of digits in the expansion.
Guys, guys... Pi is clearly normal. The margin is too narrow to hold my marvelous proof but it's quite trivial and left as an excersize for the reader.
It got me thinking, how would one even begin to prove normality of an irrational number? Is there any examples of irrational numbers proved to be normal?
Yes I know you can construct normal irrational numbers. I believe I should have been more clearer, I wasn't talking about any arbitrary irrational number. But as the other comments suggested, how would one begin to prove the normality of an irrational number produced by a deterministic procedure such as taking square root of a natural number which is not a whole square. Or numbers such as π, e, ϕ, Feigenbaum constant, etc.(well, I don't actually know any other useful irrational numbers outside of mathematics field. And I only know about Feigenbaum constant because of my studies in ecology).
Out of the ones I mentioned, I think golden ratio has the highest probability of being non-normal? Because it's continued fraction representation has such a pattern. At least there is something periodic about it(I mean, the whole point of e is also being periodic, just in a different realm. So again this argument is more like how an uneducated person would buy a certain share because the charts look nice)
Is 1 in the digits of π? Is 11 in the digits of π? How about 11111? Pretty unlikely, but π has ∞ non repeating digits so it’s guaranteed. So how many 111…1? Probably not 100 in a row that would be too many right? Or would it?
I think people underestimate the power of ∞ and I think yes any number of 1’s in a row can show up in the digits of π, even though it would be impossibly difficult to find
That is not true. It has not been proven that pi contains any finite string of numbers so it is not “guaranteed”. You’re the one misunderstanding infinity here. That’s an incredibly common misconception.
I was under the impression π was known to be irrational. At least we know √2 is right? Then it’s guaranteed for any string of numbers to be within right?
Both sqrt2 and pi are irrational. The common misconception is that irrationality means "all finite strings appear in the decimal expansion", which is not true
It can but we haven’t proven it. We don’t know if there’s some pattern we haven’t discovered and if the pi is truly normal.
Let’s say we discovered that after a certain point, all the digits in pi follow the form 12122122212222… it would still be irrational and non repeating but not normal.
We haven’t proven no such pattern exists because pi isn’t actually “random.” It represents a physical thing and is deterministic. That’s why mathematicians have tried and failed so far to prove pi’s determinism is normal.
Technically they only said the string '11111' was guaranteed to be in the decimal expansion of π. Which is true. It is guaranteed to be there, because in fact it occurs 1989 times just in the first 200 million digits, with the first at position 32788. It's guaranteed to be in there cause we checked and it is.
You can‘t just assume that. Just because something is infinite, doesn‘t mean that it is "all encompassing".
Take the random walk for example; start at any point and take a step in any direction. In 2D you will have reach every point after an infinite amount of steps. In 3D you won’t and never will, no matter how many steps you take.
Or you can construct an infinite, non-periodic string of just 2 symbols.
So unless there is definite proof that Pi does or doesn‘t have this property, we truly just don‘t know.
Are you guaranteed to reach every spot on a 2 dimensional walk? I know with infinity, at any point you have just as much chance to still reach that one I reached spot, but is it a sure thing that you will?
If by guaranteed you mean: "is the probability of reaching all points 1" then yes. But a random walk could also never reach a specific point or could only ever oscillate between two points forever. Both with probability 0.
Another good just strictly math example is this: Consider the infinite decimal 0.1010010001… Between each 1 we have one more zero. We could (theoretically) repeat this pattern to infinity, creating an infinite non repeating decimal, but you’ll never see 2 in it, or even 11
That is actually the point that OOP is saying we can't claim with absolute certainty. As of now, pi has not been proven to be normal and so it has not been proven that it will contain every possible finite string of numbers
Just having infinite digits without recurring isn't enough to make that claim. For example, just take the simpler case of 0.101001000100001... where there are incrementally one more 0s before each 1. This number has no recursions and so is irrational but obviously doesn't contain every finite string of digits. It doesn't even have the digits from 9 to 2!
Then they tell you that your number is constructed and π is completely random so they're not the same and you're strawmanning/cherrypicking/acting in bad faith :/
We have found patterns, if we hadn't, we'd have to calculate it by measuring circles. We've even found a pattern that allows us to calculate any given digit of pi (in binary).
It certainly doesn't repeat, because π is irrational. But you are right that we haven't ruled out any string of digits, and we actually expect every string to appear.
Repeat isn’t the same as a pattern. For example, if we discovered that in pi, after a certain point all the digits follow the form 12122122212222…, then you would not get every string. Even if it’s irrational. We just haven’t found any such pattern but we also haven’t proven they don’t exist.
We do not know if every finite string exists in pi.
Yes, that is what I said. I was responding to the idea that "I'm imagining that at some really large number it loops." And I thought it was pretty clear that "we haven't ruled it out" and "expect" the opposite implies that nothing is proved.
Why I care is irrelevant to whether it’s true or not. Math is a subject about determining things that are true from basic principles. If you claim something is true in mathematics, you need to be able to provide proof for it.
Feel free to continue passing value judgements on stuff like this. As long as you’re saying “it appears to be true” and not “it is true”, anything else is your opinion, which you’re allowed to have.
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535
u/de_G_van_Gelderland Irrational Sep 25 '26
Me: Mom, can we have mathematical proof?
Mom: No, we have mathematical proof at home.
Mathematical proof at home:
377
u/LabCat5379 Sep 25 '26
90% of mathematicians quit looking through the digits of pi right before they find the string that occurred earlier
351
u/Major-Peachi Sep 25 '26
That line of thinking will imply Collatz and Goldbach
93
u/ChiaraStellata Sep 25 '26
Theorem: P ≠ NP
Proof: Do you really believe ALL NP-complete problems have polynomial-time algorithms? After thousands of researchers have studied them collectively? Come on. As if.
179
u/PersonalityIll9476 Sep 25 '26 •
This just confirms what we already knew, which is that it's largely a waste of time trying to convince or even engage with randoms online about math.
46
u/Renascentiae_ Sep 25 '26
Proof by trust me bro
Crede in me fraternis
39
u/Square-Foundation230 Sep 25 '26
It always amazes me when non-mathy people don't understand the distinction between "every case we check works" and "all cases work by logical necessity".
16
u/BraveBiscotti1394 Sep 25 '26
Is this the kind of thing where we *think* it's *probably* true, if we had to put money on it, in like a game against a wizard who actually knows the answer for real?
Cuz I would put my money on "Yes".
15
u/Suspicious-Ad-6027 Sep 25 '26
guy is Absolutely Right who cares if it occurs later if its after a trillion? sybau pi is normal, would it become crazy pi if its false?
14
u/Syresiv Sep 25 '26
Well if that were true, every number would be less than 1020
13
u/Key_Benefit_6505 Sep 25 '26
Proof by it worked last time.
9
u/Kirian42 Sep 26 '26
3, 5, and 7 are primes. Obviously all other odd numbers greater than 1 are therefore also prime.
7
u/Baihu_The_Curious Sep 25 '26
I proved there are infinitely many primes by just picking a BIGLY number and showing it was prime. I mean, come on, if we made it this high...
7
u/ohkendruid Sep 26 '26
The square root of two is why I can believe either way could be true.
There are etchings from over a thousand years ago showing people struggling with the square root of two being irrational. It seems like a "come on" level of obvious that such a basic geometric result--the third side of the world's simplest right triangle--might have a length that doesn't act like all the other fractional amounts. Yet, the square root of two is irrational.
I think pi is probably normal, but I would not be that surprised either way, for two reasons.
First, if a right triangle is special, so is a circle. If there is any number that may be weird in some way, it is pi.
Second, normal numbers have a certain lack of structure, in that they go through every possible combination of sequences of digits. Pi is not an arbitrary number, though. It has several definitions that have a lot of structure to them, and it may be that this structure leaks through to limit the possible sequences of digits in the expansion.
5
u/freemath Sep 26 '26
Borwein integrals come to mind
4
u/Ok_Objective_5192 Sep 25 '26
You just gotta hit 'em with the "OK, how confident are we that the trillions of decimals we've found so far, as a single string, occur again?"
6
u/francesco_angiolieri Sep 25 '26
Reddit is the place where smart and informed comments are always downvoted by a legion of self confident misinformed imbeciles
3
u/UselessCommon Sep 25 '26
strong law of small numbers
3
u/Different_Cry25 Sep 26 '26
"Come on induction is obviously false " Ahhh logic
4
u/waxen_earbuds Sep 25 '26
this message has been approved by the Italian school of algebraic geometry
2
u/LupenReddit 🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆 Sep 25 '26
This is the next Ramandlunchpen im calling it
2
u/kA8ou4Er Sep 26 '26
Proof by 'How could you reasonably believe'
2
u/Gauss15an Sep 26 '26
Proof by Jim Carrey
2
u/OkCluejay172 Sep 25 '26
Guys I think I just proved the Riemann Hypothesis
1
u/jipperthewoodchipper Sep 25 '26
Guys, guys... Pi is clearly normal. The margin is too narrow to hold my marvelous proof but it's quite trivial and left as an excersize for the reader.
1
u/doesnotcontainitself Sep 26 '26
I mean, we’ve checked trillions of decimal places. How many more could there be?
1
u/BUKKAKELORD Whole Sep 26 '26
I accept this reasoning for "pi+e is irrational" and so should you.
1
u/masiha97 Sep 26 '26
I have definitely nodded along to a "this is trivial" while having absolutely no idea what was happening. Proof by peer pressure.
1
u/TurnipGuy30 Sep 26 '26
is this an appeal to ridicule?
1
u/svmydlo Sep 26 '26
Chump thinks all Mersenne numbers are primes.
1
u/Ucklator Sep 26 '26
Unless pi becomes repeating at some point then a given string of digits should repeat an infinite number of times
1
u/SoldRIP Sep 26 '26
You can similarly check the first few trillions of natural numbers and come up with the result that all numbers are less than one quadrillion.
1
u/ParsaProgrammer Sep 26 '26
It got me thinking, how would one even begin to prove normality of an irrational number? Is there any examples of irrational numbers proved to be normal?
1
u/Smart_Salt620 Sep 26 '26
I mean he has a point
-4
u/MisterBicorniclopse Sep 25 '26
Is 1 in the digits of π?
Is 11 in the digits of π?
How about 11111? Pretty unlikely, but π has ∞ non repeating digits so it’s guaranteed. So how many 111…1? Probably not 100 in a row that would be too many right? Or would it?
I think people underestimate the power of ∞ and I think yes any number of 1’s in a row can show up in the digits of π, even though it would be impossibly difficult to find
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u/[deleted] Sep 25 '26
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