The h Bar

Cows

12:04 AM

@0celouvsky ok, I will try to make the thoughts more precise. Give me a week

0celouvsky

@Cows why don't you read some analysis

take things slow

I don't think you're prepared for the things you're talking about

Cows

ok

0celouvsky

not that analysis would help you with that

but you're talking about poincare duality I think

I'm not sure you understand the details there

(maybe you do)

Cows

hehe neh I don't understand the details myself lol

I just ran across it today

I am definitely going to have to invest some time in analysis

before anything

Slereah

12:34 AM

hey

Simply Beautiful Art

Hey

0celouvsky

@Slereah up early?

Slereah

again

Basically I am an undead ghoul by now

Simply Beautiful Art

lol

Slereah

Current photo of me

Simply Beautiful Art

12:38 AM

@Slereah I recommend you get facial surgery

Slereah

I don't think you can do surgery on a skull

Simply Beautiful Art

Ah, then I recommend face tissue implants

0celouvsky

1:08 AM

@BernardoMeurer Kendrick has a U2 feature on his new album...

Probably going to be overly political garbage

Yashas

Don't asteroids, comets, meteors, planets, stars and galaxies sometimes move towards the earth?
And if when they do, yet're still producing redshifts (light is coming from sources moving away from the earth) as opposed to blue (from sources moving towards), than doesn't that mean this whole redshift thing is bogus?

how do I answer it?

Slereah

what

DanielSank

Hi, everybody.

Who wants to play "Understand the datasheet"?

Yashas

1:24 AM

-3

A: How high will the spring push the ball? (GCSE Physics)

i think your just a bit gay mate

:O

physics.stackexchange.com/review/low-quality-posts/171092

0celouvsky

I flagged for mods

Yashas

It is in VLQ queue

Yashas

1:37 AM

1

Q: Integrating without tricks

The integral is $\int\frac{\sqrt x -1}{\sqrt x}$. I have asked the opposite question on this site where the denominator would be the numerator and the numerator would be the denominator. I understand how that would work using ln. Is this similar to that. I am trying to do this without using any ...

lol

0celouvsky

very hard integral

@Slereah what is $\pi_1(\Bbb R^2-\text{two points})$?

the answer is tricky

Slereah

Hm

I'm guessing there's curves around one point

Curves around the other point

Curves around both point, but those can have arbitrary amounts of crossing

not sure how to write it

0celouvsky

@Slereah so you have the curve $a$ going around one hole

and the curve $b$ going around the other?

those both generate $\Bbb Z$ but you can also mix them

Slereah

yeah

0celouvsky

like you could have $abab$

or $a^2b$

Slereah

1:45 AM

and I'm guessing that a curve going around both points isn't equivalent to a figure 8 going around both points

Maybe just $\Bbb Z \times \Bbb Z$?

I dunno

0celouvsky

@Slereah Actually it is...pinch it down between the two points

you get a figure 8

the group is called the "free product" of $\Bbb Z$ with itself, denoted $\Bbb Z*\Bbb Z$

Slereah

what is the free product

0celouvsky

or: the free group on two generators

@Slereah it's a formal product where elements are like $ab^2ab$

words in $a$ and $b$

and $a$ and $b$ do not commute

it's a yuge group

Slereah

a @yuggib group?

0celouvsky

yuger

he's not that fat

Slereah

1:58 AM

is there an example of a manifold that isn't made by one of the basic manifolds and simple procedures

A manifold that isn't $\Bbb R^n$, $S^n$, isn't a cartesian product or connected sum of other manifolds, isn't made by removing closed sets or identifications

0celouvsky

2:12 AM

@Slereah E8

It's so fucked up no one knows what it is

Slereah

I'd ask for a proof but I would probably regret it if there was one

0celouvsky

All of the higher dimensional Lie groups are pretty mysterious

I don't know what $\mathrm{GL}(2,\Bbb R)$ is

Slereah

Well it's 4D

I doubt it can be that weird

0celouvsky

@Slereah It is known that the 4-manifolds cannot be classified

Slereah

Non-compact, not simply connected, fundamental group $\Bbb Z$

Hm

It's not known, actually

5-manifolds and up can't

3 and under can

4 is not currently known

0celouvsky

2:17 AM

10

Q: Why "Classification" of 4 manifolds is NOT possible?

I know classification of 2 manifolds and geometrization for 3 manifolds. Why for dimension great or equal to 4, this task become impossible? edit: Or should I ask "why geometrization won't be possible for 4 or higher dimension?"

dg.differential-geometry

Slereah

Hm

what I've read before was that but for 5 manifolds

what is this shenanigan

0celouvsky

2:28 AM

@Slereah Hörmander Vol. 1 has rigorous $i\epsilon$ stuff

Slereah

Hm, let's see

do I have charamander

"An introduction to complex analysis in several variables"?

0celouvsky

no.

the analysis of linear partial differential ops

Slereah

which part is it

hey there's a bit on the wavefront set

That's a modern book

0celouvsky

it's 1983

fairly modern

quite an amazing book tbh

I need to read it

Slereah

how many books do you have to read, currently

0celouvsky

2:34 AM

Well, currently reading Bleecker and Booss

Then there's Grubb, Chavel, Hörmander (three volumes), Gilkey

Those are the books I need to read immediately

I would like to read Besse, Li, Lieb & Loss

Chow, Lu, & Ni

These ergodic theory books can wait until next semester when I take the class

Also Choquet-Bruhat

@Slereah sound right?

Slereah

Sounds like a fair bit

0celouvsky

@Slereah I would also like to learn some cosmology

@Slereah Oh and Kosinski because my long term goal is to understand the Poincare conjecture

Slereah

Cosmology sounds like a thing where you need a lot of experimental stuff

0celouvsky

@Slereah I was planning on reading Weinberg's cosmo book

Slereah

Is there a proof that the copernician principle makes sense

0celouvsky

2:45 AM

which one is that?

Slereah

If the stress energy tensor is homogeneous on large scales, does it imp'y the metric is

0celouvsky

clearly not

Schwarzschild

> 0.455 kJ

Why not just 455J?

Slereah

Well schw. Is asymptotically flat

Good enough

But they never seem to really motivate FRW that well

0celouvsky

why not?

you just want it to be isotropic

Slereah

Yeah but

Proof?

0celouvsky

2:49 AM

what they don't motivate is why $\pi_1(\Sigma)=1$

they implicitly assume it

Slereah

They don't assume the topology

0celouvsky

uh, yeah they do

Slereah

$T^3$ is a valid cosmology

But y'know

Obviously the universe isn't homogeneous

So what is the proof that we can approximate the metric so

0celouvsky

@BernardoMeurer She wants me to pay for the ticket with labor

@Slereah the fact that it seems to work

Slereah

Plz

0celouvsky

2:55 AM

what?

Slereah

How does one even write the statement that it is homogeneous on large scales

for a sufficiently large ball of diameter $r$, $$\int_{B^1_r} T dx = \int_{B^2_r} T dx + \mathcal O(r^{-1})$$

maybe?

0celouvsky

@Slereah The isometry group is transitive on spatial slices

Slereah

or something

but it's obviously not

0celouvsky

well...it's approximately so

Slereah

yeah, but there is math to deal with approximation

0celouvsky

2:58 AM

there is or is there?

Slereah

I vaguely recall something discussing this in a book, but I forget which

0celouvsky

Lil Wayne Announces He's a Member of Roc Nation.

►

Holy shit

@BernardoMeurer

Slereah

3:11 AM

You know who wrote about that topic?

0celouvsky

duffield?

Slereah

My previous GR friend, Bela Valek

0celouvsky

previous GR friend?

Slereah

(he too wrote a book)

I have no idea where he found that bit of trivia though

Slereah

3:26 AM

valek.webs.com/mp/gr_en.pdf

if u want the book

though beware

he was kind of your opposite

All about coordinates

and explicit calculations

0celouvsky

who??

what is this?

Slereah

Are you getting jealous that I had another GR friend

0celouvsky

No

he uses $\gamma$ for $G$

Slereah

also he writes products explicitely

slightly odd

0celouvsky

do you think I am a better friend?

Slereah

3:30 AM

Well you're not talking about catholicism all the time

That's a pretty big bonus

0celouvsky

...what?

Slereah

He was pretty big into catholicism

He grew up in the soviet union so he seems to have gone way into catholicism as a reaction

Well, not soviet union proper, just Hungary

0

Q: Does the average homogeneity of the universe imply the average validity of the FRW metric

One of the big assumption of cosmology is the isotropic and homogeneous character of the matter distribution in the universe (since homogeneity implies isotropy I'll concentrate on the latter). Obviously this assumption is wrong, as any local experiment can determine. This only works on the assu...

general-relativity cosmology

plz halp

0celouvsky

I'm trying to understand relative homology

can't help

Slereah

Cosmology was a lie all along

0celouvsky

it's physics

what physics is correct?

mechanics of a single particle

Slereah

3:39 AM

electromagnetism

0celouvsky

@Slereah I doubt people take into account self-interactions to the fullest extent

Slereah

it's doable, though

I mean the full treatment came out like in 2008

but it's possible

0celouvsky

@BalarkaSen What the heck does a preferred generator of $H^n(M,M-\{p\})$ have to do with Jacobians and stuff

@Slereah The metric, being Lorentzian, does not give a notion of radius.

Slereah

@0celouvsky I'm talking about the spacelike hypersurface

Global hyperbolicity all the way baby

0celouvsky

> An exercise
or its solution may be needed for later exercises but not for the main text.

good guy Dold

Slereah

3:47 AM

I don't think HE even has exercizes

that's why it's great

0celouvsky

jesus christ, $$\bigcup_{p\in M}H^n(M,M-\{p\})$$

orientation sheaf

I'm done

I hate manifolds

Balarka Sen

@0celouvsky It's a choice of an orientation at $p$.

0celouvsky

@BalarkaSen No shit, but why

Balarka Sen

Think of a generator of that as coming from the singular chain given by a map $\Delta^n \to M$ which sends the origin (a marked point inside the simplex) to $p$.

The two generators are the orientation-preserving and reversing maps from the simplex.

0celouvsky

yeah, but why relative homology?

Balarka Sen

3:52 AM

'Cause that's where such a chain lives?

$(M, M - p)$ means you don't care about anything which goes outside $p$.

The boundary of the simplex lives in $M - p$.

0celouvsky

yeah but it would make sense that you have to orient "the area around $p$" as well

how does one talk about a consistent orientation in an open set with these homology groups?

Balarka Sen

Good question. If $B$ is a ball in a chart containing $p, q$ then there are isomorphisms $H_n(M, M - B) \to H_n(M, M - p)$ and $H_n(M, M - B) \to H_n(M, M - q)$.

If you want consistent orientation for all pairs $p, q$ in $B$, you just require the choice of generator remains the same under those maps.

(Choice of generator means choice of $\pm 1$ under a fixed isomorphism to $\Bbb Z$)

0celouvsky

does hatcher talk about this?

Balarka Sen

Ya. Chapter 3.3

0celouvsky

good to know

on the list of things to learn

@BalarkaSen do you know this notion of a difference of bundles from K-theory?

Balarka Sen

3:59 AM

Vaguely. I have forgotten what that is.

Direct sum of E with orthogonal complement of F?

0celouvsky

@BalarkaSen One mirrors the construction of $\Bbb Z$ from $\Bbb N$ but instead with $\mathrm{Vect}(X)$ and $\oplus$

Balarka Sen

Oh, Grothendieck completion. I remember several distinct notions of K-theory; one with that, another upto stable isomorphism, etc

Not sure how stuff are related.

0celouvsky

I was just wondering if you have a feeling for what it's spposed to be

I still don't understand negative numbers so I don't hope to understand negative bundles :P

Balarka Sen

It's a formal construction... vector bundles (of non-specified rank) form a monoid under direct sum.

So you just formally add inverses.

I think the geometric meaning is in what I said. Upto stable isomorphism, the vector bundles over a fixed base does form a group.

0celouvsky

Stable iso?

Balarka Sen

4:10 AM

$E, F$ are stably isomorphic if $E \oplus \Bbb R^k \cong F \oplus \Bbb R^k$.

0celouvsky

I see

I have two exams tomorrow so I'd better head to bed

cheerio

Balarka Sen

lol, crazy weird

gonna have to watch that movie

@0celouvsky Bye.

John Rennie

6:11 AM

@Kaumudi.H: I read the FBAWTFT screenplay. Good fun, but the trouble with films is that they're basically just a short story. They don't have the depth of a novel. I'm about to jump into my car and drive 200 miles, so I won't be around until later today.

user228700

6:23 AM

@JohnRennie You finished it! :-o Yes, that's true but yes, very good fun :-) Oh, you're gone but I hope you have a nice drive!

Shing

8:17 AM

although I kind of like this question and this kid, but this question seems to me opinion-based

3

Q: The importance of the Feynman Path Integral

For our middle school "Final Exhibition Paper" we had to choose a historical figure from a list to write a paper on, and I chose Richard Feynman. One of the requirements of the paper is to describe his major achievements in detail, which means describing the Feynman Path Integral. In the last fe...

quantum-mechanics path-integral

innisfree

10:03 AM

I don't think so

There's a subjective aspect, but there are surely physical insights about why the path integral is so celebrated

ACuriousMind

Well, it's a bit of an odd question, but not subjective. I mean, if the path integral didn't encode the full quantum theory then we wouldn't use it. So, in the end, I think the question isn't really asking for some metaphysical "why the path integral", but more for a argument (I dare not say "proof") that the path integral acfually does contain all that information

innisfree

Why don't you dare say proof?

ACuriousMind

Because a proof would require a rigorous definition of the path integral to begin with. I guess you could give one for the ordinary QM case (and it's probably in Glimm/Jaffe), where it is well-defined via the Wiener measure.

Slereah

heheheh

wiener

innisfree

Ah ok, yes, in QM I think it's proven

Slereah

10:13 AM

Eeeeeh

it's proven for some class of potentials

innisfree

Oh right, I didn't realize that.

ACuriousMind

@innisfree I wouldn't be surprised if there are situations where you can't obtain "all" information from the path integral. I'm not sure I've seen a QM path integral that includes spin degrees of freedom, for instance

Slereah

I think it's in what's his name

the brick book

innisfree

You wouldn't be surprised!? That'd be shocking, wouldn't it, if e.g., the Schrodinger equation were more fundamental than the path integral.

Slereah

Kleinert

ACuriousMind

10:34 AM

@innisfree It would be...interesting :)

Slereah

Physical Review, vol. 176, n°5

"A path integral for spin" by Lawrence Schulman

Emilio Pisanty

@ACuriousMind remember this?

Dec 9 '16 at 16:25, by Emilio Pisanty

$$\mathbf r = \sqrt{\frac{4\pi}{3}} r \sum_{q=-1}^{1} Y_{1q}(\hat{\mathbf r}) \hat{\mathbf e}_q^*$$

I now find myself in need of that reference again, cannot remember where I got it, and I managed to not put it on the record last go-round

ACuriousMind

Ah, yes, I remember me not knowing anything about that ;)

@EmilioPisanty That's...unfortunate

Slereah

apparently the basic idea is that you do some path integrals on $SO(3)$ or something

0celouvsky

11:55 AM

@ACuriousMind it's possible.

It's in Shankar, but I don't remember any more than that

Physics Meta

0

Q: How many up and down votes can result in this score? (18)

Today I posted an answer, which I see having a reputation of 18 so far. Also, I see that it has +1 votes. What combination of votes can result in these numbers? (For example, I imagine that two up and two down would give a reputation of 18 but zero votes. Three up and two down would give +1 vot...

support reputation voting

Secret

12:18 PM

@physicsmeta why is this important?

barlop

12:47 PM

if you have scenario A)a flat square or flat circular object, and it moves in a straight line, towardss an object e.g. a ball. vs B)A flat square or flat circular object, moves in a straight line towards an object e.g. a ball (so same so far), but also, the flat object is spinning in the Y axis too, and the centre of the spinning flat object hits the ball. So the ball still goes straight. Will the ball go faster or slower or same speed, in A, or B?

user228700

@JohnR: Eyy, you're back! Good drive? Oh no, you're gone ._.

barlop

also is my question ok for the main site, or better on the main site? Note I don't have a background in physics beyond a bit of high school.

John Doe

1:04 PM

Given state $| \psi \rangle = | \uparrow \rangle$ which spin eignstate of spin operator $\hat{S}_z = \frac{\hbar}{2} \hat{\sigma}_{z}$ viewed in the Bloch sphere, if you want to rotate the state by $\frac{\pi}{2}$ around the $x$ axis, as I understand you apply operator $\hat{U} = e^{-\frac{\pi}{2}\hat{\sigma}_x}$, would you obtain $$\hat{U}| \uparrow \rangle = e^{-\frac{\pi}{2}\hat{\sigma}_x}| \uparrow \rangle = \frac{1}{\sqrt{2}}| \uparrow \rangle + \frac{i}{\sqrt{2}}| \downarrow \rangle?$$

barlop

@JohnDoe your notation didn't come out right?

i'll ask on the main site

John Doe

@barlop What do you mean? You can't see the tex properly?

Jim

it looks like fine notation to me

@JohnDoe I'm a tad rusty on this; it's been a few years. But shouldn't a $\frac{\pi}{2}$ rotation around the $x$ axis result in $\frac{1}{\sqrt2}(|\uparrow\rangle+|\downarrow\rangle)$?

at least, if $|\uparrow\rangle$ is an eigenstate of the z-spin operator

John Doe

@Jim Yeah it is but the way I'm seeing it is that it a rotation around the x axis by $\frac{\pi}{2}$ results in the state lying along the y axis.

Jim

@JohnDoe right. That makes perfect sense. But to me, a state aligned with $y$ would be one in which all knowledge of $z$ is lost. To me that means equally up and down, not sure whether the $i$ would interfere with that. Like I said, I'm a bit rusty

and let's face it, there's cylindrical symmetry here. A rotation around $x$ should make a $z$ eigenstate act the same under $\hat S_z$ as a $y$ rotation

John Doe

1:20 PM

@Jim Yeah I see your point but I think the important point is that the modulus squared still gives $\frac{1}{2}$ up and $\frac{1}{2}$ down and that the state is distinguished from the $x$ spin eigenstate this way.

Jim

I'm pretty sure I just need to brush up on my Pauli matrices. Otherwise your form looks perfect

John Doe

@Jim Thanks for checking.

Simply Beautiful Art

Sorry to butt in, but,

in Teenage Territory on Stack Overflow Chat, 5 mins ago, by Simply Beautiful Art

@everyone would you guys like to participate in a coding contest to see who can code the largest number on a theoretical computer with infinite resources and a max character limit?

@Fawad fancy seeing you here lol

user929304

@ACuriousMind hey, hope all is well with you. This time I'm properly writing to you in chat :) let me know if you find the time later to look into this: physics.stackexchange.com/questions/326132/…

Koolman

1:35 PM

@Kenshin you there ?

barlop

@JohnDoe I see things like pastebin.com/raw/RWpaYVP8

Yashas

@SimplyBeautifulArt boost isn't allowed right? :P

Simply Beautiful Art

@Yashas I'm not sure, what is boost?

John Doe

@barlop i think the prob might be from your side...

Yashas

@SimplyBeautifulArt boost::multiprecision

It is in gmp.hpp IIRC

Simply Beautiful Art

1:38 PM

@Yashas may we continue this chat in my realm? (Check my profile page)

For those interested, please use any of these languages: repl.it/languages

Physics Meta

1:53 PM

0

Q: mathjax not working for me in chat

I've been in the chat "h-bar" physics stackexchange chat, and somebody wrote $\frac{1}{2}$ and it comes up for me like those literal characters and not as $\frac{1}{2}$ Any idea why / how I can enable mathjax in chat? The guy that posted it said he can see it and thought it might be an iss...

discussion mathjax chat

Koolman

@Kenshin I think the site it not working properly

0celouvsky

2:12 PM

@Slereah are you alive?

Yashas

How do I justify to someone who quotes person X saying "When we hear that term light-year, we need to realize it is not a measure of time but a measure of distance, telling us how far away something is. Distant stars and galaxies might be millions of light-years away, but that doesn't mean that it took millions of years for the light to get here, it just means it is really far away!"

Oh, people giving this link as proof lol

0celouvsky

amazing

I will read this and debate people

the ultimate troll >:)

they want me to buy a book

no thanks

Secret

So hard to concentrate on work in a cold weather

0celouvsky

How can it be cold on a desert island?

Secret

Sydney is entering winter now

It does not help that while I can work in 38 C fine with jacket on for 1-3 hrs, I got sick very quickly when the temperature drops to around 18 C

0celouvsky

2:25 PM

18C is just below room temp

get a long sleeve shirt

Yashas

..

now I don't even know what I can reply

Jaime Gallego

Wtf

Do they know about Doppler

Secret

They are looking for variable fuundamental constants?

Yashas

@Jamie They don't even know Newton's law of motion. When I give some meaningful explanation of something, they throw some random bullshit at me and make me unanswerable.

Two days later: "See, the round earth people cannot even answer."

0celouvsky

stop allowing yourself to get trolled

Yashas

2:29 PM

I feel so helpless

0celouvsky

failing that, no need to post everything in the chat

Yashas

@0celouvsky I don't think they are trolling; they make youtube videos of 1 hour length with random stuff.

Why would a troller spend hours making hours of troll videos?

0celouvsky

My second point stands

@Yashas Boredom

@Yashas Be nice

Yashas

"What if the universe is a paradox and ultimately WE are the ones that caused the big bang which caused everything else to come forth??? " I think that is a troll.

Secret

@Yashas aca.unsw.edu.au/profile-main/26

There are also other groups in other unis working on this, but so far, the variations are too small to draw good conclusions

arxiv.org/abs/1406.7482

journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.210801

Therefore that block of text they threw at you is not bullshit. whether the flat earthers actually understood it is another question

barlop

2:39 PM

@JohnDoe apparently I should install an extension.. to get the formulae to show in chat.. By the way i've posted my question on teh main site and you're welcome to post any answer or comment

Secret

Although technically speaking, if they don't understand it, then they are bullshitting since bullshitters don't care about the truth value of whatever they are quoting as long it serves them well

Yashas

I am also fighting with few people who claim all the governments of the world are using chemtrails as chemical bombs to reduce population.

I hope I'm not being trolled.

Secret

O, the contrail conspiracy camp. I don't have any interest of arguing with them

2

Q: How are beliefs restricted by an objective reality?

Consider the following belief A human can survive if they don't drink water for 1000 days We knew from biology that this is impossible as there are no known records of any living human who can survive for the period stated without water. However, there are also beliefs that are unreal/fals...

epistemology metaphysics belief

The comments in this question taught me that, the world is sufficiently objective enough that should someone held onto a wrong belief too long, they die

this is why I don't need to argue with them, cause if they turned out to be too wrong, the world will kill them automatically

barlop

@Secret You wrote "We knew from biology that this is impossible as there are no known records of any living human" <-- that is not a logical statement

also food contains water, some food contains more water than others. A most extreme case would be a water melon

Secret

Hmm, I am not sure how to make that more accurate, cause it is a known result that human can only survive without water for 7 days, I recall

barlop

2:54 PM

if it's just based on a lack of written records then that's not good evidence

Secret

Hmm ok, Any suggestion to make a stronger example than that?

barlop

Well, there's a lack of written record of what time King James went to the toilet on his 20th birthday, if he did at all. That doesn't mean we know it's impossible.

There's a lack of evidence that King James flew to mars, but that's not the only reason why we believe it's extremely unlikely.

Also you should talk in terms of things being likely and unlikely. Not Knowing vs not knowing..

except perhaps in the case of contradictions, for example.. I auppose a belief we know to be false would be that there are married bachelors.

So that's a negative that can be disproven because if it were true it'd be a logical contradiction.

I don't know exactly why biologists would say a human can't survive 1000 days without water.

I have heard that you can't survive 4 days.. but that's just something i've heard from non-experts who might've heard it from experts. But to what extent those experts have tested it is not clear.

Also it's not clear what you mean by "an objective reality"

e.g. do you think we are in an objective reality and what does that mean anyway.

we know you can have contradictions are some things are true and some things aren't true so that's just reality, no need to say "an objective reality".

Secret

An objective reality is one which does not depends on different perspectives, that is, there are entities that exists and is independent on how they are described, whether via science, religion or other methods

Put it in a simple way, everyone and everything agrees on some properties of said entities, regardless of what (insert word) they have

@barlop (after some googling and google scholaring) I am surprised that there are no journal articles that talk about this topic

barlop

3:20 PM

I think everybody agrees that reality has objective elements and there's no need to state "objective reality" when you mean reality.. If you mean objective parts of reality that's still ambiguous.. Also by the way, even if in every generation some humans attempted to go without water for 1000 days, and failed , it still wouldn't mean it was impossible. It'd just mean humans hadn't found a way yet.

a biological argument to back it up would make it stronger.. still be wary of saying impossible.. or knowing just based on some records.

Secret

3:37 PM

It seems when we look very carefully on it, the issue becomes even more uncertain

Anyway, I have added your married bachelor example in. It, however is not the type of false belief that will kill you when you believe in it. I am not sure if there are any examples of such false beliefs given now we look at the issue of "impossibility of something" very carefully