@user20458579510081670432 Yup. It's a wonderful feeling to be able to play a piece while "in the zone", esp. in a concert / recital setting on a good piano. Worth the weeks of practicing. I'm not a sports guy at all, but I can imagine how it feels for a sportsman in competition.

@GratefulDisciple Of course, I've been mastering Seiki No Kutou from Saint Seiya, on piano. I do play at home but sometimes I just get lazy and don't

Well more like from Saint Seiya: Heaven Chapter movie

Use it, or lose it.

:(

@冥王Hades Good for you. I got lazy too, from time to time, not touching piano for months.

2024 moderator election!

hello.

Hello everyone. How to study the behavior of the series $\sum_{n=1}^{+\infty}\frac{\ln n}{n}$? I immediately thought of the Comparison test, because if $\ln n > 1$, then $\frac{\ln n}{n} > 1/n$ and $1/n$ diverges, so $\frac{\ln n}{n}$ must also diverge. The problem is that for $n > 1$ we cannot always guarantee that $\ln n > 1$. Can anyone who can help me out on this?

bml: it is enough for the inequality ln n > 1 to hold "for all n sufficiently large" i.e. for there to be an N for which ln n > 1 holds for all n >= N.

here, N = 3 would work (because if n > e then ln(n) > ln(e) = 1)

many of the tests that you know for convergence/divergence can be reformulated in this way, only requiring a inequality or some other condition to hold only for all n "sufficiently large," even if a textbook provides a statement requiring the inequality or other condition to hold for literally all n

if you don't like messing with inequalities at this level of precision, an appropriately formulated "integral test" would also work and might only require checking a simpler inequality

@leslietownes This was my doubt. How can they be reformulated so that there is no difference between "n sufficiently large" and "literally all n"?

that requires more time than i have to explain, and possibly more knowledge about what you "already know" than it would be easy for you to explain. intuitively you might think of such a series as having two parts, a finite sum a_1 + ... + a_{N-1}, and an infinite series whose indexing starts at N

and by "re-indexing" that series or perhaps even literally applying your textbook theorem to that series (where the nth term literally satisfies the desired inequality for all n), you get what you want

or at an even higher level of abstraction, "finitely many terms don't affect convergence or divergence"

more formally and in fewer words, you could look at any textbook proof of the "all n" comparison test, and rewrite it slightly so that its hypothesis is only "all n >= N" but is otherwise mostly the same

or google around for some online resource for what a proof would look like if you were to do something like that

math.libretexts.org/Courses/Mount_Royal_University/… is one example

but again, if the things assumed by that argument aren't part of what you "already know," then the argument given there would not confront your difficulty, and only relate your difficulty to some other difficulty

@leslietownes Can you suggest a textbook in which series is addressed at this level of depth (and thus, in which the issue we are discussing is also present)? Thank you...

i don't know of one off the top of my head. the problem is that a lot of analysis books will, by this point in their discussion, have proved enough in detail that they will begin a proof of something like this with "without loss of generality, assume N = 1" or maybe not even literally state the result with the "for all n >= N" form

related but simpler: if you know that an increasing sequence of real numbers that is bounded above must be convergent, can you prove that an "eventually increasing" sequence of real numbers that is bounded above is convergent?

where "increasing" means "a_n <= a_{n+1} for all n >= 1" and "eventually increasing" means "there is N with a_n <= a_{n+1} for all n >= N"

are you aware of any book which formally states that the existence of a limit is unaffected by "removing a finite number of terms" from the front? (something like: if a_n is one sequence and there is an integer M such that b_n = a_{n + M} holds for all n >= 1, then a_n converges if and only if b_n converges)

i'm not, but it strikes me as something that might be possible to find. (it also strikes me that a lot of books might give something like this as an exercise, and then refer back to it as if they have proved it)

My highschool algebra textbook gave an exercise that required knowledge of Bezout's identity as an exercise, and then referred back to it as if they had proved it when discussing the Chinese remainder theorem. That kind of struck me as "unfair."

@leslietownes maybe we can use the theorem regarding the tail of a series? Given $M>0$, then $\sum_{k\ge 1} a_k$ and $\sum_{k\ge M}a_k$ both converge or both diverge

Reading this answer. Suppose $\Pr(Y_{(n)}\leq y)=y^n$ for $0\leq y\leq 1$. Suppose $0\leq e^{-t/n}X_{(n)}\leq 1$, then it is claimed $$\Pr\left(e^{-t/n}X_{(n)} \gt {Y_{(n)}}\right)=\mathbb{E}\left[\left(e^{-t/n}{X_{(n)}}\right)^n\right].$$Why is this true?

I'm thoroughly confused where the expectation comes from. Is $\Pr\left(e^{-t/n}X_{(n)} \gt {Y_{(n)}}\right)$ a random variable?

What assumptions are often placed on the twisting functions in a twisted Kahler metric?

I usually assume that they converge on some vertical strip in $\Bbb C$

For example, $G_t =A_t(s)drd\bar r+B_t(r)dsd\bar s$

This diagram shows how the geometry varies under a doubly warped time depend. real metric. 3 time slices shown

← 1 message moved from 2024 Mathematics Stack Exchange Moderator Election Chat

already 2 candidates

@FedericoRuck BTW the above question was also posted in this room: chat.stackexchange.com/transcript/154521

chat.stackexchange.com/rooms/154521/range-of-trig-functions

@MartinSleziak also in Chit-Chat with IB

Malicious behaviour imo

@psie finally found the answer here. Another useful identity I think.

hm

I want an espresso machine and nice pens and nice paper

I plan on walking into the pen shop and saying, "merchant fetch me your finest wares!"

@zetaspace they're expensive

That's why I have a moka pot instead of a coffee machine

I want an accurate mathematical art piece

as well

meaning not just an artists rendition of a math object

Complex dynamics might produce some really cool pictures

true

diffgeom.com/collections/softstyle-t-shirts?page=2

this is cool

reasonably priced

@zetaspace Hopefully, you have $10,000 to spend.

A decent espresso machine, however, is obtainable for $2--300.

I have one that I like.

@XanderHenderson really? Well, surely its more expensive in my country but I might consider getting one in the future now.

I could buy a 10k pen but that would be kinda like having a ferrari and no house

I could take out a loan perhaps

maybe a go fund me

@zetaspace Even a decent house costs way more than a Ferrari in states like Cali. You could get a nice Ferrari for 300K, good luck getting a good house for that

@Jakobian You can certainly spend more money (I spent close to \$500 on mine), but you can get a reasonable, capable, entry level machine for less than \$300.

wholelattelove.com/products/gaggia-espresso-evolution-lava-red

wow that looks sleek

@XanderHenderson Never thought to ask you before but, you wouldn't have happened to have watched Death Note, would you?

Is anyone in this chat from NYC (new york city)?

@冥王Hades I have seen it, yes.

@zetaspace Good g-d, no.

I wonder what the mathematical scene is like in nyc at least for amateurs looking to discuss mathematics and perhaps even socialize

I guess I would look on meetup.com or facebook. Not terribly sure how to find groups like that

@XanderHenderson Not sure if you remember but, ever wondered how L cleared the entrance exam to To-Oh without studying, in just one day?

@冥王Hades Honestly? No. The show is not high on my list, and some minor detail in the show is definitely not high on my list.

@XanderHenderson Interesting. Surely you do at least remember how incredibly brilliant L was, don't you? That's one of the hallmarks of the show.

@冥王Hades I remember how brilliant the show wanted me to believe he was, yes.

There was a lot of "telling", and not so much "showing".

@XanderHenderson That's actually a common complaint with the show. Unfortunately the animation studio missed a whole lot of lore and prequel that would've shown who L really is. Specifically the Los Angeles BB Murder Case and L:Change the WorLd novels.

@XanderHenderson my impression as well, with L as an example

Like I said, I watched it. I can't say that I really care. I am definitely not equipped to have a discussion about all of the deep lore left out from the manga. I shall exit this conversation, now.

@冥王Hades because its fiction

@Jakobian Yes but somehow far more believable than what happens in other shounen anime.

I find that to be called a genius you just need to study from an early age and then maintain that. What other perceive as "genius" is just hard work

@Jakobian Could be. L:Change the WorLd goes into more detail about that. The series does touch on this debate of "natural genius" vs hard work (or whether this is just a false dichotomy)

I think its also about methodology, but nothing really past that. Some methods of learning and problem-solving are simply inefficient. Some things might boil down to how a particular person felt at a particular day to present themselves in a particular way

A factor might also be adaptability, but its admittedly not very reliable property to have

Its much harder to adapt than to follow some kind of pattern

Are we talking about anime? I suggest watching Bungo Stray Dogs. It's extremely underrated.

@SoumikMukherjee Is it? I've heard nothing but good things about it...

I think it is, I rarely see people talking about it.

@Jakobian I'm with you on this. A lot of hard work (and lots of listening to good classical recordings as role model) to get to where I am right now, and I don't consider myself genius either. Starting early helps in that the brain is more malleable, but 99% people shouldn't rely on the "genius gene".

xander: fractal zeta functions, Jakobian: topology, Hades: high school geometry, Robjohn: analysis, sequences and series

@zetaspace me? :(

@SineoftheTime I wanna say, analysis/functional analysis

Tell me though

Hi all.

@Sahaj hi

Hi Jakobian; how’re you doing these days?

I'd say I'm struggling to be able to stay alive as I am now

This is how I see it

@Jakobian Same.

Lol.

Well, that pessimistic response wasn’t what I expected.

@Sahaj yes, its pessimistic, but also realistic

How is that^ for real?

@copper.hat

et al

@zetaspace Dunno, I suck

@user20458579510081670432 huh?

user 20458579510081670432???

What's the meaning of that long string of numbers?

@copper.hat he fought an old team mate for the gold, so the fix could've been in.

@Sahaj nothing special

@user20458579510081670432 sport has become mainly about entertainment unfortunately.

@copper.hat "Has become"? Hasn't sport always been about entertainment?

i meant in the business sense, as opposed to supporting a local team, or playing oneself

I just came up with my own space

mine is $\mathbb{R}^3$ mostly.

@copper.hat I like $\mathbb{R}^3 \times \mathbb{R}$.

I like $\Bbb R^3$

i seem to be stuck in one spot in time, especially with respect to personal finances

@XanderHenderson things like WWE have entertainment in their name.

to be fair, i do spend sometime on the $i \mathbb{R}$ axis...

@copper.hat At the age of 40-something, I feel like I adulted last Friday, when I opened a CD at 4.75%.

i liked when the Olympics pretended to be amateur

I'M AN ADULT!

:-)

:-)

:-)

:D

Oh, no! Oops!

:-)

i suggested to my 23 yo daughter that she put some of her ill gotten gains into a cd, but she was already way ahead of me

albeit, she is a bit nervous about google atm

She uses a Google ATM?

I didn't know that Google was in the business of providing automatic teller machines.

:-) they were, but belts are tightening now...

can we have $a^n \in \mathbb{R}$ for $n \in \mathbb{N}/3\mathbb{N}$ but $a^n \in \mathbb{K}$ otherwise, for some field $\mathbb{K}$ or something

basically $a,a^2,a^4,a^5,a^7,a^8,\dots$ means the number is real, but for the multiples of $3$ it's not

similar to how $i^2,i^4,\dots$ is real but the odd powers it's not

Hello

@Obliv $a^3=a\cdot a^2$

@SineoftheTime I think I messed up the order. It should be $a^n \in \mathbb{K}$ for $a,a^2,a^4,...$

does it change anything?

ok suppose instead I have $d \in \mathbb{K}$ to be of the form $a + bk$ where $a,b \in \mathbb{R}$ and $k^{3n} \in \mathbb{R}$ for $n \in \mathbb{N}$

does that make any sense

and?

what's the question

would that make any sense

does it define any structure

what you wrote makes sense, and you didn't define any structure

but what's the actual point

well in this book we're dealing with linear vector spaces over $\mathbb{C}$ and I was wondering why we choose complex field over reals

I think it has some special property to it that I was wondering if we could generalize

the fact that conjugate symmetry isn't simply real number symmetry

@Obliv I don't know

is $\mathbb{C}$ an extension of $\mathbb{R}$ or something?

what is the formal relationship between the two?

yes, $\mathbb{C}$ is the algebraic closure of $\mathbb{R}$

but this doesn't matter for your question

vector spaces over $\mathbb{C}$ are not more general than those over $\mathbb{R}$

my intro algebra class didn't have time to do field extensions and galois theory, I'll have to learn about that sometime then

who does Galois theory in an intro algebra class?

It's an exam in the master in my uni

it's in hungerford's intro textbook so theoretically if one was motivated to cover it you could

I don't find it unbelievable that some people do some basic Galois theory in an intro to algebra class

yeah galois was very young when he etched the foundation of the theory so I don't think it has to be that rigorous

we ended on sylow theorems but yeah galois theory is in "advanced topics" section so I guess it'd be pretty unlikely we cover it

we did ch 1-9. I don't think hungerford expected the whole textbook to be covered in 1 semester

@Obliv rigorous? All math should be rigorous

excuse me, I shoulda said comprehensive

at least, as rigorous as it can be, if it's not unreasonable

Can a manifold with singularities (ala singularity theory) be viewed as the local and global picture of some space?

@Obliv the reason often given for using C is that C is algebraically closed

and R (for example) isn't

so for example, an operator on a finite dimensional vector space over C is guaranteed to have an eigenvalue, and you won't have that thing you get into in R about "well, it doesn't have eigenvalues in R but if we fiddle with the field we can" you don't need to fiddle with the field you just have eigenvalues

and why do we want eigenvalues?

two short answers which might sound sarcastic but aren't: (1) maybe you don't, i don't know, a ton of linear algebra is exactly the same irrespective of the base field, or irrespective of a characteristic 0 base field at least, (2) read a linear algebra book and come back later about why eigenvalues might matter

I don't think its reasonable to reject spaces over $\mathbb{R}$ altogether, perhaps Obliv is misinterpreting their book saying that they are restricting a certain discussion to spaces over $\mathbb{C}$

it's not so much that you "want eigenvalues" in the abstract, but that in problems where they end up being useful, it simplifies the discussion (for purposes of explaining things to students) if you don't have to suddenly step outside of the field you were working over to have that discussion

the span of an eigenvector(s) is an invariant subspace right?

for example all of the stuff above about a number having powers that are sometimes real and sometimes not was really confused looking

eigenvalues matter because of the spectral theorem

and you don't run into that confusion if everything in sight is a complex number

@Obliv eigenvalues come up when solving differential equations for example. They can come up when discussing dynamics of a system. And so on

if you are doing linear algebra over $\mathbb{R}$, you'll end up with situations where there's literally no eigenvalues

so seemingly you can't discuss your problem, unless you walk into the complex numbers world

generally speaking, whether or not the underlying theory is any different or any harder, being in a situation where students are thinking "well, we have these things, and sometimes they don't exist, or they exist in some cases but not in others, and in other cases they maybe exist but only in this other world" is sub optimal for a lot of pedagogical purposes

if nothing else you avoid a bunch of case analysis if you just work over C

and the cases you get into if you specialize to R often have little to do with the underlying problem, they don't shed light on what's easy and what's hard, they're just the cases that arise if you suddenly have to worry about whether things are real or not

eigenvalues are scalars that satisfy a solution to an operator on the space?

@Obliv Roughly, yes.

no simple verbal formula is going to encapsulate any worthwhile understanding of eigenvalues, please read a book if you are interested in them

@leslietownes ...and eigenvalues are the easy part of the spectrum of an operator. :D

how about for fourier series, what is the operator in that situation?

as morpheus once said, no one can be told what the matrix is, you have to see it for yourself

are they the trig functions

@Obliv Huh?

obliv you are changing the subject in literally every question you ask. this is not how to learn anything

why are you bouncing from a topic to another?

@leslietownes I knew it, you're morpheus

this is like math via tiktok

Basically speaking, eigenvalues stem from an idea that an operator acts like multiplication by scalars on certain parts of your vector space

nah leslie is on fire today

@Jakobian when did Jakobian become so good at explaining things

@leslietownes Get used to it. Gen $\alpha$ is coming.

he doesnt miss

@SineoftheTime to incorporate my understanding of eigenvalues to places where I've seen them being used

but we didn't have time nor the motivation to go into the linear algebra aspect, we just did fourier series expansions as a stand-alone thing

for my ODE class

gtg for now

@Obliv explaining things is what I do here, so if you think I've become better at it then that's why

@onepotatotwopotato @XanderHenderson Is this election to increase the number of moderators or to replace a retiring one?

@XanderHenderson Thanks, should have looked there first.

I completely missed the Meta post too.

I usually scroll Meta with my eyes closed apparently.

Not gonna lie: I'm kind of disappointed that Harris didn't pick Mark Kelly.

@XanderHenderson Yes, he would be a fine choice, and as an Arizonan you probably know him already.

@GratefulDisciple Indeed. He's good people. And his wife was amazing. I kind of feel like he got into politics to take her place when she no longer could.

And he's a mother f*ck*ng astronaut!

He's been to space!

We need him if the Martians attack.

BTW, just looked at the moderators page, you guys have 9 already. Must be an active site. In contrast, Christianity.SE only have 4 and 1 is very, very inactive, so practically 3.

@GratefulDisciple Math SE is the second largest site on the network.

@XanderHenderson In that case, the site deserves a bigger box compared to Server Fault, Super User, etc.

@GratefulDisciple That has always annoyed me. The box needs to be bigger

that's interesting: youtu.be/m2tkEXyd_ek?si=_IsJqdErSv22F7Hm

It is. I wasn't aware of this result

@GratefulDisciple It appears to me that we have the same sized box.

And those other sites are of similar size (and Super User has, at times, been the second largest. And might be now, even. The last time I checked was several months ago.)

But if you look at questions per day, Math wins: stackexchange.com/sites?view=list#questionsperday

Is anyone comfortable with Riemannian metrics of the form $$ g_r(x,t)=f(t)^r dx^2+g(x)^r dt^2$$

SO is 1.8k / day; Math is 180 / day (an order of magnitude smaller); the next biggest site is SO in Russian, which is around 60 / day. So Math SE is still three times more active (by this metric) than any other SE site. SO just swamps everything.

Christianity, for the record, is around 3 / day. So another couple of orders of magnitude smaller. :D

@Jakobian I have actually seen this before from a Turkish Math teacher, although from the video it seems it's much older than that.

Mi Yodeya (the Jewish equivalent of Christianity) is around 6 / day. We win! :P

I love how they are all whole numbers

Yeah. Imagine if we had half a question.

@XanderHenderson Great. When considering the number of adherents, that means Christianity.SE is even less popular :-). I notice a lot more Christians go to our sister site, Hermeneutics.SE. Fewer Christians care about theology or church history.

@SineoftheTime Oh! I gave that as a problem to a bunch of grad students when I ran a qual prep seminar in grad school. I really like that result.

@GratefulDisciple Yeah, but my impression is that the Torah is not off-topic on Hermeneutics.

;)

@XanderHenderson It's not, but the discussion is definitely slanted toward Christianity. And I just wish the audience care a lot more about consulting scholarly material rather than only using other passages of the Bible to answer the questions.

@XanderHenderson Grad students? Wouldn't this be more like something undergrads would do?

I recently used a little Euclidean greek geometry to solve an integral

@冥王Hades The real analysis qualifier had an "undergraduate" section. Roughly, it was 1/4 "stuff you should have learned as an undergraduate", and 3/4 "you need to know every single result in Folland, and you maybe should have read some of these other books, too".

@GratefulDisciple Sure, I suppose. Though, honestly, I've always found the whole practice of these kinds of analyses to be suspect. Talmudic scholarship is suspect. Christian hermeneutics are suspect. It's all suspect.

@XanderHenderson what do you mean by suspect? Suspicious?

It becomes a lot less suspect when you involve real historians and archaeologists, but then it also ceases to be the thing it was before.

@Jakobian merriam-webster.com/dictionary/suspect Definition 2.

@XanderHenderson Having been exposed to how various Abrahamic religion interpret the Hebrew Bible, I can relate. One's religion (or one's agnosticism) definitely color how you would read a sacred text.

@XanderHenderson Alright, now that sounds hard. Although knowing a majority of the results in Folland wouldn't be too outrageous for most sophisticated undergrads I think

@冥王Hades That really depends on what students are expected to learn as undergraduates.

In the US, very few undergraduates are introduced to measure theory or harmonic analysis.

@XanderHenderson Starting the 19th century with German scholars' historical criticism, the Biblical texts have been analyzed as a product of culture. From this angle, there is a certain objectivity in the scholarship, although what it is (i.e. the scholarly axioms) can be debated to be subjective as well, which is why that method is in decline now in the 21st century, giving rise to the new paradigm: Ancient Near Eastern research.

At my BA / MS institution, there was an undergraduate funky anal class taught out of Kreyszig, but no harmonic analysis or measure theory. At my PhD institution real anal was taught out of baby Rudin, including the awful chapters at the end which touch on measure theory, and there was a harmonic anal class (taught out of Stein and Shakarchi, I think), but that was kind of it for undergrad anal.

The big dividing line between undergraduate and graduate analysis (in the US) is typically measure theory. Undergraduates study Riemann integrals. Graduates study Lebesgue integrals. Undergraduates study derivatives as the limits of difference quotients. Graduates use Radon-Nikodym.

@GratefulDisciple I don't see any agreement even on an individual level.

@XanderHenderson Felicitous abbreviations

I have no idea what you are talking about. ;)

@Jakobian Maybe not for practical application, but on the ground (i.e. if you go to a church / synagogue), they usually interpret major verses the same way within a particular denomination.

harmonic analysis is as one might say excellent for the young and developing mind

too bad it's not taught earlier

while the synapses are fresh

@GratefulDisciple what's your opinion on the Bible and slavery?

A healthy denomination would foster a balance between tradition, community consensus, scholarship, and practical relevancy. Difficult to achieve, and across 2,000 years, the "recipe" has changed.

"for it" and "against it," i'd imagine :)

@zetaspace I think I heard a friend mention that they very briefly went over it when they were doing PDEs. I doubt that really qualifies as actually learning it though

@冥王Hades Fourier series often show up in undergraduate DE classes, but rarely in a rigorous manner. Those classes tend to focus on computation over theory.

@Jakobian In 21st century, there is no way people can argue that the Bible supports slavery. Nor do I. It's a shame that not long ago the Bible was read as agnostic with regards to slavery.

@GratefulDisciple why not?

@XanderHenderson Ah yeah

slavery is wrong

@Jakobian Because it's an ancient text reflecting the social condition of the time where slavery was predominant. With careful scholarship, both Jews and Christians can argue that in the Torah period, the laws actually gave some protections to those who (out of economic destitute or of wars) became slaves against their will.

what if the quark feels enslaved and nobody knows

@GratefulDisciple But nowhere in the Bible there's a passage that you shall not own slaves

Bible lists exact rules for how to obtain your slaves, how to beat your slaves etc.

In the NT period, the Epistle to Philemon is an appeal by St. Paul for Philemon to release his runaway slave because in Christ both Philemon and Onesimus (the slave) are now equal in status.

I remember, that this was because the slave was his friend

or am I thinking of something else

@Jakobian Exactly. So we can definitely read the Bible condone slavery by not advocating outright abolition, but it doesn't advocate slavery either.

@GratefulDisciple that's why I don't think that passage advocates against slavery.

We all must do what is right even if what is right is difficult

But reading between the lines, we can argue that slavery is to be avoided when possible, especially about the Jubilee rules to do a "reset" every 49 years. And in the NT, the law of love and about one part of the body of Christ helping another, and about not lording over one another, can be seen to be the ideal that Christians should strive toward.

What about that passage that tells you that if you like a girl, you can kill her entire family and then take her as wife, and if you grow bored, you can divorce her for a new one?

And that was for non-Hebrew ones iirc

It was a different time back then

@zetaspace Wouldn't justify it.

@zetaspace Matthew 5:17–20

For truly I tell you, until heaven and earth disappear, not the smallest letter, not the least stroke of a pen, will by any means disappear from the Law until everything is accomplished.

@Jakobian It doesn't. But the slave wasn't Philemon's friend either; he ran away. The Letter to Philemon is very short and if you read it, basically St. Paul's exhorting Philemon to take him back under his household without being punished, and that Paul offered to pay Philemon for any damages that his running away has caused.

@GratefulDisciple yeah. I meant St. Paul's friend. Perhaps I was mistaken. It was more about the slave being a Christian

@冥王Hades yeah I was just stating something :)

of course it wouldn't justify it

@Jakobian Yes, which is why Philemon's being a Christian should be treating Onesimus the Christian slave differently.

In the new testament, Jesus seems to confirm that you should obey what's in the old testament. So even if those were different times, the passages about slaves seem relevant

@Jakobian Really? I haven't heard that you can kill her entire family. About writing someone's wife a bill of divorce, yes there is that permission, but Jesus revoked it in Matt 19:1-12.

@GratefulDisciple No its a different passage

@Jakobian About the meaning of Matt 5:17-20, there is debate, but still, the larger interpretation that all Christians agree is that the NT law of love trumps OT laws. So obviously, if the wife suffers because of her husband, the husband is wrong.

What about Mary. Was she a virgin? Or is it a bad translation of "young woman" by the Greeks that wrote the Bible?

I mean the new testament, mathew etc.

About Matt 5:17-20, notice the key phrase "I have not come to abolish them but to fulfill them" (the "I" being Jesus), which is a more modern translation (NIV). The meaning "until everything is accomplished" has Jesus as the active subject who is doing the accomplishing. And usually this is being interpreted that Jesus as the Second Adam (who doesn't sin) accomplishes fulfilling all the requirements of the law that humanity fails, thus redeeming humanity.

Actually I think that Jesus is talking here about the prophecies

Jesus, from what I know, knew the old testament very well, and seeked to fulfill the prophecies contained in it

that's why he e.g. rode the donkey into Jerusalem

@Jakobian Yes, the word is "young woman", but regardless, the context is clear: Mary didn't have sex with her betrothed Joseph but got pregnant, until Joseph wanted to get rid of her until he was told by an angel in his dream that the one responsible for the impregnation was God himself. Joseph was going to "divorce her quietly" (i.e. not wanting Mary to be stoned, by law).

Of course Divine conception of Jesus was crucial to Christianity, and early apologists like Tertullian had to defend the virginal conception to detractors, who said that it was a Roman soldier who impregnated Mary.

How do you know that the Greeks just didn't make Jesus to be more and more divine in accordance to their understanding of Christianity? There seems to be trend in that the younger parts of new testament make it seem like Jesus was more and more divine

After all, no apostles had real influence over who is writing the Bible and what would be written

@Jakobian Well, it's both. But the context of Matt 5:17-20 is only the law. The fulfillment of prophecies was treated in the Emmaus story (Luke 24:13-35). About riding the donkey into Jerusalem, that's a fulfillment of a Promise of eternal kingdom to David. And yes, Jesus must have known the Hebrew Bible inside out.

@Jakobian The major 21st century figure advocating that thesis is Bart Ehrmann. But I think Christian scholars also make a convincing case that from the very start, Jesus impressed the apostles as divine. The earlier heresy to be battled was Docetism, where Jesus is God who just appeared to be human, or Gnosticism who believes matter to be impure, so the early church fathers had to defend the humanity of Jesus.

At any rate, I believe Biblical scholarship show quite convincingly that all NT was written prior to 100 AD, with some letters of Paul around 50 AD, and letters of Paul convincingly claim Jesus is divine.

What about most prophecies that Christians claim that Jesus have fulfilled not satisfying the basic criteria to be a prophecy, or being too irrelevant to be considered seriously?

For example, the riding of the donkey. Jesus knew that he had to fulfill prophecies and that he had to ride the donkey into Jerusalem. So how is that a prophecy fulfilled?

@Jakobian What do you mean? The NT books were canonized precisely because they had been written either by an apostle or by their secretaries (like Luke / Mark).

@GratefulDisciple no. The books of NT are anonymous and were written by Greek scholars

every Bible will tell you that the books are anonymous

@Jakobian What do you mean by "Greek scholars"? Even if they are anonymous, the thought style is definitely Hebraic with different levels of Greek language sophistication. I don't think it's a controversy to say that the NT writers are Hellenistic Jew, meaning ethnically Jewish but received cosmopolitan and upper class education (meaning learning Greek), such as Paul and Luke.

@Jakobian Even the most liberal scholars will concede that Galatians, Philemon, 1 Corinthians, Thessalonians, and Roman were written by Paul (so not anonymous), most definitely Jewish, who was a Roman citizen and was very educated Hellenistically and learned his Torah from his teacher Gamaliel. The rest of his letters (Ephesians, Colossians, Hebrew, etc.) can be argued to be written by his disciples (thus anonymous).

@GratefulDisciple Huh. "Opinions vary regarding the authorship of the four Gospels of Matthew, Mark, Luke, and John. Some assert that these were the actual names of the scribes. But most scholars conclude those names are merely placeholding pseudonyms, and the Gospels were written anonymously. "

@Jakobian Sure, I can accept that. But the key point to Christians is how they are treated as authoritative testimony about Jesus by the apostles. That the gospels faithfully represent who Jesus was and what he taught.

Being pseudonymous doesn't carry as much stigma back then as nowadays.

@GratefulDisciple sure, I found online sources to confirm this

At the end of the day, we are faced with whether to trust the "apostolic testimony" as represented in the New Testament. So yes, the information is second hand and has been filtered through the minds of the authors. That's why the term is "apostolic tradition" and that heresies meant those who don't see Jesus the same way as those in that tradition.

So Christian theologians would regard NT authors as NT-era prophets, at the level similar to authors of the books in the Hebrew Bible. By definition, prophets are God's authorized messengers. That's why we Christians believe the Bible to be sacred because it's God's revelation of his mind and the process of writing itself is guided by God too.

What about that Gospel which was claiming to be a prophecy about the future, but was proved to be written after the events already happened, and the last prophecy was unfulfilled?

But unlike the Qur'an, Christians believe the Bible is also a human book, filtered through the authors' ancient culture situations, not simply dictation.

@Jakobian Do you mean the Olivet Discourse which could have been written after the fall of Jerusalem in AD 70? Or do you mean the book fo Revelation being written after Nero? Or do you mean prophecies about Jesus's Second Coming?

Or are you talking about non-canonical gospel?

Personally, I don't find it's problematic if people theorize that the gospel containing the Olivet discourse (which contained Jesus's prediction of the near future) was written after the event. I can still believe the trustworthiness of the writer of the gospel who claims that Jesus was the one who predicted it. The events happened long after Jesus's ascension to heaven (around AD 35 at the latest).

In other words, the issue is whether the gospel writer is lying because he put complete fabrication in Jesus's mouth. But we don't have to believe Jesus's words in the gospel is a verbatim word-by-word speech of Jesus. Jesus merely need to be the one who authored the message represented in the gospel.

I heard it was established custom by ancient Greek historical genre that the author preserved the message of a person, and render the person's speech in a more elegant / concise style than what was actually said. So the many speeches of Peter and Paul in the book of Acts might be heard differently, although the core message should be the same.

Christian apologists now concede that some of the authors' theological agenda were woven into how they record Jesus's words, that each gospel is a literary theology book. I personally don't have a problem with this, because what I trust is the apostolic tradition preserving Jesus's message.

Gotta go. I hope the above helps.

Well, its overwhelming, especially since I don't know the Bible very well

I don't need nor believe in the Bible other than a historical source, so I also don't need to know it that well

But your input is appreciated

@Jakobian Sure. Studying the Bible as an ancient text is fascinating on its own, at least as an identity document of the culture that produced it, similar to studying the early text of the American founders. Or as a theological reflection of that culture, of how the nation of Israel in exile in Babylon redefined their faith in God. In the NT period, the same nation saw Jesus as the fulfillment of God's promise to save them although in a mode that they didn't fully anticipate.

So as outsiders we can at least appreciate the Bible as a story of a nation's origin and history, i.e. the people of ancient Israel and how they situated himself among other nations ruled by the multiple empires that came and go (Egypt, Hittite, Assyria, Babylon, Persia, etc.).

Actually I believe the part about Israelites escaping from Egypt was a fabrication

from what I know the Biblical historians agree that Israelites are Canaanites (or however its spelled)

Similarly, Slaughter of the innocents isn't a historical event

even though such a gruesome event didn't happen, it's still quite gruesome to write about it

@Jakobian Yeah, I heard that too. Part of the nation's origin story. There's definitely a folk tale quality to the narrative, to be retold around the campfire. Again, the key question is whether there is a historical kernel to it. It might not have been literal hundreds of thousand who fled Egypt, but maybe in the thousands. That they regarded God to be behind their exodus needs to be retained though, otherwise later books written in a much later era building on it wouldn't make sense.

Its just that nations used to write myths about themselves, similar thing happened for Egyptians for example

Egyptians were telling the story of a great Egyptian king, something that inspired Alexander the Great in his conquests

but in reality no evidence of such king exists, and its considered to be a myth

@Jakobian About slaughter of the innocents, I recommend watching this video about the recovery of the original understanding of the meaning of "town" which is supposed to mean military garrison.

@Jakobian About Israelites were originally Canaanites, I heard that too, along with the theory that ancient Israelites were the Sea People or other nations.

@Jakobian I wouldn't be that hasty to discount the existence of such king, only the extent of the hyperbole recording his conquests. I think David as a historical king of Israel is quite solid.

@GratefulDisciple David?

@Jakobian Yes, what about him? He's the ultimate hero of the nation of ancient Israel. A lot of texts / stories in the Hebrew Bible are about him. His poetry is preserved in the book of Psalms. He is a model believer, and "a man after God's own heart" (Acts 13:22, 1 Samuel 17).

I was talking about the Egyptian king

@Jakobian Oh sorry. I was trying to defend against people who say that David was not historical.

Its just an example of how nations used to lie about themselves

That video is about Sesostris?

@GratefulDisciple yes

1 sec

OK. Of course we need to separate myth from history. In our history of England, don't we need to deal with the real identity of King Arthur, but we can still be relatively certain of the real kings of England, even those that predated him?

Sorry, gotta go again. TTYL.

I don't know anything about history of England, other than the split in the religion their king done once

"If $\langle \psi|A|\psi\rangle = \langle \psi|A|\psi\rangle^*$ for all $|\psi\rangle$ then it follows that $\langle\phi_1|A|\phi_2\rangle=\langle\phi_2|A|\phi_1\rangle^*$ for all $|\phi_1\rangle$ and $|\phi_2\rangle$ and hence $A=A^{\dagger}$." I can't tell if the proof should be trivial or not and where to even start

Anglicanism

and perhaps some of the ww2 and ww1 stuff

but very little, as I'm not from England

maybe I should ask the h-bar since I just realized bra-ket isn't a math thing

@Obliv it seems like you need to use polarization identity

but it's basically saying if $A$ satisfies $(A\psi,\psi)=(\psi,A\psi)^*$ then it follows for $\phi_1,\phi_2$

im not 100% on if that's how you use $A$

@Jakobian not sure what that is

its an equation that represents $\langle x, y\rangle$ as sum of norms

in here its slightly different, but following the proof of polarization identity should lead you to a proof of the fact above

like if $y = ax$ for some $a \in F$ ?

@Obliv the left side is $\langle\psi, A\psi\rangle$ I'm sure

in here it says $(A^{\dagger}\phi,\psi)=(\phi,A\psi)$

but they're also using linearity in 2nd argument and antilinearity in 1st

@Obliv by the star do you mean complex conjugation

yea, some texts they use an overline

since bra-kets together define an inner product, you can conjugate the scalar result when working in the complex scalar field

also $\langle \psi|$ defines the linear functional with fixed vector $\psi \in V$

so basically, try to expand something like $\langle \phi_1-\phi_2 | A | \phi_1-\phi_2\rangle $

You will get $\langle \phi_1|A|\phi_1\rangle + \langle \phi_2|A|\phi_2\rangle -\langle \phi_2| A|\phi_1\rangle-\langle \phi_1|A|\phi_2\rangle$

this tells you that $\langle \phi_2|A|\phi_1\rangle + \langle\phi_1 | A | \phi_2\rangle$ is a real number

oh.. I think I get it. That is an inner product $(\phi_1-\phi_2,A(\phi_1-\phi_2))$ right?

now this tells you that imaginary parts of $\langle \phi_2|A|\phi_1\rangle, \langle\phi_1 | A | \phi_2\rangle$ need to be opposite