The h Bar - 2015-03-18 (page 4 of 4)

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9:01 PM

@ACuriousMind out of curiosity what topics have u been studying recently?

Does $\psi(z,\bar{z}) = \sum_{m,n \in \mathbb{Z}}\psi_{m,n}l_{m+h-1} l_{n+\bar{h}-1}$ make sense for the mode expansion of a primary field? What about $\sum_{m,n \in \mathbb{Z}}\psi_{m,n}L_{m+h-1} L_{n+\bar{h}-1}$? This is to say the conformal field becomes an operator, but I'm winging it, probably big issues

@StanShunpike Here's my current open book list. Books I am reading en masse: BBS, Blumenhagen Basic Concepts of String Theory, Nakahara Geometry, Topology and Physics, Weigand's String Theory intro lecture notes and Bouchard's complex diffgeo lecture notes. Books I am reading intermittently or am using as references for the other stuff: Polchinski, Jost Riemannian Geometry and Geometric Analysis, Di Francesco Conformal Field Theory, Straumann General Relativity and Srednicki QFT.

That covers all the books either on my desk right now or in my "currently reading" folder.

@bolbteppa That last sum does not make sense quantumly.

@StanShunpike Last semester mainly algebraic geometry and string theory. Now, in the spare time, not much, a bit of gauge theory and I want to learn about constructive quantum field theory, but I haven't started reading Glimm/Jaffe yet.

@ACuriousMind I'm sure you also want to learn proper GR. I'm sure you just forgot to say that. We all make mistakes.

;)

right?!

Pliz GR pliz

9:11 PM

@Danu Did you recommend anything to him?

@0celo7 That happens after I've digested Henneaux and decided whether Glimm/Jaffe are worth trying to digest

I think @ACuriousMind knows what books are good (and which ones I like), but no, not explicitly

@Danu Enlighten me on which ones you like.

Carroll, Weinberg

I think Wald is good but haven't read it

Weinberg will be boring for him.

9:14 PM

Also, I think you're reading way too many books at the same time to really get much from them, but ok

I recommended Wald and Straumann.

@ACuriousMind What is constructive quantum field theory?

@DanielSank The really rigorous set-up

(it doesn't actually work thusfar)

@Danu Why not?

@Danu Perhaps, but getting the correct prerequisites is very hard to do when self studying. Hence a lot of books.

9:15 PM

@DanielSank See 'Millennium Problems'

(the Yang-Mills mass gap one)

C'mon @Danu, I think there are at least existence results in 2D and 3D

@ACuriousMind Yeah, which means 0 physical results

I'm not a fan of rigorous QFT

@Danu Aren't many condensed matter and other systems described by 2D and 3D theories instead of the full-blown 4D ones?

@ACuriousMind Not exactly, of course.

@Danu I Googled it. The description on the official web page is remarkably useless.

9:17 PM

I mean... That's totally not the idea behind CQFT

@Danu Well, SR was totally not the idea behind electromagnetism, was it ;P

Nobody cares about describing CMFT models rigorously because the approach is totally useless computationally anyways and we have perfectly good computationally viable approaches

The only reason to do CQFT is to provide a rigorous justification for the fundamental ideas of relativistic, 4D QFT as used in the Standard Model

...and it's not working

(yet)

Hmmm... @ACuriousMind and @0celo7 both reading Nakahara. I must look into it.

@StanShunpike I am not reading Nakahara!

@StanShunpike It's considered the standard reference for physicists trying to do mathematics, but I've heard it's not at all in-depth enough to really get it

9:20 PM

@ACuriousMind ah, it was @0celo7 my bad misread the namw

...I already doubted that @ACuriousMind would read it

@Danu what book would u use instead ?

Anyway, to judge whether CQFT is interesting I have to look into it, haven't I?

0

Q: Policy on disagreements about "harsh" critiscism?

Most of us have, to some degree, experienced an antagonistic situation in which we feel our answer was harshly criticized or in which we are charged with unfair criticism of an answer. In my experience, these situations rapidly escalate, and despite knowing that I shouldn't, I've been drawn into...

discussion comments ethics

I said I have to decide whether disgesting Glimm/Jaffe will be worth it

9:21 PM

@Danu When you think about it, I have three string theory books on there and two books directly related to string theory mathematics. So I don't think that's too many, because they're all quite related.

@StanShunpike Monographs on each separate topic.

I've got a math question... Let's say I have $\iiint s_{kk} \frac{\partial u_i}{\partial x_i} + s_{kl}\frac{\partial u_l}{\partial x_k} - s_{kl}\frac{\partial u_k}{\partial x_l} + 2G\left[\frac{\partial u_k}{\partial x_k} - \frac{1}{3} \frac{\partial u_i}{\partial x_i}\right]$ and I want to use the divergence theorem wherever I can to transform it into a surface integral, what is the result? The first term it's easy to see it as $s_{kk}\nabla \cdot u$ but I can't figure out how to do the rest...

@ACuriousMind Go ahead. I already found semi-rigorous QFT absolutely TERRIBLE.

Devoid of any insights

@0celo7 how do you do a mode expansion of a primary field and interpret it as an operator if that's wrong?

First I want to get through Henneaux/Teitelboim though, the idea that gauge systems are really better viewed as constrained systems is intriguing, and seeing the BRST formalism carefully developed is very good.

9:22 PM

@bolbteppa You need normal ordering for one. The modes of the fields are operators due to the OPE with the EM tensor.

@StanShunpike So rather like 6-7 books

(which will probably take ~2 years, but yeah... at least I'd know what's up haha)

@0celo7 so what I wrote is right if I use the words normal ordered?

@bolbteppa No.

@bolbteppa The expansion is something like $\phi(z)=\sum_n z^{-n-h}\phi_n$

IDK how for non-chiral fields.

Okay once you have a Chiral field, you just interpret the $\phi$'s as operators and that's it?

You can find the commutation relations of the $\phi_n$ using the OPE with $T(z)$.

Pretty much.

You create states by acting with the mode $\phi_{-h}$ on the vacuum.

9:27 PM

Woah wait, so $\phi(z) = \sum z^{-n-h} \phi_n = \sum (z^{-n-h-1}\partial) \phi_n = \sum l_{n+h} \phi_n$4 right?

@Danu do you have a list of monographs you plan to read in place of Nakahara?

You then use the negative mode Virasoro generators to create the Fock space.

@bolbteppa The modes are constant...

@StanShunpike I haven't got it all planned out---but I have quite some ideas, yes.

what's ur top 3?

So I'm asking something crazy and irrelevant because we're always just plugging in fields with fixed values into correlation functions?

9:28 PM

Like of the ones ur planning on reading

What I'll be reading next is: 1. Finishing Lee - Introduction to smooth manifolds, 2. Do Carmo - Riemannian geometry

@bolbteppa No, but the modes are, classically, the expansion coefficients, which are constant, as 0celo7 says

@Danu to quote ACuriousMind, 'As it should be' devoid of any insights ;)

@ACuriousMind ohhhh...

@Danu What did you have in mind for Kahler geometry and Chern classes?

Nothing

That's very far away

First come books on complex and symplectic geometry

9:30 PM

@Danu Complex $\sim$ Kahler

Not true, I think

Kahler is a special kind AFAIK

@0celo7 No, Kähler = complex + Riemannian + symplectic

Isn't Kähler a combination of Riemannian + Symplectic + complex?

there we go

So Danu is doing the sensible thing in first tackling the components

9:31 PM

Oh well I meant complex.

I did not know that apparently.

I think I have a completely different learning strategy from you

I like to think mine is the better one, of course

I think a Kahler manifold just lets you copy $z = x + iy$ on a manifold

But it's certainly the slower one

@Danu You're also the one getting the degree in mathematical physics.

@0celo7 mathematical physics (hehe :D)

9:32 PM

Mathematical physics to boot

Remember, I have no ambitions of being a physicist at the moment.

hahaha

xD

I got backup!

Can you hear my envy ;)

9:33 PM

Mah German homie's got my back

Dude, you should come spend some time at LMU

we have some really famous researchers, you could probably hook one of them as a MSc thesis advisor ;D

@ACuriousMind ...or just transfer to my program :D

You'd immediately have a reasonably large and close group of friends

...I even think some of em are into role playing games, lol

Phrasing! :P

In short: An offer you can't refuse!

@ACuriousMind Whaaaat?

But, well...it doesn't sound like a bad idea at all.

@StanShunpike At the same time, I'm making my way towards category theory through abstract algebra

@ACuriousMind Seriously though!

Except for Munich being even more expensive than Heidelberg

9:36 PM

My friend group is awesome fun, and I'd totally vouch for you being cool and all haha

@ACuriousMind Well... screw those rumors! :P

@Danu lol, I'm just some dude from an internet chatroom

I think housing is pretty expensive

@ACuriousMind Think of all the hours we've spent talking haha

Probably the most I've 'hung out' with someone online since I stopped playing MMO's

@0celo7 because of $\psi(z,\bar{z}) = \sum_{m,n \in \mathbb{Z}} z^{-m-h}\bar{z}^{-n-\bar{h}}\psi_{m,n}$ it's the exact same thing for a non-chiral field right, though commutation relations are probably crazy, right?

@ACuriousMind This is how people get abducted.

@0celo7 Ah, come on, I'm harmless

Just step in to my van, I've got candy!

9:38 PM

@Danu Lol I meant you'll get abducted!

The question is - who is the abductor and who's the abductee here?

Ah, lol

@bolbteppa Don't know.

Well, I've seen his face---I think

@0celo7 Thanks for caring about me

9:39 PM

I guess he's not as easily identifiable as I am (it should be completely trivial with some google-fu in my case)

@ACuriousMind Like you said, you're just some random dude on the internet :P

And I'm not? :P

@Danu You gave me Danu's PC though

That's like a google of trust and caring right there.

@0celo7 I'm such a nice guy

Also for others: I didn't actually give him a PC

I don't even know what PC is supposed to mean here

9:41 PM

Do you want to know?

Good question

@Danu Anyway, our strategies are different because we have different goals. My goal is to get reasonably comfortable with introductory superstring theory before my freshman year begins and I have no time anymore for anything.

But I think yes

@0celo7 "get reasonably comfortable with introductory superstring theory before my freshman year"

@ACuriousMind Oh self-study

@ACuriousMind Is that so unreasonable?

I have like 5 months.

9:43 PM

@0celo7 yeah that's right but Blumo doesn't mention any commutation relations in his CFT book, but just call $\psi_{m,n}$ an operator and we're happy in the non-chiral case too ;)

@bolbteppa I have convinced myself his CFT book is garbage.

His string theory book's CFT sections are far more enlightening.

@0celo7 From the perspective of an ordinary student, it sounds absolutely crazy :)

user image

@ACuriousMind OK then.

(Hopes somebody gets the reference.)

@0celo7 I think just the reverse, the string sections have some good notation but conceptually I think he explains things better in the cft book for pure cft ala DiFrancesco

For string cft maybe ur right

@0celo7 Nope

9:48 PM

@ACuriousMind oO you've never played Halo (2)?

@0celo7 Actually...no

Not a shooty type of guy

@ACuriousMind youtube.com/watch?v=gpByy8bRzyo&t=0m48s

@bolbteppa I assume the expansion would look like $\phi(z,\bar z)=\sum_n z^{-n-h}\phi_n+\sum_n\bar z^{-n-\bar h}\bar\phi_n$

Not 100% sure on that though.

No idea either

It probably never occurs you know.

Everything useful is chiral.

The chiral basis is literally completely useless for high-energy computations in QED

9:59 PM

@Danu And that has to do with CFT?

So on one level I've taken the Hamiltonian and expressed it as the $L_0 + \bar{L}_0$ operator, and for some other reason I took a field and expanded it as $\psi(z,\bar{z}) = \sum_{m,n \in \mathbb{Z}} z^{-m-h}\bar{z}^{-n-\bar{h}}\psi_{m,n}$ where $\psi_{m,n}$ are operators, they are two random facts I need to string together conceptually to explain quantization

@Jiminion Well... obviously J.K. Rowling did that on purpose

@Danu Chiral in CFT means that the field splits into a holomorphic and antiholomorphic piece.

@0celo7 Just responding to "everything useful is chiral"

Oh I meant in CFT.

@bolbteppa Actually that expansion might be right...check to see if it transforms like an $(h,\bar h)$ field.

10:04 PM

@0celo7 idk DiFrancesco says that even when you're using the $\psi(z) = \sum \phi_n$ notation the anti-holomorphic dependence is always there (p 153)

@bolbteppa Yeah (6.7) is what you wrote.

@0celo7 yeah and see below his comment in (6.12)?

All this CFT business is making me sleepy

I have to stay awake for 22 hours studying cft

It could also be due to the music I'm listening to

10:17 PM

Well about 16-17 at cft

It's almost hypnotic

Do I need to post the academia post about sleep deprivation again?

yeah

wow, tell me

haha

@bolbteppa This re me or @ACuriousMind?

academia ;)

10:19 PM

I think the only time I worked more than ten hours on anything was when I had to meet a printer's deadline the next day. And that was back in school and had nothing to do with physics.

I'm writing a high school lab report in LaTeX :D

This seems like overkill, but the Word equations look awful.

@Danu is category theory useful for physics?

@fffred: Not sure where you are, to pick you up. The mathematics in the above is not sophisticated at all. It's just Lie algebras and Lie groups. If you are at all interested in physics at a more than very superficial level, you need to familiarize yourself with the concept of a Lie algebra and a Lie group. Lie groups control the universe. But the nLab version of my discussion above (ncatlab.org/nlab/show/quantization) has all keywords hyperlinked to a dedicated page, which will give at at least pointers to standard references. Maybe try that. Else, tell me more exactly what you know. — Urs Schreiber Sep 13 '13 at 13:32

Hah, this is so wrong, but makes sense coming from a mathematician

@StanShunpike I think it is only useful in the sense that one can view physics from this perspective

But I don't think one ever has to

I do, however, think it's cool as fuck and want to learn it for its own sake, as well as possible physics applications

All hail to the Lie algebra!

...and the one and only prophet the exponential map

...the Holy Spirit being the Lie group, of course

10:30 PM

What is more fundamental, the group or the algebra?

Group

So why isn't the Father the group?

(from the symmetry perspective, at least)

And the algebra His son?

@0celo7 ...because @ACuriousMind made that linguistically impossible!

10:31 PM

Hmm

Gotta calculate some moments of inertia

So much fun

The dove is representation theory rep'rzentin' that shizzo

...that's one way of putting it :)

I just got Ryder. Very excited. Okay, question: he says that, unlike the photon, and field quantum of the strong force had to have finite mass due to the finite range of the force. how is range of force related to quantity of mass of the field quantum?

@StanShunpike chapter 2 on spinors will drive you insane

I guess if it had infinite mass then $F = ma$ would imply infinite force?

@bolbteppa :: cuts pages out of the book ::

10:45 PM

@StanShunpike The Yukawa force has finite range if the messenger particle is massive.

@bolbteppa No, he means finite as opposed to 0

@StanShunpike There is a standard handwavey argument saying 0 mass -> long range

@Danu ...called the Yukawa force.

@0celo7 I'm not sure what you're talking about

QCD != Yukawa

Yukawa potential

In particle and atomic physics, a Yukawa potential (also called a screened Coulomb potential) is a potential of the form where g is a magnitude scaling constant, i.e. is the amplitude of potential, m is the mass of the affected particle, r is the radial distance to the particle, and k is another scaling constant, which finally the product of km is the inverse scope. The potential is monotone increasing, implying that the force is always attractive. The Coulomb potential of electromagnetism is an example of a Yukawa potential with e−kmr equal to 1 everywhere; this is taken to mean that the photon...

It's the generic potential of any boson exchange.

what is the Yukawa force? I thought there were only four fundamental forces

10:47 PM

@0celo7 I don't think this is true. Why?

Scalars are not all there is, y'know

@Danu Luckily the pion is a scalar, right?

@0celo7 ...but the pion is not all there is, so this doesn't really explain anything

Also Yukawa used this potential to predict the mass of the pion given the length scale of the strong force.

My guess is that the Lagrangian has mass terms coming from the Dirac and scalar lagrangian part but the EM term does not

I think I will start with chapter 6 lol

10:51 PM

@StanShunpike The derivation I scetch here goes through for massive vector fields instead of massless as well, but we incur an extra term $\mathrm{e}^{-\mu r}$, where $\mu$ is the mass of the bson.

@ACuriousMind So I was right?

@ACuriousMind thx for the link, I will go over it

@0celo7 Yep

@Danu :P

What is $|0>$ in CFT?

10:54 PM

@bolbteppa Vacuum state.

It's annihilated by the positive Virasoro modes, the Hamiltonian and the $-1$ mode.

See chapter 4.7 in these lecture notes for a more detailed version of my answer, by the way

Yeah but where does it come from? I'm not very good with this notation tbh

Grazie

@ACuriousMind ^

@bolbteppa A PSL-invariant vacuum is assumed to exist.

On page 174, Ryder says "the act of creation may be represented as a source, and that of destruction by a sink, which is, in a manner of speaking, a source." What do we mean by source and sink? Is this in the sense of divergence?

10:58 PM

@0celo7 Right about what? It's your initial statement that it is all Yukawa is still not correct IMO

...obviously the exponential behavior should carry through, I never disputed that

@0celo7: One must be careful, though - although the form for the potential is the same for Yukawa theories and vector theories, Yukawa theories are attractive even for same charges, while vector theories produce the familiar repulsion of like charges.

If I think of $|0>$ as the exact same thing as $\psi(0,\vec{r})$ only in some new space will I get in trouble?

I really hate non-wave function stuff tbh

@ACuriousMind I'm well aware that some are attractive and some are repulsive.

@bolbteppa Typically, yes. What context are you in?

@bolbteppa What on earth is $\psi(0,r)$ in the CFT context?

You have to use the state-field correspondence to see what field the vacuum belongs to

One moment...

11:00 PM

Also, PLEASE DON'T USE WAVE FUNCTIONS, EVER :)

I <3 wave functions though, they make so much sense

That's the problem! They make you think you can get away with classical nonsense

Everything is intuitive, I mean I even found a path integral derivation using them and it makes so much more sense

Only for people who don't know what Heisenberg really means! :D

What if I told you a path integral was a green function... Morpheus look

I'm a field theory kind of person

@bolbteppa The Green's function typically appears in the path integral, if that's what you mean

Well, the vacuum is associated to a very boring field - $\phi(z,\bar z) = 1$. -.-'

11:02 PM

That is not clear with the $|sick>$ notation

@bolbteppa You appear to be thinking quantum mechanically

@Danu What is incorrect about it?

I think this is just not the way to go when doing field theory

wahoo!

@Danu only if you're scared of the Schrodinger representation ;)

I've had this conversation already, @Danu ;)

11:03 PM

@0celo7 Yukawa is only spin zero. @ACuriousMind already raised an objection in the meanwhile

@bolbteppa Lol, no you're scared of letting go of QM ;)

And want to abuse the term quantum FIELD theory by doing quantum OPERATOR theory lol

@Danu I don't see an objection...

@0celo7 You even responded to it

@Danu If it's only spin-0, then I'm confused why it works for photons.

@Danu That wasn't an objection.

@ACuriousMind woah can you shed some light on that, maybe even a link?

11:04 PM

@0celo7 It doesn't actually work exactly the same way. There's a superficial similarity between spin-0 and spin-1, yes.

On page 174, Ryder says "the act of creation may be represented as a source, and that of destruction by a sink, which is, in a manner of speaking, a source." What do we mean by source and sink? Is this in the sense of divergence?

@Danu You have to trivially fix the sign out front, yes. I still don't see what the problem is.

@StanShunpike No, in the sense of a perturbation to the vacuum

(I was well aware that the sign depends on the spin.)

@StanShunpike he's talking about creation and annihilation operators

11:05 PM

@bolbteppa In fact, I'd like to see you derive some amplitudes using your Schrödinger representation

@bolbteppa It's really boring because the state-field correspondence is derived by assuming the Hilbert space has a vacuum $\Omega$ (which one can motivate by looking at the possible unitary reps of the Virasoro algebra), and then to every state $v$ the associated field is defined by its action on that vacuum: $\phi_v(z,\bar z)\Omega := \mathrm{e}^{zL_{-1} + \bar z L_{-1}}v$

@bolbteppa @Danu oh, is this just a general term for when particles are created or destroyed at a point?

Source = creation

Sink = annihilation

Since the vacuum is assumed PSL invariant, the RHS is just the vacuum, so the field is the identity

@StanShunpike Yeah, I think the terminology just carried over for historical reasons from Schwinger's original conceptual frame

...which nobody uses anymore

yeah

11:08 PM

Oh, that's confusing. Okay, good to know. Then that makes sense

@Danu I need to sit down and do it after my project, what kind of stuff do you think is hard, I have a book that does all this stuff?

Also, PSA: There are no true wavefunctions in relativistic quantum field theory, the position representation needed to get them is subtle and hard to define.

I think you could even ask a question here on the site and Valter Moretti would come around and show why. I've seen him raise that objection in many comments

@bolbteppa Sorry, what are you talking about?

Wave functionAL ;)

You said you wanted to see me derive amplitudes in the Sch rep?

@bolbteppa Oh, yeah, no I don't think anyone actually uses that stuff in field theory... for good reasons

11:11 PM

It's complicated stuff

Hey, I think my algebra book actually ignores the existence of the word 'functional' :)

@Danu I've actually seen wavefunctionals being used in 2D gauge theories to derive that the Hamiltonian is just the quadratic Casimir

But yeah, no one uses them in usual QFT, or to compute amplitudes

@Danu These algebraists often say "linear functional" for ordinary finite-dimensional linear maps, or what do you mean?

@ACuriousMind function for everything

Well, that's also fine

"the function $\alpha(f)=$[insert integral of $f$] is a linear function on the space of continuous functions blahblah"

11:17 PM

yeah functional is just a term to say it's always real-valued

or C or whatever

No, it's typically reserved for operators from a space of functions to the field (field in the mathematical sense)

so "function of a function" is usually translated to "functional"

Where function is interpreted as "takes argument, gives element of the field"

It's also used to describe dual vectors as functions acting on vectors giving their coordinates

Weirdly, Wikipedia says: "In mathematics, and particularly in functional analysis and the Calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers."

Lol, okay, it appears there is no consistent usage of the term.

Yeah I should have specified it's defined on a vector space

11:20 PM

I'd also have gone with "A functional is a function of functions"

I'd only ever use it on a space of functions

if it's linear

I'd never call $f:\mathbb{R}^n\to \mathbb{R}$ a functional

It can be non-linear and you can call it a functional

But the algebraists also call the dual vectors "functionals on the vectors".

It's really not very consistent

11:23 PM

it's completely consistent guys, a dual vector is a functional on a vector?

A function can be thought of as a vector, and if the function maps to a field it's a functional

Like I said, I don't think functions from the vectors to the field should be called functionals personally, because it's not really a "function of a function" and that's how I've grown accustomed to it being used

But it's a personal thing

Calling the finite-dimensional things functionals gets inconsistent with things like "functional analysis" or "functional derivative", which are always meant to take place on infinite-dimensional spaces

Yeah, I think my opinion was formed by studying Reed & Simon's book which is mainly functional analysis

user54412

I'm pretty sure calling anything a functional stops making sense the moment you realize there are more than three (co)domains in Plato's heaven.

There is a difference between an algebraic dual and a topological dual en.wikipedia.org/wiki/Linear_form and you guys are focusing on the topological nature ignoring the algebraic nature

11:29 PM

@bolbteppa I have not said one word about topology in this context

@ChrisWhite I am inclined to agree ;)

"In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, although this is not true in infinite dimensions." you guys want to throw away half a big theorem for no reason basically lol

And you want to throw away the beautiful geometric duality en.wikipedia.org/wiki/… for no good reason

We don't want to throw away anything, we are just not satisfied with the nomenclature functional. Why not simply use function?

Because you want to know it's spitting out a real number specifically/conceptually

$\mathbb{R}$ is special...

I think calling anything that maps from something to the basis field a functional is just not a great idea

Why?

11:35 PM

@ChrisWhite Also, wut?!

@bolbteppa Because I prefer to call it a function.

user54412

@tpg2114 I'm not sure what's going on. Does $s_{kk}$ not depend on $x$? Otherwise how do you apply divergence?

user54412

@Danu Looking back, that is a cryptic statement, isn't it?

user54412

What I mean is this: say you have domain $A$ and codomain $B$. You can define functions $A \to B$. Then if you have another codomain $C$ lying around, you can define functions of functions $(A \to B) \to C$, and you think you're so clever for using a set of functions as a domain unto itself you call these new things functionals.

@ChrisWhite Care to elaborate?

I...got it :D

user54412

11:39 PM

But what about maps $A \to D$? or $(A \to X) \to (Z \to Y)$? or all the infinitely many other things you could define?

A function from on a vector space into the reals is a linear functional, A tensor is a multilinear functional, a determinant is an alternating linear functional, it's all beautiful...

@ChrisWhite I was hoping for more information on "Plato's heaven"

@ChrisWhite How about... NO! :)

@Danu Plato believed that there is an abstract "heaven of ideas" where there is a perfect idea of everything lying around. Mathematical objects are commonly said to live there

user54412

@Danu It's used in discussing (usually negatively) platonism

@ACuriousMind I knew about the "world of ideas", but why would we want to talk about mathematical objects in there? :P

Also, I don't think I'd ever heard anyone call it a "heaven" yet

11:42 PM

@Danu Because ideas, just as mathematical objects, are just abstract concepts?

user54412

Platonists (the modern ones at least as much as Plato himself) would say mathematical concepts like "addition" and "three" really exist, whatever that means, independent of any thinking mind

user54412

so critics would counter "oh yeah, but where do these things exist then, if not in the mind? In Plato's heaven?"

@ChrisWhite Ah, yes.

I always find it funny when mathematicians insist that mathematical ideas/structures really exist

To me, it seems very unnatural to say that

Particularly because it's so absolutely clear that we are the ones laying down the axioms

But oh well

...is it time to don the amateur philosopher's hat again?

Hey, that's my favorite hat!

Also, I took a philosophy of science course so that basically makes me omniscient in this area

11:46 PM

Using what definition of omniscient? ;)

The one that Really Exists

One can only discover it

Just like all other definitions

amirite guise?

::crickets::

You're all just standing in awe

user image

I'm not quite sure how this conversation ended up with that

@ACuriousMind 4chan is never far

11:51 PM

Evidently not :D

'4chan is never late, nor is it early, it arrives precisely when it means to.'

@Danu Which boards do you frequent?

Pretty much only /wg/ because it's the best source of wallpapers on the internet, IMO

A few years ago I did spend some time on /b/ out of boredom, but it's mostly very unrewarding haha

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Mar '1518

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