12:23 AM
I made a sphere anim in SVG, using homogeneous coordinates (in Sage) to do the 3D calculations & perspective projection. This anim has 5 frames (well, 6 frames, but frame 5 is a repeat of frame 0), and the grid cell size is 10°, so the angle step between frames is 2°. Even the 15° cell / 5° step version is pretty good, although the interpolation is a bit wobbly if you look closely.
If you see glitches in that anim, zoom in slightly.

3 hours later…
3:48 AM
@imbAF where, roughly, did you do your bachelors?

1 hour later…
5:08 AM
does anyone know of a good review on open quantum dynamics?
i am interested in the shortcomings of the current state of the art for modeling open quantum systems
5:20 AM
@imbAF I don't know what you mean by "2nd boundary condition"
do people on average use philosophy as a framework for thought?
to clarify by analogy: it seems everyone uses mathematics as a framework for physics. (however, in this case, mathematics is seemingly the only choice).
i think the answer is: no. people necessarily see a survey of mathematics to even be able to begin to do physics. however, all people think to some extent without having any formal learning in philosophy. but i am trying to understand the role of philosophy in relation to thought. i thought the model provided by the relationship between mathematics and physics would be appropriate.
@SillyGoose in general a lot of scientists think, but dont fret too much about how they think
i am wondering in the first place because i was wondering who cares to fund philosophy research at all. i took a cursory glance at the most recent issue of the philosophical review and one of the articles was on formalizing the notion of "Urges", which struck me (of all people, perhaps) as unecessarily formal
even those of us who care about the philosophy part, the epistemology part, had better concede that these issues form a minor component to how we usually operate, or else we would keep being stymied in confusions
@SillyGoose it is funded??
I thought it is mostly funded by religious organisations like Templeton foundation
@naturallyInconsistent i also was skeptical, but at the end of some papers i glanced at i saw some funding sources acknowledged
@naturallyInconsistent perhaps. the templeton also funds research in axiomatic qft as done by jaffe and such
but if not funded by other sources, these are at the least professors being paid by universities
5:58 AM
@SillyGoose The funding of philosophy and other classical arts stems mostly from the humanist model of education, about which I also talked a bit at chat.stackexchange.com/transcript/message/65551349#65551349
@SillyGoose while philosophy is indeed sometimes presented as "how to think", I think that this is, apart from the subfield of logic, not really specific enough. The topic of philosophy broadly are in the tradition of the enlightenment the three Kantian questions: What can I know? What should I do? What can I hope? Whenever one engages with these - however formal or informal - one is doing philosophy.
Much of philosophy is indeed about building some kind of framework to answer these questions, but it is not like physics and mathematics where there is one formal framework you have to learn first to then understand its applications. There isn't just one kind of philosophy, and outside of a specific kind of analytic philosophy it is much less formal
6:41 AM
hi
7:10 AM
Given a fibre bundle, using a different projection gives a different base space. Can the number of different projections that can be taken from a given total space be finite?
@Sanjana I'm not sure what you mean - part of being "given" a fibre bundle means that it comes with a fixed projection.
@ACuriousMind Yes. But then I am replacing the given projection with another different projection. Yes, that would give a new bundle. I am asking whether I can conjure up as many projections as I want getting new bundles, or are my choices limited due to some constraint?
Take the manifold of a single point $\{\bullet\}$ and consider the fiber $\mathbb{Z}_2$
Only two projections!
I guess only one projection, technically
Two embeddings of the base manifold in the bundle
how can a point have a fiber
Why not
7:17 AM
fiber is like tangent space at the point, i think
Only the tangent bundle
Bundles can be whatever
It's not uncommon to consider bundles of a single point
ok so u just took a point and assigned the vector space Z2 to it?
Sure
Although I wouldn't call Z2 a vector space
Okay... that's something. Let me give you some context. I was looking at the definition of twistor space as double fibration of projectivization of spinor bundle $PS$ where $TM \sim S \otimes \dot{S}$ and $M$ is Minkowski space. Now they consider two projections of this $PS$. One is down to ordinary Minkowski space, and another is to twistor space which is an open cover of $CP^3$.
@Slereah it is a field, i think
7:19 AM
Although this might not be necessarily true, but I was wondering whether only these two projections are special in some sense or not. Maybe, only these two "can be taken"?
a field is also a vector space
@Sanjana For any universal covering of Lie groups $G\to H$ you also have a cover $G\to G/\Gamma$ for any subgroup $\Gamma\subset \pi_1(H)$. When $\pi_1(H)$ is infinite, this should give you infinitely many subgroups, hence infinitely many coverings, and coverings are in particular fiber bundles
ok so we r talking about general fibers here.. only the tangent fiber shud b an Euclidean space
i got it
while it can happen that some of the $G/\Gamma$ are isomorphic (as for $\mathbb{R}\to S^1 = \mathbb{R}/\mathbb{Z}$), I see no reason for this to be the case in general
You can give a point a tangent bundle, but it's just $\mathbb{R}^0$
It's the set of derivatives of all curves, and all the curves are constat $\gamma : \mathbb{R} \to \bullet$
7:27 AM
@ACuriousMind I see...
btw Slereah, I gave some context. I thought of this while studying the definition of twistor space, something which you have studied as ACM says...
Double fibrations aren't uncommon in math, but I don't think they're required to be unique in general?
Is there any natural reason as to why we should consider these two projections out of possibly infinite others?
Twistor fibrations are part of the general class of fibrations representing the relationship between a curve and its destination
It's intuitively the concept of how you can describe a curve alternatively as two points or as one point and a direction
That's why you have the whole correspondence between points and lines in twistor theory
that whole thing
Sounds like Legendre transformation...
It's more related to polarity transformations
7:34 AM
What's that? I did a google search to find "polar" transformations instead which I know of...
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole. == Properties == Pole and polar have several useful properties: If a point P lies on the line l, then the pole L of the line l lies on the polar p of point P. If a point P moves along a line l, its polar p rotates about the pole L of the line l. If two tangent lines can be drawn from a pole to the...
The more boring version of this than twistor space is the path structure of a manifold
Which describes second order PDEs
It's also a double fibration similarly to that, where one fibration is the initial conditions and the other fibration is a set of two points
8:03 AM
wiki says spacetime can emerge from twistor theory
8:26 AM
@Slereah Is this related to the fact that one can efficiently solve wave equations in Minkowski space using twistor space methods?
I have read it as a motivation, although I didn't quite understand it.
No idea
They consider a $CP^1$ bundle over $C^4$ (complexified Minkowski space) because of the following. While solving 2D wave equation in complex coordinates, we complexify the coordinates to form $z=x_0+i x_1$. Similarly we can consider $\vec{x}=(x_0+i x_1 ~~ x_2+i x_2)^T$ but this isn't quite the unique choice...
The space of different norm preserving choices is $CP^1 \sim O(4)/U(2)$. Here $U(2)$ is modded out cz $U(2)$ acting on $\vec{x}$ also preserves the norm and also holomorphicity (I'm not so sure if this is the correct reasoning, though.) But, what I really don't understand is, why do the fibres become $CP^1$ after all this? Why suddenly consider the $CP^1$ bundle? Do we do an analogous thing while "under the hood" after writing the 2D wave equation in complex coordinates?
The above motivation is from pg. 20 of Nair's notes on twistors and other stuff.
It is a quick read and imo not something specific to twistor theory, so I just shared it here just in case anybody can give me a clue as to why we consider a $CP^1$ bundle in the last few sentences of the above snapshot...
@ACuriousMind so, I feel physics is made easier and enriched by (a subset of) mathematics. I am wondering how to sift through all of philosophy to find a subset of it that makes and enriches thinking. I was hoping such a "syllabus" would be existent.
separately: is magnetism purely a statistical mechanical concept? all the discussions I can find are in statistical mechanical language.
8:42 AM
@SillyGoose I think epistemology is very helpful and should be your first starting point.
@SillyGoose Practically, yes. And it only makes sense from that perspective and so you should not avoid it if you actually want to understand magnetism in materials.
9:32 AM
@SillyGoose I think this depends very much on what you mean by "enriches thinking" - personally I found it valuable to learn about different philosophical viewpoints on various things simply because seeing how other people reason (even if I don't agree with them) has an effect of broadening one's horizons.
There isn't really an objective subset of philosophy that would be "useful" and the rest is unnecessary - you ask different people and they will tell you different parts that are useful or useless, it's very much not "settled science" in the way that there's a single mainstream like in most of mathematics or physics.
when a philosophy becomes objectively true, it becomes science
but not all philosophies have the potential or intent to become science
@naturallyInconsistent i am trying to look into a microscopic dynamic process by which spin-1/2s align to be parallel. a prof. stated that the heisenberg model hamiltonian is supposed to (heuristically) do this. but as i am modeling the dynamics, I am not seeing this effect. I guess I am more interested (atm) in seeing spin-1/2 alignment rather than magnetism (though it seems the two are related)
@ACuriousMind hm have you found that people actually fit into distinct classes of "ways of thinking" as organized by established philosophy?
e.g. in morals, most people use moral absolutism or relativism or a mixture
separately, is there a text on the history of mathematics/one case study in which during a time period $X$ many widely considered "niche" or "exotic" mathematics were being developed, but at a later time period $Y$ one of the such constructions became important such that it became "household" knowledge. For example, perhaps group theory might fit this evolution?
I am wondering if on average "useful" mathematics (defined as becoming common knowledge to physicists) is recognized as it is being developed or if its usefulness is stumbled upon later.
for example, a counterexample might be dirac's own study of the already developed symplectic geometry which, to my understanding, underlies his dissertation
that there is no pattern is also an acceptable answer
some math is developed on the fly for an application, e.g. calculus
spinors were developed around the same time electrons were found to be spinors
but it is much easier to make pure math.. so, at least in the modern era, "most" math is obscure, and will not find application for long
9:48 AM
well I think my underlying question is more like: is the state of the world constant. In the sense that: yes, there is always "progress" in the sense of starting with an idea and "developing" it into a new idea that proves to be more "useful" than the original idea. However, is the feeling of "challenge" to develop a new idea from an original idea constant through history.
because i get the impression that: yes, today many researchers do their own niche, obscure thing. but is that a novel phenomena or is that how things have always been?
there is a sort of story that is told of hyper-specialization in the modern age, but I am wondering if it's really a phenomena specific to now.
@SillyGoose Well, over the last century physics got insanely broader
Physics up until the 20th century was basically calculus, geometry and algebra
i would say it is a modern era thing because of the breadth of knowledge
mirror symmetry is something mathematicians got from physicists..but these things r sparse compared to the pure mathematics line-up... pure mathematics wasnt a huge deal in the past, like before 1700s
like, Euclid's axioms were about the world we live in..
even something as simple as calculus was made directly for application
@SillyGoose it's like with all labels and categories for people - no actual human being fits 100% into some idealized category we make up to classify millions with the same labels, but they can be useful.
For instance, I think the distinction between consequentialist and deontologist morals is very useful even if it is very broad and encompasses many different specific viewpoints in each bucket - especially because modern discourse is often so thoroughly utilitarian that people don't even realize there may be non-consequentialist viewpoints
10:03 AM
philosophy helps organise thoughts.. e.g. some people may never consider deontologism in their life if they never encounter philosophy as a subject
but people have rough intuition for these ideas anyway. philosophy just helps in making things clearer
a great example is moral relativisim. many people would intuitively refrain from judging others by their morals if they r 100% a moral relativist
the above is a blockage on reasoning. it feels logical but it's a bad framework. people can know that by studying philosophy
@RyderRude people used to have mathematics debates. It is so big and important that the whole story of the cubic solution could not be told without it. Before we had modern board games, there werent that many entertainments people could have had otherwise. How many times must you be told that you are wrong before you stay out of history topics?
@SillyGoose Did your dynamics take into account Coulomb interaction? (it appears in exchange interaction) Because otherwise you will not have a thing strong enough to align the spins.
@naturallyInconsistent i am not considering spatial degrees of freedom (just spins on a lattice w/ nearest neighbor interactions, essentially)
a periodic, one-dimensional lattice of spin-1/2s
@naturallyInconsistent i know about the debates on cubic equations... it is another example to support what i said...that imaginary numbers were invented directly for an application
things have gotten far more abstract recently.. it is undeniable
@RyderRude ...what application do you think cubic equations were about?
@ACuriousMind there may not be a direct application but theyre not abstract either
10:19 AM
what does that mean? It's still "pure mathematics" if you can't point to a direct application!
@ACuriousMind "application" in my comment refers to cubic equations
imaginary numbers were invented to solve equations, instead of "let's define this obscure pure math thing"
"solving equations" is a pure math task unless the equations come from a particular application!
@SillyGoose If you include too few things in your model, then it should not be a wonder when you discover that it does not work properly.
i think it need not be either "pure math" or "abstract math". solving equations is definitely somewhere in between.. e.g. people used to relate quadratics to areas
so cubics could be interpreted as volumes
@naturallyInconsistent hm i didn't know the spatial d.o.f. were necessary to see alignment
10:22 AM
@RyderRude but did anyone do that? can you point to anyone in history using that as the motivation?
do you know a reference that goes into more detail?
@SillyGoose Again, I don't know how you are getting into that, but it seems like you are not consulting a standard textbook on the subject. It feels like you are jumping into the topic from qubits?
@ACuriousMind i dont know any person using volumes as a motivation, yes
@naturallyInconsistent yes
@RyderRude so where do you get off claiming that the history of cubic equations is "another example to support what i said" where what you said is that there wasn't a lot of pure mathematics?
10:23 AM
Well, if you really want to understand magnetism, you should consult a solid state textbook, and be utterly destroyed by the 9823475 models discussing each particular type of magnetism.
but now we've clarified that you don't have any reason to claim that cubic equations weren't primarily "pure"
i more just want to see one quantum mechanical model that shows that spins align to be parallel (to some degree)
@ACuriousMind i have clarified this... it need no be either "pure" or "applied". but solving equations is definitely something people would consider close to applied
@RyderRude [citation needed]
If you want to understand the Heisenberg model alone, there are those that somewhat specialise to that topic, in which case they would still start with Curie-Weiss mean field theory, in which case they kinda proved that the mean field had to be way too big, before quantum theory via Heisenberg model showed the exchange interaction can produce such huge mean fields.
10:25 AM
the entire field of number theory (very much "pure mathematics" outside of its very recent cryptography applications) is about "solving equations"
"solving equations" is something that has its roots in applications, not in "let's define this and see what happens". cubic equations are a generalisation of what people initially developed equations for
@SillyGoose To this I would recommend something covering the topics I just mentioned. They might even appear as slides. You can ignore the itinerant magnetism concept as being something else entirely.
that is precisely what i mean by "it is closely related to applications"
@RyderRude you just mean "I define words in such a way that I cannot be wrong, even if nobody else uses words that way."
by "pure math", people mean the kind of math where u go "let's define these abstract operations and see what happens"
10:29 AM
I dont even know why ACM tolerates your kind of discourse here.
@RyderRude Like Yang-Mills?
indeed if an idea is pure or applied is dynamic
(Yang-Mills being an example)
but i get ur point of view that the definition of "pure math" evolves over time
it's an interesting point. definitely also worth considering in the discussion
e.g. something i consider applied math may be considered pure math relative to the past
11:13 AM
@RyderRude Once again: Do you have any evidence for these claims? The oldest book that is arguably about equations is Diophantus' Arithmetica and it contains no trace of these "applications"
sure, you can apply this kind of math to stuff, but there is no evidence that the mathematicians of the past were different from the modern ones in that at least for some of them the reason they do math is entirely decoupled from its potential application (consider also the ancient obsession with compass-and-straightedge constructions in geometry when any real application would have been able to use a marked ruler instead of the unmarked straightedge)
11:37 AM
@ACuriousMind yes. the broader point i was making is that application-related math is more sparse now compared not older times. but maybe that has less to do with the nature of the math itself (what i initially thought), and more to do with the breadth of the math (as there is far more amount of math now, so application-related math would be sparse)
@RyderRude but your broader point is still not true! Ancient math consisted of nothing more than algebra and geometry, and I've provided you with evidence that both of these fields were not primarily driven by applications.
another important thing is that people understand the nature of math better today compared to the past, so they are now comfortable with extremely abstract ideas. in the past, people would be arguing over meaningless questions like whether or not irrationals, negatives or complexes really "exist"
@ACuriousMind i was about to comment on the "application" part. what do we define as an application? when i made the cubic equation comment, i wouldve counted "solving cubic equations" in itself as an application. do we only count engineering as applications?
i also commented on Euclid's axioms as being about the real world as "application" even though Euclid's axioms may have been useless for actual applications @ACuriousMind
maybe im defining "real world interpretability" as applied and "non interpretability" as "abstract"
in the past, people had a tendency to relate math to the real world, so they would work on stuff they thought "really existed". the compass-and-straightedge geometry is an example of that
12:27 PM
@ACuriousMind I mean a lot of them thought that the fundamental nature of the universe was numbers
And that such things had probably intrinsic magical properties
But then that points more towards the difficulty of interpreting the work of ancient people :p
They just thought that prime numbers were cool to study because reality was constructed from them in some sense that they were not able to define
Although ancient mathematicians were most certainly aware of some things being without intended applications, ie
> 'Herein', says Proclus, 'I emulate the Pythagoreans who even had a conventional phrase to express what I mean, "a figure and a platform, not a figure and sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among sensible objects and so become subservient to the common needs of this mortal life"
Although this kind of thing is kind of specific to western tradition sort of stuff I get the impression?
Like doing math for its own sake seems to be a weird obsession deriving from Pythagoreanism
Sounds like western theology.
It's hard to really find something equivalent in India or China, or at least I haven't been able to
Well Pythagoreanism was a weird math religion essentially
12:43 PM
It was virtuous to do somethings for their own sake.
@user85795 Was it? Do you have any evidence for that?
what is it with people thinking they can just assert whatever they want about what ancient people must have thought instead of looking at the evidence?
Art.
@Slereah so your assertion is modern math-for-its-own-sake derives from an ancient cult strangely obsessed with beans? I hate that this is the most evidence-backed suggestion so far :P
@ACuriousMind I tried hard to find something equivalent abroad, but slim pickings!
But then again I'm not a Sanskrit scholar, so who knows
The closest I could find was Mohism in China but they lost popularity early on so it never went anywhere
@Slereah I would suggest that people still think there is something "intrinsically magical" about this kind of math, we just don't use the same language anymore ;)
12:47 PM
@Slereah Yeah, the language is a huge barrier.
i would say people only really understood math around the time non euclidean geometry got accepted. then people got comfortable with math unrelated to reality
before that, people didnt know the point of math
They most certainly did (because math is cool)
non euclidean geometry was also met with some resistance
@Slereah they thought about it in terms of magic and stuff
Nah not really
I am just exaggerating for comic effect
There wasn't really a clear distinction between magic and science until the renaissance/modern era
oh :P
12:50 PM
They just thought it was related to "reality" in some sense
Or at least most of them did
It's not like nobody thought "What if math was just imaginary stuff" before the 19th century
It just wasn't a popular position
But yeah you don't really see a good delineation of math and its applications until like Kant and so on
@Slereah I mean, there's plenty of platonists still around that think even the purest math is "real" :P
"real" vs. "invented" is a bit of a different conceptual distinction than "pure" vs. "applied"
yeah but these days it is at least fashionable to claim that it's all Fake and about Models and whatnot
Even if in practice they rarely do :p
Instrumentalism is one of the fashionable abstraction people pretend is Good and Modern and then never do it
1:05 PM
How's the whooping cough?
Still whooping
Most articles about the topic seem to not really explain what the "whooping" part is outside of "a high-pitched whoop sound or gasp may occur as the person breathes in", but as far as I can experience it seems to be struggling to breathe in again as your throat seems to close
Very unpleasant
It's fine
Not a particularly threatening disease if you're not a baby
@Slereah it's cool to pretend to be an existentialist only accepting the subjective meaning we forge for ourselves, it's hard to actually be one
Also spacetime is really just a tablecloth imo
Instrumentalism is one of those idea that sounds cool until you look at people actually doing it
It's basically unworkable in practice because of the wide variety of instrumental devices
It might have been more appealing in the 1930's before widespread complex electronic gizmos :p
You're technically not even allowed to talk about length except as some (probably) equivalence between the measurements by a ruler, laser, theodolite, etc
1:15 PM
is instrumentalism similar to constructivism
Instrumentalism is the practice of favoring physics in terms of the actual physical operations that one does rather than abstracted ideas
instrumentalism is exahausting
platonism is much more practical for physicists to work, even though technically wrong
Some disciplines are more inclined to it than others :p
Typically the ones where they are keenly aware that the instruments are not perfect tools
that is the correct mindset to have. physics is not about rulers lol.. even tho all models are wrong about the universe, the universe is still inherently mathematical
1:57 PM
i currently believe in a philosophical framework where the nature of the universe as experienced by the mind is indeed mathematical. But independent of the mind, it is not mathematical.
anyone know this framework?

4 hours later…
5:49 PM
@Slereah dont worry, most cultures had enough fun with their alchemical mysticisms, often with mathematics imbued into them too. Ever heard of the I-ching? Not even agreed whether it is a cryptographical text, or a historical text, or a calculation text, or a tarot deck, or ...