4denthusiast

[4denthusiast]

Things are looking dire in the "4D" tag. Some of us are looking for actual maths content, not superstitious nonsense.

[4denthusiast]

I made a little graphics demo which I'm quite proud of so I felt like sharing. It generates a 4D mesh from 4D voxel data using the dual contouring method, then renders a rotating 3D slice. The live version is animated, but here's a screenshot of what it looks like.

[4denthusiast]

New theory of consciousness: panpsychism used to be true, but sapient brains are so much more attractive to souls than any other sort of matter that everything else stopped having conscious experiences when humans evolved, at least in the vicinity of Earth. Fresh souls continue to slowly accrete onto the planet from ever more distant regions of the universe, allowing us to keep up with population growth.

[4denthusiast]

older lesbian at a bar: [lights cigarete] i was your age when you were your age isnt that weird

other lesbian the same age as her: can yuo put that out on me

her: i just lit it though

[4denthusiast]

To elaborate on why every homeomorphism is given by the right action of some element: Take a finite boring space S of size n. There are n homeomorphisms given by the right action of elements of S on the space. This gives an injection S -> Aut(S) (where Aut here is topological automorphisms, not just topological group automorphisms). A and Aut(S) are finite and equinumerous (because S is boring), therefore this is a bijection, i.e. every homeomorphism on S is given by the right action of some element of S.

For abelianness, let the identity be e and take some element g. The left action of g on S is a homeomorphism, therefore it is the right action of some element h on S. eh = ge, therefore h=g, and the right action of g is equal to its left action. Since g was arbitrary, S is abelian.

I don't think the automorphism group of a circle can be O(2), right? Either you mean topological automorphisms, in which case the automorphism group is infinite-dimensional (I think homeomorphic to (ℝ/ℤ) × (0,1)^ℕ or something like that), or topological group automorphisms of SO(2), in which case it's the group of order 2.

I suspect that SO(2n+1) is isomorphic (via the conjugation map) to its (topological group) isomorphism group. The proof seems kind of fiddly though, so I don't feel like doing it. That still wouldn't make it boring or course, because the topological automorphism group is much larger. For some other Lie groups the symmetries of the Dynkin diagram give outer automorphisms, but the Dynkin diagram for SO(2n+1) has no non-trivial symmetries, as don't Cn, F4, G2, E7 and E8.

Call a space boring if it's homeomorphic to its homeomorphism group. Call it whoring if it's homotopy equivalent to its homeomorphism group.

(Easy hopefully?) Classify all the boring spaces.

(Probably incredibly hard) Classify all the whoring spaces.

I don't really have an instinct for how many whoring spaces there should be. Probably not many?

[4denthusiast]

Some thoughts on boring spaces:

They're homeomorphic to a group, so we may as well consider them to be equipped with that group structure. The left and right actions of this group's elements on itself are all homeomorphisms, and there are as many homeomorphisms given by the right action of some element on the set as there are elements. Assuming for now the set is finite, every homeomorphism is therefore given by the right action of some element (and similarly for left action, which implies it's abelian, but this fact won't be necessary).

If it can be partitioned into 3 or more non-empty clopen subsets, it can be partitioned into 3 or more non-empty clopen subsets which are homeomorphic to each other, and a homeomorphism of the space as a whole can be constructed swapping two of the subsets which don't contain the identity, therefore there is a homeomorphism not given by the group action, which is a contradiction, therefore the space can be partitioned into at most 2 non-empty clopen subsets.

If there is a partition into 2 non-trivial clopen subsets, each containing more than one element, then applying a non-identity element of the identity component to only the identity component, and leaving the other one unchanged, gives a homeomorphism other than the group action, which is a contradiction, therefore if the space has 2 non-trivial clopen subsets, each of them contains only a single element. The only disconnected finite boring space is therefore the discrete topology on 2 points.

If on the other hand the space is connected, take the intersection X of all the open sets containing the identity. Assume there is an element g not in X, and further assume there is h∈X ∩ gX, then X ∩h⁻¹gX is smaller than X and still contains the identity, which is a contradiction. X ∩ gX is therefore empty for all g ∉X, therefore the space is not connected, another contradiction, so there is no such g and the space is indiscrete. Every bijection from an indiscrete space to itself is a homeomorphism, therefore if the space is indiscrete it has at most 2 elements (because n! > n for n > 2).

There are only 3 spaces meeting these constraints, and all of them are indeed boring: the singleton, the discrete space of 2 points, and the indiscrete space of 2 points. These are therefore all of the finite boring spaces.

As for infinite boring spaces, I don't have a classification but I do have a few examples. ℤ, with [n,∞) as its open sets, is a boring space, as are I believe at least all finite powers of this space. These are abelian topological groups all of whose homeomorphisms are given by the group action, but the proof used for the finite case fails because the fact that X was finite was implicitly used to prove that X ∩h⁻¹gX was smaller than it.

I haven't found any more boring spaces yet, but I suspect they exist. As for whether there are any infinite boring spaces that are actually Hausdorff (let alone CW complexes) I wouldn't be surprised if not. It seems like it would be hard to make the points distinguishable enough for an infinite space to be boring without the asymmetry non-T1ness allows.

Call a space boring if it's homeomorphic to its homeomorphism group. Call it whoring if it's homotopy equivalent to its homeomorphism group.

(Easy hopefully?) Classify all the boring spaces.

(Probably incredibly hard) Classify all the whoring spaces.

I don't really have an instinct for how many whoring spaces there should be. Probably not many?

[4denthusiast]

A lot of the times when people complain about bad reading comprehension on here, it seems like it's actually just that people are disagreeing with them. When someone reblogs and says something totally contrary to the spirit of the original post, it's entirely possible they comprehended what it said fine, they just don't think it's true.

[4denthusiast]

#I've got this fictional “magic” system that I really want to talk about but it's really really really weird and also technical tbh

@lilith-hazel-mathematics Do tell.

Non-Narrative Fiction

Fiction, in general, is a detailed depiction of a hypothetical for the purposes of art or entertainment. Fiction is dominated by narrative fiction, sequences of hypothetical events. I don't have anything against narratives, narratives can be great, but I'm interested in what else there is.

If we describe fiction by what field it would fall under if it were non-fiction, we have:

Narrative fiction could be described roughly as fictional biography/history.

Fictional sociology/anthropology/the less personal side of history is the main part of what's generally called "worldbuilding", and is pretty well established. There are even some published books of this sort, such as Gormenghast and some by Ursula K. Le Guin and Kafka.

Fictional geography often overlaps with the previous category, but there are some works of purely physical fictional geography too, such as the excellent Worldbuilding Pasta blog.

Fictional linguistics: those parts of conlanging which are not intended as languages to be used in real life. Many such examples.

Fictional biology (usually called "speculative biology" I think) is moderately commmon. Jay Eaton here on Tumblr and Biblaridion's Alien Biospheres project on Youtube are good examples. There's even a Netflix series on fictional biology, (I didn't watch it because the trailer did not give me the impression it was actually any good, but it exists). Mostly this tends to focus on the high level biology, anatomy and behaviour and stuff.

Greg Egan does fictional physics. This is usually associated with narratives but some of the physics has been published on its own too.

I do fictional chemistry. I know a few other people who talk about this but don't publish.

Fictional maths is still just maths, at least if it's considered in the proper level of detail.

The Codex Seraphinianus is a fictional encyclopedia of sorts, and the Vonyich Manuscript may be too though we don't know its author's intention. These are rather vague and unclear.

Hard magic systems can work as standalone works of fiction (totally soft magic systems not so much because they lack detail). Depending on the style this may be fictional physics. Worldbuilding Notes on Youtube has some examples.

Other artforms like painting and comedy often include little bits of non-narrative fiction, but the emphasis isn't usually on the fiction itself enough that I'd generally consider these interesting examples.

There is of course often a lot of overlap between these, because a large part of the interest of one aspect of the world may be the ways in which it impacts other aspects of the world. Physics influences chemistry, which influences geology and biology in turn. Magic influences society and history. Having any of these things in the background of a narrative is basically the whole genre of speculative fiction.

I'd be interested to hear what other examples of non-narrative fiction people know of, especially in the hard sciences and hard magic, or any fields I haven't listed here.

[4denthusiast]

A while ago I was going through some obscure browser settings or something, and came across one for language. Not for the browser's interface, but to send as a hint to websites. My first language is English but I know a bit of French so I figured I might as well put that as a second preference, in case for some reason a website has multiple languages including French but not English. What actually happened though is that some websites, seeing I've listed English then French, ignore the order and decide I must actually be a French speaker who knows some English, and present the content to me in French instead, even when they do actually have an English version available. I could remove the setting I guess but I just find it kind of funny when this happens.

[4denthusiast]

I think there are a few reasons why the anomaloscope thing isn't obvious when just viewing screens normally. Firstly, depictions on screens of real-life things don't tend to actually match the real colours anyway. There's a reason things like the sky are generally thought of as hard to get good photographs of. Screens are limited both in the range of brightnesses they can produce and the range of chromas (there are a lot of colours that can't be depicted on screens without being desaturated even to normal colour vision). The colours displayed therefore have to be a compromise, with some attempt made to make them roughly right at least relative to each other. Even beyond what's necessary due to technical restrictions, a lot of the time cameras and screens aren't perfectly callibrated anyway. I assume this is easy to subconsciously filter out by the same methods that we filter out lighting conditions to perceive colours of objects.

Even for a camera/screen combination that someone has put effort into callibrating, the particular wavelength dependence of the sensitivity of each of the camera's primaries will almost certainly not actually be the same as that of any human eye, colourblind or not. The camera's anomaloscope results, for various triples of wavelengths, won't in general match a human viewer. It is therefore necessary to compromise again when doing the callibration, as unless the camera actually does have a human-like spectral response, setting which match the colours correctly for one spectrum won't in general match for a different, perceptually equivalent spectrum.

On top of that, I think the anomaloscope thing in colourblindness is just quite subtle. It's certainly way less obvious than the general lack of sensitivity to red/green distinctions (as far as I know, red/blue and green/blue partial colourblindness don't exist). Most people with partial red/green colourblindness (myself included) don't even know if they have deuteranomaly or protanomaly, despite these conditions giving, I believe, opposite anomaloscope readings. Precise matching of perceived colours using different spectra just doesn't come up that much in everyday life.

I was watching a video that touched on the difference between "True Yellow", which is ~580nm, and the "Fake Yellow" that RGB monitors and screens use, which is a combination of different strengths of red plus green (~650nm plus ~540nm). And of course this video was presented to me on a screen that only shows RGB, and it was captured using a camera that only stores RGB, so really it was just Fake Yellow twice for the viewer at home.

But this got me thinking, "wait, are there people who can distinguish Fake Yellow from True Yellow?"

My initial thought was that maybe some forms of colorblindness would allow for it, but everything that I've read seems to say that this isn't the case. Yellow is a combination of activation of L-cones (red) and M-cones (green), and even if your M-cones are more sensitive to yellow light (which can happen) the activation pattern seems like it's going to remain the same.

I'm searching for some way to translate it into mathematics that makes more sense to me, but having trouble with that, and maybe it's simply impossible that with only two cones being activated that they can tell the difference between spectral yellow and monitor yellow. I've certainly failed to find much in the way of reports from colorblind people about the difference between what they see on their monitor and what they see in real life, but there are a fair number of colorblind people who are simply missing cones, rather than having cones that respond to the spectrum differently.

(There are extremely rare tetrachromats who have a fourth cone cell, and they can tell the difference, but that's not what I was looking for.)

[4denthusiast]

#i mean i would also like the US to ditch fptp #but it matters less when you have an effective duopoly

Isn't the fact that the US only has 2 major parties a consequence of FPTP, and also a bad thing? If they switched to another system that didn't disadvantage small parties so much, then sure the other parties would take a while to develop but the country would still benefit eventually.

but canada and britain both desperately need to ditch fptp. it's genuinely nuts to have that as your electoral system in a 21st century parliamentary democracy. even a non-proportional system like single-member districts with like ranked choice voting would be better than this!

[4denthusiast]

Artificial Immaculate Conception

If original sin is specifically transmitted from parent to child by the act of reproduction, it stands to reason that if a human could be created by some other method, they might be entirely free of it. How might we go about this?

IVF and cloning are presumably insufficient. Although they are not natural reproduction, they still generate the child from the parent(s)'s cells.

Synthesizing the genome from a computerized record is better, if perhaps a bit beyond current technology. It would be odd for a digital file to be able to hold a spiritual quality like original sin after all. Still, the person whose genome was scanned to generate the file might reasonably count as the parent, so there's a risk of the original sin being transmitted from them.

If you could reconstruct the genome of someone who already lacked original sin (i.e. Mary, Jesus, or Adam or Eve pre-fall), that should work. There have been claims of various relics that ought to have traces of Jesus's DNA, but it will all be thoroughly degraded by now. You could try the eucharist, but that only works in certain denominations, and even those that claim the real presence of Christ in the eucharist presumably count DNA as the sort of material quality that remains bread and wine like.

Synthesizing a genome which is the average of sufficiently many measured genomes has the quality that a change in any one input isn't actually sufficient to change the child genome, so perhaps none of them quite count as parents. If every input is corrupted by original sin though, perhaps the output still will be too.

With a sufficiently advanced understanding of genetics, it may in principle be possible to create a genome for someone with a basically human phenotype without any direct human inputs. As animals lack original sin, you could use ape genomes as a basis, so you don't have to start from scratch.

So there are difficulties yes, but technical difficulties that as science progresses we might one day overcome. There's one final issue though. This entire enterprise bears quite some resemblance to the fall of man, given it involves knowledge of life and death and good and evil, and attempting to become as God, creating life in your own image. As such, even if you did succeed, it's quite possible you'd just create a fresh new strain of original sin and infect your creation with that instead.

[4denthusiast]

Anyone do any fun sins during the papal interregnum? Invent any interesting heresies?

[4denthusiast]

It’s always funny to me how الله‎ is translated as “Allah” instead of “God” so that intentionally or not (and I suspect more intentionally than not) it ends up sounding like Muslims worship some other, alternative monotheistic God of Abraham. A different one.

Anyway what I actually think we should do is just start doing this for every language. Have news stories talk about how Notre Dame is a cathedral where French Catholics pray to their god, Dieu. There’s a schism between the Russian and Ukrainian churches even though they both worship Bog. Etc.

[4denthusiast]

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[4denthusiast]

Here's a thought experiment related to the previous post. I'm very much not sure of the logic here, so do correct me if you disagree.

What would a universe built on the hyperreals instead of the reals look like? Just take all the laws of physics, and apply the same differential equations over the hyperreals. I'm assuming this is consistent. I don't know the theory of everything so I can't be sure.

It seems likely that the results of any physical experiment can be approximated to any degree of accuracy using an approximation to the laws of physics that's a first-order formula (as you can encode countably many reals, and therefore a continuous function, in a single real). Therefore the experiences of any observer whose coordinates are finite ought to be indistinguishable from the corresponding observer in the real world.

What about observers living infinitely far after the Big Bang? Note that by "infinite" I mean a hyperreal greater than any real number, not the usual sorts of infinities in physics that are at the limit of the system of numbers used. I think the first-order approximation of physics thing fails here because the confidence intervals on the approximation become infinitely wide at infinite times. One possibility is that the heat death occurs at a finite time so there are no observers at infinite times other than Boltzmann brains whose experiences are nonsense not worth interpreting. The other possibility, and I'm not totally sure this makes sense, is that the initial load of negentropy was infinite, the heat death occurs at an infinite time, and there are observers at infinite times who, due to the infinite history leading up to their existence, have the potential to have experiences no finite observer ever would.

In particular, I think it's possible for them to construct infinitely long mathematical proofs and, coming to the end of the proofs, deem them valid. This is assuming that, in addition to the heat death not occurring at a finite time, mathematical research doesn't permanently stall at a finite time either. Perhaps the infinite proofs, being very long, would only be practical as computer-verified proofs, if there's a finite limit to human patience.

Using such proofs, they could look at the evidence for the age of the universe, and think that it's finite. They'd have different mathematics than us, mostly extending ours with their infinite proofs, but also rejecting some theories we might use entirely, having observed that they have contradictions. I don't know if they'd have any reason to suspect that their universe was in some sense more infinite than the mere infinite future and sideways extent the universe normally has.

Even if a proof is extremely long, you'd still be able to count the steps, and if there's a short proof that that number is finite, that should give you more confidence. Maybe this would be enough to tell the difference? It feels vaguely suspicious though. "My computer says P can be proven in 8000 lines of CoC. 8000=4*4*4*5*5*5. 4=SSSS0. 5=S4." would be a short semi-empirical proof of P, but it relies on CoC+soundness(CoC), which is stronger than just CoC. People do maths all the time that's based on theorems whose proofs it would be impractical for them to check.

Given all this, how sure can we be that the age of the universe (or, to avoid cosmological complications, the age of our civilization) is finite? It seems absurd to deny, but as far as I can tell it would still seem absurd to deny even if it were false, so how can we tell? I'm... not really convinced this should be taken seriously. Still, it does make me slightly concerned we can't just ground our ideas of what's finite in the physical universe.

[4denthusiast]

I've just noticed an odd combination of beliefs I hold about how maths works. I'm not exactly convinced that every statement in arithmetic (i.e. statements that can be written in terms of 0, the successor function, =, +, ×, and, not, and quantification over natural numbers) is either true or false. (My previously rather Platonist views were shaken by some university courses on topics like model theory and Gödel's incompleteness theorems. In order to understand that stuff you have to entertain the possibility that certain seemingly obvious things aren't true, and I then sort of never stopped.) At the same time, I accept the law of the excluded middle (that P∨¬P is always true). I generally wouldn't describe myself as an intuitinist, even if I am interested in the applications of some intuitionistic logics in computer science.

I think the way I resolve this apparent contradiction is that the reason I don't feel like all arithmetic statements are true or false is that I'm not sure the natural numbers, as a set, are uniquely defined. Any definition of what is and what isn't in ℕ tends to involve some degree of circularity. "It's 0, and S0, and SS0, and so on.", but "so on" for how many steps? A natural number of steps. Hopefully you see the issue. So then, an arithmetic statement may be true of one model of the naturals, but not another. Within any one model, P∨¬P is true (or, more to the point, (∀x. Px)∨¬(∀x. Px) is true), so if it's true in every model it's true, but we can't ever pin down quite which model we're talking about, so the individual statement (∀x. Px) can remain indeterminate.

All this sort of implicitly relies on a separation of the language and meta-language, even though I didn't set out to have a separate meta-language in the first place. I'm not quite sure whether what I'm thinking here even makes sense. Perhaps what I mean is that the meta-language does have logical connectives (and, or, not), so you can form a claim like "(∀x. Px) is true, or ¬(∀x. Px) is true.", but it doesn't have quantification over the naturals, at least not always, because in the meta-language there isn't a unique ℕ, and you can't specify which one you mean because there's no way to totally pin it down. At least I think. But then the semantics of A∨B is meant to be that A∨B is true iff A is true or B is true. I guess we can still recover this by saying that any statement in the language that includes any quantifiers is implicitly with reference to a particular model of ℕ, and a statement is true iff it's true for all models, but then that requires that the meta-language can quantify over models of ℕ, which should be way less possible than quantifying over individual naturals. I don't know how to resolve this, if it even can be resolved. I'm kind of confused.

The true ℕ, if it exists, ought to be the smallest one of course. The trouble is you can't define "smallest" properly without discussing the whole class, which is a less basic concept than the numbers themselves. Also, not every ordered set or class has a smallest element. I think probably if you allow yourself sufficient expressiveness you can prove that in this case there is a smallest (take the intersection or something), but again I don't think you can prove that without making assumptions at least as strong as the conclusion.

The same thing happens with set theory, but there it all feels clearer. In contrast to the naturals where I'm not sure, I feel somewhat more confident that there isn't a single true set-theoretic universe V. There ought to be sets that can't be named (there are only countably many names after all), which makes the universe much trickier to pin down than the naturals. I know there are countable models of ZFC, but they don't feel like they're the real model, and ZFC is itself kind of vague. It leaves a lot of room for rather natural variation in what sets are allowed (e.g. the continuum hypothesis), while non-standard naturals seem much more exotic. If you assume that there's some particular ℕ that's the real ℕ, in the meta-language, this gives you much more solid foundation to use when talking about potential uncertainty in V. You can do induction, talk about constructible sets and stuff. It seems quite likely that the continuum hypothesis doesn't have a definite truth value, even though CH∨¬CH does, but it feels like quite an ordinary sort of indefiniteness, like "He has brown hair." when it isn't clear who "he" refers to. "Is there a cardinal between ω and 2^ω?" What version of the class of cardinals are you talking about?