Oh wow, I wasn't expecting to see this on Hacker News again!
This remains my most popular post. I'm very glad about the interest in mathematics it continues to generate!
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To that one criticism, yes, there is no real "why" to the animations other than I thought they looked cool.
The post is not meant to be comprehensive, or teach anything more than bare basics meant to enjoy the visualizations.
I disagree that math visualizations must have clear pedagogical goals. Math visualizations can be purely exploratory.
The curves the poles trace out over time, are they significant somehow? Perhaps. Perhaps not. That's the exciting part of exploring new concepts. And part of the reason I chose linear over geometric interpolation.
Exploring those curves and alternate interpolations/animations was going to be part two, but it never happened.
I try to make posts accessible to as many people as possible. There is plenty of rigorous content already out there for learning more.
The focus for my blog is exploration and curiosity.
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Perhaps I'll get around to part 2, and make it interactive with a compute shader.
Apologies for the code, it was never meant to be reused. I'm sure you can improve it!
Yeah, I can't confirm it, but the weirdest I think I've seen is in r/science about research that was funded by the meat industry saying that a diet with meat is healthier than vegetarianism or veganism.
The study was poorly done and there were tons of comments pushing the same message: "vegetarians/vegans are annoying hipsters who will lecture you for eating meat and they'll be so deservedly upset by this."
Found it and most of those comments are deleted now (https://redd.it/qskxol). Is the meat industry losing a sizable chunk of profits to more people swearing off meat for moral reasons, or ditching meat as a financial decision?
It's actually very common for subreddits to eventually get owned or controlled by the industry leaders and then begin censorship campaigns. One of the most blatant examples of this is /r/bitcoin where any discussion of alternative scaling strategies except 1MB blocks with Layer 2 on top will earn you a ban.
I don't disagree with you, but Occam's razor is a principle; a guideline. Not a hard law. From the panpsychic perspective all matter is conscious making everything a "subjective observer," which solves the same problem of humans being arbitrary special observers. Unprovable philosophical dead-end? Sure. Woo? Nah.
this is why I said "implies". I fully acknowledge that in principle QM and human minds might be intimately boundm but that, like panpsychism is effectively ipossible to prove or disprove in our current scientific framework.
Thank you. I really like the point you are making, but I am unsure about a lack of cause and effect. When I think of cause and effect, I think of it as taking the state of the universe from state t+0 to t+1 using the laws of physics. All of it being deterministic, except for quantum mechanics. This indeterminism in QM creating the branches allowing one to escape death.
Couldn't believing in quantum immortality be compatible with this? You could do things like firing a gun to your head, and it would misfire every time. You could jump off a building and would land in a passing open-topped garbage truck full of soft material. You'd experience all sorts of crazy coincidences, but all of them plausibly deniable and within the framework of cause and effect as we know it.
With that said, I feel like what you are saying is true but can not fully formulate my thoughts on it yet. I agree completely that you can never have logical certainty of it's truth, but I'm not certain it's contingent on it being true. Perhaps we should instead say that QI is either true or false, but we can never prove which? This removes the implication but still makes sense in the context of Godel's first incompleteness theorem.
I find the whole QI following so weird... Do people really argue that QM can provide cartoon-like effects like a convenient truck under the bridge?
Do people really argue that conscience is so special that the universe refuses to end one?
The universe don't differentiate between a dead human and a living one. From a QM point of view, both body masses, correctly placed, can work as an "observer" to cause waveform collapse.
Well, yes. Those examples were chosen to be cartoonish stereotypes. Extremely statistically unlikely events would look... strange, to say the least. Sort of like the flapping of a butterfly's wings resulting in a hurricane.
Also there has been some discussion about how quantum events would only affect the microscopic and the macroscopic would remain the same. This is false, quantum events can only occur in the microscopic, but their effects can propagate to the macroscopic in ways beyond our comprehension. Take the experiment itself, the outcome of a quantum event propagates to the life or death of a human being.
Lastly there have also been remarks about how death needs to be guaranteed because QI only promises consciousness not health or a lack of suffering. I will address that here too. That is a perfectly fine way of viewing QI but there is a distinction we should formalize to make it clear.
Greedy QI and Perfect QI.
Greedy QI just branches from the present moment to whichever consciousness will persist. This can result in a local optimum where you are maimed but alive.
Perfect QI always picks the best branch point of every present moment to account for all future possibilities. I argue this in assumption #2. Once the bullet has say, pierced the brain stem, there is no greedy choice from there that will save you. Perfect QI likely won't maim you as that would limit your survival possibilities. It's the global optimum.
There is still a problem with this though. What if there is more than one timeline where you always live? How is one chosen? My guess would be either it's randomly chosen, or there is only one timeline. Let me elaborate on the latter with a mathematical analogy.
You have an infinite list of real numbers. These numbers represent different timelines, the value being how long you survive. If you order them, you can always find a larger finite number looking at the next one, but none of them are infinite in value. Since QI assumes immortality not longevity we will assume the limit of this ordered list is infinity and not a finite number. Thus there is one reality in which you live infinitely, and that's the limit of this ordered set. The only chance for approaching this limit is through a mechanism like perfect QI.
"Do people really argue that conscience is so special that the universe refuses to end one?"
QI doesn't protect consciousness. Its argument applies to any quantum state, described by any arbitrarily complicated acceptance function of that state. The QI argument implies that you can pick up a rock, and that there exists some set of universes in which that rock will survive indefinitely, exactly as it is now, to any degree of accuracy you care to specify. It's just the more precise your specification gets, the lower the probability mass is.
That's why in my other message, I say QI only promises consciousness. Technically, there are universes in which by sheer quantum chance you are in fact healthy and happy indefinitely. It's just that "a healthy and happy body" is incomprehensibly more orders of magnitude more unlikely than being stuck in whatever constitutes the minimally conscious body. And in fact, the latter isn't a unique construct either. It's actually a function of the selectivity you apply to the acceptance function; you have the universes in which you are conscious enough to realize your plight, but they're dominated by the universes in which you are too minimally conscious to even realize that, which are in turn dominated by the universes in which you are just not conscious. But those are excluded from your consciousness' future states, which means your consciousness is left over with whatever else is left.
The distinctive thing that consciousness adds to the argument is merely that if you metaphorically drew out the future universes in which your consciousness survives, there is something meaningful (to us, anyhow!) "inside" that set. You can do the same thing to the rock as I described above, but there isn't really anything "inside" that set; it's just the rock and nothing else. Excepting of course that vanishing fraction of the universes in which quantum processes drive the rock to become conscious by any definition of your choice, of course.
(Another thing I'd point out is that for this discussion, I have no concern about what your definition of "consciousness" is. The argument works regardless of your definition, it just tweaks the exact non-zero numbers that come out.)
Why should it bring more comfort than knowing, e.g. that there are alternative universes where my deceased loved ones are alive?
There seems to be a belief that our consciousness will "jump" to an alive alternative. I think that's a strong misconception of what many-worlds interpretation says.
Talking about misconceptions, without having more than superficial background in QM, I doubt that being able to consider even very unlikely outcomes allows you to discard macroscopic causality altogether, which QI basically requires.
"Why should it bring more comfort than knowing, e.g. that there are alternative universes where my deceased loved ones are alive?"
I'm not sure you're reading my posts for what they say, rather than what you expect them to say. I think QI is horrible, as in, more suited to horror stories, than something that offers comfort. Nor do I discuss any sort of "jumping".
"Couldn't believing in quantum immortality be compatible with this? You could do things like firing a gun to your head, and it would misfire every time. You could jump off a building and would land in a passing open-topped garbage truck full of soft material. You'd experience all sorts of crazy coincidences, but all of them plausibly deniable and within the framework of cause and effect as we know it."
I think one of the several issues I have with this article is that you're measuring survival in human terms, not quantum terms. One of the secret constraints in the quantum suicide thought experiment is that your method of death must be literally 100% guaranteed, and ideally for the argument, instantaneous from the point of view of your consciousness. (That is, for our consciousness's sake, I don't think Planck time is particularly relevant; more human time frames are relevant.) Sometimes this is expressed as sitting next to an atom bomb rather than just a gun.
But, even then, as a pragmatic matter, a real person trying to perform this experiment would need to be concerned about the quantum probability that the bomb does indeed go off, but leaves them maimed because it didn't exactly explode fully, but the explosives half-detonate and the uranium goes only a bit critical, spraying you with radiation and seriously injuring you but not actually killing you. As you sit there next to your quantum number generator, you are indeed systematically pruning the quantum state space of the worlds in you do actually die, but the remaining quantum state space has a great deal more than just "you get up and turn off the quantum number generator once you are satisfied".
Similarly, a gun can do a lot more than just fire and kill you, or not fire and leave you harmless. It could fire, and put what should be a fatal hole in your head, but the bare minimum quantum events occur to keep you just barely conscious.
As you prune away all the likelihoods, you're amplifying the probably of everything other than what you pruned away, not just the human-conceivable events. The harder and more successfully you prune, the weirder what's left over gets. Given that you've set up a situation in which you've got a gun aimed at your head or an atom bomb or what have you, those "weird" situations are likely to be unpleasant.
Of course, if QI and the multiple worlds hypothesis is correct, this all happens anyhow all the time, and seeing our family or friends just suddenly explode for no reason is something we are literally experiencing all the time, just with such a low probability that it hasn't happened yet. But, somewhere in the great multiverse, there's a set of people who read this message and it is immediately followed by, say, their left hand just disintegrating. Take heart... for some yet smaller fraction of you, it will also immediately fix itself.
But there must be a mechanism for the immortality shouldn't there be? Let's go back to the gun example quickly. (assuming a perfect gun, no misfires, or lucky non-fatal shots)
If quantum immortality holds and the experiment is run 8 times, you should see the sequence 00000000 right?
This is a message too, because it deviates from expectation to shovel you down a reality you can experience. Sort of like Zipf's law where the less likely the result the more information it contains https://en.wikipedia.org/wiki/Zipf%27s_law#Applications
My reasoning behind this was that there can be multiple futures where you could exist and experience, but some of them shorter than others. E.g.
A------B
\ --------C
If you are travelling towards B, once you reach it, you die. But you can't magically jump last second to parallel reality C, you can only branch from the present. Thus you would have needed to branch down to C in advance. Thanks for pointing this out, I will make my reasoning more clear in a future edit.
2. Getting shot in the head isn't always fatal. The bullet could deflect off your skull, or even go through your brain and just not hit you. The bullet could experience spontaneous existence failure, or simply fail to interact with the matter in your body and pass through you. My limited understanding of physics says that things like these are possible but massively unlikely at large scale. However, if some set of things are possible but rare, and the only alternative things are impossible, then something rare will happen.
True, the doctor example is really just supposed to be segue into thinking about how a direct message could work. Although, perhaps facing death and recovering turns you into the kind of fellow who will live their life in a vastly different way leading to a longer life?
This is something I very recently was wondering existed. I was thinking of creating a vector logo for a project, and was curious if vector software was capable of defining relationships like you have done here. It would allow for easily determining what ratios between elements look most visually appealing. I'll definitely take a look at this and let you know if I have any feedback.
Is there actually no other existing tools that have a similar functionality?
I have done a similar tool but based on geometric relations instead of constraints, you can see links in my profile (it's only on mac app store at the moment)
Only because you have chosen for the learning rate to decrease over time so the model converges. You could reset it.
I would even argue the LR is not relevant to plasticity, it's a meta-variable for training the model and not to do with the model itself.
A plastic model would be one that could accomplish a given task after being trained on it, and then trained on a second task while maintaining the ability to accomplish the first task.
Use non spatial dimensions as well, like color, size, and shape. You can also use clever projections like t-sne to reduce the number of dimensions while retaining some of the structure of the data (https://www.youtube.com/watch?v=RJVL80Gg3lA)
Tangential but I've always thought it may be possible for a human brain to experience higher spatial dimensions, it's just that all it ever experiences is 3. Imagine a video game is designed in 4 spatial dimensions and is fed directly into the brain through a neuralink type interface. The brain is pretty adaptable, would the recipient correctly experience those 4 dimensions?
I've seen more than three dimensions with LSD. At least, I've seen what looks a lot like those rotating hypercube images. Except more complicated, and looking somewhat like Mandelbrot sets.
Extraordinary claims require extraordinary evidence. With your mind in a suggestible state, it's much easier to convince yourself that you're seeing in 4D than is to actually see in 4D.
It's pretty easy to work out what they actually saw from the words used to describe them. A hypercube rotation is a 4d object projected into 3 dimensions. Nothing extraordinary at all about that.
Yes, it was a visual hallucination, and so constrained to two dimensions plus perspective. But I clearly recall the sense that I was seeing a 2D projection of a higher-dimensional image.
Yeah, visualizing 2D/3D shapes is not extraordinary. "The sense" of an extra dimension is meaningless woo, unless someone with sufficient mathematical experience is able to observe consistent properties of a shape's higher-dimensional structure.
Obviously anything experiential can be dismissed as "meaningless woo", but as someone who's done a fair amount of LSD and is a lifelong skeptic, it's an unmistakable experience that the hallucinations are taking place in a higher ordered space.
The process of hallucinating on LSD feels like you're sliding a window through a higher dimensional structure projected onto a 2D plane. LSD hallucinations aren't "3D" in the same way that mushroom hallucinations are. LSD feels like infinite 2D planes stacked on top of each other, rather than a 3D, if that makes any sense.
If you pause to meditate and quiet your mind, you can get the hallucinations to briefly stall or at least retain their current character. If you focus on one area and let your mind explore the hallucinating you're looking at and just sort of "let it happen", or perhaps change music/environments/stimulus, you can push the hallucinations in a new direction.
The 2D nature of the hallucinations and the lens-like focusing mechanic does give the impression that you're viewing a projection of a higher dimensional space.
I understand it sounds like woo, but the experience is remarkably similar to the feeling of playing with something like http://4dtoys.com/ and I can very much understand what mirimir is describing.
I don't have any science-based understanding of how the brain works, but maybe I've added enough color that you can understand what the experience feels like.
I hope some day to encounter a situation where a non-psychedelic-experienced person lecturing the experienced about how silly they are agrees to "put their money where their mouth is" so to speak and actually try it out for themselves, and then return to the previous argument and reflect upon it with the new insight they've gained.
Of course they wouldn't suddenly "get it" entirely like in the movies, but I think it would have to be somewhat interesting to see their reaction to going from absolute confidence in their perceptions and knowledge, to.....well, whatever you call the state one is in after having had such experiences. I know for me, even being fairly experienced, returning to a psychedelic state is rather shocking....it seems like it is literally impossible to store even remotely proper memories of it in one's mind, at least mine. Ineffable doesn't seem like the right word for it.
I have a lot of meditation experience, and deep trance states are also extremely hard to hold on to. Meditating on the phenomena has revealed that it's a combination of unfamiliarity with the subject matter along with the sheer complexity of the experience that leads to inability to remember all the details.
People also tend to forget that ordinary life is crazy complicated in its own right. We don't have trouble remembering it because our brains have tuned itself to compress and operate on those kinds of experiences symbolically.
But since we don't have appropriate symbolic representation for 'deep experience' it gets shoved out of mind before we can form memories.
> But since we don't have appropriate symbolic representation, our (or, most people's at least) mind is literally unable to store the experience.
This is my armchair theory on it. I imagine someone smarter than I has speculated on it, but I've yet to encounter anything on the subject.
I'd also like to read anything on comparisons of the two from people who are experienced in both, although I suspect interactions between the two may skew things a bit.
I'm not a fan of the "you have to experience it to understand it" mantra.
I understand experiencing something yourself can be a very easy/efficient way for humans to learn. But the ability to transfer knowledge and understanding in other ways even for complicated thoughts is one of the things that sets us apart from lesser beings on this planet.
Don't get me wrong, I understand there are effects. If alcohol weakens myelin and thus boosts unusual interactions between neurons (my layman understanding based on little education on the topic) it might be a good tool to boost creativity, within reason and with a bunch of asterisks attached. Meditation helping you to understand and control your mental state sounds very useful.
But yeah, for psychedelic I've not yet encountered a convincing elevator pitch of it providing lasting benefits beyond being an interesting experience. At least benefits in areas I consider myself lacking. At the same time I've read enough about the AI control problem that messing around with your thinking should only be done with a lot of caution. And an online personality I follow had a shroom trip, including a limited existential crisis and following panic attacks. Just that seems like a high price to pay, considering I don't see it having made him a better person.
So yeah, I'll hold off on such an experience until I get a plausible lecture from some psychedelic advocate or at least encounter statistical evidence of it being beneficial.
That's a great wiki. I'm talking about 8A/8B. And it's hard to describe, because you forget most of it so fast. What I remember most vividly is close to the image "Abstract by Matt W. Moore", but not so hard edged. More like "Untitled by Luke Brown". Like a bunch of strings of 3D characters, rotating and writhing through higher dimensions. And they moved and changed so quickly that I could never make out what they meant. Plus the fact that I had virtually no memory left.
I really did have the sense that I was somehow seeing my thought process. As the wiki says:
> At the lower end of level 8B geometry, the experience manifests itself as being able to perceive the supposed organization and structure behind one's current conscious thought stream. This is typically presented in the form of a complex, multisensory, and fast-moving network that contains innately understood and relevant geometric representations of specific and abstract concepts. The experience of these innately understandable geometric representations consistently triggers one to visualize and physically feel the concept through highly detailed conceptual thinking.
> At the higher end of level 8B geometry, the effect retains its lower levels but expands itself to include the experience of subjectively perceiving, through innately understandable geometric representations, the architecture of subconscious neurological processes which are usually outside of one's normal daily perception or understanding. These processes are often interpreted to include concepts such as the structure of one's neurology, memories, perspectives, emotions, and general cognitive functions.
It would have to be higher dimensional properties observed by someone who doesn't have mathematical experience relayed to and confirmed by someone who does.
Visually we perceive 2 dimensions (i.e. you can't see behind things), the third is just something we imagine as being closer to or further from the camera without being able to directly perceive it. So maybe the fourth would be the same, but just orthogonal to the third.
You're right that our vision gives us a 3D perception, but only in the same way that a surface in a 3D space may be stretched and squashed and even have discontinuities, but is still fundamentally 2-dimensional, it has no volume. That still doesn't give us volumetric vision.
If you had true 3D/volumetric vision, you would find painting over objects aesthetically pointless, because you see behind not just the paint, but also through every layer and sub-component of the object all at once.
For example, most "3D" video games would seem very strangely empty and hollow to a being that could perceive volumetric space directly, because games are implemented by manipulating (many) fundamentally 2D surfaces in 3D space. This implementation technique works to suspend our disbelief because we don't see 3D volumes, we see 3D surfaces.
I would also be very keen to know if the human brain could directly perceive a volumetric space.
Stereoscopic vision doesn't let you see a whole 3D scene at once, it just gives some clues as to how far away things are in the 2D scene which you can perceive visually.
I'd lika to be able to have a vr app that let's me move in 4D space. I guess at least some kind of sensoring for 4D orientation and 4d acceleration should be developed.
> it may be possible for a human brain to experience higher spatial dimensions
The visual system has evolved in a 3D world, in vision science actually the human visual system is considered "2.5D". I don't think we can "visualize" higher dimensions even though we can reason and extrapolate about them just fine.
I've been wanting to make a vr application for a while that lets you navigate around n dimensional structures with ability to control which dimensions you are moving in.
Also the concept of perceptual extension is interesting, feed in a bunch of additional information and present it in a consistent way until the brain begins to incorporate it in it's world model. e.g. add additional senses.
That sounds like it would be very interesting. And while we can't extend our senses just yet, we may be able to swap them. I'm very curious if anyone has tried to use a space-filling Hilbert curve to transform vision into audio and let their brain adjust to it like suggested in this video https://youtu.be/3s7h2MHQtxc
Interesting idea. My understanding is that some of the seeing impaired community does use echo location tools to help navigate. There were also some pin grid tests in the past that effectively gave the seeing impaired the ability to see again. Which is one of the things that got me to thinking about interesting ways we could extend our senses.
This remains my most popular post. I'm very glad about the interest in mathematics it continues to generate!
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To that one criticism, yes, there is no real "why" to the animations other than I thought they looked cool.
The post is not meant to be comprehensive, or teach anything more than bare basics meant to enjoy the visualizations.
I disagree that math visualizations must have clear pedagogical goals. Math visualizations can be purely exploratory.
The curves the poles trace out over time, are they significant somehow? Perhaps. Perhaps not. That's the exciting part of exploring new concepts. And part of the reason I chose linear over geometric interpolation.
Exploring those curves and alternate interpolations/animations was going to be part two, but it never happened.
I try to make posts accessible to as many people as possible. There is plenty of rigorous content already out there for learning more.
The focus for my blog is exploration and curiosity.
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Perhaps I'll get around to part 2, and make it interactive with a compute shader.
Apologies for the code, it was never meant to be reused. I'm sure you can improve it!
Thank you for reading :)