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!
---
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.
---
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?
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.
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.
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.
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
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?
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.
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?
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.
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
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 personally think that the split-octonions[1] hold promise for physics. The subset of the split-octonions where the i, j, k, and l components are 0 have a metric that is similar to that of Minkowski space[2].
On the surface I like the idea of the knowledgeable and intelligent having more say, but I do not think it would work in practice. It is misleading to frame it as there being a "correct" weighing out there and that we just need to find it. Any non-equal metric used for weighing an individual's vote will be gamed and exploited. And there is the possibility of creating a class divide of intellectuals if the metric is intelligence.
I believe a safer solution is to overhaul education. This way we get both the equality of democracy and the better decision making of an educated populace.
Are you serious? That's a terrible idea. Every vote should counted equally as everyone should get to represent their own interests. This is the point of democracy. Taking rights away from those you view as "undesirables" is wrong.
I'll expand my own take on this thought, borrowing a mathematical analogy I read recently.
Take the real and complex numbers. The real numbers have the nice property that they are easy to order since they form a line. Given any two you can tell which is larger. When you go up the complex numbers you lose this obvious ordering, you can define a new ordering, but it is no longer trivial since you are comparing points on a plane.
Now, some people see complex numbers as an extension of the real numbers, but it would be more accurate to call them a generalization of the real numbers. The real numbers are a subset of the complex numbers that satisfy the property of having this obvious ordering.
This again extends to the complex numbers and quaternions. A quaternion is a point on a 4D plane that does not necessarily satisfy commutativity, AKA ab != ba. The subset of quaternions that do satisfy commutativity are the complex numbers.
Every time you go up a level you lose an axiom, which is an assumption on how things work for that system. In a sense an axiom is a useful limitation that gives a certain structure to things. So what if there are 0 axioms? This would mean everything is possible, no limitations. 2 + 2 = 5. Obviously everything being possible is not useful.
Now how does this apply to physics? Physical laws are our axioms. But what if there are actually no physical laws, we are just witnessing the subset of "everything" that appears to have structure? If we are only capable of understanding things that are rational, we would be inherently unable to process events that are irrational. We would project this irrational observation down to the rational subset that we can make sense of. Let me finish with an example.
Take a uniform quantum superposition. It has an equally random chance of being measured 0 or 1. There is no hidden information determining this, it is truly random. This is irrational because there seems like there should be a reason for one final outcome being measured over the other due to our familiarity with cause-and-effect, but this ultimately appears to have no cause. We project this phenomena down to the rational and explain it the best we can with quantum states and probability.
Thank you for the feedback. That is a very valid complaint and one that is hard to avoid because the complex numbers are basically derived by trying to solve equations like x^2 + 1 = 0. It may be a bit more geometrically intuitive to describe complex numbers as a system of making 2D vectors work with multiplication and division which I sort of touch on later. The explanation is there to provide a brief overview and support the following visualizations, but I will see if I can improve that section.
This remains my most popular post. I'm very glad about the interest in mathematics it continues to generate!
---
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.
---
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 :)