All of Joseph Van Name's Comments + Replies

 I am going to share an algorithm that I came up with that tends to produce the same result when we run it multiple times with a different initialization. The iteration is not even guaranteed convergence since we are not using gradient ascent, but it typically converges as long as the algorithm is given a reasonable input. This suggests that the algorithm behaves mathematically and may be useful for things such as quantum error correction. After analyzing the algorithm, I shall use the algorithm to solve a computational problem.

We say that an algorith... (read more)

I would have thought that a fitness function that is maximized using something other than gradient ascent and which can solve NP-complete problems at least in the average case would be worth reading since that means that it can perform well on some tasks but it also behaves mathematically in a way that is needed for interpretability. The quality of the content is inversely proportional to the number of views since people don't think the same way as I do.

Wheels on the Bus | @CoComelon Nursery Rhymes & Kids Songs

Stuff that is popular is usually garbage.

B... (read more)

In this post, the existence of a non-gradient based algorithm for computing LSRDRs is a sign that LSRDRs behave mathematically and are quite interpretable. Gradient ascent is a general purpose optimization algorithm that works in the case when there is no other way to solve the optimization problem, but when there are multiple ways of obtaining a solution to an optimization problem, the optimization problem is behaving in a way that should be appealing to mathematicians.

LSRDRs and similar algorithms are pseudodeterministic in the sense that if we train the... (read more)

1Logan Riggs
That does clarify a lot of things for me, thanks! Looking at your posts, there’s no hooks or trying to sell your work, which is a shame cause LSRDR’s seem useful. Since they are you useful, you should be able to show it. For example, you trained an LSRDR for text embedding, which you could show at the beginning of the post. Then showing the cool properties of pseudo-determinism & lack of noise compared to NN’s. THEN all the maths. So the math folks know if the post is worth their time, and the non-math folks can upvote and share with their mathy friends.   I am assuming that you care about [engagement, useful feedback, connections to other work, possible collaborators] here. If not, then sorry for the unwanted advice! I’m still a little fuzzy on your work, but possible related papers that come to mind are on tensor networks. 1. Compositionality Unlocks Deep Interpretable Models - they efficiently train tensor networks on [harder MNIST], showing approximately equivalent loss to NN’s, and show the inherent interpretability in their model. 2. Tensorization is [Cool essentially] - https://arxiv.org/pdf/2505.20132 - mostly a position and theoretical paper arguing why tensorization is great and what limitations. Im pretty sure both sets of authors here read LW as well.

In this post, we shall describe 3 related fitness functions with discrete domains where the process of maximizing these functions is pseudodeterministic in the sense that if we locally maximize the fitness function multiple times, then we typically attain the same local maximum; this appears to be an important aspect of AI safety. These fitness functions are my own. While these functions are far from deep neural networks, I think they are still related to AI safety since they are closely related to other fitness functions that are locally maximized pseudod... (read more)

This is a post about some of the machine learning algorithms that I have been doing experiments with. These machine learning models behave quite mathematically which seems to be very helpful for AI interpretability and AI safety.

Sequences of matrices generally cannot be approximated by sequences of Hermitian matrices.

Suppose that  are -complex matrices and  are -complex matrices. Then define a mapping  by   for all&nb... (read more)

In this post, I will post some observations that I have made about the octonions that demonstrate that the machine learning algorithms that I have been looking at recently behave mathematically and such machine learning algorithms seem to be highly interpretable. The good behavior of these machine learning algorithms is in part due to the mathematical nature of the octonions and also the compatibility with the octonions and the machine learning algorithm. To be specific, one should think of the octonions as encoding a mixed unitary quantum channel that loo... (read more)

Here are some observations about the kind of fitness functions that I have been running experiments on for AI interpretability. The phenomena that I state in this post are determined experimentally without a rigorous mathematical proof and they only occur some of the time.

Suppose that  is a continuous fitness function. In an ideal universe, we would like for the function  to have just one local maximum. If  has just one local maximum, we say that  is maximized pseudodeterministically (or simply pseudodet... (read more)

This post gives an example of some calculations that I did using my own machine learning algorithm. These calculations work out nicely which indicates that the machine learning algorithm I am using is interpretable (and it is much more interpretable than any neural network would be). These calculations show that one can begin with old mathematical structures and produce new mathematical structures, and it seems feasible to completely automate this process to continue to produce more mathematical structures. The machine learning models that I use are linear... (read more)

It is time for us to interpret some linear machine learning models that I have been working on. These models are linear, but I can generalize these algorithms to produce multilinear models which have stronger capabilities while still behaving mathematically. Since one can stack the layers to make non-linear models, these types of machine learning algorithms seem to have enough performance to be more relevant for AI safety.

Our goal is to transform a list of -matrices  into a new and simplified list of -matrices ... (read more)

7dirk
Your grievance with your former employer seems to me to have little relevance to how would-be college students should plan to spend their time, and even if it had, you haven't shared enough detail for people to judge your report as accurate (assuming this is in fact the case).

Universities are altogether unprofessional, so it is probably best for everyone to shame them and regard the degrees from these universities are completely worthless. Universities promote violence and they refuse to apologize or acknowledge that there is any problem whatsoever.

Since AI interpretability is a big issue for AI safety, let's completely interpret the results of evolutionary computation. 

Disclaimer: This interpretation of the results of AI does not generalize to interpreting deep neural networks. This is a result for interpreting a solution to a very specific problem that is far less complicated than deep learning, and by interpreting, I mean that we iterate a mathematical operation hundreds of times to get an object that is simpler than our original object, so don't get your hopes up too much.

A basis matroid is ... (read more)

I have originally developed a machine learning notion which I call an LSRDR (

-spectral radius dimensionality reduction), and LSRDRs (and similar machine learning models) behave mathematically and they have a high level of interpretability which should be good for AI safety. Here, I am giving an example of how LSRDRs behave mathematically and how one can get the most out of interpreting an LSRDR.

Suppose that  is a natural number. Let  denote the quantum channel that takes an  qubit quantum state and selects one of those ... (read more)

In this note, I will continue to demonstrate not only the ways in which LSRDRs (-spectral radius dimensionality reduction) are mathematical but also how one can get the most out of LSRDRs. LSRDRs are one of the types of machine learning that I have been working on, and LSRDRs have characteristics that tell us that LSRDRs are often inherently interpretable which should be good for AI safety.

Suppose that  is the quantum channel that maps a  qubit state to a  qubit state where we select one of the 6 qubits at random and se... (read more)

I personally like my machine learning algorithms to behave mathematically especially when I give them mathematical data. For example, a fitness function with apparently one local maximum value is a mathematical fitness function. It is even more mathematical if one can prove mathematical theorems about such a fitness function or if one can completely describe the local maxima of such a fitness function. It seems like fitness functions that satisfy these mathematical properties are more interpretable than the fitness functions which do not, so people should ... (read more)

Here is an example of what might happen. Suppose that for each , we select a orthonormal basis  of unit vectors for . Let . Then

Then for each quantum channel , by the concavity of the logarithm function (which is the arithmetic-geometric mean inequality), we have 

. Here, equality is reached if and only if 

 for each , but this equality... (read more)

The notion of the linear regression is an interesting machine learning algorithm in the sense that it can be studied mathematically, but the notion of a linear regression is a quite limited machine learning algorithm as most relations are non-linear. In particular, the linear regression does not give us any notion of any uncertainty in the output.

One way to extend the notion of the linear regression to encapsulate uncertainty in the outputs is to regress a function not to a linear transformation mapping vectors to vectors, but to regress the function to a ... (read more)

1Joseph Van Name
Here is an example of what might happen. Suppose that for each uj, we select a orthonormal basis ej,1,…,ej,s of unit vectors for V. Let R={(uj,ej,k):1≤j≤n,1≤k≤s}. Then Then for each quantum channel E, by the concavity of the logarithm function (which is the arithmetic-geometric mean inequality), we have  L(R,E)=∑nj=1∑nk=1−log(E(uju∗j)ej,k,ej,k⟩) ≤∑nj=1−log(∑nk=1⟨E(uju∗j)ej,k,ej,k⟩) =∑nj=1−log(Tr(E)). Here, equality is reached if and only if  E(uju∗j)ej,k,ej,k⟩=E(uju∗j)ej,l,ej,l⟩ for each j,k,l, but this equality can be achieved by the channel defined by E(X)=Tr(X)⋅I/s which is known as the completely depolarizing channel. This is the channel that always takes a quantum state and returns the completely mixed state. On the other hand, the channel E has maximum Choi rank since the Choi representation of E is just the identity function divided by the rank. This example is not unexpected since for each input of R the possible outputs span the entire space V evenly, so one does not have any information about the output from any particular input except that we know that the output could be anything. This example shows that the channels that locally minimize the loss function L(R,E) are the channels that give us a sort of linear regression of R but where this linear regression takes into consideration uncertainty in the output so the regression of a output of a state is a mixed state rather than a pure state.

There are some cases where we have a complete description for the local optima for an optimization problem. This is a case of such an optimization problem. 

Such optimization problems are useful for AI safety since a loss/fitness function where we have a complete description of all local or global optima is a highly interpretable loss/fitness function, and so one should consider using these loss/fitness functions to construct AI algorithms.

Theorem: Suppose that  is a real,complex, or quaternionic -matrix that minimizes the quantity&n... (read more)

I do not care to share much more of my reasoning because I have shared enough and also because there is a reason that I have vowed to no longer discuss except possibly with lots of obfuscation. This discussion that we are having is just convincing me more that the entities here are not the entities I want to have around me at all. It does not do much good to say that the community here is acting well or to question my judgment about this community. It will do good for the people here to act better so that I will naturally have a positive judgment about this community.

-1FireStormOOO
There's a presumption you're open to discussing on a discussion forum, not just grandstanding.  Strong downvoted much of this thread for the amount of my time you've wasted trolling.

You are judging my reasoning without knowing all that went into my reasoning. That is not good.

2FireStormOOO
Again you're saying that without engaging with any of my arguments or giving me any more of your reasoning to consider.  Unless you care to share substantially more of your reasoning, I don't see much point continuing this?

I will work with whatever data I have, and I will make a value judgment based on the information that I have. The fact that Karma relies on very small amounts of information is a testament to a fault of Karma, and that is further evidence of how the people on this site do not want to deal with mathematics.  And the information that I have indicates that there are many people here who are likely to fall for more scams like FTX. Not all of the people here are so bad, but I am making a judgment based on the general atmosphere here. If you do not like my judgment, then the best thing would be to try to do better. If this site has made a mediocre impression on me, then I am not at fault for the mediocrity here.

Let's see whether the notions that I have talked about are sensible mathematical notions for machine learning.

Tensor product-Sometimes data in a neural network has tensor structure. In this case, the weight matrices should be tensor products or tensor sums. Regarding the structure of the data works well with convolutional neural networks, and it should also work well for data with tensor structure to it.

Trace-The trace of a matrix measures how much the matrix maps vectors onto themselves since

 where  follows the multivariat... (read more)

0FireStormOOO
You're still hammering on stuff I never disagreed with in the first place.  In so far as I don't already understand all the math (or math notation) I'd need to follow this, that's a me problem not a you problem, and having a pile of cool papers I want to grok is prime motivation for brushing up on some more math.  I'm definitely not down-voting merely on that. What I'm mostly trying to get across is just how large of a leap of logic you're making from [post got 2 or 3 downvotes] => [everyone here hates math].  There's got to be at least 3 or 4 major inferences there you haven't articulated here and I'm still not sure what you're reacting so strongly to.  Your post with the lowest karma is the first one and it's sitting at neutral, based on a grand total of 3 votes besides yours.  You are definitely sophisticated enough on math to understand the hazards of reasoning from a sample size that small.

Talking about whining and my loss of status is a good way to get me to dislike the LW community and consider them to be anti-intellectuals who fall for garbage like FTX. Do you honestly think the people here should try to interpret large sections of LLMs while simultaneously being afraid of quaternions?

It is better to comment on threads where we are interacting in a more positive manner.

I thought apologizing and recognizing inadequacies was a core rationalist skill. And I thought rationalists were supposed to like mathematics. The lack of mathematical appr... (read more)

0FireStormOOO
Any conversation about karma would necessarily involve talking about what does and doesn't factor into votes, likely both here and in the internet or society at large.  Not thinking we're getting anywhere on that point. I've already said clearly and repeatedly I don't have a problem with math posts and I don't think others do either.  You're not going to get what you want by continuing to straw-man myself and others.  I disagree with your premise you've thus far failed to acknowledge or engage with any of those points.

I usually think of the field of complex numbers algebraically, but one can also think of the real numbers, complex numbers, and quaternions geometrically. The real numbers are good with dealing with 1 dimensional space, and the complex numbers are good for dealing with 2 dimensional space geometrically. While the division ring of quaternions is a 4 dimensional algebra over the field of real numbers, the quaternions are best used for dealing with 3 dimensional space geometrically. 

For example, if  are open subsets of some Euclidean space, ... (read more)

Answer by Joseph Van Name100

Um. If you want to convince a mathematician like Terry Tao to be interested in AI alignment, you will need to present yourself as a reasonably competent mathematician or related expert and actually formulate an AI problem in such a way so that someone like Terry Tao would be interested in it. If you yourself are not interested in the problem, then Terry Tao will not be interested in it either.

Terry Tao is interested in random matrix theory (he wrote the book on it), and random matrix theory is somewhat related to my approach to AI interpretability and alig... (read more)

We can use the spectral radius similarity to measure more complicated similarities between data sets.

Suppose that  are -real matrices and  are -real matrices. Let  denote the spectral radius of  and let  denote the tensor product of  with . Define the -spectral radius by setting , Define the -spectral radius similarity between  and  as

... (read more)

I am curious about your statement that all large neural networks are isomorphic or nearly isomorphic and therefore have identical loss values. This should not be too hard to test.

Let  be training data sets. Let  be neural networks. First train  on  and  on . Then slowly switch the training sets, so that we eventually train both  and  on just . After fully training  and , one should be able to train an isomorphism between the networks  ... (read more)

1RogerDearnaley
I gather node permutation is only one of the symmetries involved, which include both discrete symmetries like permutations and continuous ones such as symmetries involving shifting sets of parameters in ways that produce equivalent network outputs. As I understand it (and I'm still studying this stuff), the prediction from Singular Learning Theory is that there are large sets of local minima, each set internally isomorphic to each other so having the same loss function value (modulo rounding errors or not having quite settled to the optimum). But the prediction of SLT is that there are generically multiple of these sets, whose loss functions are not the same. The ones whose network representation is simplest (i.e. with lowest Kolmogorov complexity when expressed in this NN architecture) will have the largest isometry symmetry group, so are the easiest to find/are most numerous and densely packed in the space of all NN configurations. So we get Occam's Razor for free. However, typically the best ones with the best loss values will be larger/more complex, so harder to locate. That is about my current level of understanding of SLT, but I gather that with a large enough training set and suitable SGD learning metaparameter annealing one can avoid settling in a less good lower-complexity minimum and attempt to find a better one, thus improving your loss function result, and there is some theoretical mathematical understanding of how well one an expect to do based on training set size.

I have made a few minor and mostly cosmetic edits to the post about the dimensionality reduction of tensors that produces so many trace free matrices and also to the post about using LSRDRs to solve a combinatorial graph theory problem.

"What's the problem?"-Neural networks are horribly uninterpretable, so it would be nice if we could use more interpretable AI models or at least better interpretability tools. Neural networks seem to include a lot of random information, so it would be good to use AI models that do not include so much random information. Do y... (read more)

2FireStormOOO
Wouldn't be engaging at all if I didn't think there was some truth to what you're saying about the math being important and folks needing to be persuaded to "take their medicine" as it were and use some rigor.  You are not the first person to make such an observation and you can find posts on point from several established/respected members of the community. That said, I think "convincing people to take their medicine" mostly looks like those answers you gave just being at the intro of the post(s) by default (and/or the intro to the series if that makes more sense).  Alongside other misc readability improvements.  Might also try tagging the title as [math heavy] or some such. I think you're taking too narrow a view on what sorts of things people vote on and thus what sort of signal karma is.  If that theory of mind is wrong, any of the inferences that flow from it are likely wrong too.  Keep in mind also (especially when parsing karma in comments) that anything that parses as whining costs you status even if you're right (not just a LW thing).  And complaining about internet points almost always parses that way. I don't think it necessarily follows that math heavy post got some downvotes therefore everyone hates math and will downvote math in the future.  As opposed to something like people care a lot about readability and about being able to prioritize their reading to the subjects they find relevant, neither of which scores well if the post is math to the exclusion of all else. I didn't find any of those answers surprising but it's an interesting line of inquiry all the same.  I don't have a good sense of how it's simultaneously true that LLMs keep finding it helpful to make everything bigger, but also large sections of the model don't seem to do anything useful, and increasingly so in the largest models.

I would go further than this. Future architectures will not only be designed for improved performance, but they will be (hopefully) increasingly designed to optimize safety and interpretability as well, so they will likely be much different than the architectures we see today. It seems to me (this is my personal opinion based on my own research for cryptocurrency technologies, so my opinion does not match anyone else's opinion) that non-neural network machine learning models (but which are probably still trained by moving in the direction of a vector field... (read more)

The -spectral radius similarity is not transitive. Suppose that  are -matrices and  are real -matrices. Then define . Then the generalized Cauchy-Schwarz inequality is satisfied:

.

We therefore define the -spectral radius similarity between  and  as . One should think of the -spectral radius similarity as a gene... (read more)

I appreciate your input.  I plan on making more posts like this one with a similar level of technical depth. Since I included a proof with this post, this post contained a bit more mathematics than usual. With that being said, others have stated that I should be aware of the mathematical prerequisites for posts like this, so I will keep the mathematical prerequisites in mind.

Here are some more technical thoughts about this.

  1. We would all agree that the problem of machine learning interpretability is a quite difficult problem; I believe that the solution
... (read more)

If you have any questions about the notation or definitions that I have used, you should ask about it in the mathematical posts that I have made and not here. Talking about it here is unhelpful, condescending, and it just shows that you did not even attempt to read my posts. That will not win you any favors with me or with anyone who cares about decency. 

Karma is not only imperfect, but Karma has absolutely no relevance whatsoever because Karma can only be as good as the community here.

P.S. Asking a question about the notation does not even signify an... (read more)

2FireStormOOO
I did go pull up a couple of your posts as that much is a fair critique: That first post is only the middle section of what would already be a dense post and is missing the motivating "what's the problem?", "what does this get us?"; without understanding substantially all of the math and spending hours I don't think I could even ask anything meaningful.  That first post in particular is suffering from an approachable-ish sounding title then wall of math, so you're getting laypeople who expected to at least get an intro paragraph for their trouble. The August 19th post piqued my interest substantially more on account of including intro and summary sections, and enough text to let me follow along only understanding part of the math.  A key feature of good math text is I should be able to gloss over challenging proofs on a first pass, take your word for it, and still get something out of it.  Definitely don't lose the rigor, but have mercy on those of us not cut out for a math PhD.  If you had specific toy examples your were playing with while figuring out the post, those can also help make posts more aproachable.  That post seemed well received just not viewed much; my money says the title is scaring off everyone but the full time researchers (which I'm not, I'm in software). I think I and most other interested members not in the field default to staying out of the way when people open up with a wall of post-grad math or something that otherwise looks like a research paper, unless specifically invited to chime in.  And then same story with meta; this whole thread is something most folks aren't going to start under your post uninvited, especially when you didn't solicit this flavor of feedback. I bring up notation specifically as the software crowd is very well represented here, and frequently learn advanced math concepts without bothering to learn any of the notation common in math texts.  So not like, 1 or 2 notation questions, but more like you can have people w

I am pointing out something wrong with the community here. The name of this site is LessWrong. On this site, it is better to acknowledge wrongdoing so that the people here do not fall into traps like FTX again. If you read the article, you would know that it is better to acknowledge wrongdoing or a community weakness than to double down.

2FireStormOOO
It is still a forum, all the usual norms about avoid off-topic, don't hijack threads apply.  Perhaps a Q&A on how to get more engagement with math-heavy posts would be more constructive?  Speaking just for myself, a cheat-sheet on notation would do wonders. Nobody is under any illusions that karma is perfect AFAICT, though much discussion has already been had on to what extent it just mirrors the flaws in people's underlying rating choices.

I already did that. But it seems like the people here simply do not want to get into much mathematics regardless of how closely related to interpretability it is. 

 P.S. If anyone wants me to apply my techniques to GPT, I would much rather see the embedding spaces as more organized objects. I cannot deal very well with words that are represented as vectors of length 4096 very well. I would rather deal with words that are represented as 64 by 64 matrices (or with some other dimensions). If we want better interpretability, the data needs to be structured in a more organized fashion so that it is easier to apply interpretability tools to the data.

"Lesswrong has a convenient numerical proxy-metric of social status: site karma."-As long as I get massive downvotes for talking correctly about mathematics and using it to create interpretable AI systems, we should all regard karma as a joke. Karma can only be as good as the community here.

3Algon
If I could give some advice: Show that you can do something interesting from an interpretability perspective using your methodology[1], rather than something interesting from a mathematical perspective.  By "something interesting form an interpretability perspective" I mean things like explaining some of the strange goings on in GPT-embedding spaces. Or looking at some weird regularity in AI systems and pointing out that it isn't obviously explained by other theories but explained by some theory you have/ 1. ^ Presumably there is some intuition, some angle of attack, some perspective that's driving all of that mathematics. Frankly, I'd rather hear what that intuition is before reading a whole bunch of mathematics I frankly haven't used in a while. 
4Simon Fischer
(I downvoted your comment because it's just complaining about downvotes to unrelated comments/posts and not meaningfully engaging with the topic at hand)

Let's compute some inner products and gradients.

Set up: Let  denote either the field of real or the field of complex numbers. Suppose that  are positive integers. Let  be a sequence of positive integers with . Suppose that  is an -matrix whenever . Then from the matrices , we can define a -tensor . I have been doing computer experiments where I use this tensor to approximate other tensors by minimizing the... (read more)

So in my research into machine learning algorithms, I have stumbled upon a dimensionality reduction algorithm for tensors, and my computer experiments have so far yielded interesting results. I am not sure that this dimensionality reduction is new, but I plan on generalizing this dimensionality reduction to more complicated constructions that I am pretty sure are new and am confident would work well.

Suppose that  is either the field of real numbers or the field of complex numbers. Suppose that  are positive integers and ... (read more)

So in my research into machine learning algorithms that I can use to evaluate small block ciphers for cryptocurrency technologies, I have just stumbled upon a dimensionality reduction for tensors in tensor products of inner product spaces that according to my computer experiments exists, is unique, and which reduces a real tensor to another real tensor even when the underlying field is the field of complex numbers. I would not be too surprised if someone else came up with this tensor dimensionality reduction before since it has a rather simple description ... (read more)

Thanks for pointing that out. I have corrected the typo.  I simply used the symbol  for two different quantities, but now the probability is denoted by the symbol .

Every entry in a matrix counts for the -spectral radius similarity. Suppose that  are real -matrices. Set . Define the -spectral radius similarity between  and  to be the number

. Then the -spectral radius similarity is always a real number in the interval , so one can think of the -spectral radius similarity as a generalization of the value  where &nbs... (read more)

2Algon
Your notation is confusing me. If r is the size of the list of matrices, then how can you have a probability of 1-r for r>=2? Maybe you mean 1-1/r and sqrt{1/r} instead of 1-r and sqrt{r} respectively?

The problem of unlearning would be solved (or kind of solved) if we just used machine learning models that optimize fitness functions that always converged to the same local optimum regardless of the initial conditions (pseudodeterministic training) or at least has very few local optima. But this means that we will have to use something other than neural networks for this and instead use something that behaves much more mathematically. Here the difficulty is to construct pseudodeterministically trained machine learning models that can perform fancy tasks a... (read more)

I think that all that happened here was the matrices  just ended up being diagonal matrices. This means that this is probably an uninteresting observation in this case, but I need to do more tests before commenting any further.

Suppose that  are natural numbers. Let . Let  be a complex number whenever . Let  be the fitness function defined by letting . Here,  denotes the spectral radius of a matrix  while  denotes the Schatten -norm of .

Now suppose that  is a tuple that maximizes . Let  be the fitness functio... (read more)

3Joseph Van Name
I think that all that happened here was the matrices A1,…,Ar just ended up being diagonal matrices. This means that this is probably an uninteresting observation in this case, but I need to do more tests before commenting any further.

I forgot to mention another source of difficulty in getting the energy efficiency of the computation down to Landauer's limit at the CMB temperature.

Recall that the Stefan Boltzmann equation states that the power being emitted from an object by thermal radiation is equal to . Here,  stands for power,  is the surface area of the object,  is the emissivity of the object ( is a real number with ), is the temperature, and  is the Stefan-Boltzmann constant. Here, ... (read more)

This post uses the highly questionable assumption that we will be able to produce a Dyson sphere that can maintain a temperature at the level of the cosmic microwave background before we will be able to use energy efficient reversible computation to perform operations that cost much less than  energy. And this post also makes the assumption that we will achieve computation at the level of about  per bit deletion before we will be able to achieve reversible computation. And it gets difficult to overcome thermal noise at an en... (read more)

1William the Kiwi
This post makes a range of assumptions, and looks at what is possible rather than what is feasible. You are correct that this post is attempting to approximate the computational power of a Dyson sphere and compare this to the approximation of the computational power of all humans alive. After posting this, the author has been made aware that there are multiple ways to break the Landauer Limit. I agree that these calculations may be off by an order of magnitude, but this being true doesn't break the conclusion that "the limit of computation, and therefore intelligence, is far above all humans combined".

Let \(X,Y\) be topological spaces. Then a function \(f:X\rightarrow Y\) is continuous if and only if whenever \((x_d)_{d\in D}\) is a net that converges to the point \(x\), the net \((f(x_d))_{d\in D}\) also converges to the point \(f(x)\). This is not very hard to prove. This means that we do not have to discuss as to whether continuity should be defined in terms of open sets instead of limits because both notions apply to all topological spaces. If anything, one should define continuity in terms of closed... (read more)

I have heard of filters and ultrafilters, but I have never heard of anyone calling any sort of filter a hyperfilter. Perhaps it is because the ultrafilters are used to make fields of hyperreal numbers, so we can blame this on the terminology. Similarly, the uniform spaces where the hyperspace is complete are called supercomplete instead of hypercomplete.

But the reason why we need to use a filter instead of a collection of sets is that we need to obtain an equivalence relation.

Suppose that  is an index set and  is a set with ... (read more)

1Valdes
Oops, my bad. I re-read the post as I was typing to make sure I hadn't missed any explanation. That can sometimes cause me to type what I read instead of what I intended. I probably interverted the prefixes because they feel similar. Thank you for the math. I am not sure everything is right with your notations in the second half, it seems to me there must be a typo either for the intersection case or the superset one. But the ideas are clear enough to let me complete the proof.
Load More