1,201 karma · joined February 8, 2013
https://base-1-srrnmbhh3rkmnk8ygcxhvb.streamlit.app/
It seems that waking up the demo from the iframe doesn't always work, but directly visiting the embed url can wake it up.
By the way I just finished completely rewriting the entire blog post.
starting with 3=1+2 we have (1+x)P(x)=3P(x) when x=2. so we lift the problem from n to P(2)=n. this is a known technique of lifting the problem to a polynomial setting. after each itteration of the Collatz map i make sure all coeffients are either 0 or 1 by applying carry operations when a coefficient overflows. since the coefficients are unary strings, this makes it like a fluid dynamics problem (each character in a unary string is analogous to one unit of mass in a list of buckets where the buckets can overflow and spill unary characters over into their left neihgbor)
when x=2, multiplying x by P(x) is a left shift, whereas dividing by x, P(x)/x, is a right shift. (when P(2)=n is even the constant term in P(x) is zero)
the +1 term in 3n+1 effectively induces a non linear carry propoagation.
the new technique i used is based on a realization that the polynomial representation of the Collatz map behaves like an LFSR implementation of a finite field with a missing modulus. in LFSR a finite field is implemented where each element is an array of bits of fixed size corresponding to a polynomial and multiplication of elements is polynomial multiplication taken mod Q(x) where Q(x) is an irreducible polynomial. unlike the finite field LFSR the Collatz map in polynomial form as i have described allows the degree of the polynomial (size of the array of bits) to grow unbounded.
the surprise is when i subtract these two objects the sierpinski gasket appears and this fractal is not destroyed by itterations of the collatz map
this document[1] is a prior result showing a connection between fractals and collatz that i found after posting the OP
[1] https://upcommons.upc.edu/server/api/core/bitstreams/9bad675...
lesson learned! i will never post an ai slop blog post on here ever again. thanks for the feedback i needed to hear it.
there is more to this post than just ai slop. there is a real experimental result here.
if you or anyone would like to see the non ai slop version i posted over on math stack exchange without any ai at all
https://math.stackexchange.com/questions/5121753/why-does-th...
I recently prompted Gemini Deep Research to “solve the Riemann Hypothesis” using a specific strategy and it just lied and fabricated the result of a theorem in its output, which otherwise looked very professional.
i believe it makes a substantial difference. the reason is that a short query contains a small number of tokens, whereas a large “wall of text” contains a very large number of tokens.
I strongly suspect that a large wall of text implicitly activates the models persona behavior along the lines of the single sentence “if you aren't sure of the answer say you don't know” but the lengthy argument version of that is a form of in-context learning that more effectively constrains the models output because you used more tokens.
> Is there any way to deduce which invariance gives which conservation?
Yes. See Landau vol 1 chapter 2 [1].
> I'm looking for the fundamental reason, as well as how to tell what will be paired with some invariance when looking at some other new invariance
I'm not sure there is such a "fundamental reason", since energy, momentum, and angular momentum are by definition the names we give to the conserved quantities associated with time, translation, and rotation.
You are asking "how to tell what will be paired with some invariance" but this is not at all obvious in the case of conservation of charge, which is related to the fact that the results of measurements do not change when all the wavefunctions are shifted by a global phase factor (which in general can depend on position).
I am not aware of any way to guess or understand which invariance is tied to which conserved quantity other than just calculating it out, at least not in a way that is intuitive to me.
[1] https://ia803206.us.archive.org/4/items/landau-and-lifshitz-...
A neural network is a type of machine that solves non linear optimization problems, and the principle of least action is also a non linear optimization problem that nature solves by some kind of natural law.
This is the one thing that chatgpt mentioned which surpised me the most and which I had not previously considered.
> Eigenvalues of the Hamiltonian in quantum mechanics correspond to energy states. In neural networks, the eigenvalues (principal components) of certain matrices, like the weight matrices in certain layers, can provide information about the dominant features or patterns. The notion of states or dominant features might be loosely analogous between the two domains.
I am skeptical that any conserved quantity besides energy would have a corresponding conserved quantity in ML, and the Reynolds operator will likely be relevant for understanding any correspondence like this.
iirc the Reynolds operator plays an important role in Noethers theorem, and it involves an averaging operation similar to what is described in the linked article.
Nevertheless, a sensible definition for self awareness is some kind of neural network that becomes aware of its own activity and is in some way able to influence its own function.
After considering these issues for a long time, I came to the conclusions that
1. It's impossible for a program running on a normal computer to have self awareness (or consciousness), because those things are essentially on the hardware level and not the software level
2. In order to create a machine that is capable of self awareness (and consciousness) it is necessary to invent a new type of computer chip which is capable of modifying its own electrical structure during operation.
In other words, I believe that a computer program which models a neural network can never be self aware, but that a physical neural network (even if artificially made) can in principle achieve self awareness.
My view is that fingerprinting is a set of tools which can be used for "good or evil" if that makes sense. If you are gathering meta-data to determine the capabilities of the device, then this is part of the wider framework of data points which can, in principle, be used for fingerprinting a user. This data can be imported into a completely different system by a sophisticated adversary, so it needs to be treated as a security vector, imho
Therefore, precautionary measures which respond dynamically to trends detected in newly available data, is the logical, ethical, and scientifically correct thing to do, imo.
One of the many unknowns about these vaccines is the length of time they give immunity, this can only be determined with confidence by looking at the data after a sufficient amount of time has passed. If the immunity only lasts for say 10-20 weeks, then getting it one week early, would mean the immunity goes away one week early as well. So in this case, I'm not sure there would be a major measurable impact. If the immunity lasts for say 50 weeks, that would be a different story.
As opposed to programming languages, where the goal is to do something practical, in pure mathematics, our goal is to create a language capable of helping our brains understand the infinite complexities of nature.
https://math.stackexchange.com/questions/24241/why-do-mathem...
I think this comment really sums up very well what is at the core of our discussion: the future of mathematics and science.
My strong belief is that thousands of years from now, Archimedes and Gauss will still be remembered, and everything we think is great now will be forgotten while they are not. That tells me that they were much farther ahead of their times than us, even though they didn't have modern computers.
Mathematicians and computer scientists both have it totally backwards imo. On the one hand, mathematicians think they have something to teach us about computer science, but they refuse to use technology properly. On the other hand, when we write code, it's all governed by mathematical laws and there are many questions (but maybe not you know, coding standards or the philosophy of writing good code) we could really use the guiding hand of mathematicians with, and they need to catch up with the times and we programmers need to accept they have something valuable to offer and to teach us.
I think our points of view are actually very strongly aligned. However I believe the next big idea is likely to come from outside of computer science.
Personally, I am betting on biology. So many of the most sophisticated techniques are based on biology, e.g. neural nets and genetic algorithms. I have done a lot of work on extending the theory of computation with a new axiom which gives Turing machines a self replicating axiom[1], [2]
In many parts of science, there is a cross pollination, where new ways of thinking about subject X come from a new discovery in subject Y. Typically, research will follow a group think pattern until it hits a brick wall, then you need that really big breakthrough idea. This line of reasoning leads to the conclusion, imo, that it's approximately equally likely to come from either pure computer science, or pure mathematics, or somewhere else.
[1] https://math.stackexchange.com/questions/3605352/what-is-the...
[2] https://medium.com/swlh/self-replicating-computer-programs-8...
I agree with this. I believe pure mathematics is suffering greatly because many mathematicians refuse to fully embrace the computational power of modern technology.
My belief is the age of pretty formulas is coming to an end, and that the future of mathematics will be it focuses more and more on computational aspects of the subject, and problem sets in pure math courses will be done using programs that are much more advanced than anything which exists today, and everyone will think nothing more of those programs than we do about calculators.
Apologies for the self plug, but this has been my vision with mathinspector[1]. I've been working very hard on that, and this is why I got so interested in your statement. Thank you for clarifying your thinking here. Makes sense to me, and you could be right
Same. It makes me kind of sad that pure math is not more helpful here! I think its possible that coding standards today are really high quality, and that any improvements are going to require a revolutionary paradigm shift in how we think about programming at an extreme level of generality. That's where pure math comes in, its a good place to shop around for new ideas, since we are kind of at a local maximum now in terms of best practices, imo.
This is how I use math while coding: Instead of tinkering with numbers, just calculate the thing and get all the numbers right the first time. When trying to solve a problem, it's super helpful to have spent years proving theorems in pure math, it helped me compile programs in my head. When I am suuuuuuper stuck and need to figure out if what I am trying to do has already been done in math somewhere. When constructing an overall plan for a project, using abstract concepts in mathematics to properly anticipate the limits of what various implementations will be in order to pick the best one.
When Faraday was asked about what the practical use of his discovery of electromagnetic induction was, he famously replied, "what is the use of a newborn baby?"
Which is how I read the OP
I don't think this is a fair comparison. The modern computer is really distinct from everything that came before. That's because it was built according to the theory of Turing Machines.
One of the most important historical papers for the development of modern computers was the Report on the ENIAC by Von Neumann[1].
Von Neumann took the idea's of others working in the field, and was able to apply his understanding of mathematical logic to formulate the principles which led to the first working modern computer, which Von Neumann built in the basement of IAS. At that time, there was a major debate at IAS between Einstein and Von Neumann, which centered around whether or not to only do pure math at IAS, with the idea that building a computer was part of experimental science. [2]
> Regarding the arrow of influence: a Fields medalist spent a decade coming up with a new foundation of mathematics (or at least algebraic topology), only to realise that the computer science department already teaches Coq to undergraduates!
LOL! That is an interesting and funny story. However, I don't think this example demonstrates that in the future, mathematics will not be the source of improvements to code writing standards.
Question: if code writing standards improve, where else will these improvements come from other than pure mathematics? I consider this question to be a problem type similar to maximum compression algorithms, i.e. its a question whose solution can only be verified using the language of pure mathematics. Therefore it seems likely these improvements can also have their roots in pure mathematics as well. At least, I would not say it "seems unlikely"
[1] https://en.wikipedia.org/wiki/First_Draft_of_a_Report_on_the...
[2] https://www.amazon.com/Turings-Cathedral-Origins-Digital-Uni...
I strongly disagree with this. To me it does not seem "unlikely", quite the opposite. The reason I feel so strongly about this is that computer science is a subset of pure mathematics, its the part that deals with algorithms and computation. There are many questions in computer science which cannot be answered without an appeal to pure math (e.g., what is the best possible compression algorithm)
All of computer science began as a result of the work of Hilbert on the foundations of logic[1]. This is what led to Turing's paper on Turing Machines. Since that time, mathematicians have done a lot of exciting work on foundations.
Category theory has completely transformed the way we think about abstract mathematical objects; and type theory is very exciting in that it brings mathematics closer to a programming language which is compiled in the human brain and is done with pen and paper.
It's seems to me very likely that in the future, code writing standards will change and get better, and will not remain the same. Towards that end, it makes sense that improvements will come from a better understanding of foundations, since that is where it all started.
Curious for your thoughts?
[1] https://en.wikipedia.org/wiki/Hilbert%27s_second_problem#:~:....
Ever since then, I have considered type theory to be my favorite version of the foundations of math. I have often played around in theoretical foundations theory to discover as many connections as possible between foundations of math and computer science. That being said, I think the perspectives offered by set theory and category theory are extremely valuable, and that these three approaches compliment each other, and sometimes one is better than another for a specific purpose.
This is the first time I had really seen the possibility to reformulate computer science to bring it closer to mathematics. How exciting!
[1] https://math.stackexchange.com/questions/1519330/is-it-possi...
[2] https://medium.com/@calhoun137/a-programmers-way-of-thinking...
Wanted to share this trick with HN because I think it's really cool, and also am not sure if anyone has ever done this before.