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‘Alien calculus’ could save particle physics from infinities
101–110 of 132 posts
Re: ‘Alien calculus’ could save particle physics from infinities
#102Using different math approach is super fruitful. For example, alternating current in presense of capaciators and induction coils is well handled by switching to imaginary (complex) resistance/current/voltage calculation, and transition processes in electric chains are handled by operational calculus. If we used differential equations to solve these, which indeed look natural for these tasks, we will not be able to ac…
I learned how to use differential equations to solve circuits as an undergraduate in electrical engineering, and also how to derive the more efficient methods from the differential equations by assuming solutions are (possibly complex) exponentials. No PhD required.
Re: ‘Alien calculus’ could save particle physics from infinities
#103Earlier quoted context omitted.
Traditional calculus is not well-suited to describing the behavior of particles at extremely small scales because it leads to infinite values in calculations: Traditional calculus is based on real numbers, which are continuous and can take on any value. However, at the subatomic level, particles behave more like waves than like particles with a definite position, and the value of some physical quantities, like energy…
Wait, is this new? This is exactly how I'd assumed quantum calculations worked already
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Re: ‘Alien calculus’ could save particle physics from infinities
#104Earlier quoted context omitted.
Everything is simple, given the right notation (and the concepts underlying it). The original Maxwell theory of electromagnetism is about 10 rather involved equations. Maxwell-Heaviside form is 4 simpler equations. A formulation using differential 3-forms is 2 simple equations. A formulation using geometric algebra / Clifford algebra is one utterly simple equation.
Number of equations != How simple something is
Humans have a limited capacity of holding something before their "mind's eye", like 5-7 items. The simpler the equation is, the easier it is to understand and operate. Of course, it only works provided that the parts of the equation are well-understood, too.
This way, the nabla symbol is very helpful in turning a group of (usually) 3 related PDEs into one pretty understandable one. Same for vector and matrix forms of common 2D and 3D transforms. Once you understand how these symbols work (they have a very regular structure), you can think at a bit higher level, and juggle with transforms that won't fit in your head in the scalar form, requiring toilsome operations on paper (or equivalent).
Say, general relativity is hard as it is; without various notational tricks which abstract away some complexity, it would likely be completely unwieldy.
A change of notation is like a refactoring in programming: a good set of powerful and well-defined functions makes the code to solve a problem significantly easier to produce and to understand, and harder to make an error.
Re: ‘Alien calculus’ could save particle physics from infinities
#105What I find fascinating about this, is the question of "What other applications (besides saving particle physics from infinities) could this branch of mathematics have"? Every since I read The Universe Speaks in Numbers [1] I've been fascinated by those scenarios where there's some hard problem in science, and it turns out that the math needed to solve it was invented years before, to solve some other problem. And th…
Would be cool to train GPT with all math books and have it a go at a few unsolved math problems
The issue is that GPT is very bad at logical reasoning. Unfortunately, already for humans it is very difficult sometimes to recognize tiny (but devastating) logical errors in mathematical proofs.
Re: ‘Alien calculus’ could save particle physics from infinities
#106So one question I have in the introduction section is that it seems the article misses the difference between the number of Feynman diagrams to calculate a_n and the value of a_n. They point out that the the number of Feyman diagrams grows ~n!, which is much larger than the rate x^n shrinks (given 0 Based on my limited knowledge of particle physics, physicists are currently able to calculate using Feynman diagrams be…
It is generally believed that the perturbation expansion that we see in realistic quantum field theories are what are known as asymptomatic expansions. These are series that have a radius of convergence of zero (i.e. they only converge when the expansion parameter is exactly zero and diverge for all non zero values). There are then two natural questions: 1. if the perturbation series diverges, why doesn't the univers…
That's such a great comment; you have answered many of the questions I had after reading the article! Thank you!
Re: ‘Alien calculus’ could save particle physics from infinities
#107Earlier quoted context omitted.
I've only explored ChatGPT in the most cursory way, but wouldn't this be a function of the prompt it's given? E.g. "Summarize this idea in a way that uses novel comparisons to everyday phenomena. The target reader is someone who doesn't have domain knowledge of the field...etc." Or something like that.
Yes, you're correct. People who complain that chatGPT is bland don't realize that they have to specify a style to not get the average of all content. It's just like if you ask an image AI for a "woman" you will get the average of all artists women and it will look very generic and bland. But with the right stylistic qualifiers in your prompt you will get something so captivating that it wins awards for its creativity…
Prompting works very well, but I have still not seen any output from ChatGPT that captivated me in a way my favorite writing has.
I used to work in the field of creative text generation for fiction. I’m genuinely very curious and I go out of my way to find compelling examples. There is definitely “good” output that’s on the right track. GPT-4 also does way better, but it still falls short.
This is also difficult to just evaluate objectively. If you find that GPT models have generated the best prose you’ve seen. That’s wonderful! I understand that my standards are quite high (high does not equate to “better” either)
Re: ‘Alien calculus’ could save particle physics from infinities
#108Earlier quoted context omitted.
Sarcasm aside. Even the output from GPT-4 is quite bland and generic. Very good for performing tasks (I.e convert a blob of text into a knowledge graph or generate X code), but quite awful at the elegant prose we see at play Wonder how long before that’s solved. Hard to believe training on such large swathes of the web doesn’t result a compressed generic representation of language within its weights.
>Wonder how long before that’s solved. creative prompting gets most of the way there, already. 'Tell me the current weather as if you were Marcel Proust'
I replied to another comment with my thoughts on prompting. I will add that I don’t consider mimicking another writers style to matter much. Seems like an easy cop out (just my opinion though).
That’s not to say it isn’t impressive. It is and it accomplishes the job very well. We’re looking for progress not perfection, but I personally have very high standards from creative writing and GPT doesn’t meet my personal bar. However not everyone shares that bar and personal evals of GPT’s output are equally valid. Plenty of people find it to be great and at the end of the day that’s all that matters
Re: ‘Alien calculus’ could save particle physics from infinities
#109Earlier quoted context omitted.
Sarcasm aside. Even the output from GPT-4 is quite bland and generic. Very good for performing tasks (I.e convert a blob of text into a knowledge graph or generate X code), but quite awful at the elegant prose we see at play Wonder how long before that’s solved. Hard to believe training on such large swathes of the web doesn’t result a compressed generic representation of language within its weights.
Nobody outside OpenAI/Microsoft has access to the GPT-4 base model yet. Use Bing Creative for a differently finetuned GPT-4 than ChatGPT or "gpt-4".
GPT-4 comment isn’t true
Regarding Bing Creative. It is delightful! I do like it a bit and whatever they’ve done to the system does make for some of the better output I’ve seen from LLMs
Re: ‘Alien calculus’ could save particle physics from infinities
#110Earlier quoted context omitted.
It seems like there’s a tension. On the one hand generating the most likely sequence of tokens maximises the chance the response will make sense and be relevant. On the other hand it also guarantees you will get the most bland and unimaginative response.
Counterargument: "explain linked lists in the style of edgar allan poe"
My personal favorite is in the style of Yoda ;)
A tangent. Bard also confuses Yoda with Jar Jar Binks quite often. Try it out!