mathlib
Pluto.jl
mathlib | Pluto.jl | |
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36 | 78 | |
1,639 | 4,871 | |
1.2% | - | |
8.8 | 9.5 | |
12 days ago | 7 days ago | |
Lean | JavaScript | |
Apache License 2.0 | MIT License |
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mathlib
- An Easy-Sounding Problem Yields Numbers Too Big for Our Universe
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Towards a new SymPy: part 2 – Polynomials
It's been on my mind lately as well. I was trying out `symbolics.jl` (a CAS written in Julia), and it turned out that it didn't support symbolic integration beyond simple linear functions or polynomials (at least back then, things have changed now it seems). Implementing a generic algorithm for finding integrals is hard, but I was expecting more from that CAS since this seems to be implemented in most other CASs. The thing is that every single CAS that covers general maths knowledge will have to implement the same algorithm, while it's hard to do it even once!
I feel like at least a large part of the functionality of a general purpose CAS can be written down once, and every CAS out there could benefit from it, similar to what the Language Server Protocol did for programming tools. They also had to rewrite the same tool for some language multiple times because there are lots of editors out there, and the LSP cut the time investment down a lot. They did have to invest a large amount of time to get LSP up and running, and it'll have to be maintained, but I think it's orders of magnitudes more efficient than having every tool developed and maintained for every single (programming language, editor) pair out there.
Main problem is like you said how to write down mathematical knowledge in a way that all CASs can understand it. I've been learning about Mathlib lately [0], which seems like a great starting point for this. It is as far as I know one of the first machine readable libraries of mathematical knowledge; it has a large community which has been pushing it continuously forward for years into research-level mathematics and covering the entire undergraduate maths curriculum and it's still accelerating. If some kind of protocol can be designed to read from libraries like this and turn it into CAS code, that would be a major step towards making the CAS ecosystem more sustainable I think.
It's not exactly what you were talking about, as in, this would allow multiple CASs to co-exist and benefit from each other, but I think that's better than having one massive CAS that has a monopoly. No software is perfect, but having a diverse set of choices that are open source would be more than enough to satisfy everyone.
(I have posted about this before on the Lean Zulip forum, it's open to everyone to read without an account [1])
[0] https://leanprover-community.github.io/
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Lean 4.0.0, first official lean4 release
Kinda agree but Mathlib and its documentation makes for a big corpus to learn by example from. Not ideal but it helps.
https://github.com/leanprover-community/mathlib
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It's not mathematics that you need to contribute to (2010)
https://github.com/leanprover-community/mathlib
https://1lab.dev/
You can watch the next generation, or participate, right now.
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If given a list of properties/definitions and relationship between them, could a machine come up with (mostly senseless, but) true implications?
Still, there are many useful tools based on these ideas, used by programmers and mathematicians alike. What you describe sounds rather like Datalog (e.g. Soufflé Datalog), where you supply some rules and an initial fact, and the system repeatedly expands out the set of facts until nothing new can be derived. (This has to be finite, if you want to get anywhere.) In Prolog (e.g. SWI Prolog) you also supply a set of rules and facts, but instead of a fact as your starting point, you give a query containing some unknown variables, and the system tries to find an assignment of the variables that proves the query. And finally there is a rich array of theorem provers and proof assistants such as Agda, Coq, Lean, and Twelf, which can all be used to help check your reasoning or explore new ideas.
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Will Computers Redefine the Roots of Math?
For the math that you mention, I would suggest looking at mathlib (https://github.com/leanprover-community/mathlib). I agree that the foundations of Coq are somewhat distanced from the foundations most mathematicians are trained in. Lean/mathlib might be a bit more familiar, not sure. That said, I don't see any obstacles to developing classical real analysis or linear algebra in Coq, once you've gotten used to writing proofs in it.
- Did studying proof based math topics e.g. analysis make you a better programmer?
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Which proof assistant is the best to formalize real analysis/probability/statistics?
At this point I would go with Lean because of mathlib. Mathlib's goal is to formalize modern mathematics, so many of the theorems you would need for analysis should already be there for you.
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[R] Large Language Models trained on code reason better, even on benchmarks that have nothing to do with code
I think about that every day. Lean's mathlib is a gigantic (with respect to this kind of project) code base and each function, each definition has a precise and rigorous natural language counterpart (in a maths book, somewhere).
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Is there a paid service where someone can explain a paper to me like I am 15?
It's been around since 2013, although there are LLM that interact with Lean to do automated theorem proving. Anyway, you can learn more about Lean here. I enjoyed their natural numbers game (which reminds, me I should finish the last two levels)
Pluto.jl
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Potential of the Julia programming language for high energy physics computing
I thought that notebook based development and package based development were diametrically opposed in the past, but Pluto.jl notebooks have changed my mind about this.
A Pluto.jl notebook is a human readable Julia source file. The Pluto.jl package is itself developed via Pluto.jl notebooks.
https://github.com/fonsp/Pluto.jl
Also, the VSCode Julia plugin tooling has really expanded in functionality and usability for me in the past year. The integrated debugging took some work to setup, but is fast enough to drop into a local frame.
https://code.visualstudio.com/docs/languages/julia
Julia is the first language I have achieved full life cycle integration between exploratory code to sharable package. It even runs quite well on my Android. 2023 is the first year I was able to solve a differential equation or render a 3D surface from a calculated mesh with the hardware in my pocket.
- Pluto.jl: Simple, reactive programming environment for Julia
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Ask HN: Why don't other languages have Jupyter style notebooks?
Re Julia there is also pluto.jl that is another notebook-like environment for julia. It's been a few years since I played with it but it looked cool, for example it handles state differently so you don't get into the same messes as with ipython notebooks. https://plutojl.org/
- Pluto: Simple Reactive Notebooks for Julia
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Looking for a Julia gui framework with a demo like EGUI
For this, Notebooks are often used. Julia offers a uniquely nice and interactive Pluto notebook for the web https://github.com/fonsp/Pluto.jl
- Excel Labs, a Microsoft Garage Project
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IPyflow: Reactive Python Notebooks in Jupyter(Lab)
I believe this is what Pluto sets out to do for Julia.
I used it as part of the “Computational Thinking” with Julia course a year or two back. Even then the beta software was very good and some of the demos the Pluto dev showed were nothing short of amazing
https://plutojl.org/
- For Julia is there some thing like VSCode's python interactive window?
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What have you "washed your hands of" in Python?
I think what you want is Pluto!
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Show HN: Out of order execution in Jupyter notebooks is a solved problem
I like how Pluto.jl handles this:
> Pluto offers an environment where changed code takes effect instantly and where deleted code leaves no trace. Unlike Jupyter or Matlab, there is no mutable workspace, but rather, an important guarantee:
> At any instant, the program state is completely described by the code you see.
[1] https://github.com/fonsp/Pluto.jl
What are some alternatives?
coq - Coq is a formal proof management system. It provides a formal language to write mathematical definitions, executable algorithms and theorems together with an environment for semi-interactive development of machine-checked proofs.
vim-slime - A vim plugin to give you some slime. (Emacs)
Coq-Equations - A function definition package for Coq
rmarkdown - Dynamic Documents for R
mathquill - Easily type math in your webapp
Weave.jl - Scientific reports/literate programming for Julia
fricas - Official repository of the FriCAS computer algebra system
Dash.jl - Dash for Julia - A Julia interface to the Dash ecosystem for creating analytic web applications in Julia. No JavaScript required.
polynomial-algebra - polynomial-algebra Haskell library
IJulia.jl - Julia kernel for Jupyter
lean-liquid - 💧 Liquid Tensor Experiment
Tables.jl - An interface for tables in Julia