Fitting an elephant with four non-zero parameters
The paper discusses addressing Fermi's critique of Dyson's model by fitting an elephant with four non-zero parameters, aiming to define the problem accurately and proposing a new solution. It provides insights into mathematical modeling challenges.
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The paper titled "Fitting an Elephant with Four non-Zero Parameters" by Dian Jin and Junze Yuan addresses Enrico Fermi's critique of Dyson's model, referencing Johnny von Neumann's statement about fitting an elephant with parameters. While previous attempts have been made to fit an elephant using four parameters, the problem lacked a clear definition, leading to unsatisfactory results. This paper aims to define the problem accurately and presents a new attempt to fit an elephant with four non-zero parameters. The authors highlight the challenges faced in previous methods and propose a solution in their work. The study falls under the subject of History and Overview in mathematics. The paper provides insights into the mathematical modeling of complex shapes and the significance of parameter selection in fitting models accurately.
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Fitting an Elephant with Four Non-Zero Parameters
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- Many appreciate the humor and irony in the article, noting its well-written and entertaining nature.
- There is a discussion on the importance of minimizing parameters in mathematical models to ensure they reflect true aspects of reality rather than just fitting data.
- Some comments reference historical anecdotes and quotes from Fermi and Neumann, emphasizing the challenge of creating accurate models with minimal parameters.
- Several comments provide additional resources and references, such as Freeman Dyson's interviews and related scientific papers.
- There is a debate on the technical aspects of the model, including the necessity and interpretation of the parameters used.
22 commentsI'm always making the joke (observation) that ML (AI) is just curve-fitting. Whether "just curve-fitting" is enough to produce something "intelligent" is, IMO, currently unanswered, largely due to differing viewpoints on the meaning of "intelligent".
In this case they're demonstrating some very clean, easy-to-understand curve-fitting, but it's really the same process -- come up with a target, optimize over a loss function, and hope that it generalizes, (this one, obviously, does not. But the elephant is cute.)
This raises the question Neumann was asking -- why have so many parameters? Ironically (or maybe just interestingly), we've done a lot with a ton of parameters recently, answering it with "well, with a lot of parameters you can do cool things".
(Seriously, though, this was a lot of fun!)
This was a lovely passage from Dyson’s Web of Stories interview, and it struck a chord with me, like it clearly did with the authors too.
It happened when Dyson took the preliminary results of his work on the Pseudoscalar theory of Pions to Fermi and Fermi very quickly dismissed the whole thing. It was a shock to Dyson but freed him from wasting more time on it.
Fermi: When one does a theoretical calculation, either you have a clear physical module in mind or a rigorous mathematical basis. You have neither. How many free parameters did you use for your fitting?
Dyson: 4
Fermi: You know, Johnny Von Neumann always used to say ‘with four parameters I can fit an elephant; and with five I can make him wiggle his trunk’.
If I could make a discovery in my own time without using company resources I would absolutely publish it in the most humorous way possible.
He tried for a while to get an aerodynamics paper published on the flight of dinosaurs. http://levenspiel.com/wp-content/uploads/2016/02/DinosaurW.p...
This intellectual curiosity reminded me a bit of Feynman and his plate spinning.
A real-parameter (r(theta) = sum(r_k cos(k theta))) Fourier series can only draw a "wiggly circle" figure with one point on each radial ray from the origin.
A compex parameter (z(theta) = sum(e^(z_ theta))) can draw more squiggly figures (epicycles) -- the pen can backtrack as the drawing arm rotates, as each parameter can move a point somewhere on a small circle around the point computed from the previous parameter (and recursively).
Obligatory 3B1B https://m.youtube.com/watch?v=r6sGWTCMz2k
Since a complex parameter is 2 real parameters, we should compare the best 4-cosine curve to the best 2-complex-exponential curve.
Maybe this sort of thing would be a really good tradition. Everyone must write a very silly article with some mathematical arguments in it. Then, we can all go forward with the comfort of knowing that we aren’t really at risk of breaking new grounds in appearing unserious.
It is well written and very understandable!
Why would that be a harder problem? In the case that you get a zero parameter, you could inflate it by some epsilon and the solution would basically be the same.
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The Banach-Tarski theorem challenges common sense by showing a solid ball can be split into pieces to form two identical balls in 3D space. It questions measurement principles and the Axiom of Choice's role in resolving mathematical paradoxes.
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Fitting an Elephant with Four Non-Zero Parameters
The paper discusses addressing Fermi's critique of Dyson's model by fitting an elephant with four non-zero parameters, aiming to define the problem accurately and proposing a new approach. It contributes to mathematical discussions.
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Quantization in machine learning involves reducing model parameters to lower precision for efficiency. Methods like GGUF are explored, impacting model size and performance. Extreme quantization to 1-bit values is discussed, along with practical steps using tools like Llama.cpp for optimizing deployment on various hardware.