Limitations of Mathematics

Category: Technical

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Book recommendation request! I’m struggling to find a good overview of fundamental limitations of mathematics. Examples of what I mean include:

1. Computational Irreducibility
2. Chaotic systems that vary dramatically under perturbation, rendered unpredictable
3. n-body problems, the unrestricted 3-body problem
4. True random number generation
5. Squaring the circle impossibility proof (impossibility proofs in general)
6. Self-reference (Russel’s Paradox, Godel’s Incompleteness Theorem, Halting Problem)
7. Heisenberg’s Uncertainty Principle (measurement limits on momentum and position) Questions I’d like answers to include? What physical phenomena have we failed to find formalisms for? What systems did we believe were unmodelable / intractable before but eventually discovered methods for? What are the important outstanding problems in mathematics and physics that seem like they should be straightforward? What are the major categories of limitations? What properties do they share with one another? What paths to partial resolution may they have in common?

Practical problems:

1. Finding analytical solutions
   1. Higher order / Partial Differential Equations
   2. Normalization in bayes rule
2. Non-convex optimization

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