Yes, there are numerous cases where a field in “pure” mathematics proved interesting theorems that mathematicians undertook because of its challenging and elegant nature (like certain theorems possess generality and elegance) which were then to be found to be practically useful, which are called “applied” mathematics. Frankly, this distinction is blurred as pure mathematics are so useful (see Eugene Wigner’s “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”) that the abstract nature of mathematics has huge extensibility and general applications in multiple domains. For instance, Einstein’s GR was based on the pure mathematics of Riemannian manifolds, which is an abstract topological structure, not tied to reality in any way initially. Or how algebraic topology is used for data mining, how number theory is used for cryptography, how linear algebra is used for machine learning, group theory is used for particle physics… and even how Bayesian probability theory is used for LW rationality.
Stephen Wolfram has great resources on rulial spaces and the nature of computation for the universe’s fundamental ontology (the territory not the map) in which these networks of theorems can correspond to our empirical reality. (psst I am a very new LW user, and I am deciding if I should do a Sequence for this idea of “rulial cover” which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction, would be great if someone thinks this is interesting to explore so I can be motivated)
To link back to Eliezer’s post, “floating beliefs” in a Bayesian net can be connected through adjusting the “weights” of the edges that connects that belief using Bayesian inference, and mathematics make these robust inferences from axioms (deductively validity as 100% in weight and 0% in prior). Therefore, anticipation becomes certain under a set of idealized axioms.
‘I am deciding if I should do a Sequence for this idea of “rulial cover” which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction’
I don’t really know what you mean, but if it’s something unseen you can expect it to be useful!
Yes, there are numerous cases where a field in “pure” mathematics proved interesting theorems that mathematicians undertook because of its challenging and elegant nature (like certain theorems possess generality and elegance) which were then to be found to be practically useful, which are called “applied” mathematics. Frankly, this distinction is blurred as pure mathematics are so useful (see Eugene Wigner’s “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”) that the abstract nature of mathematics has huge extensibility and general applications in multiple domains. For instance, Einstein’s GR was based on the pure mathematics of Riemannian manifolds, which is an abstract topological structure, not tied to reality in any way initially. Or how algebraic topology is used for data mining, how number theory is used for cryptography, how linear algebra is used for machine learning, group theory is used for particle physics… and even how Bayesian probability theory is used for LW rationality.
Stephen Wolfram has great resources on rulial spaces and the nature of computation for the universe’s fundamental ontology (the territory not the map) in which these networks of theorems can correspond to our empirical reality. (psst I am a very new LW user, and I am deciding if I should do a Sequence for this idea of “rulial cover” which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction, would be great if someone thinks this is interesting to explore so I can be motivated)
To link back to Eliezer’s post, “floating beliefs” in a Bayesian net can be connected through adjusting the “weights” of the edges that connects that belief using Bayesian inference, and mathematics make these robust inferences from axioms (deductively validity as 100% in weight and 0% in prior). Therefore, anticipation becomes certain under a set of idealized axioms.
‘I am deciding if I should do a Sequence for this idea of “rulial cover” which is how rulial deduction can be applied to Solomonoff induction and Bayesian abduction’
I don’t really know what you mean, but if it’s something unseen you can expect it to be useful!