[1] Decidability

Foundational Questions Institute: Undecidability, Uncomputability, and Unpredictability Essay Contest (2019-2020) Support materials for the submission from Bruce Camber in April 2020.

Determinant becomes unpredictable, uncomputable, and undecidable (PDF)

[1] Decidability
[2] Computability
[3] Predictability
[4] Undecidability
[5] Uncomputability
[6] Unpredictability
[7] First units
[8] Grand reductionism
[9] Triangulation
[10] Fourier
[11] Lorentz
[12] Poincaré spheres
[13] Planckspheres
One second: 299,792± km
[14] Automorphic forms
[15] Base-2 and Prime Numbers
Aristotle’s Mistake
[17] Fuzzy Universe
[18] Scholars
Background: The 2020 FQXi challenge is to engage the study of the concept of “Undecidability, Uncomputability, and Unpredictability.”

This is the start of what we hope will become a rigorous analysis and study. We will build off of our introductory comments and first impressions.

[1] Decidability (Subject). We have a lot to learn about the coherence of logic and mathematics with physics. As we study the dimensionless constants, and the basic-basics —  c with special relativity, G with general relativity, ħ with quantum mechanics, ε0 (vacuum permittivity) with electromagnetism, and kB (Boltzmann constant) with temperature-to-energy — we also begin to understand more deeply first-order theory of Euclidean geometry. We’ll take a much deeper dive with Tarski (1949) and all those mentioned by Hodges (Stanford Encyclopedia of Philosophy, 2018), i.e. Abelian groups, hyperbolic geometries, the historic work of Schwabhäuser and W.M. Szmielew and more  recent work of Petra Wolf.

FQXi statement:  “For a brief time in history, it was possible to imagine that a sufficiently advanced intellect could, given sufficient time and resources, in principle understand how to mathematically prove everything that was true. They could discern what math corresponds to physical laws, and use those laws to predict anything that happens before it happens. That time has passed. Gödel’s undecidability results (the incompleteness theorems), Turing’s proof of non-computable values, the formulation of quantum theory, chaos, and other developments over the past century have shown that there are rigorous arguments limiting what we can prove, compute, and predict. While some connections between these results have come to light, many remain obscure, and the implications are unclear. Are there, for example, real consequences for physics — including quantum mechanics — of undecidability and non-computability? Are there implications for our understanding of the relations between agency, intelligence, mind, and the physical world?”

“In this essay contest, we open the floor for investigations of such connections, implications, and speculations. We invite rigorous but bold and open-minded investigation of the meaning of these impossibilities for reality, and for us, its residents.”