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September 8, 2018 ~ squarishbracket

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Recent Posts

The Theory of Knots
A Compact Proof of the Compactness Theorem
ZFC, and getting the right answer by accident
Polish Notation and Garden-Path Sentences
Defining uncountability
Who would win in a fight, logic or computation?
Computing truth values of sentences of arithmetic, or: Math is hard
How uncomputable are the Busy Beaver numbers?
The Arithmetic Hierarchy and Computability
A Self-Interpreting Book
Epsilon-induction and the cumulative hierarchy
Nonstandard integers, rationals, and reals
Transfinite Nim: uncomputable games and games whose winner depends on the Continuum Hypothesis
A Coloring Problem Equivalent to the Continuum Hypothesis
No known unknowns in Peano arithmetic
The subtlety of Gödel’s second incompleteness theorem
Undecidability results on lambda diagrams
The Anti-Set Program
Pictures of some extremely large numbers
How to draw lambda diagrams
Fundamentals of Logic: Syntax, Semantics, and Proof
The Pictorial Mathematics of Klak-Adbmal
The Rise of Anti-Set Theory
The Transfinite Factorial
On Self-Hating Theories of Arithmetic
Updated Introduction to Mathematical Logic
Finiteness can’t be captured in a sound, complete, finitary proof system
The Mindblowing Goodstein Sequences
Wacky Non-Standard Models of PA
The Continuum Hypothesis is False
Kolmogorov Complexity, Undecidability, and Incompleteness
Constructing the Church-Kleene Ordinal
What ordinals can be embedded in ℚ and ℝ?
Visualizing the rationals
The order type of the rational numbers
Can you compute all the countable ordinals?
Immoral or Inconsistent?
Constructing ordinal numbers in ZFC
Meaning ain’t in the brain
On philosophical progress
Logic on Planet Zorko
A result on the incompleteness of mathematics
A Gödelian Logic Puzzle
Crazy conditionals
A Dice Puzzle
Proving the Completeness of Propositional Logic
Sum and Product Puzzle
Four Pre-Gödelian Limitations on Mathematics
A proof of the Compactness Theorem
A challenge to constructivists
Describing the world
The full solution to the dog puzzle
The Surprise-Response Heuristic
Three paradoxes of self-reference
A logic puzzle about dogs
A suspicious proof of God’s existence
Fractal Trees
In defense of collateralized debt obligations (CDOs)
Six Case Studies in Consequentialist Reasoning
Will we ever build a Galactic Empire?
Earning to give or waiting to give
Diversification is magic
Paradoxical Precommitment
Solving the Linear Cournot Model
Building an analog calculator
Complex numbers in physics
The laugh-hospital of constructive mathematics
Can chess be solved?
For Loops and Bounded Quantifiers in Lambda Calculus
There’s a problem with infinity
A Supertask Puzzle
Can an irrational number raised to an irrational power be rational?
Are the Busy Beaver numbers independent of mathematics?
Introduction to Mathematical Logic (Part 4: Zermelo-Fraenkel Set Theory)
The Weirdest Consequence of the Axiom of Choice
Introduction to Mathematical Logic (Part 3: Reconciling Gödel’s Completeness And Incompleteness Theorems)
Introduction to Mathematical Logic (Part 2: The Natural Numbers)
Introduction to Mathematical Logic (Part 1)
Thinking in first order: Are the natural numbers countable?
The Central Paradox of Statistical Mechanics: The Problem of The Past
Is The Fundamental Postulate of Statistical Mechanics A Priori?
The Necessity of Statistical Mechanics for Getting Macro From Micro
A Cognitive Instability Puzzle, Part 2
Logic, Theism, and Boltzmann Brains: On Cognitively Unstable Beliefs
Philosophers of religion are religious. Why?
Why do prediction markets work?
A Talmudic Probability Puzzle
Galois’ Unsolvable Polynomials
x^5 – x – 1
Group Theory: The Mathematics of Symmetry?
Symmetries of Platonic Solids
Some Curious Group Isomorphisms
Extending Cauchy’s Theorem
Strong Induction Proofs of Cauchy’s Theorem and Sylow’s First Theorem
Table of Small Finite Groups
Some Group Factoids
Group Theory Toolkit
The sweet spot on the structure spectrum
Proving Cauchy’s Theorem
Group Theory: Lagrange’s Theorem and the Sudoku Principle
The Most Irrational of Numbers
All About Adultery, BDSM, and Divorce
Collatz Tree
Record-breaking Collatz chains!
Producing all numbers using just four fours
A sequence prediction puzzle
Buddhabrot, Mandelbrot’s older sibling
Hopping Midpoints
Exploring Julia Sets
Mandelbrot Exploration
How will quantum computing impact the world?
Hopping Midpoints and Mathematical Snowflakes
Backwards induction and rationality
Firing Squads and The Fine Tuning Argument
The EPR Paradox
Measurement without interaction in quantum mechanics
Is the double slit experiment evidence that consciousness causes collapse?
Decoherence is not wave function collapse
On decoherence
The problem with the many worlds interpretation of quantum mechanics
Consistently reflecting on decision theory
Decision Theory
Rationality in the face of improbability
A probability puzzle
Solving the common knowledge puzzle
A common knowledge puzzle
The history of lighting technology
Kant’s attempt to save metaphysics and causality from Hume
Deriving the Lorentz transformation
Swapping the past and future
Visualizing Special Relativity
Fractals and Epicycles
How to Learn From Data, Part 2: Evaluating Models
How to Learn From Data, Part I: Evaluating Simple Hypotheses
The Match Problem
Gödel’s Second Incompleteness Theorem: Explained in Words of Only One Syllable
Anti-inductive priors
Making sense of improbability
A simple probability puzzle
The end goal of epistemology
Deciphering conditional probabilities
The Problem of Logical Omniscience
Sapiens: How Shared Myths Change the World
Anthropic reasoning in everyday life
Adam and Eve’s Anthropic Superpowers
Sleeping Beauty Problem
Infinities in the anthropic dice killer thought experiment
Not a solution to the anthropic dice killer puzzle
Pushing anti-anthropic intuitions
An expected value puzzle
Dialogue from Kafka On The Shore
A closer look at anthropic tests for consciousness
Two principles of Bayesian reasoning
Explaining anthropic reasoning
Predicting changes in belief
In favor of anthropic reasoning
Clarifying self-defeating beliefs
Getting empirical evidence for different theories of consciousness
Bad Science Reporting
The Anthropic Dice Killer
What do I find conceptually puzzling?
More on quantum entanglement and irreducibility
Ramanujan’s infinite radicals
Communication through entanglement
Quantum mechanics, reductionism, and irreducibility
Building the diffusion operator
Building the quantum oracle
Grover’s algorithm
Deutch-Josza Algorithm
More on quantum gates
Quantum Computing in a Nutshell
Matter and interactions in quantum mechanics
100 prisoners problem
Some simple probability puzzles
Deriving the Schrodinger equation
On existence
Concepts we keep and concepts we toss out
Seeing, Doing, Imagining
Against moral realism
Explanation is asymmetric
Wave function entropy
Query sensitivity of evidential updates
Summarizing conscious experience
Simple induction
The North Korea problem isn’t solved
Priors in the supernatural
Patterns of inductive inference
If all truths are knowable, then all truths are known
Variational Bayesian inference
The value of the personal
Confirmation bias, or different priors?
Objective Bayesianism and choices of concepts
A shameful history
“Beating the drum for justice”
Overemphasizing disagreement
Regularization as approximately Bayesian inference
Against the status quo
Why minimizing sum of squares is equivalent to frequentist inference
History is Lamarckian
Short and sweet proof of the f(xy) = f(x) + f(y) logarithmic property
Constructing the world
Moving Naturalism Forward: Eliminating the macroscopic
Some social justice factoids
What is integrated information?
How to sort a list of numbers
Defining racism
Utter confusion about consciousness
Getting evidence for a theory of consciousness
The Scourge of Our Time
The problem with philosophy
Argument screens off identity
Facts about guns
“You don’t believe in the God you want to, and I won’t believe in the God I want to”
Galileo and the Schelling point improbability principle
Why “number of parameters” isn’t good enough
Bayesian Occam’s Razor
All about IQ
A model selection puzzle: Why is BIC ≠ AIC?
Bayes and overfitting
The basics of information divergence
More model selection visualizations
Where I am with utilitarianism
The Monty Hall non-paradox
Akaike, epicycles, and falsifiability
On complexity and information geometry
A note of ambiguity regarding model selection
Gibbs’ inequality
Some simple visual comparisons of model selection techniques
Bayes and beyond
Racism and identity
What is bias?
Value beyond ethics
Taxonomy of infinity catastrophes for expected utility theory
Two-factor markets for sex
A simple explanation of Bell’s inequality
Cults, tribes, states, and markets
Bayesian experimental design
Why relative entropy
A survey of entropy and entropy variants
Entropy is expected surprise
The Rival-Expert Heuristic
Inference as a balance of accommodation, prediction, and simplicity
Pascal’s mugging
Those who have forgotten words
Why systematize epistemology?
Journey into abstraction
Timeless decision theory and homogeneity
Timeless ethics and Kant
A failure of Bayesian reasoning
Bayesianism as natural selection of ideas
Against falsifiability
Metaphysics and fuzziness: Why tables don’t exist and nobody’s tall
Is [insert here] a religion?
Hyperreal decision theory
Infinite ethics
More on random sampling from Nature’s Urn
Free will and decision theory (Part 2)
Paradoxes of infinite decision theory
Does fine-tuning give evidence for God?
Noisy Evidence
Nature’s Urn and Ignoring the Super Persuader
Entropy vs relative entropy
Fun with Akaike
Anthropic argument for common priors
Why you should be a Bayesian
Consistency and priors
Dutch book arguments
Cox’s theorem
Maximum Entropy and Bayes
Advanced two-envelopes paradox
Sam Harris and the is-ought distinction
Two envelopes paradox
Principle of Maximum Entropy
Consciousness
Inequality and free markets
40 papers on inequality in one sentence each
Gregory Watson (and how to change the world)
Statistical mechanics is wonderful
Solution: How change arises in QM
TIMN view of social evolution
Race, Ethnicity, and Labels
Is quantum mechanics simpler than classical physics?
Iterated Simpson’s Paradox
Causal decision theory
Free will and decision theory
Simpson’s paradox
Screening off and explaining away
Societal Failure Modes
Dialogue: Why you should one-box in Newcomb’s problem
Opt-out organ donation
The spiritual and the scientific
Comprehensibility of the Complex
Correlation and causation
Causal Intervention
Causal Arrows
Causality: Preliminaries
Low-hanging policy fruit

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