Summary: The Tower of Hanoi

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Check out the first post from my friend Thomas Colclough, in which he talks about very large but finite numbers and a group of priests trying to destroy the world.

Exercise for the reader

I first read Hugh Woodin’s Tower of Hanoi years ago early in my undergraduate. I remember getting bogged down by the math, and finding it more difficult to keep hold of the main thread as I got further into the paper. A few years later, I read the paper again upon starting grad school, and was pleasantly surprised to find it an easier read. No longer slowed down by technical obscurities, ideas that remained concealed before now seemed clearer, and the paper was much more enjoyable as a result. One of my reasons for starting this blog was to condense those years into a single exercise — to try and elicit the key ideas in papers involving technically challenging or dense mathematical material, where (I think) often those technical details can overshadow what is really being said. The Tower of Hanoi, then, seemed like a natural place to start. 

The…

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Paper Review: The Leibniz-Carnap Program for Inductive Logic

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Hacking’s broad claim in this paper is that there are deep connections between Leibniz’s program for a “new kind of logic” and Carnap’s inductive logic program. His more specific thesis is that inductive logic as Carnap imagined it would only be possible if some theory like Leibniz’s were true.  ***You can find the original paper… Continue reading Paper Review: The Leibniz-Carnap Program for Inductive Logic

Paper Review: Is the Dream Solution of the Continuum Hypothesis Attainable?

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One central notion in mathematics is cardinality (I've given an intuitive of cardinality in this post). In particular, there are different types of infinity, with some being larger than others. The smallest infinity is the cardinality of the natural numbers. We say that any set with such a cardinality is countable, since its elements can… Continue reading Paper Review: Is the Dream Solution of the Continuum Hypothesis Attainable?

Paper Review: From a Cosmic Fine-Tuner to a Perfect Being

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Many arguments for the existence of God face the gap problem. In general, the gap problem is the problem of moving from some premiss of the form "there is a being with some perfections" to a conclusion of the form "there is a being with all perfections". Justin Mooney, following an approach by T. Ryan… Continue reading Paper Review: From a Cosmic Fine-Tuner to a Perfect Being

Paper Review: New Semantics for Bayesian Inference: The Interpretive Problem

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The standard Bayesian story goes something like this: probabilities represent a rational agent's degrees of belief. When the agent learns something new she conditions on it, meaning that she updates her probabilities according to Bayes' rule. Importantly, the interpretation of the probability function is that it represents the agent's degrees of belief about how likely… Continue reading Paper Review: New Semantics for Bayesian Inference: The Interpretive Problem

Paper Review: The Transmission Sense of Information

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We're taught in school something like the following: DNA is the blueprint for an organism. We can sharpen up this kind of idea using ideas from information theory. In particular, biologists tend to say things like "DNA transmits information" or "DNA sequences are coded instructions". However, some philosophers of biology have claimed that when biologists… Continue reading Paper Review: The Transmission Sense of Information

Can one rationally choose to have a child?

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Is it rational for me to want to have a child if I cannot know what it will be like to have a child? Laurie Paul, in her book Transformative Experience, argues that this question poses a significant problem for traditional theories of decision-making. Paul holds that many major life decisions are ones in which… Continue reading Can one rationally choose to have a child?

Paper Review: Subjective Probability and the Theory of Games: Comments on Kadane and Larkey’s Paper

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Last week I reviewed Kadane and Larkey's paper. In short, the main claim of their paper was that since the solution concepts used in game theory do not depend on the beliefs of the players they are irrelevant to game theory. Haranyi wrote a response to this paper, which it what I will review today.… Continue reading Paper Review: Subjective Probability and the Theory of Games: Comments on Kadane and Larkey’s Paper

Paper Review: Subjective Probability and the Theory of Games

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Game theory is the study of the strategic interaction of rational agents. Decision theory is the study of the decision making of a rational agent. Clearly there is something similar about these two fields. What, exactly, is the relationship though? Do they study different aspects of rational action, or do they overlap? If they overlap,… Continue reading Paper Review: Subjective Probability and the Theory of Games

Paper Review: The Rational Status of Quantum Cognition

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We've seen before on this blog that we often take probability theory to be the core of rational belief. What we haven't seen before is that our beliefs tend to violate probability theory not just in random ways, which is what you would expect if you thought we were making random mistakes, but systematically, in… Continue reading Paper Review: The Rational Status of Quantum Cognition