A discourse on epistemology, height and language

The formalism I presented so far has avoided the issue of choosing the free parameters in the model, namely the height function and the measure \( \mu \). I said that given an underlying logic we can use the notion of complexity to get a canonical height map, but this map, obviously, is only canonical in a specific signature/language. If we change the language of our theory to add new symbols for some additional operations which already exist in the theory, thus reducing their complexity, we have a more powerful language which can express more ideas in lower complexity. However, by doing so we also lose a lot of information - we put different functions which may "ought to" be in different complexity classes together. In practice, this means there is a trade-off between the expressive power of a language and the information its complexity classes contain. It's important to note at this point that the asymptotic properties of the formalism do not depend on the specific cho…

A discourse on epistemology, duality

In the previous two posts, I focused on how observation of a phenomenon constrains its possible explanations - to make inferences about a phenomenon, we first explore the low height space of characterizations for ones which match observations, and then we extrapolate each of them to make predictions about the future behavior of the phenomenon. In this post, I will examine the dual problem, which is quite important to the study of knowledge: how knowledge constrains possible phenomena which it can explain, and I will explain how they are connected.

In part 1, I defined knowledge within a fixed theory in terms of a formula with one free variable in the language of the theory. Afterwards, the framework focused almost exclusively on a special kind of knowledge - a characterization. However, we do not have to restrict ourselves to characterizations. To proceed further, there's a definition we need to make:

Definition. The (truth value) pairing map is a map \( I : F \times M \to 2 = \{ 0…

A discourse on epistemology, height and time

In the last post, I laid out some of the formalism for what will come next in this series. While I worked with the explicit context of maps from \( \mathbb Z \) to \( \mathbb Z \), it's easy to see that the concepts I defined can generalize to other settings easily, and I will not repeat the definitions in the general context since they are identical to the specific case I already covered.

In this post I will talk a bit more about the concept of height. As we saw in the last post, the asymptotic behavior of the tentative truth value \( T_{A_k}(S) \) does not depend on the choice of pruning map \( f \) or the choice of weights \( \alpha \). However, in practice we care about the behavior in a relatively small initial segment of \( \mathbb N \), since that's the part we can obtain information about through finite measurement. Therefore, the specific choice of the pair \( (f, \alpha) \) is very important in practice. How do we make the "right" choice to obtain the corre…

A discourse on epistemology, the basics

Epistemology is the branch of philosophy that concerns it with answering questions about knowledge, explanations, understanding, etc. Unfortunately, like other disciplines of philosophy, professional philosophers with little experience in mathematics have caused the field to become a vast collection of nonsense. The few philosophers of science and knowledge are spread thin, and their works are hard to find among the massive body of low quality work produced by incompetent academics. Therefore I ask the reader to forget whatever they might've heard about this field and start from scratch when beginning to read this post.

I will start with a toy model in which we try to acquire knowledge about a function \( f : \mathbb Z \to \mathbb Z \) defined on integers. What should be the definition of knowledge in this context? We will say that a piece of knowledge about \( f \) is a formula with one free variable which is true when specialized to the case of the function \( f \). (There are m…

The stationary score distribution

I occasionally play (because I like wasting my time), and today I was curious if it's possible to derive a model for the score distribution in the game at any point in time. I won't explain the rules of the game in this post, so you may look them up if you want, I assume that the reader is familiar with them.

Say that the score distribution density at time \( t \) is given by \( f_t(x) \). I make some assumptions: new players arrive at a constant rate \( \alpha \) and start with initial score \( 0 \), and players are chosen uniformly at random at a rate \( \beta \) and are killed, their score is distributed uniformly across the entire player base. Players leave the game at a rate \( \delta \), and every player in the game acquires score at a constant rate \( \gamma \). This gives the law of motion

\[ f_{t + dt}(x) = (1 - \beta dt - \delta dt) f_t(x - \gamma dt - \beta \mu_t dt) \]

where \( \mu_t \) is the mean score at time \( t \). We solve for the stationary distr…

The end of civilization

Talking about the end of civilization in a modern context brings up two ideas: one involving climate change, scarcity of resources on Earth, pollution, ocean acidification, and "the abuse of planet Earth" by the human species; and one involving immigration of undesirable elements into a society, bringing genes and culture which will over time eat the society from within and regress it to a metaphorical Dark Age. Other scenarios are considered (say, an impact from a sufficiently large asteroid), but considered unlikely and therefore dismissed. However, the clear and present danger comes from within, and in a way that most appear unable to appreciate.

To understand this, it is necessary to first define the rather ambiguous word "civilization". How do we measure how "civilized" a given society is? Though there is dispute about this within the historical community, the general definition looks like a checklist, much like the definition of "life" in …

Simple labor search model (a la Pissarides)

In this post I will outline a search model of the labor market, similar to some found in the literature on labor economics. I think macroeconomists have focused too much on sticky prices and wages as a mechanism for explaining fluctuations in unemployment, and not nearly enough on mechanisms of economic disruption and search frictions in the labor market. You can view this simple model as a module that may be put into a larger model incorporating various kinds of shocks.

The model has infinite periods. There are a continuum of firms with measure \( 1 \) and workers of measure \( N \).

For simplicity, I assume that workers have a fixed reservation wage \( w \), above which they are willing to supply labor to any firm, and firms make take-it-or-leave-it offers to workers in bilateral meetings. Labor is indivisible, i.e each worker can choose between supplying 0 or 1 unit of labor to one firm. This assumption simplifies the analysis without changing the resulting dynamics very much.