A mathematical theory of knowledge, science, bias and pseudoscience

This essay proposes mathematical answers to meta-scientific questions including "how much knowledge is produced by research?", "how rapidly is a field making progress?", "what is the expected reproducibility of a result?", "what do we mean by soft science?", "what demarcates a pseudoscience?", and many others. From two simple postulates - 1) information is finite; 2) knowledge is information compression - we derive a function \(K(y;x\tau)=\frac{T(y)-T(y|x \tau)}{(T(y)+T(x)+T(\tau)}\), in which the total information \(T()\) contained in an explanandum \(y\) is lossless or lossy compressed via an explanans composed of an information input \(x\) and a "theory" component \(\tau\). The latter is a factor that conditions the relationship between \(y\) and \(x\), with an information "cost" equivalent to the description length of the relationship itself. This function is proposed as a simple and universal tool to understand and analyse knowledge dynamics, scientific or otherwise. Soft sciences are shown to be simply fields that yield relatively low K values. Bias turns out to be information that is concealed in methodological choices, thereby reducing K. Disciplines typically classified as pseudosciences are suggested to be sciences that suffer from extreme bias: their informational input is greater than their output, yielding \(K(y;x\tau) < 0\). The essay derives numerous general results, some of which may be counter-intuitive. For example, it suggests that reproducibility failures are inevitable, and that the value of publishing negative results may vary across fields and within a field over time. Therefore, there may be conditions in which the costs of reproducible research practices such as publishing negative results and sharing data may outweigh the benefits. The theory makes several testable predictions concerning science and cognition in general, and it may have numerous applications that future research could develop, test and implement to foster progress on all frontiers of knowledge.