This article proposes quantitative answers to meta-scientific questions including "how much knowledge is attained by a research field?","how rapidly is a field making progress?", "what is the expected reproducibility of a result?", "how much knowledge is lost from scientific bias and misconduct?" "what do we mean by soft science?", "what demarcates a pseudoscience?".

Knowledge is suggested to be a system-specific property measured by K, a quantity determined by how much the information contained in an

Examples from a variety of fields are given to illustrate the possible uses of K. These examples include quantifying: the knowledge value of proving Fermat's last theorem; the accuracy of measurements of the electron's mass; the half life of eclipse predictions; the usefulness of evolutionary models of reproductive skew; the significance of gender differences in personality; the sources of irreproducibility in psychology; the impact of scientific misconduct and QRP; the knowledge value of astrology. Furthermore, a cumulative K may complement ordinary meta-analysis and may give rise to a universal classification of sciences and pseudosciences.

Simple and memorable mathematical formulae summarize the theory's key results and implications. In addition to practical uses in meta-research, these formulae may have conceptual applications in philosophy and in research policy, and may guide scientists to make progress on all frontiers of knowledge.

This is an entirely new version of the manuscript, simplified in the theory and giving examples of applications. The theory is maximally simplified and stripped of all non-essential elements. New arguments are presented for the validity of K, new properties are discussed, and proofs are generally made tighter. The Results secition now offers numerous practical examples, using real data or simulations, of how K can be used to answer meta-scientific questions.

R code used to generate all figures and analyses.

The author declares that they have no competing interests.

The following information was supplied regarding data availability:

The R code used to generate all figures and analyses is in the supplementary file "S11 text". The few empirical data sets analyzed in the study were obtained from published articles or online repositories that are publicly accessible and duly referenced.