Frequently Asked Questions

Q.:

Sir, I’m not involved in the issues your site addresses, but I am very involved in philosophical/religious Gnosticism. My question is how you are defining Gnostics in the context of your site & business. I tried to find an answer on the site, but couldn’t really locate one. If you could enlighten me on this topic I would really appreciate it.

A.:

What we call „Mathematical Gnostics“ (MG) is related to quantitative recognition of the reality. The goal is thus similar to that of mathematical statistics. But according to the approach of MG, it is agnostic in its resignation to search for the cause of uncertainty disturbing the observations or measuring. Instead, statistics creates its mathematical model of uncertainty based on the indeterministic notion of randomness, which is subjected only to the Law of Large Numbers. Unlike this, we consider the uncertainty as a lack of knowledge, which motivates us to improve the state of our information by further attempts to get closer to the true quantity masked by uncertainty. The uncertainty in our observations (data) is not something mysterious, but it is an effect of real causes, which can be revealed by further cognitive effort. The (e.g. physical) nature of uncertainty is thus as real as the nature of the true quantity we want to quantify. One of colleagues in Academy of Sciences formulated once a bon mot: „MG is a deterministic theory of indeterminism“. Anyway, MG is an approach opposite to the statistical agnosticism, it therefore deserves to be called gnostic.

We were frequently asked if there is a relation to the philosophical/religious Gnosticism. The answer was that there is no religious relation, but two important similarities:

  1. Relations of statisticians to MG appeared to be as hostile as that of the ancient „official“ Church to Gnostics. The MG as the consistent and robust mathematical theory of individual uncertain data is not acceptable for an orthodox statistician. Such a „strange“ paradigm mathematically supported by mathematics, which is different from that of used in statistics as well as its results, software, is not welcome in „orderly“ oriented statistical journals. But it can be shown, that this software outperforms the statistical functions in both robustness and efficiency.
  2. 2) We both are looking for the truth: the religious Gnostics for the essence of God, mathematical gnostics for the true value of an uncertain observation.

It may be interesting to recall the Kuhn’s criteria for judging a new paradigm: it should delimit the boarders of the old paradigm and reveal its hidden assumptions. MG satisfies this requirement by proving, that its estimating formulae converge to the classical statistical ones when the data uncertainty is very weak. It means, that statistics can be thought of as an approach valid for sufficiently „clean“/“good“ data and MG as a more general extension of statistics also valid for individual uncertain data and small samples of highly disturbed data . By the way, the Central Limit Theorem holds only under the assumption that the data to be treated have a mean and a variance. This condition is not universally satisfied.

Another Kuhn’s criterion relates to links of the new paradigm to the established scientific theories. MG proves close relations between uncertain data and Einstein’s relativistic mechanics, between data uncertainty and classical (Clausius‘) thermodynamic entropy. Moreover, it derives the equation of mutual conversion of information and entropy, of the problem long disconcerting the scientists (see von Baeyer H.Ch.: Maxwell’s Demon, Random House, Inc., New York, 1998).

Q.:

Could you characterize the most fundamental difference between mathematical gnostics and statistics?

A.:

The fundamental difference is of geometric nature. Metric of a space determines the way of measuring all distances and angles, the way of aggregating these quantities, the invariants of space transformations and extremals of the space, i.e. special lines along which some functionals are minimized or maximized.

Classical statistics is based on the Euclidean geometry: errors (distances) are evaluated by formulae linear with respect to data. Errors, their squares and products are aggregated additively, space shift and orthogonal rotations leave the relations between geometrical objects invariant and the extremal is the straight line connecting some points. All this is simply a tradition: there is no proof, that the actual space of data contaminated by uncertainty is just Euclidean.

Experience resulted from mass applications of statistics led to development of many statistical methods to improve the robustness by implementation of non-linear operations on data. Some special features on statistical properties of data were assumed resulting in special geometries applied. Such methods work when applied to the assumed kind of data but fail otherwise. This approach can be characterized as subjective.

Unlike this, methods of mathematical gnostics derive the metric suitable for some given data objectively by getting all necessary information from data only. The gnostic theory proves, that the space of real data is curved due to uncertain data components. The stronger uncertainty, the more considerable the curvature. The suitable geometry is of Riemann type, the curvature being determined by parameters estimated from the particular data. The curvature makes the data treatment not only linear, but also robust resulting in maximization of information content of results. In case of very weak uncertainty, the gnostic methods give results converging to those of classical statistics, because the Euclidean geometry is in such a case the proper one for the tangential plane.

© 2017,  Pavel Kovanic and Zdeněk Wagner