Review: Theoriae causalitatis principia mathematica

Theoriae causalitatis principia mathematica by Ilija Barukcic
My rating: 1 of 5 stars

Philosophiæ Naturalis Principia Mathematica, regarded as one of the most important works in the history of science, is Newton's opus on his laws of motion and universal gravitation. Principia mathematica is a similarly regarded achievement on the foundations of mathematics by Alfred North Whitehead and Bertrand Russell. Ambitiously, this Theoriae causalitatis principia mathematica (as capitalized by the author) has a similar lofty goal for causality through a set of axioms “to characterize the relationship between cause and effect while using the tools of probability theory.” From the preface comes the succinct aim: “This book is designed to provide both, a new mathematical methodology for making causal inferences from experimental and nonexperimental data and the underlying (philosophical) theory.”

At least some readers will find this presentation containing unfortunate obstacles to an effective perusal. Aristotle and Hume are quoted in German without translation. There are quotes in English and others are translated into “broken English”, as the author declares. Still, readers not multilingual will miss some meanings here and there. This inconsistency is an unnecessary inconvenience. Typographical difficulties also come in from extensive, lengthy 4-pt. footnotes, long italicized passages, unusual notation such as “+1” as a special value, and symbols often using underscores with left- and right-subscripts all at the same time. I had to resort to a magnifying glass to see that this definition of joint probability includes the statement p(dt)=p(ict.ꓵ jet). An example of opacity from the +-prefix notation is when contrasting “today’s rules of algebra” the awkward statement +1 – 1 becomes equivalent to +i2 + i2 during a definition of |0| as the “square root of zero”.

These aspects seem either unnecessarily complex or in need of an improved layout. Added to this are unclear connections between definitions and results. The introduced +1 is defined as the product of the speed of light in vacuum squared, magnetic constant (vacuum permeability), electric constant (vacuum permittivity) without discussion of units or why these particular physical constants should be combined in this way. Later, +1 is treated as a unit-less value and no further mention is made of this basis in physical constants making opaque the motivation to define +1 in this peculiar manner. Further, infinity is treated as it were a real number, or least subject to the typical operations of multiplication, division, etc. as if it were a real number in such claims as 1 x 0 = ∞, etc. The book makes no acknowledgement that infinity is being treated unexpectedly nor is the concept of limits brought in when expected. Combining this approach with introduced concepts of “anti-number”, “anti-finite”, etc. the reader is confronted with such equations as:

(+1)/(+∞)+((+∞-1))/(+∞)=+1

This author, who has been publishing in this area since at least 1989, bundles redefined operations and new notation into “proofs” based on applying algebraic operations to false statements in a way that appears to avoid the material conditional that a false statement can be used to support false or true statement. This will cause the eyebrow to arch for some. Here is one example, including some representative grammar:

Given
+2=+3
as the starting point of this proof, which is incorrect, we obtain immediately
2×0=3×0
or due to the definition before
2_0×∞=3_0×∞
Multiplying be positive infinity, we obtain
2_0×∞=3_0×∞
which is equal to
2×1=3×1
At the end, a logical contradiction does not follow since from something incorrect (+2 = +3) something incorrect is derived as
+2=+3
QUOD ERAT DEMONSTRANDUM

Some facts strike one as incorrect or at best not accurately stated. For example, “A definite value of Archimedes’ constant π is still not known…” Also, some definitions lead to apparent contradictions when later employed. What is most unsettling is that these issues are crowded around the core ‘Axioms’.
The Pythagorean Theorem gets introduced in what appears the typical, planar interpretation with hypotenuse c defined as “the longest side…” Later, Axiom II is stated:

(+0)/(+0)=+1

Support for this (here, all ‘axioms’ have supporting proofs) comes from “In general according to Pythagorean theorem it is a2 + b2= c2. Under conditions where b2= c2 it is a2 + c2= c2 or a2 = c2 - c2 = 0.” So, from this specific case apparently excluded by the stated definition, it follows from a normalized Pythagorean equation that this ‘axiom’ arises. Similarly, a Pythagorean case where “b2= c2 = +∞” contributes to a general ‘proof of Axiom III

(+1)/(+∞)=+0

This book contains a thin history of formal approaches to causality spotty fundamentals for probability, statistics, set theory, probability, and more. Much of this content comes across as needing an editor. For instance, the Lorentz factor is defined after variance and before Bernoulli trial. The basics presented are to support philosophical arguments. However, often the suddenness and unclear linkage weakens the presentation. For instance, preceding the definition of an equivalence relationship is the pronouncement:

EQUIVALENCE – THE UNITY OF IDENTITY AND DIFFERENCE
The one is not the other, the other is not the one, both are different. Still, there are circumstances where the one has within itself the relation to the other and vice versa. Two, which are different and separated in the same relation, are still united or identical. Thus far, difference as such is a kind of contradiction for it is the unity of two which are not one.”

This is a typical arrangement here: some mathematics lies adjacent to a philosophical statement with a connection not clear to this reader. The evolution from arithmetic to philosophy of mathematics is missing some steps resulting in steep jumps. The result is a series of assertions, not developments; a gallery of fundamentals loosely coupled to a philosophical worldview. Along the way, other generally accepted principals are rejected: The gravity constant is not a constant and fuzzy logic is refuted. However, while such things are taken away, this does add to world’s store of proofs of the existence of God: “The notion of God is mathematically justified only if God is equivalent with nature or objective reality itself.”

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