The Nature of Order in Analysis of Asymmetrical Relations - Essay on Math

In the book Principles of Mathematics, Bertrand Russel (1903) argues that the concept of order should be given priority when analyzing asymmetrical relations, and that general logic should be disregarded in the reduction of non-monadic to monadic relations to reduce, or altogether avoid contradictions. First, though, he acknowledges that all order in mathematics depends on transitive asymmetrical relations, but also points out the stark differences between them and the traditional system of logic. He makes a claim that the refusal of the traditional system of logic to admit the very nature of transitive asymmetrical relations is one of the factors that led to the rise of contradictions in mathematics. Thus, Russel sets out to demonstrate why traditional systems of logic are unsuitable for the analysis of asymmetrical relations and the need to do away with them completely in the process of reducing non-monadic (monistic) relations to monadic relations

Russel used two primary attributes; transitiveness and symmetry to group various relations into four broad classes. He then used three factors, x, y, z and an operator R to define the different relationships as follows (Russel, 1903):

Symmetrical relations, being those where the occurrence of xRy always implies that yRx is also true and of equal value. Any relation that does not satisfy this requirement is termed as not-symmetrical.

Transitive relations, being those where the values of xRy and yRz, when considered together, always implies a resulting value xRz. All relations that do not conform to this requirement are designated not-transitive.

Asymmetrical relations, being those where the occurrence of xRy always excludes the possibility of yRx. In essence, these are mutually exclusive possibilities

Intransitive relations where the occurrence of xRy and yRz together always excludes the possibility of xRz. These are also mutually exclusive possibilities.

A further illustration of these relationships was provided using examples from a social perspective whereby brother was classified as a transitive relationship but not symmetrical, or transitive if a person could be his brother. This is in direct contradiction of general logic which maintains that a person cannot be his brother. Similarly, the spouse was classified as an intransitive but symmetrical relationship, half-sister was classified as symmetrical but not transitive and half-brother classified as a symmetrical relationship which is not transitive, and yet at the same time neither symmetrical nor transitive (Russel, 1903). This draws a stark contrast with normal logic in the sense that if something is symmetrical, then it automatically excludes all other possibilities that would negate that quality and render it as not-symmetrical while it is in the same state.

Monastic theory and quantitative relations

To further exemplify his argument, Russel mentions the proposition of a variable, A, being greater than another variable, B. He notes that if A and B were assumed to be quantities, then a suitable adjective to describe them would be their magnitude. However, for A to be greater than B would necessitate that their magnitudes be mutually exclusive, which necessarily results in an asymmetrical relation similar to the relation between the original quantities and an infinite process of analyzing the relation (Russel, 1903). As a solution, therefore, he proposes that only adjectives that have a mutual reference be used for such an analysis to ensure an asymmetrical relation ensues. He also suggests that at least one of the selected adjectives should be intrinsic, that is, should have an indication of complexity with which to analyze the other terms (Russel, 1903).

Monadistic theory and asymmetrical relations

Russel (1903) points out that identity and diversity are both symmetrical. The import of this is that the mere existence of an identity implies that there could be several other items of a similar or dissimilar nature that need to be differentiated, hence diversity. In a different illustration using two variables again, a and b, an operator, R, and two adjectives a and b to describe the former and latter variables respectively, he demonstrates the asymmetrical relation that results when an analysis of the relations aRb and bRa is attempted. Since the adjectives a and b have a mutual reference to each other but no intrinsic difference corresponding to the operator R, Russel argues that the difference between the adjectives cannot be used to determine an intrinsic difference between the variables a and b because it still results in a comparison devoid of an indicator of complexity in either term. To ensure an indicator of complexity. Thus, he argues that each element of a relation should be treated as a discrete part, rather than resolving it into smaller relations which, although making up the whole, are each considered to have internal diversities which lead to an infinite cycle of resolution of relations during analysis. In fact, he asserts that such contradictions are borne out of general philosophy and should therefore not be considered in an analysis of asymmetrical relations. Russel holds that the monistic theory is flawed because in proposing that each entity has an internal diversity within itself which when split synthesizes diversity, it gives rise to contradictions on all propositions.

Russel proposes that the notion that all relations have some intrinsic diversity be discarded wholly to achieve a satisfactory philosophy of mathematics (Russel, 1903). He postulates that the logical puzzles posed by the said notion are in fact man-made. Once again, he uses the example of simultaneity and equality to illustrate his point. Equality and simultaneity are both considered transitive and symmetrical. Both can be reduced to achieve a so-called identity of content, which in turn is to be analyzed on another term to establish similarity in their relation. He argues that the properties of one term are in fact just other terms to which the term in context is related, and if two or three terms share a common property, then that property is also just one term to which all the other terms are related. To avoid this cyclic redundancy thus, Russel argues that in the analysis of asymmetrical relations, terms should be considered as discrete because all relations are all based on time, motion, number, quantity, space, and order. Even in general logical terms, his argument is validated simply by the savings of time and effort that can be realized by avoiding an infinite cycle of resolving relations in an analysis.

Russel makes the conclusion that order is of utmost importance in the analysis of asymmetrical relations. Since the concept of order is mutually exclusive to established precepts of general philosophy, the latter must be abandoned utterly before any satisfactory explanation can be rendered (Russel, 1903). Specifically, he mentions that traditional logical thinking fails to reconcile the concept of a difference of sense and the concept of asymmetry, which are the fundamentals of order.

Reference

Ch 26 Asymmetrical Relations Principles of Mathematics, Bertrand Russel N.p., n.d. Web. 09 Apr. 2017