14. Convenient abbreviation

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

Convenient abbreviation

The following definitions give alternative and often more convenient notations:

φx . É . yx : = : (x): φx . É . yx  Df,

φx . º_x . yx : = : (x) : φx . º . yx  Df.

This notation " φx . É_x . yx " is due to (19^th century Italian mathematician) Peano, who, however, has no notation for the general idea " (x) . φx." It may the general idea " (x) . φx." It may be noticed as an exercise in the use of dots as brackets that we might have written

φx É_x yx . = . (x) . φx É yx  Df,  

φx . º_x yx . = . (x) . φx º yx  Df.

In practice however, when φx^ and yx^ are special functions, it is not possible to employ fewer dots than in the first form, and often more are required.

The following definitions give abbreviated notations for functions of two or more variables:

(x, y) . φ(x, y). = : (x): (y). φ(x, y)  Df,

and so on for any number of variables;

φ(x, y) . É_xy . y(x, y) : = : (x, y) : φ(x, y). É . y(x, y)  Df,

and so on for any number of variables.

 Script:Richard Hughes Pencils and inks: Reed Crandall  © Respective copyright holders.

The irony of definitions in this system is that they are actually meaningless in terms of applying this process in thought or expression, still they seem to be most important starting out, and are the focus of much attention. Just the same with everything offered in this work; and indeed all metaphysical, theological, philosophical or mathematical works, fields or systems are very much the same in nature. One studies them, learns their nature and process, but leaves the text behind in favor of the work of the mind. Barring teachers and preachers of course, these inevitably acquire attachments and agendas within their subjects, and feel a need to direct a straightforward and nearly always neutral system of thought into directions not necessarily intended by the authors.

Principia Mathematica was devised by Whitehead and Russell as a system of categorizing and prioritizing propositions so that they may be treated and combined in equations so that complex and longwinded strings of often diverse information can be expressed concisely, and in a way that error is easily detected within the structures of the equations. After publication of the first edition of this work, the authors received a great deal of feedback, and agreed that a number of useful changes were offered that, not pointed out errors in reasoning, but simplifications of the process offered. All along they predicted that their work must evolve, but were confident that these changes would be of citations of improvement, rather than error.

Script: Albert B. Feldstein  Pencils and Inks: Joe Orlando  © Respective copyright/trademark holders.

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.