An Introduction to Axiom Systems

Philosophy - "1. (a) Love and pursuit of wisdom by intellectual means and moral self-discipline. (b) The investigation of causes and laws underlying reality. (c) A system of philosophical inquiry or demonstration. 2. Inquiry into the nature of things based on logical reasoning rather than empirical methods. 3. The critique and analysis of fundamental beliefs as they come to be conceptualised and formulated. 7. The science comprising logic, ethics, aesthetics, metaphysics, and epistemology. 10. The system of values by which one lives: a philosophy of life."(1)


Definition of Axiom Systems

Proposition - "(Logic) a. A statement in which the subject is affirmed or denied by the predicate; b. Something that is expressed in a statement, as opposed to the way it is expressed; c. A statement containing only logical constants and having a fixed truth-value."

Postulate - "1. Something assumed without proof as being self-evident or generally accepted, especially when used as a basis for an argument; 2. A fundamental element; a basic principle; 3. (Mathematics) An axiom; 4. A requirement; a prerequisite."

Axiom - "A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate."

Hypothesis - "1. A tentative explanation that accounts for a set of facts and can be tested by further investigation; a theory; 2. Something taken to be true for the purpose of argument or investigation; an assumption."

Theorem - "1. An idea that is demonstrably true or is assumed to be so; 2. A proposition that has been or is to be proved on the basis of explicit assumptions."

Theory - "1. a. Systematically organized knowledge applicable in a relatively wide variety of circumstances, especially a system of assumptions, accepted principles, and rules of procedure devised to analyze, predict, or otherwise explain the nature or behaviour of a specified set of phenomena. b. Such knowledge or such a system; 3. A belief that guides action or assists comprehension or judgment."

Law - "a. A formulation describing a relationship observed to be invariable between or among phenomena for all cases in which the specified conditions are met; b. A generalization based on consistent experience or results."

Logic - "1. The study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning; 2. a. A system of reasoning. b. A mode of reasoning."

Logical - "1. Of, relating to, in accordance with, or of the nature of logic; 2. Based on earlier or otherwise known statements, events, or conditions; reasonable; 3. Reasoning or capable of reasoning in a clear and consistent manner."

Reason - "(verb) 1. To use the faculty of reason; think logically; 2. To talk or argue logically and persuasively."

It is unfortunate that the introduction to a book on Philosophy demands so many definitions at one time. You will find the chapters of this text littered with numerous definitions for otherwise commonly used terms. I feel it is necessary if I am to adhere to a commitment to define my terms before I use them. I am convinced that many philosophical disagreements and argument result from a difference in the understanding of what is meant by the words employed to communicate.

"I know that you believe you understand what you though you heard me say.
But what I don't think you realize is that
what I said is not what I meant."  
- Anonymous

Another of the great causes for argument (and not merely philosophical argument either), stems from real or imagined or misunderstood differences in the basic assumptions upon which a discussion is based. Two people will find it very difficult to communicate productively, if they do not share a common language. As a Canadian, I would find it impossible to communicate with a Russian, for example, if we shared no common language. Especially if the communications was taking place over the telephone, where even sign language or body language would not be possible. The trouble starts in many arguments, when the parties to the discussion appear to speak the same language. Consider English, for example. Many pundits suggest, not entirely in humour, that the greatest difficulties in Anglo-American relations stem from the fact that the two nations speak what is supposed to be the same language. The problem of missed communication is hidden because everybody is using the same words. Just because the audio sounds are the same, everybody assumes that the meanings are the same. It is frequently discovered, after much fruitless argument and disagreements, that the two "opposing" positions in the argument are not as far apart as was initially believed. When the actual meanings are made clear, the fog created by the words used can be more easily cleared. This is one reason why good diplomats and mediators are so often successful. They make a special effort to get past the words to the meaning.

For this reason, the introduction to Evolutionary Pragmatism must begin with detailed definitions of the meanings behind the words being employed. But before getting into the philosophical discussion, I think it would be advantageous to digress a little, and talk a bit more about the nature and consequences of Axiom Systems.

The Impact of "A Priori" Postulates

To prevent any misunderstanding, in the discussion that follows, the terms "Starting axioms", "a priori postulates", "initial postulates", and "underlying basic assumptions" are going to be used relatively interchangeably. They all mean roughly the same thing. They all refer to the basic starting propositions upon which the rest of a discussion or logical analysis is based. Frequently, these starting positions are left unsaid. And there in lays the problem. Since they are left unspoken, the assumptions that you start with, may not be the same assumptions that I start with. In most cases, this may not cause any more than temporary difficulties. But in the case of the development of an entire system of philosophical argument, the starting points are critical.

Every system of reasoning has a set of basic axioms. Mathematics (of which the discipline of Deductive Logic is a part) is the only reasoning system that makes it a basic fundamental rule to detail and document the axioms before starting the reasoning. But all systems of reasoning do have their axioms.

The most fundamental and significant aspect of starting axioms, is that they are not "provable". Axioms are axioms because there is no way to derive them from more fundamental principles, and no way to "prove" them by using information drawn from experience. Axioms, by their nature, can be neither 'deduced', nor 'induced'. If there were a way to deduce them from other principles, then those other principles would become the axioms, and the statement that was just proved, would become one of the theorems or consequences of the more basic axioms. If there was a way to induce them from experience, then they would become a 'law' in the sense of the definition provided at the start of this chapter, rather than an axiom.

It is also in the nature of starting axioms, that the structure of any reasoning built upon them is intimately dependent upon them. To use an example from Mathematics, let us consider the field of Geometry. Most of us are familiar with the geometry of the plane, called "Euclidean Geometry"(2). Many of us studied this geometry in high school. I am sure that many of you can remember some of the Theorems that you laboured to prove during the course of your studies. Remember the struggle to use a set of limited deductive rules to prove a "New Idea"? Do you recall the geometric proof that the interior angles of any triangle (drawn on an Euclidean plane) sum to 180 degrees?

All of Euclidean Geometry is based on 5 basic assumptions that Euclid made about the nature of the plane upon which he played his geometry game. It also makes use of a carefully defined and quite limited set of rules of logical deduction. By employing these rules, new Theorems can be deduced from the basic axioms. Once deduced, these theorems can be used as part of the proof of another new Theorem. In this way, an entire set of knowledge about lines and polygons can be developed. All from a set of deduction rules, and 5 basic axioms.

Although most people are at least passing familiar with Euclidean Geometry, not many are familiar with Riemannian Geometry(3). Riemann (Georg Friedrich Bernhard Riemann, 1826-1866) was a mathematician of the 19th century. He examined Euclid's five basic axioms and decided to see what would result if he modified them. The result of his "relaxation" of only a single one of Euclid's five axioms was his development of Riemannian Geometry. Euclid's 5th axiom specified (approximately, in readable English)

'through any given point, only one line can be drawn parallel to another line"

What Riemann did, was eliminate this axiom, and allow a variable number of lines to be drawn parallel to a given line. When he examined the result of this change, he found that there were only three answers that would result in a consistent set of Theorems. (The importance of consistency we will examine later.) Riemann found that through any given point, zero, one or an infinity of lines can be drawn parallel to another line. What he had discovered, was that there is a different kind of geometry, that yields different theorems, and different answers to simple questions, depending upon the detailed specification of the "Fifth Axiom". These geometries together form the body of mathematics known as Riemannian Geometry. Euclidean Geometry is now understood by mathematicians to be a "Special Case" of Riemannian Geometry. That is to say, by setting a free variable (the number of lines that can be drawn parallel to another - more properly expressed as the curvature of the plane) to a specific value, Riemannian Geometry looks like Euclidean Geometry. The difference is that to Euclid, Euclidean Geometry was all there was, while to modern mathematicians, the geometry of the plane is but one case of the more generalized Riemannian Geometry. Riemannian Geometry is said to "contain" Euclidean Geometry. The fact that Riemannian Geometry contains Euclidean Geometry, does not make Euclidean geometry wrong. Given the starting assumptions of Euclidean geometry, the rest of the discipline is properly deduced, and self-consistent. It is merely that the starting assumptions have a determining impact on the resulting structure.

Two aspects of this comparison between Euclidean and Riemannian Geometry are of significance to our discussion of starting axioms. The first thing to understand from this example, is that the resulting body of knowledge deducible from the starting axioms, even following the same rules of deduction, can look quite different as a result of seemingly small changes in the axioms. The second important thing to understand, is that the answers that the two Axiom Systems give to seemingly simple questions can be quite different. To an Euclidean geometer, the sum of the interior angles of a triangle is a constant (180 degrees). To a Riemannian geometer, the sum is a function of the curvature of the plane. Since the Euclidean geometer cannot even comprehend that the plane can curve, the two cannot properly hold a meaningful discussion on this subject. It is but one example of the confusion that results when two people attempt to communicate, but do not share a common ground. They are building on different postulates.

So too with philosophy. Unless there is an agreement on starting postulates, philosophical discussion will be fruitless. The necessary agreement need not be complete, merely sufficient to allow a common basis upon which to found the discussion. The two geometers could, for example, agree to discuss the geometry of the plane. The Euclidean geometer would regard this as a discussion of "The whole shebang", while the Riemannian geometer would regard it as a discussion of only a special subset of "The whole shebang". This might cause some confusion if the argument digressed into the nature of "The whole shebang", but would be sufficient to allow specific discussion on, say, the sum of the interior angles of a triangle.

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(1)   Unless otherwise specified, all dictionary definitions are quoted from The American Heritage® Dictionary of the English Language, Third Edition copyright � 1992 by Houghton Mifflin Company.

(2)   Coxeter, H.S.M.   Introduction to Geometry, 2nd Ed.   John Wiley & Sons; 1989.

(3)   Farkas, Hershel M.; Kra, I.   Riemann Surfaces, 2nd Ed. (Graduate Texts in Mathematics, Vol 71).   Springer Verlag. 1992.