Math form Words
Chapter One - Introduction
James Adrian
      At an early age, and by rote, most of us learned to count and calculate. We can tally our earnings and make change. The world agrees with our conclusions. All is right with the universe; but where did these methods come from?

      For centuries, mathematicians have tried, in various ways, to explain the foundations of mathematics so that it could be developed further at any time to meet any new quantitative challenge. The present development of math is yet another way. It makes a contribution that otherwise would be absent:

      Axioms and postulates are assumptions. Calculus and complex numbers can be developed without them. Without at least a single axiom or postulate, infinity cannot be mathematically established. If there are desirable results that depend on axioms, first knowing at least one way to develop math without axioms is an aid to the development of an individual's mathematical reasoning.

      In this development, the root source of mathematical meanings is the common language. Our languages describe the most basic features of the world. Every human language includes many words that form and represent the foundation of math. The meanings of the most basic of these words are learned through repeated exposure associating them with the context in which they are used. Please contemplate the following sentence:

      A pair of things is a single thing together with another single thing.

      This sentence is a definition of the term pair of things. Of course, we don't need that definition because we all know, from our experience, what pair of things means.

      This definition is very understandable, partly because we know the meanings of thing, single thing, and together with.

      Throughout this writing, the term thing is intended to be thoroughly indefinite. An idea, an object, a sentence, a mark, an action, a description, a time, a location, or anything else can be referred to as a thing.

      Here, the term item is used as a synonym for the term thing.

      A rational number is the quotient of two integers, p/q, where q is not zero; but this statement contains a few terms that have not yet been defined. A formal development must present meanings in an order that allows every newly defined term to be defined by terms that are already understood.

      Any statement that is offered as a formal definition (as in science, math, law, and other disciplines) must have all of the following properties:

1.   A definition must name the term being defined and provide a description of that term.

2.   The term being defined must not appear in the description of that term. Using the term being defined in the description of that term is circular.

3.   Other than the term being defined, the meaning of each term used in a definition must be known to the audience before that definition is stated.

4.   The term being defined and the description of that term must be interchangeable. It must be clear that the term and its description have exactly the same meaning.

      Here is an example: A hydrocarbon is a compound consisting only of carbon and hydrogen.

      "A hydrocarbon"is the term being defined, and "a compound consisting only of carbon and hydrogen" is the description. Other sentence structures are possible, but both the term being defined and its description must be somewhere in the statement.

      The term "hydrocarbon" does not appear in the description.

      The meaning of each of the terms in the description are known before the definition is composed.

      The term being defined and the description of that term are interchangeable. They mean the same thing.

      A term is well defined if and only if it is described by a statement that has all of the properties required of a formal definition.

      A formal system cannot consist of formal definitions alone. In such a hypothetical case, there would be no formally defined terms with which to formally define the first term. It is the learning that we do by conditioning, trial and error, repetition, and association with contexts, that allows us to clearly understand words like "If," "the," "we," "were," "to," "accept," "this," "as," and "true." These words, and a few thousand others, are learned through experience. They may be called empirical terms. They are called undefined terms.

      In a formal context, the meaning of each undefined term must be widely known for its intended meaning. If a word has more than one meaning, then the context, or a specific explanation preceding its use, must clearly select the intended meaning.

      The common language gives us many terms that are mathematical in nature.

      You can tell the difference between a single thing and a pair of things as surely as you can tell night from day or red from blue. Children perceive, reason, and communicate with such distinctions as soon as they learn which words other people use to refer to them. We all have the ability to perceive a single thing, a pair of things, and many things. The terms amount, quantity, more than, less than, at least, no more than, at most, any, some, collection, and few refer to perceptions that we share. These terms are too basic to be defined by more basic terms. These and related meanings are the starting point for mathematical definitions. These terms refer to empirical observations.

      There is a centuries-old reluctance among mathematicians to use the passage of time as an element in the proof of mathematical statements. However, this development recognizes our experience with time and includes terms such as before and after as empirical terms. The order of time is thoroughly conspicuous. Events of any nameable kind come to pass in an order, with some events happening before others, and other events coming next in order relative to the event preceding it. Like the terms mentioned earlier, they have meanings that are too basic to be described by more basic terms and their attempted formal definitions, are circular. These terms can be used in formal definitions because they refer to reality in ways that are beyond dispute.

      Perception of order occurs in our perception of space. Rocks placed in a line give us next and immediately previous or immediately neighboring rocks.

      There are many kinds of order. We often refer to an order of succession, or an order of authority, or an order of importance, and other types of order. Defining mathematical entities as having order, or having an order, or having been ordered, becomes more straight forward if these meanings (that we know so well) are acknowledged.

      This is a system of formal definitions in which rational numbers play a dominant role. The number line consists of rational numbers together with procedures that produce rational numbers. For instance, π would be the name of a procedure that stands for the circumference of a circle divided by its diameter. A number on the number line is either a rational number or a procedure.