For centuries, mathematicians have tried, in various ways, to explain the foundations of mathematics so that math 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. Knowing at least one way to develop math without assumptions 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 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 indicate 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.
A number names an amount.
There are ancient words and phrases for amounts. If we had a single word for the phrase a single, as in a single thing, it would be a name of an amount. We do have such a word. The word is one. The sentence "I have one thing" indicates the number of things that I have. When used in this sense, the term one is the name of an amount; therefore one is a number. (One can observe that this term also has another meaning that is not of interest at the moment.)
There is also a single-word term for the phrase a pair, as in a pair of things. This word is the word two. If I say I have two things, the word two is being used as the name of the amount of things that I have. This makes it a number.
There is yet another common word that is the name of an amount. It is the word none.
If you are thinking "I will have none of this," please think again. Habits of English usage have placed none behind something else in the line of popular acknowledgement as a number.
Centuries ago, special single-character symbols called numerals were devised to stand for certain specific amounts. The Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) were introduced to most countries of the world by the 17th century. The 0 (in English it is spelled zero) was the last to be fully accepted as a numeral on a par with the other numerals. (This development does not yet tell us what most of these numerals mean.)
Eventually, 0 (or zero) firmly became the name of an amount. This amount also has the name none, but rarely do English-speaking people count with the words none, one, two. Nonetheless, and not withstanding historical confusions and verbal habits, none is the name of an amount.
Very often, the definition of numbers helps us answer questions like "How many plates are on the table?" Such questions do not involve directions along dimensions such as north versus south, right versus left, hot versus cold, or forces in opposite directions; but the need to answer such questions is so common that a system of numbers must address the reality of dimensions, each of which has exactly a pair of directions.
For the purpose of defining numbers, selecting a name for each of the directions in any dimension is thoroughly arbitrary. The world has almost universally settled on positive and negative. They could just as well be called some other pair of words. The name of the amount is a number. The direction in a dimension is an additional characteristic of the number. This often prompts a definition of a kind of number that has a magnitude (an amount) and a direction (east versus west; up versus down, positive versus negative, etc.).
A number always names an amount; and in any dimension, it names a magnitude in one of the two directions of the dimension.
The term equal is used in a few ways. If A is the name of an amount, and B is the name of that same amount, it is said that A equals B. This does not mean that these names are the same names. It means that both names refer to the same amount. This is the equality of a pair of amounts. If a collection contains all of the same things as another named collection, these collections are said to be equal, not because their names are the same, but because their names refer to the same collection. This is the equality of a pair of collections. When set is defined, the foregoing sentences will apply to the term set as well as it does to the undefined term collection.
Because this system does not assume the existence of infinity, the common roles of infinite sets are displace by the used of well-defined sets called unbounded sets. The common belief that calculous and other branches necessary to engineering cannot be developed without infinity is a false belief. Rates of change can be calculated through the use of unbounded sets.
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