Mathematics is a language. Like any other language it is written by combining characters into words and words into grammatical sentences. In this post we use plain English to describes this grammar and situations where the language can be useful.
Mathematics make statements about elements of a Set. It does not matter what Sets are exactly. What matters is presence of one or many compositions. A Compositon is ability to use elements, to produce references to other elements. Compositions can be unary, binary, ternary, e.t.c. depending on how many elements we start with. A Mathematical statement is an assertion that a given element can be referenced by applying compositions in different ways. Few of these statements are designated as fundamental rules. Others are derived from these rules by logical reasoning. To formulate a statement we first assign names (labels) to the elements involved. By convention, the names are short, often 1-letter long, with optional numeric subscripts or accents. The statement is a pair of expressions separated by the = sign. Each expression is a sequence of element names separated by some fixed, non-alphanumeric, symbols identifying compositions and delimited by parentheses to clarify the order in which the compositions are applied. Sometimes the symbol and/or the parentheses can be inferred from the context and skipped. For example we may agree that one composition has precedence over another. The number of compositions used in a particular theory is usually quite small and the selection of the symbol often carries a hint about the fundamental rules in-effect. Some fundamental rules are valid universally. Sometimes we postulate presence of ‘special’ elements or subsets and of certain rules that apply only to them. Sometimes we may derive additional ways to combine elements stated in terms of the compositions specified initially. We may then assign additional symbols to these ‘secondary’ compositions.
Where do Composition Rules come from?
Sometimes we are guided by simplicity and elegance. Here are the most common examples of fundamental rules used in binary compositions
- Postulates that the order in which elements are written does not matter
- Postulates that placements of parentheses does not matter
- neutral element
- Postulates presence of an element which, when used in the composition, does not produce anything new. (example of a ‘special’ element)
- inverse element
- Postulates that when an element is a result of the composition then the knowledge of one of the elements used implies the knowledge of the second. If the result of the composition is fixed to be the neutral element then that second element is called the inverse of the first (example of a derived unary composition)
- Postulates that, if we are dealing with 2 compositions, it does not matter if we first combine 2 elements using one rule and then combine the result with a third element or if we first combine the third element with each of the 2 elements using the second rule and then combine the results using the first rule
- Shorthand for a combination of the following postulates: composition is associative, has a neutral element and every element has an inverse.
Sometimes we take a cue from Real Life. We may encounter a category of objects which allow us to manipulate them to produce other objects. Applying these manipulations in different order may sometimes produce identical outcome. A statement to that effect is an experimental fact, a law of Nature if you will. But that statement, written more concisely using letters and symbols as described above, can be used to define a mathematical theory. We then refer to the objects as if they were elements of a set. There is a difference of emphasis. In an experiment often new objects are produced. In Mathematics, the result of a composition is a reference to an element already there. It is as if the set, which the theory talks about, is an outcome of unlimited applications of experimental compositions.
In the following we will show few examples. In each we will start with formal definition of a theory and then give examples of Natural compositions which satisfy fundamental rules of the theory.