A Formal System
In general, in mathematics a formal system or "formal theory" consists of "objects" in a structure:
- The symbols to be concatenated (adjoined),
- The formation-rules (completely specified, i.e. formal rules of syntax) that dictate how the symbols and the assemblies of symbols are to be formed into assemblies (e.g. sequences) of symbols (called terms, formulas, sentences, propositions, theorems, etc.) so that they are in "well-formed" patterns (e.g. can a symbol be concatenated at its left end only, at its right end only, or both ends simultaneously? Can a collection of symbols be substituted for (put in place of) one or more symbols that may appear anywhere in the target symbol-string?),
- Well-formed "propositions" (called "theorems" or assertions or sentences) assembled per the formation rules,
- A few axioms that are stated up front and may include "undefinable notions" (examples: "set", "element", "belonging" in set theory; "0" and " ' " (successor) in number theory),
- At least one rule of deductive inference (e.g. modus ponens) that allow one to pass from one or more of the axioms and/or propositions to another proposition.
Read more about this topic: Object Theory
Famous quotes containing the words formal and/or system:
“True variety is in that plenitude of real and unexpected elements, in the branch charged with blue flowers thrusting itself, against all expectations, from the springtime hedge which seems already too full, while the purely formal imitation of variety ... is but void and uniformity, that is, that which is most opposed to variety....”
—Marcel Proust (1871–1922)
“Columbus stood in his age as the pioneer of progress and enlightenment. The system of universal education is in our age the most prominent and salutary feature of the spirit of enlightenment, and it is peculiarly appropriate that the schools be made by the people the center of the day’s demonstration. Let the national flag float over every schoolhouse in the country and the exercises be such as shall impress upon our youth the patriotic duties of American citizenship.”
—Benjamin Harrison (1833–1901)