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:
“Two clergymen disputing whether ordination would be valid without the imposition of both hands, the more formal one said, “Do you think the Holy Dove could fly down with only one wing?””
—Horace Walpole (1717–1797)
“When the finishing stroke was put to his work, it suddenly expanded before the eyes of the astonished artist into the fairest of all the creations of Brahma. He had made a new system in making a staff, a world with full and fair proportions; in which, though the old cities and dynasties had passed away, fairer and more glorious ones had taken their places.”
—Henry David Thoreau (1817–1862)