Formalized Account of Theorems
A theorem may be expressed in a formal language (or "formalized"). A formal theorem is the purely formal analogue of a theorem. In general, a formal theorem is a type of well-formed formula that satisfies certain logical and syntactic conditions. The notation is often used to indicate that is a theorem.
Formal theorems consist of formulas of a formal language and the transformation rules of a formal system. Specifically, a formal theorem is always the last formula of a derivation in some formal system each formula of which is a logical consequence of the formulas which came before it in the derivation. The initially accepted formulas in the derivation are called its axioms, and are the basis on which the theorem is derived. A set of theorems is called a theory.
What makes formal theorems useful and of interest is that they can be interpreted as true propositions and their derivations may be interpreted as a proof of the truth of the resulting expression. A set of formal theorems may be referred to as a formal theory. A theorem whose interpretation is a true statement about a formal system is called a metatheorem.
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Famous quotes containing the words formalized and/or account:
“In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
—Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)
“It is an agreeable change to cross a lake, after you have been shut up in the woods, not only on account of the greater expanse of water, but also of sky. It is one of the surprises which Nature has in store for the traveler in the forest. To look down, in this case, over eighteen miles of water, was liberating and civilizing even.”
—Henry David Thoreau (1817–1862)