Introduction To Metamathematicsby Stephen Cole Kleene, Michael Beeson
Pub. Date: 03/04/2009
Publisher: Ishi Press
Metamathematics is 'mathematics used to study mathematics', or it involves the application of a philosophy of mathematics. The first part of this general description appears tautological, or is perhaps open to Bertrand Russell's and Alfred Whitehead's types of antimonies (e.g., "the <set> of all sets is not a set"), as described in their famous "Principia Mathematica".
An alternative, non-circular definition is as follows: Metamathematics is the study of metatheories of standard theories in mathematics, or <theories> about mathematical--not 'purely logical'-- theories. Thus, in Encyclopædia Britannica, metatheory is defined as a "<theory> ,MT, the subject matter of which is another theory, T . A finding proved in the former (MT) that deals with the latter (T) is known as a metatheorem " (cited from Metatheory-Encyclopædia Britannica Online). Thus, a major part of metamathematics deals with: metatheorems, that is "<theorems> about theorems", meta-propositions about propositions, metatheories about mathematical proofs (that of course utilize logic, but also are based upon fundamental mathematics concepts), and so on.
Meta-mathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century, to focus on what was then called the foundational crisis of mathematics. Richard's paradox concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions which can easily occur if one fails to distinguish between mathematics and metamathematics. Bertrand Russell's and Alfred Whitehead's type of paradoxes is yet another important example of possible contradictions due to such failures in the 'old' set theory.
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