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 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 about mathematical--not
purely logical'-- theories. Thus, in Encyclop dia
Britannica, metatheory is defined as a ,'' 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-Encyclop dia Britannica Online). Thus, a
major part of metamathematics deals with: metatheorems,
that is '' 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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