Metamathematics

To oversimplify the definition of the term metamathematics, one can say it is the logical analysis of mathematical reasoning. It is used as a tool in the study of mathematics. The study of metamathematics dates back to the 19th century when it was differentiated from regular mathematical study in that it focused on what was then called foundation problems in mathematics. There are many sub-branches associated with metamathematics such as model theory, mathematical logic and mathematical theory and it is synonymous with various parts of propositional logic, predicate logic and formal logic.

Metamathematics is the application and analysis of mathematics using mathematical methods that produce metatheories, which are types of mathematical theories about other types of mathematical theories. Metamathematical metatheories about mathematics are different from mathematical theorems, which focus upon the foundational crisis of mathematics. In 1905, the theory called Richard's paradox was offered to explain the phenomenon. It concerned itself with contradictions that can occur if one cannot distinguish and pinpoint the exact difference between metamathematics and ordinary mathematics.

During the late 19th and early 20th centuries, metamathematics and mathematical logic were closely interconnected with many overlapping theories. However, as the study of metamathematics became more developed in the late 20th century the gap between the two widened. Mathematical logic centered around and included the study of pure mathematics, with theories that include recursion theory, model theory and set theory which in no way are related to metamathematics. When studying and analyzing well-accepted principles and derivation rules associated with set theory, that is what is known as mathematics; when looking for acceptable principles and derivation rules associated with set theory, that is what is known as metamathematics. The total collection of facts about the universe, regardless of whether they fall under the category of mathematics or metamathematics, is referred to as reality. The one factor that distinguishes man from other species is the capability of rational thinking and the best way to utilize that capacity of rational thinking is in the study of metamathematics.

There are many issues with regard to the foundations of ordinary mathematics and the philosophy of mathematics that touch upon or borrow ideas from metamathematics. The working hypothesis of metamathematics is that all mathematical material and content can be encapsulated into a formal system. On the other hand the cognitive science of mathematics, ethno-cultural studies of mathematics and the quasi-empiricism in mathematics are non-mathematical ways in which one should study mathematics. The reason is that those genres focus on empirical methods and mathematical practice.

Formalism is the name given to various philosophical views with respect to mathematics. These philosophical rules focus on the magnitude to which mathematical proof can be understood or modeled as a succession of mechanical rules about the sequence of numbers. As far as the philosophy is concerned, the formula may have no meaning at all. The aim is supply a traceable origin for mathematics while avoiding an assurance of the presumption of doubtful principles.

Those who oppose formalism claim that mathematics is generally informal and they describe it as non-mechanical. They claim that the language of mathematics has meaning and it is a gross exaggeration to think otherwise. At best, the theory of formalism places emphasis on a small aspect of mathematics, intentionally leaving out what is essential to the study. One type of formalism which is known as "game formalism" claims that the essence of mathematics is all about following meaningless rules. They compare mathematics to the game of chess, where the numbers written down on a piece of paper play the part of the pieces to be moved. All that has to be done is to follow the rules correctly.

Many of the formalist programs that exist today had their beginnings in the early 20th century. Deductive systems and formal languages were formulated with mathematical accuracy and the actual systems became part of mathematical study. Those efforts and studies later became known as metamathematics, which goes beyond just following meaningless rules. Its aim and purpose is to shed light on subject matter, such as deductive systems and formal language. Therefore, at this point a game formalist would not disagree, or else would believe that metamathematics is not mathematics. However, not all who consider themselves formalists are game formalists.

Metamathematics: Selected full-text books and articles

Logic, Semantics, Metamathematics: Papers from 1923 to 1938
Alfred Tarski; J. H. Woodger.
Clarendon Press, 1956
Recursion Theory for Metamathematics
Raymond M. Smullyan.
Oxford University Press, 1993
Metamagical Themas: Questing for the Essence of Mind and Pattern
Douglas R. Hofstadter.
Basic Books, 1985
Librarian’s tip: Chap. 13 "Metafont, Metamathematics, and Metaphysics: Comments on Donald Knuth's Article 'The Concept of a Meta-Font' "
A Survey of Mathematical Logic
Hao Wang.
Science Press, 1963
Librarian’s tip: "Metamathematics" begins on p. 97
The Philosophy of Mathematics Today
Matthias Schirn.
Clarendon Press, 1998
Librarian’s tip: "Metamathematics in Philosophy" begins on p. 58 and "Hilbert's Conception of Metamathematics" begins on p. 273
Natural Deduction: The Logical Basis of Axiom Systems
John M. Anderson; Henry W. Johnstone Jr.
Wadsworth Publishing, 1962
In the Light of Logic
Solomon Feferman.
Oxford University Press, 1998
Librarian’s tip: Part IV "Proof Theory"
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