Russell, Whitehead - Principia Mathematica to 56

Metamathematics is the study of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

attempt An attempt to commit a crime occurs if a criminal has an intent to commit a crime and takes a substantial step toward completing the crime, but for reasons not intended by the criminal, the final resulting crime does not occur.''Criminal Law - ...

to secure the

foundations of mathematics Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...

in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

" (Kleene 1952, p. 59). An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.

History

Metamathematical

metatheorem In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory ...

s about mathematics itself were originally differentiated from ordinary

mathematical theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the ...

s in the 19th century to focus on what was then called the

foundational crisis of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathem ...

Richard's paradox In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully betwee ...

(Richard 1905) concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions that can easily occur if one fails to distinguish between mathematics and metamathematics. Something similar can be said around the well-known

Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...

(Does the set of all those sets that do not contain themselves contain itself?). Metamathematics was intimately connected to

mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

, so that the early histories of the two fields, during the late 19th and early 20th centuries, largely overlap. More recently, mathematical logic has often included the study of new pure mathematics, such as

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

recursion theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since e ...

and pure

model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

, which is not directly related to metamathematics. Serious metamathematical reflection began with the work of Gottlob Frege, especially his '' Begriffsschrift'', published in 1879.

was the first to invoke the term "metamathematics" with regularity (see Hilbert's program), in the early 20th century. In his hands, it meant something akin to contemporary

proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four correspo ...

, in which finitary methods are used to study various axiomatized mathematical theorems (Kleene 1952, p. 55). Other prominent figures in the field include Bertrand Russell, Thoralf Skolem, Emil Post, Alonzo Church, Alan Turing, Stephen Kleene, Willard Quine, Paul Benacerraf,

Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century. He made significant contributions ...

, Gregory Chaitin, Alfred Tarski and

Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...

. Today, metalogic and metamathematics broadly overlap, and both have been substantially subsumed by mathematical logic in academia.

Milestones

The discovery of hyperbolic geometry

The discovery of hyperbolic geometry had important philosophical consequences for metamathematics. Before its discovery there was just one geometry and mathematics; the idea that another geometry existed was considered improbable. When Gauss discovered hyperbolic geometry, it is said that he did not publish anything about it out of fear of the "uproar of the Boeotians", which would ruin his status as ''princeps mathematicorum'' (Latin, "the Prince of Mathematicians"). The "uproar of the Boeotians" came and went, and gave an impetus to metamathematics and great improvements in

mathematical rigour Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as m ...

, analytical philosophy and

Begriffsschrift

''Begriffsschrift'' (German for, roughly, "concept-script") is a book on

by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notation''; the full title of the book identifies it as "a

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...

language, modeled on that of

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

, of pure

thought In their most common sense, the terms thought and thinking refer to conscious cognitive processes that can happen independently of sensory stimulation. Their most paradigmatic forms are judging, reasoning, concept formation, problem solving, a ...

." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in his ''Foreword'' Frege clearly denies that he reached this aim, and also that his main aim would be constructing an ideal language like Leibniz's, what Frege declares to be quite hard and idealistic, however, not impossible task). Frege went on to employ his logical calculus in his research on the

, carried out over the next quarter century.

Principia Mathematica

Principia Mathematica, or "PM" as it is often abbreviated, was an attempt to describe a set of

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

s and inference rules in

symbolic logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, there would in fact be some truths of mathematics which could not be deduced from them. One of the main inspirations and motivations for ''PM'' was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. ''PM'' sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with notion of a hierarchy of sets of different ' types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.

Gödel's incompleteness theorem

Gödel's incompleteness theorems are two theorems of

that establish inherent limitations of all but the most trivial axiomatic systems capable of doing

. The theorems, proven by

in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of

s for all

is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "

effective procedure In logic, mathematics and computer science, especially metalogic and computability theory, an effective methodGeoffrey Hunter (logician), Hunter, Geoffrey, ''Metalogic: An Introduction to the Metatheory of Standard First-Order Logic'', University o ...

" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the

natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

(

). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

Tarski's definition of model-theoretic satisfaction

The T-schema or truth

schema The word schema comes from the Greek word ('), which means ''shape'', or more generally, ''plan''. The plural is ('). In English, both ''schemas'' and ''schemata'' are used as plural forms. Schema may refer to: Science and technology * SCHEMA ...

(not to be confused with ' Convention T') is used to give an

inductive definition In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively-definable objects include fact ...

of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth. Some authors refer to it as the "Equivalence Schema", a synonym introduced by Michael Dummett. The T-schema is often expressed in

natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

, but it can be formalized in many-sorted predicate logic or

modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...

; such a formalisation is called a ''T-theory''. T-theories form the basis of much fundamental work in

philosophical logic Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...

, where they are applied in several important controversies in

analytic philosophy Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United Sta ...

. As expressed in semi-natural language (where 'S' is the name of the sentence abbreviated to S): 'S' is true if and only if S Example: 'snow is white' is true if and only if snow is white.

The undecidability of the Entscheidungsproblem

The (

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

for ' decision problem') is a challenge posed by

in 1928. The asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of

s beyond the usual axioms of first-order logic) and answers "Yes" or "No" according to whether the statement is ''universally valid'', i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic. In 1936, Alonzo Church and Alan Turing published independent papersChurch's paper was presented to the American Mathematical Society on 19 April 1935 and published on 15 April 1936. Turing, who had made substantial progress in writing up his own results, was disappointed to learn of Church's proof upon its publication (see correspondence between Max Newman and Church i
Alonzo Church papers
). Turing quickly completed his paper and rushed it to publication; it was received by the ''Proceedings of the London Mathematical Society'' on 28 May 1936, read on 12 November 1936, and published in series 2, volume 42 (1936-7); it appeared in two sections: in Part 3 (pages 230-240), issued on Nov 30, 1936 and in Part 4 (pages 241-265), issued on Dec 23, 1936; Turing added corrections in volume 43(1937) pp. 544–546. See the footnote at the end of Soare:1996. showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notation of "

effectively calculable In logic, mathematics and computer science, especially metalogic and computability theory, an effective method Hunter, Geoffrey, ''Metalogic: An Introduction to the Metatheory of Standard First-Order Logic'', University of California Press, 1971 ...

" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the

lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation ...

). This assumption is now known as the

Church–Turing thesis In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of comp ...