Finitary

In mathematics or logic, a finitary operation is one, like those of arithmetic, that takes a finite number of input values to produce an output. An operation such as taking an integral of a function, in calculus, is defined in such a way as to depend on all the values of the function (infinitely many of them, in general), and is so not "prima facie" finitary. In the logic proposed for quantum mechanics, depending on the use of subspaces of Hilbert space as propositions, operations such as taking the intersection of subspaces are used; this in general cannot be considered a finitary operation. What fails to be finitary can be called "infinitary".

A finitary argument is one which can be translated into a finite set of symbolic propositions starting from a finite [The number of axioms "referenced" in the argument will necessarily be finite since the proof is finite, but the number of axioms from which these are "chosen" is infinite when the system has axiom schemes, as for example the axiom schemes of propositional calculus.] set of axioms. In other words, it is a proof that can be written on a large enough sheet of paper (including all assumptions).

The emphasis on finitary methods has historical roots. Infinitary logic studies logics that allow infinitely long statements and proofs. In such a logic, one can regard the existential quantifier, for instance, as derived from an infinitary disjunction.

In the early 20th century, logicians aimed to solve the problem of foundations; that is, answer the question: "What is the true base of mathematics?" The program was to be able to rewrite all mathematics starting using an entirely syntactical language "without semantics". In the words of David Hilbert (referring to geometry), "it does not matter if we call the things "chairs", "tables" and "beer mugs" or "points", "lines" and "planes"."

The stress on finiteness came from the idea that human "mathematical" thought is based on a finite number of principles and all the reasonings follow essentially one rule: the "modus ponens". The project was to fix a finite number of symbols (essentially the numerals 1,2,3,... the letters of alphabet and some special symbols like "+", "->", "(", ")", etc.), give a finite number of propositions expressed in those symbols, which were to be taken as "foundations" (the axioms), and some rules of inference which would model the way humans make conclusions. From these, "regardless of the semantic interpretation of the symbols" the remaining theorems should follow "formally" using only the stated rules (which make mathematics look like a "game with symbols" more than a "science") without the need to rely on ingenuity. The hope was to prove that from these axioms and rules "all" the theorems of mathematics could be deduced.

The aim itself was proved impossible by Kurt Gödel in 1931, with his Incompleteness Theorem, but the general mathematical trend is to use a finitary approach, arguing that this avoids considering mathematical objects that cannot be fully defined.

Notes

External links

* [http://plato.stanford.edu/entries/logic-infinitary/ Stanford Encyclopedia of Philosophy entry on Infinitary Logic]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • finitary — adjective Of a function, taking a finite number of arguments to produce an output …   Wiktionary

  • finitary — fi·ni·tary …   English syllables

  • finitary — ˈfīnəˌterē, ˈfin adjective Etymology: finite + ary : having a finite character; specifically : capable of being completed in a finite number of steps used of a proof or other logical procedure …   Useful english dictionary

  • Finitary relation — This article sets out the set theoretic notion of relation. For a more elementary point of view, see Binary relation. For a combinatorial viewpoint, see Theory of relations. For other uses, see Relation (disambiguation). In set theory and logic,… …   Wikipedia

  • finitary methods — See finitism …   Philosophy dictionary

  • Closure operator — In mathematics, a closure operator on a set S is a function cl: P(S) → P(S) from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S. X ⊆ cl(X) (cl is extensive) X ⊆ Y implies cl(X) ⊆ cl(Y)   (cl… …   Wikipedia

  • Boolean algebras canonically defined — Boolean algebras have been formally defined variously as a kind of lattice and as a kind of ring. This article presents them more neutrally but equally formally as simply the models of the equational theory of two values, and observes the… …   Wikipedia

  • Hilbert's program — Hilbert s program, formulated by German mathematician David Hilbert in the 1920s, was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent.Hilbert proposed that the… …   Wikipedia

  • Infinitary logic — Those unfamiliar with mathematical logic or the concept of ordinals are advised to consult those articles first. An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have… …   Wikipedia

  • Matroid — In combinatorics, a branch of mathematics, a matroid (  /ˈmeɪ …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.