Finitary
Encyclopedia
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 proposition
s, 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 set of axiom
s. 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 statement
s 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, logic
ians 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
or logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, a finitary operation is one, like those of arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
, that takes a finite number of input values to produce an output. An operation such as taking an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, 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
Prima facie
Prima facie is a Latin expression meaning on its first encounter, first blush, or at first sight. The literal translation would be "at first face", from the feminine form of primus and facies , both in the ablative case. It is used in modern legal English to signify that on first examination, a...
finitary. In the logic proposed for quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, depending on the use of subspaces of Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
as proposition
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
s, operations such as taking the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
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 set of axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s. In other words, it is a proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
that can be written on a large enough sheet of paper (including all assumptions).
The emphasis on finitary methods has historical roots. Infinitary logic
Infinitary logic
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and...
studies logics that allow infinitely long statement
Statement
Statement may refer to:* A kind of expression in language *Statement , declarative sentence that is either true or false*Statement , the smallest standalone element of an imperative programming language...
s 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, logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
ians aimed to solve the problem of foundations
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
; 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
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
(referring to geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
), "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
Modus ponens
In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
. The project was to fix a finite number of symbols (essentially the numerals
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
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
Rule of inference
In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion...
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
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
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.