Finitism
Encyclopedia
In the philosophy of mathematics
, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural number
s in a finite number of steps. Another form of finitism was pursued by Hilbert
and Bernays
.
In her book Philosophy of Set Theory, Mary Tiles
characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists. Historically, the written history of mathematics was thus classically finitist until Cantor invented the hierarchy of transfinite cardinals in the end of the 19th century. Leopold Kronecker
is remembered as Cantor's opponent:
In 1923, Thoralf Skolem
published a paper in which he presented a semi-formal system, which is now known as primitive recursive arithmetic
, that is widely taken to be a suitable background for finitist mathematics. This was adopted by David Hilbert
and Paul Bernays
as the "contentual" finitist system for metamathematics
, in which a proof of the consistency
of other mathematical systems (e.g. full Peano Arithmetic) was to be given. (See Hilbert's program
.)
Reuben Goodstein
is another proponent of finitism. Some of his work involved building up to analysis
from finitist foundations. Although he denied it, much of Ludwig Wittgenstein
's writing on mathematics has a strong affinity with finitism. If finitists are contrasted with transfinitists
(proponents of e.g. Georg Cantor
's hierarchy of infinities), then also Aristotle
may be characterized as a strict finitist. Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity
. (Note that Aristotle's actual infinity means simply an actualization of something never-ending in nature, when in contrast the Cantorist actual infinity means the transfinite cardinal
and ordinal
numbers, that have nothing to do with the things in nature):
Even stronger than finitism is ultrafinitism
(also known as ultraintuitionism), associated primarily with Alexander Esenin-Volpin
.
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s in a finite number of steps. Another form of finitism was pursued by 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...
and Bernays
Paul Bernays
Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert.-Biography:Bernays spent his childhood in Berlin. Bernays attended the...
.
In her book Philosophy of Set Theory, Mary Tiles
Mary Tiles
Mary Tiles is a philosopher and historian of mathematics and science. she is professor and chair in the philosophy department of the University of Hawaii at Manoa.-Life:...
characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists. Historically, the written history of mathematics was thus classically finitist until Cantor invented the hierarchy of transfinite cardinals in the end of the 19th century. Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
is remembered as Cantor's opponent:
In 1923, Thoralf Skolem
Thoralf Skolem
Thoralf Albert Skolem was a Norwegian mathematician known mainly for his work on mathematical logic and set theory.-Life:...
published a paper in which he presented a semi-formal system, which is now known as primitive recursive arithmetic
Primitive recursive arithmetic
Primitive recursive arithmetic, or PRA, is a quantifier-free formalization of the natural numbers. It was first proposed by Skolem as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist...
, that is widely taken to be a suitable background for finitist mathematics. This was adopted by 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...
and Paul Bernays
Paul Bernays
Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert.-Biography:Bernays spent his childhood in Berlin. Bernays attended the...
as the "contentual" finitist system for metamathematics
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
, in which a proof of the consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
of other mathematical systems (e.g. full Peano Arithmetic) was to be given. (See Hilbert's program
Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...
.)
Reuben Goodstein
Reuben Goodstein
Reuben Louis Goodstein was an English mathematician with a strong interest in the philosophy and teaching of mathematics....
is another proponent of finitism. Some of his work involved building up to analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
from finitist foundations. Although he denied it, much of Ludwig Wittgenstein
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
's writing on mathematics has a strong affinity with finitism. If finitists are contrasted with transfinitists
Transfinite number
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these...
(proponents of e.g. Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
's hierarchy of infinities), then also Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
may be characterized as a strict finitist. Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity
Actual infinity
Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural...
. (Note that Aristotle's actual infinity means simply an actualization of something never-ending in nature, when in contrast the Cantorist actual infinity means the transfinite cardinal
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
and ordinal
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
numbers, that have nothing to do with the things in nature):
Even stronger than finitism is ultrafinitism
Ultrafinitism
In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism is a form of finitism. There are various philosophies of mathematics which are called ultrafinitism...
(also known as ultraintuitionism), associated primarily with Alexander Esenin-Volpin
Alexander Esenin-Volpin
Alexander Sergeyevich Esenin-Volpin is a prominent Russian-American poet and mathematician.Born on May 12, 1924 in the former Soviet Union, he was a notable dissident, political prisoner, poet, and mathematician...
.
External links
- Finitism in Geometry, entry in the Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...