Infinity is a concept in many fields, most predominantly
mathematicsMathematics 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...
and
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
infinitas or "unboundedness".
In mathematics, "infinity" is often treated as if it were a
numberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
(i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. In number systems incorporating
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s, the reciprocal of an infinitesimal is an infinite number, i.e. a number greater than any real number.
Georg CantorGeorg 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,...
formalized many ideas related to infinity and
infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called
cardinalities). For example, the set of
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is
countably infiniteIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
, while the set of real numbers is
uncountably infiniteIn mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
.
History
Ancient cultures had various ideas about the nature of infinity. The
ancient IndiansThe Maurya Empire was a geographically extensive Iron Age historical power in ancient India, ruled by the Mauryan dynasty from 321 to 185 BC...
and
GreeksAncient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
, unable to codify infinity in terms of a formalized mathematical system approached infinity as a philosophical concept.
Early Greek
The earliest attestable accounts of mathematical infinity come from
Zeno of EleaZeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...
(ca. 490 BCE? – ca. 430 BCE?), a
pre-SocraticPre-Socratic philosophy is Greek philosophy before Socrates . In Classical antiquity, the Presocratic philosophers were called physiologoi...
Greek philosopher of southern Italy and member of the Eleatic School founded by
ParmenidesParmenides of Elea was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy. The single known work of Parmenides is a poem, On Nature, which has survived only in fragmentary form. In this poem, Parmenides...
.
AristotleAristotle 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...
called him the inventor of the
dialecticDialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...
. He is best known for his
paradoxSimilar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es, which
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
has described as "immeasurably subtle and profound".
In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the
actual infinityActual 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...
; for example, instead of saying that there are an infinity of primes,
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, Book IX, Proposition 20).
However, recent readings of the
Archimedes PalimpsestThe Archimedes Palimpsest is a palimpsest on parchment in the form of a codex. It originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors, which was overwritten with a religious text.Archimedes lived in the...
have hinted that at least Archimedes had an intuition about actual infinite quantities.
Early Indian
The
Isha UpanishadThe Isha Upanishad is one of the shortest of the Upanishads, consisting of 17 or 18 verses in total; like other core texts of the vedanta, it is considered revealed scripture by diverse traditions within Hinduism...
of the
YajurvedaThe Yajurveda, a tatpurusha compound of "sacrificial formula', + ) is the third of the four canonical texts of Hinduism, the Vedas. By some, it is estimated to have been composed between 1400 and 1000 BC, the Yajurveda 'Samhita', or 'compilation', contains the liturgy needed to perform the...
(c. 4th to 3rd century BCE?) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".
The
Indian mathematicalIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
text
Surya Prajnapti (c. 400 BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
- Enumerable: lowest, intermediate, and highest
- Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
- Infinite: nearly infinite, truly infinite, infinitely infinite
In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and
ontologicalOntology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
grounds, a distinction was made between
{{pp-move-indef}}{{pp-semi|small=yes}}
{{other uses}}
Infinity (symbol:
∞In algebraic geometry, a lemniscate refers to any of several figure-eight or ∞ shaped curves. It may refer to:*The lemniscate of Bernoulli, often simply called the lemniscate, the locus of points whose product of distances from two foci equals the square of half the interfocal distance*The...
) is a concept in many fields, most predominantly
mathematicsMathematics 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...
and
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
infinitas or "unboundedness".
In mathematics, "infinity" is often treated as if it were a
numberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
(i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. In number systems incorporating
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s, the reciprocal of an infinitesimal is an infinite number, i.e. a number greater than any real number.
Georg CantorGeorg 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,...
formalized many ideas related to infinity and
infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called
cardinalities). For example, the set of
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is
countably infiniteIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
, while the set of real numbers is
uncountably infiniteIn mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
.
History
{{Main|Infinity (philosophy)}}
Ancient cultures had various ideas about the nature of infinity. The
ancient IndiansThe Maurya Empire was a geographically extensive Iron Age historical power in ancient India, ruled by the Mauryan dynasty from 321 to 185 BC...
and
GreeksAncient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
, unable to codify infinity in terms of a formalized mathematical system approached infinity as a philosophical concept.
Early Greek
The earliest attestable accounts of mathematical infinity come from
Zeno of EleaZeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...
(ca. 490 BCE? – ca. 430 BCE?), a
pre-SocraticPre-Socratic philosophy is Greek philosophy before Socrates . In Classical antiquity, the Presocratic philosophers were called physiologoi...
Greek philosopher of southern Italy and member of the Eleatic School founded by
ParmenidesParmenides of Elea was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy. The single known work of Parmenides is a poem, On Nature, which has survived only in fragmentary form. In this poem, Parmenides...
.
AristotleAristotle 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...
called him the inventor of the
dialecticDialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...
. He is best known for his
paradoxSimilar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es, which
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
has described as "immeasurably subtle and profound".
In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the
actual infinityActual 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...
; for example, instead of saying that there are an infinity of primes,
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, Book IX, Proposition 20).
However, recent readings of the
Archimedes PalimpsestThe Archimedes Palimpsest is a palimpsest on parchment in the form of a codex. It originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors, which was overwritten with a religious text.Archimedes lived in the...
have hinted that at least Archimedes had an intuition about actual infinite quantities.
Early Indian
The
Isha UpanishadThe Isha Upanishad is one of the shortest of the Upanishads, consisting of 17 or 18 verses in total; like other core texts of the vedanta, it is considered revealed scripture by diverse traditions within Hinduism...
of the
YajurvedaThe Yajurveda, a tatpurusha compound of "sacrificial formula', + ) is the third of the four canonical texts of Hinduism, the Vedas. By some, it is estimated to have been composed between 1400 and 1000 BC, the Yajurveda 'Samhita', or 'compilation', contains the liturgy needed to perform the...
(c. 4th to 3rd century BCE?) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".
The
Indian mathematicalIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
text
Surya Prajnapti (c. 400 BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
- Enumerable: lowest, intermediate, and highest
- Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
- Infinite: nearly infinite, truly infinite, infinitely infinite
In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and
ontologicalOntology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
grounds, a distinction was made between
{{pp-move-indef}}{{pp-semi|small=yes}}
{{other uses}}
Infinity (symbol:
∞In algebraic geometry, a lemniscate refers to any of several figure-eight or ∞ shaped curves. It may refer to:*The lemniscate of Bernoulli, often simply called the lemniscate, the locus of points whose product of distances from two foci equals the square of half the interfocal distance*The...
) is a concept in many fields, most predominantly
mathematicsMathematics 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...
and
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
infinitas or "unboundedness".
In mathematics, "infinity" is often treated as if it were a
numberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
(i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. In number systems incorporating
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s, the reciprocal of an infinitesimal is an infinite number, i.e. a number greater than any real number.
Georg CantorGeorg 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,...
formalized many ideas related to infinity and
infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called
cardinalities). For example, the set of
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is
countably infiniteIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
, while the set of real numbers is
uncountably infiniteIn mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
.
History
{{Main|Infinity (philosophy)}}
Ancient cultures had various ideas about the nature of infinity. The
ancient IndiansThe Maurya Empire was a geographically extensive Iron Age historical power in ancient India, ruled by the Mauryan dynasty from 321 to 185 BC...
and
GreeksAncient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
, unable to codify infinity in terms of a formalized mathematical system approached infinity as a philosophical concept.
Early Greek
The earliest attestable accounts of mathematical infinity come from
Zeno of EleaZeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...
(ca. 490 BCE? – ca. 430 BCE?), a
pre-SocraticPre-Socratic philosophy is Greek philosophy before Socrates . In Classical antiquity, the Presocratic philosophers were called physiologoi...
Greek philosopher of southern Italy and member of the Eleatic School founded by
ParmenidesParmenides of Elea was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy. The single known work of Parmenides is a poem, On Nature, which has survived only in fragmentary form. In this poem, Parmenides...
.
AristotleAristotle 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...
called him the inventor of the
dialecticDialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...
. He is best known for his
paradoxSimilar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es, which
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
has described as "immeasurably subtle and profound".
In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the
actual infinityActual 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...
; for example, instead of saying that there are an infinity of primes,
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, Book IX, Proposition 20).
However, recent readings of the
Archimedes PalimpsestThe Archimedes Palimpsest is a palimpsest on parchment in the form of a codex. It originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors, which was overwritten with a religious text.Archimedes lived in the...
have hinted that at least Archimedes had an intuition about actual infinite quantities.
Early Indian
The
Isha UpanishadThe Isha Upanishad is one of the shortest of the Upanishads, consisting of 17 or 18 verses in total; like other core texts of the vedanta, it is considered revealed scripture by diverse traditions within Hinduism...
of the
YajurvedaThe Yajurveda, a tatpurusha compound of "sacrificial formula', + ) is the third of the four canonical texts of Hinduism, the Vedas. By some, it is estimated to have been composed between 1400 and 1000 BC, the Yajurveda 'Samhita', or 'compilation', contains the liturgy needed to perform the...
(c. 4th to 3rd century BCE?) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".
The
Indian mathematicalIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
text
Surya Prajnapti (c. 400 BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
- Enumerable: lowest, intermediate, and highest
- Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
- Infinite: nearly infinite, truly infinite, infinitely infinite
In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and
ontologicalOntology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
grounds, a distinction was made between {{IAST ("countless, innumerable") and
ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
The infinity symbol
The infinity symbol
is sometimes called the
lemniscateIn algebraic geometry, a lemniscate refers to any of several figure-eight or ∞ shaped curves. It may refer to:*The lemniscate of Bernoulli, often simply called the lemniscate, the locus of points whose product of distances from two foci equals the square of half the interfocal distance*The...
, from the
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
lemniscus, meaning "ribbon". John Wallis is credited with introducing the symbol in 1655 in his
De sectionibus conicis. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the
Etruscan numeralThe Etruscan numerals were used by the ancient Etruscans. The system was adapted from the Greek Attic numerals and formed the inspiration for the later Roman numerals.There is very little surviving evidence of these numerals...
for 1000, which looked somewhat like
CIƆ and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω (
omegaOmega is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" , as opposed to omicron, which means "little O"...
), the last letter in the
Greek alphabetThe Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
.
The infinity symbol is also sometimes depicted as a special variation of the ancient
ouroborosThe Ouroboros is an ancient symbol depicting a serpent or dragon eating its own tail. The name originates from within Greek language; οὐρά meaning "tail" and βόρος meaning "eating", thus "he who eats the tail"....
snake symbol. The snake is twisted into the horizontal eight configuration while engaged in eating its own tail, a uniquely suitable symbol for endlessness.
The symbol is encoded in
UnicodeUnicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...
at {{unichar|221E|infinity|html=|size=200%}} and in
LaTeXLaTeX is a document markup language and document preparation system for the TeX typesetting program. Within the typesetting system, its name is styled as . The term LaTeX refers only to the language in which documents are written, not to the editor used to write those documents. In order to...
as
\infty.
Also, but less available in fonts, are encoded: {{unichar|29DC|INCOMPLETE INFINITY|html=|size=100%|note=ISOtech entity
⧜}}, {{unichar|29DD|TIE OVER INFINITY|html=|size=100%}} and {{unichar|29DE|INFINITY NEGATED WITH VERTICAL BAR|html=|size=100%}} in block Miscellaneous Mathematical Symbols-B.
Calculus
Leibniz, one of the co-inventors of
infinitesimal calculusInfinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties.
Real analysis
In
real analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
, the symbol
, called "infinity", denotes an unbounded
limitIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
.
means that
x grows without bound, and
means the value of x is decreasing without bound. If
f(
t) ≥ 0 for every
t, then
-
means that f(t) does not bound a finite area from a to b
-
means that the area under f(t) is infinite.
-
means that the total area under f(t) is finite, and equals n
Infinity is also used to describe infinite series:
-
means that the sum of the infinite series converges to some real value a.
-
means that the sum of the infinite series divergesIn mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....
in the specific sense that the partial sums grow without bound.
Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled
and
can be added to the
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
of the real numbers, producing the two-point
compactificationIn mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat
and
as the same, leading to the one-point compactification of the real numbers, which is the
real projective line.
Projective geometryIn mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
also introduces a
line at infinityIn geometry and topology, the line at infinity is a line that is added to the real plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.-Geometric formulation:In...
in plane geometry, and so forth for higher
dimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s.
Complex analysis
As in real analysis, in
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
the symbol
, called "infinity", denotes an unsigned infinite
limitIn mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
.
means that the magnitude
of
x grows beyond any assigned value. A
point labeled 
can be added to the complex plane as a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
giving the one-point
compactificationIn mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of the complex plane. When this is done, the resulting space is a one-dimensional
complex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
, or
Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
, called the extended complex plane or the
Riemann sphereIn mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
. Arithmetic operations similar to those given below for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely
for any nonzero complex number
z. In this context it is often useful to consider
meromorphic functionIn complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s as maps into the Riemann sphere taking the value of
at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of
Möbius transformations.
Nonstandard analysis
The original formulation of
infinitesimal calculusInfinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
by
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and
Gottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
used
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
quantities. In the twentieth century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including
smooth infinitesimal analysisSmooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete...
and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a
hyperreal fieldThe system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to
non-standard calculusIn mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus...
is fully developed in
Howard Jerome KeislerH. Jerome Keisler is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis.His Ph.D...
's book (see below).
Set theory
{{Main|Cardinality|Ordinal number}}
A different form of "infinity" are the
ordinalIn 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...
and
cardinalIn 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...
infinities of set theory.
Georg CantorGeorg 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,...
developed a system of
transfinite numberTransfinite 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...
s, in which the first transfinite cardinal is aleph-null
, the cardinality of the set of
natural numberIn 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. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor,
Gottlob FregeFriedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
,
Richard DedekindJulius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
and others, using the idea of collections, or sets.
Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite.
Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite
sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s which are maps from the positive integers leads to
mappingsIn most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
from ordinal numbers, and transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is
countably infiniteIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
. If a set is too large to be put in one to one correspondence with the positive integers, it is called
uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the
hyperreal numberThe system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
s, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
Cardinality of the continuum
{{Main|Cardinality of the continuum}}
One of Cantor's most important results was that the cardinality of the continuum
is greater than that of the natural numbers
; that is, there are more real numbers
R than natural numbers
N. Namely, Cantor showed that
(see
Cantor's diagonal argumentCantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...
or
Cantor's first uncountability proofGeorg Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountable. This proof differs from the more familiar proof that uses his diagonal argument...
).
The
continuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
states that there is no
cardinal numberIn 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...
between the cardinality of the reals and the cardinality of the natural numbers, that is,
(see Beth one). However, this hypothesis can neither be proved nor disproved within the widely accepted Zermelo-Fraenkel set theory, even assuming the
Axiom of Choice.
Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any
segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the
intervalIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
(−π/2, π/2) and
R (see also
Hilbert's paradox of the Grand HotelHilbert's paradox of the Grand Hotel is a mathematical veridical paradox about infinite sets presented by German mathematician David Hilbert .-The paradox:...
). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when
Giuseppe PeanoGiuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...
introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, or
hypercubeIn geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points in the side of a square and those in the square.
Geometry and topology
{{Main|Dimension (vector space)}}
Infinite-
dimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al spaces are widely used in
geometryGeometry 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 ....
and
topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, particularly as
classifying spaceIn mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
s, notably
Eilenberg−MacLane spaceIn mathematics, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" , and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. In mathematics, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without...
s. Common examples are the infinite-dimensional
complex projective spaceIn mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
K(Z,2) and the infinite-dimensional
real projective spaceIn mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
K(Z/2Z,1).
Fractals
The structure of a
fractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the
Koch snowflakeThe Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described...
.
Mathematics without infinity
Leopold KroneckerLeopold 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"...
was skeptical of the notion of infinity and how his fellow mathematicians were using it in 1870s and 1880s. This skepticism was developed in the
philosophy of mathematicsThe 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...
called
finitismIn 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 numbers in a finite number of steps...
, an extreme form of the philosophical and mathematical schools of constructivism and
IntuitionismIn the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
.
Physics
{{Unreferenced section|date=December 2009}}
In
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, approximations of
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s are used for
continuousContinuum theories or models explain variation as involving a gradual quantitative transition without abrupt changes or discontinuities. It can be contrasted with 'categorical' models which propose qualitatively different states.-In physics:...
measurements and
natural numberIn 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 are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no
measurable quantityIn physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
could have an infinite value,{{Citation needed|date=February 2008}} for instance by taking an infinite value in an
extended real numberIn mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
system, or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite
plane waveIn the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
exist, but there are no experimental means to generate them.{{Citation needed|date=May 2010}}
Theoretical applications of physical infinity
The practice of refusing infinite values for measurable quantities does not come from
a prioriThe terms a priori and a posteriori are used in philosophy to distinguish two types of knowledge, justifications or arguments...
or ideological motivations, but rather from more methodological and pragmatic motivations.{{Citation needed|date=February 2008}} One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality. Sometimes infinite result of a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory.
This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series, unbounded
functionIn 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...
s, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called
renormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
.
However, there are some theoretical circumstances where the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a
mathematical singularityIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
, or a point where a physical theory breaks down. This does not necessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and
Coulomb's lawCoulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
of electrostatics. At r=0 these equations evaluate to infinities.
Cosmology
In ancient cosmologies, the sky was perceived as a solid dome, or
firmamentThe firmament is the vault or expanse of the sky. According to Genesis, God created the firmament to separate the oceans from other waters above.-Etymology:...
. In 1584,
BrunoGiordano Bruno , born Filippo Bruno, was an Italian Dominican friar, philosopher, mathematician and astronomer. His cosmological theories went beyond the Copernican model in proposing that the Sun was essentially a star, and moreover, that the universe contained an infinite number of inhabited...
proposed an unbounded universe in
On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
CosmologistsCosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...
have long sought to discover whether infinity exists in our physical
universeThe Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
: Are there an infinite number of stars? Does the universe have infinite volume? Does space
"go on forever"The shape of the universe is a matter of debate in physical cosmology over the local and global geometry of the universe which considers both curvature and topology, though, strictly speaking, it goes beyond both...
? This is an open question of
cosmologyPhysical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar
topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
; if one travelled in a straight line through the universe perhaps one would eventually revisit one's starting point.
If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through
multipole momentsIn mathematics, especially as applied to physics, multipole moments are the coefficients of a series expansion of a potential due to continuous or discrete sources . A multipole moment usually involves powers of the distance to the origin, as well as some angular dependence...
in the spectrum of the
cosmic background radiationIn cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....
. As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe. The Planck spacecraft launched in 2009 is expected to record the cosmic background radiation with 10 times higher precision, and will give more insight into the question of whether the universe is infinite or not.
Logic
In
logicIn 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...
an
infinite regressAn infinite regress in a series of propositions arises if the truth of proposition P1 requires the support of proposition P2, the truth of proposition P2 requires the support of proposition P3, .....
argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
Computing
The
IEEE floating-point standardIEEE 754–1985 was an industry standard for representingfloating-pointnumbers in computers, officially adopted in 1985 and superseded in 2008 byIEEE 754-2008. During its 23 years, it was the most widely used format for...
specifies positive and negative infinity values; these can be the result of
arithmetic overflowThe term arithmetic overflow or simply overflow has the following meanings.# In a computer, the condition that occurs when a calculation produces a result that is greater in magnitude than that which a given register or storage location can store or represent.# In a computer, the amount by which a...
,
division by zeroIn mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...
, or other exceptional operations.
Some
programming languageA programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....
s (for example, J and
UNITYUNITY is a programming language that was constructed by K. Mani Chandy and Jayadev Misra for their book Parallel Program Design: A Foundation. It is a rather theoretical language, which tries to focus on what, instead of where, when or how. The peculiar thing about the language is that it has no...
) specify
greatest and least elementsIn mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
, i.e. values that compare (respectively) greater than or less than all other values. These may also be termed
top and
bottom, or
plus infinity and
minus infinity; they are useful as
sentinel valueIn computer programming, a sentinel value is a special value whose presence guarantees termination of a loop that processes structured data...
s in
algorithmIn mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s involving
sortingSorting is any process of arranging items in some sequence and/or in different sets, and accordingly, it has two common, yet distinct meanings:# ordering: arranging items of the same kind, class, nature, etc...
,
searchingIn computer science, a search algorithm is an algorithm for finding an item with specified properties among a collection of items. The items may be stored individually as records in a database; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots...
or
windowingIn signal processing, a window function is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation...
. In languages that do not have greatest and least elements, but do allow
overloadingIn object oriented computer programming, operator overloading—less commonly known as operator ad-hoc polymorphism—is a specific case of polymorphism, where different operators have different implementations depending on their arguments...
of
relational operatorIn computer science, a relational operator is a programming language construct or operator that tests or defines some kind of relation between two entities. These include numerical equality and inequalities...
s, it is possible to
create greatest and least elements.
The arts and cognitive sciences
PerspectivePerspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is seen by the eye...
artwork utilizes the concept of imaginary
vanishing pointA vanishing point is a point in a perspective drawing to which parallel lines not parallel to the image plane appear to converge. The number and placement of the vanishing points determines which perspective technique is being used...
s, or
points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms. Artist
M. C. EscherMaurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
is specifically known for employing the concept of infinity in his work in this and other ways.
From the perspective of cognitive scientists
George LakoffGeorge P. Lakoff is an American cognitive linguist and professor of linguistics at the University of California, Berkeley, where he has taught since 1972...
, concepts of infinity in mathematics and the sciences are metaphors, based on what they term the Basic Metaphor of Infinity (BMI), namely the ever-increasing sequence <1,2,3,...>.
See also
- 0.999...
In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number...
- Aleph number
In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph...
- Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare....
- Paradoxes of infinity
- Surreal number
In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number...
External links
{{Wikibooks|Infinity is not a number}}
{{Wiktionary}}
- A Crash Course in the Mathematics of Infinite Sets, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1-59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
- Infinite Reflections, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1-59.
- Infinity, Principia Cybernetica
- Hotel Infinity
- Source page on medieval and modern writing on Infinity
- The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
- Dictionary of the Infinite (compilation of articles about infinity in physics, mathematics, and philosophy)
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