Infinity

For the symbol, see Infinity symbol. For other uses of "Infinity" and "Infinite", see Infinity (disambiguation). Not to be confused with Infiniti.

Due to the constant light reflection between opposing mirrors, it seems that there is a boundless amount of space and repetition inside of them.

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol $\infty$.

Since the time of the ancient Greeks, the philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of Grothendieck universes, very large infinite sets, for solving a long-standing problem that is stated in terms of elementary arithmetic.

In physics and cosmology, whether the universe is spatially infinite is an open question.

History

Further information: Infinity (philosophy)

Ancient cultures had various ideas about the nature of infinity. The ancient Indians and the Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

Early Greek

The earliest recorded idea of infinity in Greece may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".

Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving the infinitude of the prime numbers, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated:

If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.

Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.

Zeno: Achilles and the tortoise

Main article: Zeno's paradoxes § Achilles and the tortoise

Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".

Achilles races a tortoise, giving the latter a head start.

• Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
• Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
• Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
• Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.

Etc.

Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.

Zeno was not attempting to make a point about infinity. As a member of the Eleatics school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.

Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,

$$a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.$$

Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him

$$10+0.1+0.001+0.00001+\cdots ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.$$

Early Indian

The Jain mathematical text Surya Prajnapti (c. 4th–3rd century 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

17th century

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation $\infty$ for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of ${\tfrac {1}{\infty }}.$ But in Arithmetica infinitorum (1656), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."

In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.

Mathematics

Hermann Weyl opened a mathematico-philosophic address given in 1930 with:

Mathematics is the science of the infinite.

Symbol

Main article: Infinity symbol

The infinity symbol $\infty$ (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (&infin;) and in LaTeX as \infty.

It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.

Calculus

Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, 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 in accordance with the Law of continuity.

Real analysis

In real analysis, the symbol $\infty$, called "infinity", is used to denote an unbounded limit. The notation $x\rightarrow \infty$ means that $x$ increases without bound, and $x\to -\infty$ means that $x$ decreases without bound. For example, if $f(t)\geq 0$ for every $t$, then

• $\int _{a}^{b}f(t)\,dt=\infty$ means that $f(t)$ does not bound a finite area from $a$ to $b.$
• $\int _{-\infty }^{\infty }f(t)\,dt=\infty$ means that the area under $f(t)$ is infinite.
• $\int _{-\infty }^{\infty }f(t)\,dt=a$ means that the total area under $f(t)$ is finite, and is equal to $a.$

Infinity can also be used to describe infinite series, as follows:

• $\sum _{i=0}^{\infty }f(i)=a$ means that the sum of the infinite series converges to some real value $a.$
• $\sum _{i=0}^{\infty }f(i)=\infty$ means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.

In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled $+\infty$ and $-\infty$ can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat $+\infty$ and $-\infty$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.

Complex analysis

By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.

In complex analysis the symbol $\infty$, called "infinity", denotes an unsigned infinite limit. The expression $x\rightarrow \infty$ means that the magnitude $|x|$ of $x$ grows beyond any assigned value. A point labeled $\infty$ can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely $z/0=\infty$ for any nonzero complex number $z$. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of $\infty$ 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 (see Möbius transformation § Overview).

Nonstandard analysis

Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)

The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis 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 field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).

Set theory

Main articles: Cardinality and Ordinal number

One-to-one correspondence between an infinite set and its proper subset

A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ₀), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind 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 (derived from Euclid) that the whole cannot be the same size as the part. (However, see Galileo's paradox where Galileo concludes that positive integers cannot be compared to the subset of positive square integers since both are infinite sets.) 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. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".

Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to 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 infinite. 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 as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Cardinality of the continuum

Main article: Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum $\mathbf {c}$ is greater than that of the natural numbers ${\aleph _{0}}$; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that $\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}$.

Further information: Cantor's diagonal argument and Cantor's first set theory article

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, $\mathbf {c} =\aleph _{1}=\beth _{1}$.

Further information: Beth number § Beth one

This hypothesis cannot be proved or 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 segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R.

See also: Hilbert's paradox of the Grand Hotel

The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.

Geometry

Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).

One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.

Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).

In particular, in modern mathematics, lines are infinite sets.

Infinite dimension

The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.

In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.

Fractals

The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.

Mathematics without infinity

Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.

Physics

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.

Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno 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."

Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is still an open question of cosmology. 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 with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.

The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. 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.

However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.

The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes. Also, cyclic models posit an infinite amount of Big Bangs, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.

Logic

In logic, an infinite regress 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 standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.

Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting, searching, or windowing.

In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point data type, the infinity values may still be accessible and usable as the result of certain operations.

In programming, an infinite loop is a loop whose exit condition is never satisfied, thus executing indefinitely.

In philosophy

Ancient Indian philosophy

The Isha Upanishad of the 4th and 3rd centuries BCE reveals the idea that adding or subtracting a part to something infinite leaves it infinite. The Jain treatise Surya Prajnapti Sutra (English Sūryaprajñapti), dated to 400 BC. e. , all amounts are divided into three categories and three subcategories - uncountable (small, medium and large), uncountable ("almost uncountable", "really uncountable" and "count- innumerable") and infinite ("almost infinite", "truly" infinite" and "infinitely infinite"), this division is the first attempt not only to distinguish between types of infinity, but also to measure the relationship between them and infinite the idea of ​​defining subcategories of quantities and their arrangement is close to Cantor's concept of transfinite numbers.

Ancient Greek philosophy

Among ancient Greek philosophers, infinity is usually seen as something unformed, imperfect, close to chaos, or even identified with it, so in the Pythagorean list of opposites, infinity is classified as the aspect of evil. Among the ancient Greek philosophers who positively use the category of the infinite, Anaximander stands out, who introduces the cosmological principle as an infinite container - apeiron (ancient Greek Αττερερνν) and atomists (Democritus, Leucippus), who, according to their opinion, have a number. worlds formed from an infinite number of atoms located in infinite empty space. At the same time, the atomistic concept opposed the continuist approach, which considered space and time to be infinitely divisible, while the atomists assumed primary indivisible elements, and Zeno's aporia [⇨] was intended to show the logical incompatibility of the two approaches.

But the dominant thought in ancient Greek philosophy was the denial of real infinity, the most characteristic reflection of these views is presented in Aristotle's work "Physics", where he denies the infinity of the cosmos, the infinity of the sequence of causes, and speaks of the possibility. the infinite growth of the natural series and the infinity of dividing the segment into small components is only about potential infinity [⇨] . Aristotle refers to the infinite that arises from the infinite addition of objects to a collection to the extensive, and to the intensive that arises from the infinite deepening of the structure of the object. Ancient geometries, notably Euclid's position that denying actual infinity and working only with potential infinity, the second postulate in the "Principles" states that a straight line can continue for an arbitrary length of time, but that straight lines and planes themselves is finite even though it is almost infinitely "large".

In the works of the Neoplatonists, first of all, Plotinus, in connection with the introduction of Eastern mystical ideas and mainly under the influence of the works of Philo of Alexandria, gave a Hellenistic interpretation of the Christian God. an actual infinity of infinitely powerful and unified consciousness and a potential infinity of infinite matter are formed.

European medieval philosophy

In early Christianity and early medieval philosophy (Origen, Augustine, Albert Magnus, Thomas Aquinas), the denial of the true infinity of the world was inherited from Aristotle, who in one form or another recognized the Christian God as actually infinite.

In the works of scholastics of the 13th and 14th centuries (William of Sherwood, Heytesbury, Gregory of Rimini) the difference between the concepts of potential and real infinity is clearly indicated (in the early works, potential and real infinity are syncategorema and categorical infinities), but the divine is actually a relation to the infinite or a complete real infinity. negation is postulated (lat. infinitum actu non datur). At the same time, Occam draws attention to and supports the possibility of recognizing the continuum and its parts as actually existing, while maintaining the possibility of infinite properties - the possibility of infinite division into components. mathematically proves the idea about the infinite divisibility of the continuum, the sum of an infinite number series [⇨]. Oresme, developing Swineshead's constructions, constructs a system of geometrical proofs of the convergence of infinite series, constructing an example of a plane figure of infinite extension but finite area.

In the 15th century, Nicholas Cusa created the doctrine of the "absolute maximum", which he believed to be the infinite measure of all finite things, thereby giving an idea that is completely incompatible with the ancient: everything is considered finite. the limitation of the divine infinity existing in reality (lat. possest), on the contrary, the idea of the existence of finite things and the potentiality of the infinite prevails.

New age philosophy

Nicolaus Cusa's ideas were developed by Spinoza, according to which things assume their own destiny by denying their existence within the boundless divine substance. From these ideas comes the recognition of the idea of ​​the infinity of the universe established in the 16th and 17th centuries due to the heliocentric system of Copernicus, the educational works of Bruno, and the studies of Kepler and Galileo. Kepler and Galileo began to use the methods of infinity in mathematical practice, for example, Kepler, relying on the ideas of Nicholas Cusa, approximated the circle with a regular polygon with the number of sides tending to infinity, while Galileo. , paying attention to the correspondence between numbers and their squares, noted that the thesis "greater than the whole" cannot be applied to infinite things.

An important role in the idea of ​​the nature and essence of the continuum was brought by a pupil of Galileo Cavalieri, who wrote in his treatise "Geometry, defined in a new way by means of an indivisible continuum." (1635) considered plane figures to be an infinite set of segments that fill them, and volume bodies to be composed of an infinite number of parallel plane figures, and used the following metaphors: a line is like a pearl necklace, made of flat points. the figure is made of lines, just as fabric is made of threads, the body is made of planes, and a book is made of pages; With this "method of indivisibles" Cavalieri achieved important mathematical results.

Descartes argues that it is impossible to know God from the existence of the world he created with its finite and in fact infinite immensity, the incomprehensibility of which, in his opinion, lies in the very formal definition of infinity. Accordingly, Descartes recognizes only the Almighty God as truly infinite, and considers such manifestations of infinity as the "infinity of human will" to be the manifestation of the divine image in man. Leibniz, the most consistent supporter of the existence of real infinity, in his work "Monadology", he consistently continues the idea of ​​the infinity of monads in the universe and expresses it in the form of matter; The stability of these parts by a predetermined law of harmony and special principles of subordination to monads, as well as the consideration of monads, in turn, as an infinite universe in space and time. These ideas of Leibniz were reflected in his fundamental work on infinitesimal calculus, which represented infinitesimals as monads [⇨]. The differential calculus created by Newton and Leibniz, who clearly updated the infinitesimals, was Berkeley's most consistent opponent of infinitesimals among philosophers of the 17th and 18th centuries, and these debates were reflected in the culture in plots. From Swift's Gulliver's Travels and Voltaire's Micromegas.

In "Critique of Pure Reason", Kant denies the possibility of considering both infinite numbers and infinite quantities; Based on the analysis of the antinomies of pure reason, Kant characterizes the world as neither finite nor infinite, but "indeterminate".

Hegel develops the idea of ​​the closest connection, the almost similarity, between the infinite and the absolute, especially considering the "bad infinity" as the negation of the finite, and introduces the "true infinity" as a dialectical overcoming of antagonism; According to Hegel, only Absolute Spirit is truly infinite. The philosophy of dialectical materialism emphasizes the idea of ​​infinity as a dialectical process, in which the concept of infinity itself has different meanings: the simplest, practical infinity; infinity as absoluteness, universality, completeness; the infinity of the intellectual world; true infinity. Engels sees the infinity of space and time as an example of "bad infinity".

The most important 19th-century work on infinity, philosophical rather than mathematical, is Bolzano's monograph Infinite Paradoxes [English] (published posthumously in 1851), which contains infinite sets. systematically studied numerical, logical, and mathematical arguments are presented in favor of considering true infinity, and tools are offered for studying types of infinity using the concept of one-to-one correspondences.

Modern philosophy

In the philosophy of the 20th century, the main content of the study of problems related to infinity is closely related to the foundations of mathematics and, first of all, to the problems of set theory.

In Principia Mathematica, the system Russell built with Whitehead to overcome the paradoxes of set theory [⇨] assumed the existence of infinity by introducing the axiom of infinity, which also precluded the possibility of inferring infinity from other things. A priori concepts, the concept of infinity, cannot be excluded purely analytically from the principle of avoiding contradictions. Russell also did not believe that it was possible to find a posteriori justification for infinity based on common sense and experience, especially noting that there was no reason to believe in the infinity of space, the infinity of time, or the infinite divisibility of objects. Thus, according to Russell, infinity is a hypothetical imperative that may or may not apply in different systems, but cannot be justified or disproved.

Pursuing a program to overcome the paradoxes of set theory, Gilbert and Bernays formulated principles known as "Gilbert's finitism", according to which statements about properties made for all elements of an infinite set can only be repeated for each specific element. ladi limiting the possible abstractions of the infinite, including transfinite induction. Wittgenstein, who most fundamentally developed the concept of finitism in analytic philosophy, believed that the infinite can be considered only as a record of a recursive process and fundamentally rejected the possibility of considering different classes of infinity.

The issues of infinity have also been explored in the schools originating from neo-Kantianism and phenomenology, for example, Cassirer in his conversation with Heidegger ("Davos discussion", 1929) introduces immanent infinity, which appears as an objectification of the field. experiments, programming works dedicated to the infinite in 1950-1960 were written by Koyré and Levinas.

Induction

Induction is a classic logical method that allows you to go from certain statements to universal, including an infinite set of objects. Induction on natural series, without any formalization, is even mentioned in Proclus and Euclid, and its concept as a method of mathematical induction goes back to Pascal and Gersonides. In modern notation, mathematical induction consists of a syllogism:

$P(1),\forall n\in \mathbb {N} (P(n)\rightarrow P(n+1))\vdash \forall n\in \mathbb {N} (P(n))$i.e. property deduction

𝑃 subtraction for the whole set of natural numbers based on the fact that it fulfills one and for each subsequent number based on the fulfillment of the property for the previous one.

The method of mathematical induction is reliable, but it can only be extended to enumerable well-ordered sets.  An attempt to extend induction to arbitrarily ordered sets was Cantor's creation of a method of transfinite induction within set theory [⇨] using the idea of ​​transfinite (ordered) numbers.

In intuitionistic logic, bar induction is used to apply inductive reasoning to uncountable sets (described as flow in intuitionism).

Arts, games, and cognitive sciences

Perspective artwork uses the concept of vanishing points, roughly corresponding to mathematical 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. Escher is specifically known for employing the concept of infinity in his work in this and other ways.

Variations of chess played on an unbounded board are called infinite chess.

Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1,2,3,...>.

See also

• 0.999...
• Aleph number
• Ananta
• Exponentiation
• Indeterminate form
• Names of large numbers
• Infinite monkey theorem
• Infinite set
• Infinitesimal
• Paradoxes of infinity
• Supertask
• Surreal number

References

Bibliography

• Cajori, Florian (1993) [1928 & 1929], A History of Mathematical Notations (Two Volumes Bound as One), Dover, ISBN 978-0-486-67766-8
• Gemignani, Michael C. (1990), Elementary Topology (2nd ed.), Dover, ISBN 978-0-486-66522-1
• Keisler, H. Jerome (1986), Elementary Calculus: An Approach Using Infinitesimals (2nd ed.)
• Maddox, Randall B. (2002), Mathematical Thinking and Writing: A Transition to Abstract Mathematics, Academic Press, ISBN 978-0-12-464976-7
• Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, New York: Oxford University Press, pp. 1197–1198, ISBN 978-0-19-506135-2
• Russell, Bertrand (1996) [1903], The Principles of Mathematics, New York: Norton, ISBN 978-0-393-31404-5, OCLC 247299160
• Sagan, Hans (1994), Space-Filling Curves, Springer, ISBN 978-1-4612-0871-6
• Swokowski, Earl W. (1983), Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber & Schmidt, ISBN 978-0-87150-341-1
• Taylor, Angus E. (1955), Advanced Calculus, Blaisdell Publishing Company
• Wallace, David Foster (2004), Everything and More: A Compact History of Infinity, Norton, W.W. & Company, Inc., ISBN 978-0-393-32629-1

Sources

• Aczel, Amir D. (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. New York: Pocket Books. ISBN 978-0-7434-2299-4.
• D.P. Agrawal (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
• Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
• Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", Proceedings of the National Academy of Sciences of the United States of America, 50 (6): 1143–1148, Bibcode:1963PNAS...50.1143C, doi:10.1073/pnas.50.6.1143, PMC 221287, PMID 16578557.
• Jain, L.C. (1982). Exact Sciences from Jaina Sources.
• Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
• Joseph, George G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.). Penguin Books. ISBN 978-0-14-027778-4.
• H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
• Eli Maor (1991). To Infinity and Beyond. Princeton University Press. ISBN 978-0-691-02511-7.
• O'Connor, John J. and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor' Archived 2006-09-16 at the Wayback Machine, MacTutor History of Mathematics archive.
• O'Connor, John J. and Edmund F. Robertson (2000). 'Jaina mathematics' Archived 2008-12-20 at the Wayback Machine, MacTutor History of Mathematics archive.
• Pearce, Ian. (2002). 'Jainism', MacTutor History of Mathematics archive.
• Rucker, Rudy (1995). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press. ISBN 978-0-691-00172-2.
• Singh, Navjyoti (1988). "Jaina Theory of Actual Infinity and Transfinite Numbers". Journal of the Asiatic Society. 30.

External links

• "The Infinite". Internet Encyclopedia of Philosophy.
• Infinity on In Our Time at the BBC
• A Crash Course in the Mathematics of Infinite Sets Archived 2010-02-27 at the Wayback Machine, 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 Archived 2009-11-05 at the Wayback Machine, 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.
• Grime, James. "Infinity is bigger than you think". Numberphile. Brady Haran. Archived from the original on 2017-10-22. Retrieved 2013-04-06.
• Hotel Infinity
• John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor' Archived 2006-09-16 at the Wayback Machine, MacTutor History of Mathematics archive.
• John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics' Archived 2008-12-20 at the Wayback Machine, MacTutor History of Mathematics archive.
• Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive.
• 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)