Actual infinity
Encyclopedia
Actual infinity is the idea that numbers, or some other type of mathematical object
, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics
, the abstraction
of actual infinity
involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects.
handled the topic of infinity in Physics and in Metaphysics. Aristotle distinguished between infinity in respect to addition and in respect to division.
Aristotle also distinguished between actual and potential infinities. An actual infinity is something which is completed and definite and consists of infinitely many elements, and according to Aristotle, a paradoxical idea, both in theory and in nature. In respect to addition, a potentially infinite sequence or a series is potentially endless; being a potentially endless series means that one element can always be added to the series after another, and this process of adding elements is never exhausted.
As an example of a potentially infinite series in respect to increase, one number can always be added after another in the series that starts 1,2,3,... but the process of adding more and more numbers cannot be exhausted or completed, because there is no end to the process.
In respect to division, a potentially infinite series of divisions is e.g. the one that starts as 1, 0.5, 0.25, 0.125, 0.0625. According to Aristotle, the process of division never comes to an end, and the limit value 0 is never reached, although the division can be continued as long as one wants. This is a crucial difference to the transfinitists, who start with the very notion that the limit value exists and is reached (this is not to say that 0 would not exist; zero is at our disposal).
In contrast to the potential infinity, all the elements of an actually infinite (= transfinite) set are assumed to exist together simultaneously as a completed totality. The term 'transfinite' ought to be used instead of 'actually infinite' to denote the transfinite sets, because the set-theoretical notion of actual infinity has nothing to do with actualization in nature.
.
Proponents of intuitionism
, from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets. (Also, according to Aristotle, a completed infinity cannot exist even as an idea in the mind of a human.) Consequently, they reconstruct the foundations of mathematics in a way that does not assume the existence of actual infinities. On the other hand, constructive analysis does accept the existence of the completed infinity of the integers.
For intuitionists, infinity is described as potential; terms synonymous with this notion are becoming or constructive. For example, Stephen Kleene describes the notion of a Turing machine
tape as "a linear tape, (potentially) infinite in both directions." To access memory on the tape, a Turing machine moves a read head along it in finitely many steps: the tape is therefore only "potentially" infinite, since while there is always the ability to take another step, infinity itself is never actually reached.
Classical mathematicians generally accept actual infinities. Georg Cantor
is the most significant mathematician who defended actual infinities, equating the Absolute Infinite
with God. He decided that it is possible for natural and real numbers to be definite sets, and that if one rejects the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then one is not involved in any contradiction
.
The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound.
(unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon. Apeiron stands opposed to that which has a peras (limit). These notions are today denoted by potentially infinite and actually infinite, respectively.
Aristotle
sums up the views of his predecessors on infinity thus:
The theme was brought forward by Aristotle's consideration of the apeiron in the context of mathematics and physics (the study of nature).
adhered to the motto Infinitum actu non datur. This means there is only a (developing, improper, "syncategorematic") potential infinity but not a (fixed, proper, "categorematic") actual infinity. There were exceptions, however, for example in England.
During the Renaissance and by early modern times the voices in favor of actual infinity were rather rare.
The majority agreed with the well-known quote of Gauss:
The drastic change was initialized by Bolzano and Cantor in the 19th century.
Bernard Bolzano
who introduced the notion of set (in German: Menge) and Georg Cantor who introduced set theory
opposed the general attitude. Cantor distinguished three realms of infinity: (1) the infinity of God (which he called the "absolutum"), (2) the infinity of reality (which he called "nature") and (3) the transfinite numbers and sets of mathematics.
Mathematical object
In mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...
, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
, the abstraction
Abstraction
Abstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....
of actual infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects.
Aristotle's potential–actual distinction
AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
handled the topic of infinity in Physics and in Metaphysics. Aristotle distinguished between infinity in respect to addition and in respect to division.
Aristotle also distinguished between actual and potential infinities. An actual infinity is something which is completed and definite and consists of infinitely many elements, and according to Aristotle, a paradoxical idea, both in theory and in nature. In respect to addition, a potentially infinite sequence or a series is potentially endless; being a potentially endless series means that one element can always be added to the series after another, and this process of adding elements is never exhausted.
As an example of a potentially infinite series in respect to increase, one number can always be added after another in the series that starts 1,2,3,... but the process of adding more and more numbers cannot be exhausted or completed, because there is no end to the process.
In respect to division, a potentially infinite series of divisions is e.g. the one that starts as 1, 0.5, 0.25, 0.125, 0.0625. According to Aristotle, the process of division never comes to an end, and the limit value 0 is never reached, although the division can be continued as long as one wants. This is a crucial difference to the transfinitists, who start with the very notion that the limit value exists and is reached (this is not to say that 0 would not exist; zero is at our disposal).
In contrast to the potential infinity, all the elements of an actually infinite (= transfinite) set are assumed to exist together simultaneously as a completed totality. The term 'transfinite' ought to be used instead of 'actually infinite' to denote the transfinite sets, because the set-theoretical notion of actual infinity has nothing to do with actualization in nature.
Opposition from the Intuitionist school
The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential, but not to be mistaken for physically existing. The question of whether natural or real numbers form definite sets is therefore independent of the question of whether infinite things exist physically in natureNature
Nature, in the broadest sense, is equivalent to the natural world, physical world, or material world. "Nature" refers to the phenomena of the physical world, and also to life in general...
.
Proponents of intuitionism
Intuitionism
In 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...
, from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets. (Also, according to Aristotle, a completed infinity cannot exist even as an idea in the mind of a human.) Consequently, they reconstruct the foundations of mathematics in a way that does not assume the existence of actual infinities. On the other hand, constructive analysis does accept the existence of the completed infinity of the integers.
For intuitionists, infinity is described as potential; terms synonymous with this notion are becoming or constructive. For example, Stephen Kleene describes the notion of a Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
tape as "a linear tape, (potentially) infinite in both directions." To access memory on the tape, a Turing machine moves a read head along it in finitely many steps: the tape is therefore only "potentially" infinite, since while there is always the ability to take another step, infinity itself is never actually reached.
Classical mathematicians generally accept actual infinities. Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
is the most significant mathematician who defended actual infinities, equating the Absolute Infinite
Absolute Infinite
The Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God...
with God. He decided that it is possible for natural and real numbers to be definite sets, and that if one rejects the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then one is not involved in any contradiction
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
.
The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound.
History
The ancient Greek term for the potential or improper infinite was apeironApeiron (cosmology)
Apeiron is a Greek word meaning unlimited, infinite or indefinite from ἀ- a-, "without" and πεῖραρ peirar, "end, limit", the Ionic Greek form of πέρας peras, "end, limit, boundary".-Apeiron as an origin:...
(unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon. Apeiron stands opposed to that which has a peras (limit). These notions are today denoted by potentially infinite and actually infinite, respectively.
AnaximanderAnaximanderAnaximander was a pre-Socratic Greek philosopher who lived in Miletus, a city of Ionia; Milet in modern Turkey. He belonged to the Milesian school and learned the teachings of his master Thales...
(610-546 BC) held that the apeiron was the principle or main element composing all things. Clearly, the 'apeiron' was some sort of basic substance. PlatoPlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
's notion of the apeiron is more abstract, having to do with indefinite variability. The main dialogues where Plato discusses the 'apeiron' are the late dialogues Parmenides http://www.greektexts.com/library/Plato/parmenides/eng/index.html and the Philebus http://www.greektexts.com/library/Plato/philebus/eng/index.html.
Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
sums up the views of his predecessors on infinity thus:
Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also. (Aristotle)
The theme was brought forward by Aristotle's consideration of the apeiron in the context of mathematics and physics (the study of nature).
Infinity turns out to be the opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'. (Aristotle [1])
Belief in the existence of the infinite comes mainly from five considerations:
- From the nature of time - for it is infinite.
- From the division of magnitudes - for the mathematicians also use the notion of the infinite.
- If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.
- Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.
- Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody - not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought. (Aristotle [1])
With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens. (Aristotle [1])
Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish. (Aristotle [1])
Scholastic philosophers
The overwhelming majority of scholastic philosophersScholasticism
Scholasticism is a method of critical thought which dominated teaching by the academics of medieval universities in Europe from about 1100–1500, and a program of employing that method in articulating and defending orthodoxy in an increasingly pluralistic context...
adhered to the motto Infinitum actu non datur. This means there is only a (developing, improper, "syncategorematic") potential infinity but not a (fixed, proper, "categorematic") actual infinity. There were exceptions, however, for example in England.
It is well known that in the Middle Ages all scholastic philosophers advocate Aristotle's "infinitum actu non datur" as an irrefutable principle. (G. CantorGeorg 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,...
[3, p. 174])
The number of points in a segment one ell long is its true measure. (R. GrossetesteRobert GrossetesteRobert Grosseteste or Grossetete was an English statesman, scholastic philosopher, theologian and Bishop of Lincoln. He was born of humble parents at Stradbroke in Suffolk. A.C...
[9, p. 96])
Actual infinity exists in number, time and quantity. (J. Baconthorpe [9, p. 96])
During the Renaissance and by early modern times the voices in favor of actual infinity were rather rare.
The continuum actually consists of infinitely many indivisibles (G. GalileiGalileo GalileiGalileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...
[9, p. 97])
I am so in favour of actual infinity. (G.W. Leibniz [9, p. 97])
The majority agreed with the well-known quote of Gauss:
I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction. (C.F. GaussCarl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
[in a letter to Schumacher, 12 July 1831])
The drastic change was initialized by Bolzano and Cantor in the 19th century.
Bernard Bolzano
Bernard Bolzano
Bernhard Placidus Johann Nepomuk Bolzano , Bernard Bolzano in English, was a Bohemian mathematician, logician, philosopher, theologian, Catholic priest and antimilitarist of German mother tongue.-Family:Bolzano was the son of two pious Catholics...
who introduced the notion of set (in German: Menge) and Georg Cantor who introduced set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
opposed the general attitude. Cantor distinguished three realms of infinity: (1) the infinity of God (which he called the "absolutum"), (2) the infinity of reality (which he called "nature") and (3) the transfinite numbers and sets of mathematics.
A multitude which is larger than any finite multitude, i.e., a multitude with the property that every finite set [of members of the kind in question] is only a part of it, I will call an infinite multitude. (B. Bolzano [2, p. 6])
There are twice as many focuses as centres of ellipses. (B. Bolzano [2a, § 93])
Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. (G. Cantor [3, p. 399; 8, p. 252])
One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor [3, p. 400])
The numbers are a free creation of human mind. (R. DedekindRichard 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:...
[3a, p. III])
Classical set theory
Classical set theory accepts the notion of actual, completed infinities. However, some finitist philosophers of mathematics and constructivists object to the notion.If the positive number n becomes infinitely great, the expression 1/n goes to naught (or gets infinitely small). In this sense one speaks of the improper or potential infinite. In sharp and clear contrast the set just considered is a readily finished, locked infinite set, fixed in itself, containing infinitely many exactly defined elements (the natural numbers) none more and none less. (A. Fraenkel [4, p. 6])
Thus the conquest of actual infinity may be considered an expansion of our scientific horizon no less revolutionary than the Copernican system or than the theory of relativity, or even of quantum and nuclear physics. (A. Fraenkel [4, p. 245])
To look at the universe of all sets not as a fixed entity but as an entity capable of "growing", i.e., we are able to "produce" bigger and bigger sets. (A. Fraenkel et al. [5, p. 118])
(BrouwerLuitzen Egbertus Jan BrouwerLuitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
) maintains that a veritable continuum which is not denumerable can be obtained as a medium of free development; that is to say, besides the points which exist (are ready) on account of their definition by laws, such as e, pi, etc. other points of the continuum are not ready but develop as so-called choice sequenceChoice sequenceIn intuitionistic mathematics, a choice sequence is a constructive formulation of a sequence. Since the Intuitionistic school of mathematics, as formulated by L. E. J...
s. (A. Fraenkel et al. [5, p. 255])
Intuitionists reject the very notion of an arbitrary sequence of integers, as denoting something finished and definite as illegitimate. Such a sequence is considered to be a growing object only and not a finished one. (A. Fraenkel et al. [5, p. 236])
Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for “actual infinity.” The arguments using infinity, including the Differential CalculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
of NewtonIsaac 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 LeibnizGottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
, do not require the use of infinite sets. (T. Jech http://plato.stanford.edu/entries/set-theory/#1)
Owing to the gigantic simultaneous efforts of FregeGottlob 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...
, DedekindRichard 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 Cantor, the infinite was set on a throne and revelled in its total triumph. In its daring flight the infinite reached dizzying heights of success. (D. HilbertDavid HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
[6, p. 169])
One of the most vigorous and fruitful branches of mathematics [...] a paradise created by Cantor from which nobody shall ever expel us [...] the most admirable blossom of the mathematical mind and altogether one of the outstanding achievements of man's purely intellectual activity. (D. Hilbert on set theory [6])
Finally, let us return to our original topic, and let us draw the conclusion from all our reflections on the infinite. The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking - a remarkable harmony between being and thinking. (D. Hilbert [6, 190])
Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. (A. RobinsonAbraham RobinsonAbraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
[10, p. 507])
Indeed, I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world. (A. Robinson)
Georg Cantor's grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory. (Y. Manin http://arxiv.org/pdf/math/0209244)
Thus, exquisite minimalism of expressive means is used by Cantor to achieve a sublime goal: understanding infinity, or rather infinity of infinities. (Y. Manin http://arxiv.org/pdf/math/0209244)
There is no actual infinity, that the Cantorians have forgotten and have been trapped by contradictions. (H. PoincaréHenri PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
[Les mathématiques et la logique III, Rev. métaphys. morale 14 (1906) p. 316])
When the objects of discussion are linguistic entities [...] then that collection of entities may vary as a result of discussion about them. A consequence of this is that the "natural numbers" of today are not the same as the "natural numbers" of yesterday. (D. Isles http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndjfl/1093634481&abstract=)
There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity. (E. Nelson http://www.math.princeton.edu/~nelson/papers/e.pdf)
A viable and interesting alternative to regarding the numbers as a completed infinity, one that leads to great simplifications in some areas of mathematics and that has strong connections with problems of computational complexity. (E. Nelson http://www.math.princeton.edu/~nelson/papers/e.pdf)
During the renaissance, particularly with BrunoGiordano 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...
, actual infinity transfers from God to the world. The finite world models of contemporary science clearly show how this power of the idea of actual infinity has ceased with classical (modern) physics. Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing. Within the intellectual overall picture of our century ... actual infinity brings about an impression of anachronism. (P. LorenzenPaul LorenzenPaul Lorenzen was a philosopher andmathematician.As a founder of the Erlangen School and the inventor of game semantics he was a famous German philosopher of the 20th century.-Biography:Lorenzen studied with David Hilbert as a schoolboy and he was one of Hasse's...
http://www.sgipt.org/wisms/geswis/mathe/lor01.gif)
