In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity,[1] involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. This type of process occurs in mathematics, for instance, in standard formalizations of the notions of an infinite series, infinite product, or limit.[2]

Anaximander edit

The ancient Greek term for the potential or improper infinite was apeiron (unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon.[3] Apeiron stands opposed to that which has a peras (limit). These notions are today denoted by potentially infinite and actually infinite, respectively.

Anaximander (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. Plato'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 and the Philebus.

Aristotle edit

Aristotle sums up the views of his predecessors on infinity as follows:

"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)[4]

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)[5]

Belief in the existence of the infinite comes mainly from five considerations:[6]

  1. From the nature of time – for it is infinite.
  2. From the division of magnitudes – for the mathematicians also use the notion of the infinite.
  3. If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.
  4. 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.
  5. 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)

Aristotle postulated that an actual infinity was impossible, because if it were possible, then something would have attained infinite magnitude, and would be "bigger than the heavens." However, he said, mathematics relating to infinity was not deprived of its applicability by this impossibility, because mathematicians did not need the infinite for their theorems, just a finite, arbitrarily large magnitude.[7]

Aristotle's potential–actual distinction edit

Aristotle handled the topic of infinity in Physics and in Metaphysics. He distinguished between actual and potential infinity. Actual infinity is completed and definite, and consists of infinitely many elements. Potential infinity is never complete: elements can be always added, but never infinitely many.

"For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different."

— Aristotle, Physics, book 3, chapter 6.

Aristotle distinguished between infinity with respect to addition and division.

But Plato has two infinities, the Great and the Small.

— Physics, book 3, chapter 4.

"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."[citation needed]

With respect to division, a potentially infinite sequence of divisions might start, for example, 1, 1/2, 1/4, 1/8, 1/16, but the process of division cannot be exhausted or completed.

"For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately."

— Metaphysics, book 9, chapter 6.

Aristotle also argued that Greek mathematicians knew the difference among the actual infinite and a potential one, but they "do not need the [actual] infinite and do not use it" (Phys. III 2079 29).[8]

Scholastic, Renaissance and Enlightenment thinkers edit

The overwhelming majority of scholastic philosophers 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. Cantor)[9]

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. Galilei [9, p. 97])

I am so in favour of actual infinity. (G.W. Leibniz [9, p. 97])

However, the majority of pre-modern thinkers[citation needed] 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.[10] (C.F. Gauss [in a letter to Schumacher, 12 July 1831])

Modern era edit

Actual infinity is now commonly accepted. 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.

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])

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. (Georg Cantor)[11] (G. Cantor [8, p. 252])

The numbers are a free creation of human mind. (R. Dedekind [3a, p. III])

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])

Cantor distinguished two types of actual infinity, the transfinite and the absolute, about which he affirmed:

These concepts are to be strictly differentiated, insofar the former is, to be sure, infinite, yet capable of increase, whereas the latter is incapable of increase and is therefore indeterminable as a mathematical concept. This mistake we find, for example, in Pantheism. (G. Cantor, Über verschiedene Standpunkte in bezug auf das aktuelle Unendliche, in Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, pp. 375, 378)[12]

Current mathematical practice edit

Actual infinity is now commonly accepted, because mathematicians have learned how to construct algebraic statements using it. For example, one may write down a symbol,  , with the verbal description that "  stands for completed (countable) infinity". This symbol may be added as an ur-element to any set. One may also provide axioms that define addition, multiplication and inequality; specifically, ordinal arithmetic, such that expressions like   can be interpreted as "any natural number is less than completed infinity". Even "common sense" statements such as   are possible and consistent. The theory is sufficiently well developed, that rather complex algebraic expressions, such as  ,   and even   can be interpreted as valid algebraic expressions, can be given a verbal description, and can be used in a wide variety of theorems and claims in a consistent and meaningful fashion. The ability to define ordinal numbers in a consistent, meaningful way, renders much of the debate moot; whatever personal opinion one may hold about infinity or constructability, the existence of a rich theory for working with infinities using the tools of algebra and logic is clearly in hand.

Opposition from the Intuitionist school edit

The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential,[13] 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 nature.

Proponents of intuitionism, from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets. 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.[13] For example, Stephen Kleene describes the notion of a Turing machine tape as "a linear 'tape', (potentially) infinite in both directions."[14] 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.[15]

Mathematicians generally accept actual infinities.[16] Georg Cantor is the most significant mathematician who defended actual infinities. 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 present-day conventional finitist interpretation of ordinal and cardinal numbers is that they consist of a collection of special symbols, and an associated formal language, within which statements may be made. All such statements are necessarily finite in length. The soundness of the manipulations is founded only on the basic principles of a formal language: term algebras, term rewriting, and so on. More abstractly, both (finite) model theory and proof theory offer the needed tools to work with infinities. One does not have to "believe" in infinity in order to write down algebraically valid expressions employing symbols for infinity.

Classical set theory edit

The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound.

Classical set theory accepts the notion of actual, completed infinities. However, some finitist philosophers of mathematics and constructivists object to the notion.

See also edit

References edit

  1. ^ Strogatz, Steven H. (2019). Infinite powers: how calculus reveals the secrets of the universe. Boston: Houghton Mifflin Harcourt. ISBN 978-1-328-87998-1.
  2. ^ Fletcher, Peter (2007). "Infinity". Philosophy of Logic. Handbook of the Philosophy of Science. Elsevier. pp. 523–585. doi:10.1016/b978-044451541-4/50017-8. ISBN 9780444515414.
  3. ^ Fenves, Peter David (2001). Arresting Language: From Leibniz to Benjamin. Stanford University Press. p. 331. ISBN 9780804739603.
  4. ^ Thomas, Kenneth W.; Thomas, Thomas, Aquinas (2003-06-01). Commentary on Aristotle's Physics. A&C Black. p. 163. ISBN 9781843715450.{{cite book}}: CS1 maint: multiple names: authors list (link)
  5. ^ Padovan, Richard (2002-09-11). Proportion: Science, Philosophy, Architecture. Taylor & Francis. p. 123. ISBN 9781135811112.
  6. ^ Thomas, Kenneth W.; Thomas, Thomas, Aquinas (2003-06-01). Commentary on Aristotle's Physics. A&C Black. ISBN 9781843715450.{{cite book}}: CS1 maint: multiple names: authors list (link)
  7. ^ "Logos Virtual Library: Aristotle: Physics, III, 7". Retrieved 2017-11-14.
  8. ^ Allen, Reginald E. (1998). Plato's Parmenides. The Dialogues of Plato. Vol. 4. New Haven: Yale University Press. p. 256. ISBN 9780300138030. OCLC 47008500.
  9. ^ Cantor, Georg (1966). Zermelo, Ernst (ed.). Gesammelte abhandlungen: Mathematischen und philosophischen inhalts. Georg Olms Verlag. p. 174.
  10. ^ Stephen Kleene 1952 (1971 edition):48 attributes the first sentence of this quote to (Werke VIII p. 216).
  11. ^ Cantor, Georg (1966). Zermelo, Ernst (ed.). Gesammelte abhandlungen: Mathematischen und philosophischen inhalts. Georg Olms Verlag. p. 399.
  12. ^ Kohanski, Alexander Sissel (June 6, 2021). The Greek Mode of Thought in Western Philosophy. Fairleigh Dickinson University Press. p. 271. ISBN 9780838631393. OCLC 230508222.
  13. ^ a b Kleene 1952/1971:48.
  14. ^ Kleene 1952/1971:48 p. 357; also "the machine ... is supplied with a tape having a (potentially) infinite printing ..." (p. 363).
  15. ^ Or, the "tape" may be fixed and the reading "head" may move. Roger Penrose suggests this because: "For my own part, I feel a little uncomfortable about having our finite device moving a potentially infinite tape backwards and forwards. No matter how lightweight its material, an infinite tape might be hard to shift!" Penrose's drawing shows a fixed tape head labelled "TM" reading limp tape from boxes extending to the visual vanishing point. (Cf page 36 in Roger Penrose, 1989, The Emperor's New Mind, Oxford University Press, Oxford UK, ISBN 0-19-851973-7). Other authors[who?] solve this problem by tacking on more tape when the machine is about to run out.
  16. ^ Actual infinity follows from, for example, the acceptance of the notion of the integers as a set, see J J O'Connor and E F Robertson, "Infinity".

Sources edit

  • "Infinity" at The MacTutor History of Mathematics archive, treating the history of the notion of infinity, including the problem of actual infinity.
  • Aristotle, Physics [1]
  • Bernard Bolzano, 1851, Paradoxien des Unendlichen, Reclam, Leipzig.
  • Bernard Bolzano 1837, Wissenschaftslehre, Sulzbach.
  • Georg Cantor in E. Zermelo (ed.) 1966, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Olms, Hildesheim.
  • Richard Dedekind in 1960 Was sind und was sollen die Zahlen?, Vieweg, Braunschweig.
  • Adolf Abraham Fraenkel 1923, Einleitung in die Mengenlehre, Springer, Berlin.
  • Adolf Abraham Fraenkel, Y. Bar-Hillel, A. Levy 1984, Foundations of Set Theory, 2nd edn., North Holland, Amsterdam New York.
  • Stephen C. Kleene 1952 (1971 edition, 10th printing), Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam New York. ISBN 0-444-10088-1.
  • H. Meschkowski 1981, Georg Cantor: Leben, Werk und Wirkung (2. Aufl.), BI, Mannheim.
  • H. Meschkowski, W. Nilson (Hrsg.) 1991, Georg Cantor – Briefe, Springer, Berlin.
  • Abraham Robinson 1979, Selected Papers, Vol. 2, W.A.J. Luxemburg, S. Koerner (Hrsg.), North Holland, Amsterdam.