FIX EQUATIONS

 

A response to the paper by Bruce A. Headman

The Infinite and the Finite

by ADAM DROZDEK

Department of Mathematics and Computer Science
Duquesne University
Pittsburgh, PA 15282

From Perspectives on Science and Christian Faith 45 (September 1993):187-191.
A very interesting paper by Bruce A. Hedman on the concept of infinity appeared recently in Perspectives (45:1). However, the importance of this concept, for Cantor in particular and for philosophy in general, does not seem to have been sufficiently stressed by the author.

First, Hedman treats Kant a bit too harshly by blaming him for pushing the incontingent world view to the extreme by screening the mind from noumena and making it occupy itself only with phenomena using its "innate patterns," an a priori knowledge. This, however, suggests a limiting view of the Kantian philosophy, according to which Kant was primarily interested in cognition. Analysis of cognition constituted to him a stepping stone for analysis of the problem of morality, which was the core of Kant's philosophy.

Kant wanted to defend man's dignity by showing that he is not one of the cogs of the universe ruled deterministically by mechanical laws. In order to defeat this world view, Kant had to first explore the nature and limitations of scientific cognition, thereby showing that science is unable to encompass everything within the confines of its categories. Science only has a limited scope, and scientists are just "artificers in the field of reason" (Critique of Pure Reason, A839/B867), and for "the necessary practical employment of reason" it is needed "to take away from this reason its pretensions to transcendent insight" (Kant, Bxxx). The highest goal of human reason should be the study of the "whole vocation of man," the study of morality (Kant, A840/B868). Thus, the study of theoretical reason precedes the study of practical reason. Theoretical and practical reasons, however, may have conflicting interests which are resolved under the guidance of practical reason; that is, practical reason (ethics) has priority over pure reason (cognition). Kant even "argued that of itself theoretical activity is neither unconditionally nor intrinsically good; it is valuable only insofar as it enhances moral practice and offers morally permissible maxims of happiness!"1 Therefore, practical reason is the highest arbiter in the case of any conflict, and its decisions can nullify even decisions of theoretical reason. Hence, Kant very clearly embraces contingent rationality by seeing that science has no final answers; these answers can be found in other domains than of practical reason.

On the other hand, Kant clearly spoke about a priori knowledge. But this knowledge is acquired using an a priori cognitive apparatus, and this is, in fact, what makes it a priori. The design of such an apparatus was based by Kant on Newtonian physics, but it seems that Kant would not insist that it is the only solution which can be given, especially if practical reason would have its say. Newtonian physics turned out to have a non-universal validity, but it does not devoid Kant's approach of all reasonableness. There are inherited elements in the human mind which allow man to live, although we may be mistaken with regard to their nature. Chomsky would mention here linguistic competency, which allows man to acquire any language. Similarly, the apparatus to process sense perceptions can be flexible enough to accommodate to any physics or geometry, but such an apparatus is an a priori. This view is also corroborated by Cantor, although, in the process of proving Cantor's contingent rationality, Hedman does not emphasize this point sufficiently. The point in question is the relationship between actual and potential infinity, and between the infinite and the finite.

Can the infinite be derived from the finite? The question may seem to have an obvious answer, since by an endless repetition of simple operations endless entities can be derived. But in this statement an idem per idem explanation is being used, since generating an endless entity uses the ability to make endless repetitions as a presumption. Infinity is assumed before it is even proved.

This implicit assumption is also made to establish the validity of some claims. One of its earliest uses was made by Zeno of Elea (5th century B.C.) who says in one of his paradoxes that a person moving from one point to another has first to reach half the distance; before he can do this, a half of the half must be reached, and before he can do that, he must traverse a half of that half's half, and so on to infinity. Since there is an infinite number of such divisions, the person, according to Zeno, is not able to cover the distance in a finite time and simply does not move. It may be easily proved that this series is convergent, i.e., Zeno's argument is formally (and also practically) wrong, but for us it is important that the alleged impossibility of covering an infinite number of intervals was proved in finite time by virtue of what can be called the and so on principle. It was unquestionable that one could, theoretically speaking, construct successive divisions, although nobody would be able to actually accomplish it. Thus the concept of potential infinity was freely used in the ancient thought and was taken for granted, although actual infinity was denied implicitly by Plato and explicitly by Aristotle up until the time of Cantor.

However, the and so on principle was used from then on not only in philosophy and common sense reasoning, but also in mathematics. But as Gottlob Frege pointed out, this was a formally incorrect method, and could not constitute the ultimate basis of proving mathematical theorems. According to Frege, nobody could, "to take a crude example," decide by means of that principle, "whether the number Julius Caesar belongs to a concept, or whether that famous conqueror of Gaul is a number or not."2 Therefore, a formally correct formulation of the principle of complete induction is needed. The principle was implicitly applied by Levi ben Gerson in Maase Choteb (1321) and by Blaise Pascal in Trait╚ du Triangle Arithmetique (1654) and it was explicitly formulated for the first time by Giuseppe Peano in Aritmetices Principia Nova Methodo Exposita (1889) as the fifth axiom of his theory of natural numbers.3 The axiom states that if a subset A of natural numbers contains zero and every number only if its predecessor belongs to A, then A contains each natural number, ie.,

 

copy section from text 188

 

This means that the set N of integers is the smallest set among all such sets which contain 0, and along with any element n, they also contain its successor, whereby N will not include "the famous conqueror of Gaul," although some sets A may include him. Thus, N is composed of integers only and nothing else. This axiom can be presented in an equivalent form (under modus ponens) as a rule used for proving theorems by induction, as in

 

 copy section from text 188

which means that if the sentence      copy fig      has been proven and, on the inductive hypothesis n<F128M>*<F255MI>A (for arbitrary n<F128M>*<F255MI>N), sentence (n+1)<F128M>*<F255MI>A has been proven, then it can be concluded that any integer is in A. The idea is that the proof of `(n+1)<F128M>*<F255MI>A' from `n<F128M><|>*<F255MI>A' can be repeated ad infinitum to produce the universal sentence <$E pile {down 30 bold ~ above {{size 8 m ~ symbol * ~italic N}}} ~~m ~symbol * ~ A>. Thus, induction compresses an infinite number of steps, or it tacitly assumes that an infinite repetition is feasible. But even assuming that such a repetition is possible, induction does not prove that an infinite set exists. It only states that if zero is in an A, and if for each n it includes both n and its successor, then N is a subset of A, that is, A is infinite. But the principle of infinite induction by itself is powerless to create an infinite set. It coveys certain ideas of how to do it, but it may only lead from proven statements to conclusions. This, so to speak, sets the tone for, to use Thoralf Post's expression, Athe recursive mode of reasoning"4 to constructively approach infinity, but it is not the solution for the construction proper. What is needed is an axiom of infinity, later assumed explicitly in set theory. It can be claimed that the use of recursive definitions would be sufficient and existence of no infinite set would have to be made. Peano, in fact, used such definitions in his system, in particular to define addition and multiplication, thereby substantially extending his system.

Recursive definitions are needed to show how to generate elements of a set (or how to generate new values for a function) being defined, and also to show that such a function exists in toto. This is what Richard Dedekind did in his Was Sind und Was Sollen die Zahlen (1888). However, in his system, recursive definitions are provable, but at the cost of "proving" that an infinite set exists that is a set for which a 1-1 mapping into its proper subset can be determined. Dedekind referred in this attempt to his own Gedankenwelt, a realm of thought. His proof (of theorem 66), however, did not guarantee uniqueness of elements in the sequence S = {thought t, thought on thought t, thought on thought on thought t, ... }. On the other hand, if man is able to think only a finite number of thoughts, can reference to thought be convincing? "So the validity of Dedekind's proof rests on the assumption that thoughts obtain independently of our thinking,"5 which is another blow against Dedekind's proof, which he considered "clear." But what is important is that while generating sequence S, Dedekind implicitly used recursive definition. Thus, before proving it, he used it, if only outside his "science of arithmetics."

Would the use of recursive definitions at the beginning of the system solve the problem of infinity? As indicated, Dedekind did not succeed. But can we succeed? First, recursive definitions can generate an infinite set if applied and reapplied endlessly by an indefinite repetition which derives new elements from those already existing. This possibility of repeating some operations an infinite number of times should not be limited, even in theory, otherwise it would be only verbal infinity, a pretense of an infinite creation. The concept of infinity is, therefore, assumed before any generating process starts. Secondly, recursive definitions cannot create anything ex nihilo: they have to presume an existence of infinite resources, even if these resources are only in our creative mind. This latter problem was recognized by Cantor, who wrote in his Briefbuch (1886):

In order for a changing quantity [the potential infinity] to be usable in any mathematical analysis, there must, strictly speaking, be known by definition the "area" of its changeability; this "area" cannot be anything changeable, otherwise a solid basis for the analysis would be missing; this "area" of values is then a certain actually infinite set. And hence, any potential infinity presupposes an actual infinity to be strictly usable in mathematics.6

The concept of infinity already exists and all efforts are made to hide it behind potential infinity in the form of allowing a possibility to infinitely apply some procedure. Thus Cantor was justified in saying: "the potential infinity has only a borrowed reality, insofar as it constantly points towards the actual infinity, through which it is possible in the first place."7 Actual infinity precedes potential infinity, the latter being a result of our limited comprehension and limited generation powers rather than a result of the ontological nature of the world.

Set theory solves the problem of infinity rather simply by introducing an axiom of infinity which Ernest Zermelo formulated as:                                                                  which allowed the generation of an infinite sequence                (although, by this axiom, X can have more elements than just these). The axiom uses a generation rule established by recursive definitions, hence no pretense was made that an infinite set can be generated outside theory, as Dedekind attempted to do. Zermelo introduces such an infinite set by a simple fiat, indicating how new elements can be derived from those already in X; but the existence of such an infinite set is ascertained from the outset. He needs an actual infinity in his set theory, doing it in "the recursive mode of reasoning." It is an assumption of actual infinity with a bow towards potential infinity.

This solution is made in the Cantorian spirit. Cantor very clearly realized that talk about potential infinity is either paradoxical or untruthful. It is paradoxical, since potential infinity is not infinite at any time. It is a non-existent entity assumed for the sake of argument and only approximated by something finite. On the other hand, discussion of potential infinity may be considered untruthful, since if extending the finite indefinitely is actually possible, then this infinity exists, if only in the future, if only in the mind of the beholder.

It is interesting to notice that we may go even further, in particular when we refer to Turing machines. The concept of Turing machines is entangled in the problem of infinity. Turing machines allow us to perform very complex operations using extremely simple steps defined on a finite number of states. They can use only 0's and 1's and yet perform very impressive operations. However, there is an underlying assumption that an infinite tape is available and also infinite time. The tape is not potentially existing, but actually. Operations of Turing machines are interesting and useful, if eventually ended. But regardless of the number of transitions, the tape should be infinite (linear-bounded automata have a limited power of dealing only with context-sensitive languages). Hence, infinity has to be known and at hand in order to grapple with the finite. There has to be set an infinite stage to allow finite actors to perform. The finite presupposes infinity, infinity is prior to the finite C a paradoxical if not surprising result. In cases like this, we may agree with Descartes, who said in the third meditation that "in some way I have in me the notion of the infinite earlier that the finite."8

This result might have been expected after we realized that potential infinity is a dignified name for the finite which is in the process of extension. Therefore, because potential infinity is simply extending indefinitely the finite, we may strengthen Cantor's statement on the priority of actual infinity over potential infinity by stating that the infinite precedes the finite.

And the infinite? The infinite is simply given to us: it is a synthetic a priori datum. God implanted it in our minds. "He has put eternity into man's mind" (Eccl. 3:11), the infinite "even inhabits our minds."9 This infinite is an endowment which we bring to this world, it is an endowment with which the world can be understood. It is an a priori which enables the cognitive a posteriori, an incontingent tool which allows us to grasp the contingent rationality. It is truly a great achievement of Cantor that he ceased to hide the assumption of infinity behind ever-extending the finite and made a scientific concept out of what was allowed to have existed only in theology.

NOTES

1Roger J. Sullivan, Immanuel Kant's Moral Theory, Cambridge: Cambridge University Press 1989, p. 97.

2Gottlob Frege, The Foundations of Arithmetic, London: Basil Blackwell 1950, p. 68.

3Joseph Carlebach, Lewi ben Gerson als Mathematiker, Heidelberg 1908, p. 56; Blaise Pascal, Oeuvres Compl╦tes, Paris 1963, pp. 56-57; Giuseppe Peano, Selected Works, London 1973, p. 113.

4Quoted from G.T. Kneebone, Mathematical Logic and the Foundations of Mathematics, London: Van Norstrand 1963, p. 266.

5Gottlob Frege, Posthumous Writings, Chicago: Chicago University Press 1979, p. 136.

6Herbert Meschkowski, Probleme des Undendlichen: Werk und Leben Georg Cantors, Braunschweig: Vieweg 1967, p. 250.

7Georg Cantor, Gesammelte Abhandlungen, Berlin 1932, p. 404.

8Ren╚ Descartes, The Philosophical Works, Cambridge: Cambridge University Press 1967, v.1, p. 166.

9Georg Cantor, op. cit., 375.