The Infinite and the Finite

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 Ainnate 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 Aartificers in the field of reason@ (Critique of Pure Reason, A839/B867), and for Athe necessary practical employment of reason@ it is needed Ato take away from this reason its pretensions to transcendent insight@ (Kant, Bxxx). The highest goal of human reason should be the study of the Awhole 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 Aargued 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.

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, Ato take a crude example,@ decide by means of that principle, Awhether 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 Traité 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.,

<R><R> <$E pile {bold ~ above {size 8 { A ~symbol E~ N}}} ~~(0 ~symbol * ~italic A ~ sub symbol L ~pile {bold ~ above {size 6 italic n}} ~( italic n ~symbol * ~italic A ~symbol 7 ~italic succ italic ( n ) ~symbol * ~ italic A ~) ~ symbol 7 ~italic N ~symbol E ~italic A )>. <R>

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 Athe 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

@CENTERED = <$E ccol {pile { down 30 {0 ~symbol * ~italic A} above {pile {down 30 bold ~~ above {size 6 italic n}}( italic n ~symbol * ~italic A ~ symbol 7 ~italic succ~(n) ~symbol * ~italic A~)} over down 30 {italic N ~symbol E ~~italic A}}}>

which means that if the sentence 0<F128M>*<F255MI>A 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 Aproving@ 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? ASo 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 Aclear.@ 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 Ascience of arithmetics.@

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: Athe 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:<R><R> <$Epile {{down 30 font 2 symbol $} above {italic X}} ~(~ 0 ~symbol * ~X~ sub font 2 symbol L ~(~ roman z~symbol * ~X~symbol 7 ~ "{" ~ font 21 roman z ~ "}" ~ symbol * ~X~)~) >, <R><R>which allowed the generation of an infinite sequence {0, {0}, {{0}}, ... } (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 Athe recursive mode of reasoning.@ It is an assumption of actual infinity with a bow towards potential infinity.

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 Ain some way I have in me the notion of the infinite earlier that the finite.@<P8MJ243>8

And the infinite? The infinite is simply given to us: it is a synthetic a priori datum. God implanted it in our minds. AHe has put eternity into man's mind@ (Eccl. 3:11), the infinite Aeven 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.