REASONING ABOUT INFINITY: MATHEMATICAL CONCEPTS AND
IMPLICATIONS ON THE LIMITATIONS OF COMPUTING DEVICES
D.F.M. STRAUSS ABSTRACT
Within the context of a finite number of entities or even an endless succession of numbers – like the sequence of natural numbers or a convergent sequence of fractions (such as n/n+1 that converges towards the limit 1) the capacities of mechanical or electronic computing devices match those of the human intellect (of course the former are much quicker!). Whereas the nature of infinity in the sense of endlessness (the successive infinite) opens up the possibilities of computing devices this same property highlights their inherent limitations, because no machine can switch from the successive infinite – providing the foundation for every construction and computation – to the at once infinite (the actual infinite). Cantor's proof of the non-denumerability of the real numbers (1874; the famous diagonal proof 1889), found at the cradle of modern mathematics, employs the at once infinite transcending the limited nature of the successive infinite.
1. THE ‘UNCOUNTABLE’
Even before I went to school I was baffled by the awareness that one can extend the act of counting indefinitely. Sometimes the older kids played a game by challenging each other to see who can count further than anyone else. As I was listening to them I realized that this exercise could be extended for an indefinite length of time. As a result I joined the contest ‘ahead’ of everyone else by starting with “one uncountable,” “two uncountable” and so on.
2. TRANSFINITE ARITHMETIC
Of course what I did not realize at the time was that owing to the German mathematician Georg Cantor (1845-1918) modern mathematics already developed an impressive theory of ‘transfinite’ numbers. This development was exceptional in many ways, amongst them particularly because the awareness of infinity prompted the sharpest spirits in the domains of mathematics, physics and philosophy throughout the history of scholarly reflection to master the nature of the infinite. 3. THE CONTRIBUTION OF GREEK CULTURE
Greek culture made significant contributions to an understanding of the infinite. Although the Pythagoreans truly admired the explanatory power of number and numerical relations above anything else, the school of Parmenides (apparently borne in 510 B.C.) soon challenged our ordinary understanding of multiplicity and movement. In response to the challenge entailed in the Paradoxes of Zeno we later on find that Aristotle rejected what was known as the “actual infinite.” If there would
be something like that it would imply that each one of its parts is also actually infinite – but then such parts would be equal (or: equivalent) to the whole (see Aristotle,
Physica, 204 a 25; 2001:260-261), contradicting Aristotle's belief that the whole is
prior to (or: more than the sum of) its parts. What is finite and limited was complete,1
but what is infinite was incomplete. Descartes turns this classical view upside down for according to him the infinite is complete and the finite incomplete, so that the finite should actually be referred to as the non-infinite. Since Spinoza identified God with nature (Deus sive natura), he also saw the universe as an instance of completed infinitude.
4. GALILEO ON SQUARES AND THE NATURAL NUMBERS
But it is to Galileo that we owe a discussion of the remarkable relation between square numbers and the sequence of all numbers in dialogue form in March 1638: All numbers are not square numbers (like 1, 4, 9, 16, 25... ). All numbers, i.e. square and non-square numbers, are certainly more than the square numbers on their own. From 0-100 there are only ten squares (100 = 102); from 0-10000 there are hundred squares; from 0-10000 there are 1000 squares (10000 = 1002); from 0-1000000 there are only a 1000 squares (1000000 = 10002); and so forth. However, if we ask: how many square numbers exist? we can answer: just as many as there are square roots, since every square has a root and every root is the root of a square. Then, however, there are just as many squares as there are (natural) numbers!
12 22 32 42... 1 2 3 4...
Bernard Bolzano built on this in a posthumously published work by considering an infinite set as being characterized by the fact that the whole set can be matched element by element (in the case of Galileo's example: 1 with 12, 2 with 22, 3 with 32, and so forth) with a true subset (the set of squares is a subset of the set of natural numbers) (Bolzano, 1920, par.20:27ff.). In the case of an infinite set the whole is therefore equal (or: equivalent) to a part, thereby showing that Aristotle's objection against the “actual infinite” turned out to highlight a characteristic feature of it! 5. BACK TO ARISTOTLE ON THE POTENTIAL AND ACTUAL INFINITE
Let us return for a moment to the bipolar nature of the infinite as it is strikingly en-countered in Zeno's arguments against motion. Aristotle mentions Zeno's four arguments in his Physics (cf. 233 a 13ff. and 239 b 5ff.). We will refer to only two particularly illustrative arguments, namely that of Achilles who can never catch up with the tortoise (since the tortoise constantly establishes a lead while Achilles moves towards the previous position of the tortoise), and the argument that it is impossible to move from point A to point B for in order to do so, after all, it is first necessary to traverse half the distance, thereafter half of the remaining distance, and thereafter again a half of the remaining distance – ad infinitum (cf. Dielz-Kranz, 1959-60 B Fr.3). Zeno concludes: an infinite number of spatial sub-intervals must be
crossed to move from A to B and this is impossible in a finite period of time. It is after all impossible to actually exhaust the infinite. Therefore motion is impossible. This is an instance of the apparent contradiction between the uncompleted infinite and the completed infinite.2
6. THE BASIC MEANING OF ENDLESSNESS
Surely we are all familiar with the most basic nature of the infinite – understood in the literal sense of being without an end. Words like endlessness and indefinitely echo the same core meaning of the infinite. In order to conform to our basic intuition it is preferable to substitute the phrase “potential infinite” by using the phrase “the successive infinite.”
In the ordinary theory of numbers the acknowledgement of more and less in the first place fully depends on the succession of natural numbers: 1, 2, 3, 4, and so on, indefinitely, endlessly. As a first year student in mathematics I have asked the professor (Prof A.P Malan) what is the infinite, to which he replied: “infinity is a number larger than any number regardless how large such a number may be.”3 Surely this definition takes the succession of natural numbers (1, 2, 3, 4, …) to its ultimate consequence: endlessness.
7. THE COMPLICATION CAUSED BY IRRATIONAL NUMBERS
As simple and uncomplicated as this notion of the infinite may appear, it gets entangled in difficulties as soon as we extend the concept of number beyond natural numbers, integers and fractions. Let us look at the sketch 1 below.
1
1
2 or = 2 ?
=
Sketch 1
If the number of ‘steps’ approximate infinity will the length of the diagonal be
2
We must therefore differ from Titze's statement that the “numerically-infinite was inconceivable in Greek philosophy” (1984:141). Anaxagoras without doubt already had a conception of the potentially-infinite, while we obviously find an initial conception of the infinite divisibility of a continuum in Zeno's thought (cf. his B Fr.3).
3
The sketch captures 5 steps – each constituted by a horizontal and a vertical line. The horizontal lines divide the basis of the rectangular triangle in five equal parts and combined they add up to the length 1 – and the same applies to the vertical line (their combined lengths also add up to 1). Suppose now we increase the number of steps and make them 10 – will anything change in respect of the length of the combined horizontal or vertical lines? The answer seems to be: no. Suppose now that we proceed by increasing the number of steps indefinitely – then the same would hold, i.e. the respective (horizontal and vertical) sums of the lines will still add up to 1 + 1 = 2. Yet we know from the theorem of Pythagoras that the diagonal is the limit of this constantly increasing number of steps (ever diminishing intervals) – and we also know that its length is not 2 but the square root of 2 (
We may have an intellectual representation of this limiting case (where the length of the diagonal equals 2) and at the same time be aware of the fact that any physical succession of events (’steps –’ to play with the ambiguity of the word /step/) can never be completed since it has to remain open to the future. Apparently physical reality only allows for infinite possibilities without being able to ‘close them off’ into an infinite totality, a completed infinite whole.
Greek mathematics had an implicit awareness of the idea of a limit as can be seen from the following example:
The sequence is calculated as follows: the denominator (under the line) of every subsequent fraction equals the sum of the numerator (above the line) and denominator of the previous fraction, while the numerator of every subsequent fraction equals the sum of its own denominator and that of the previous fraction. The sum of the numerator and denominator of 1/1 equals 2– the denominator of the second fraction– while the sum of the first two denominators (i.e. 1+2) equals 3 the numerator of the second fraction. In the same way the denominator of the third fraction equals the sum of the numerator and denominator of the second fraction (i.e. 2+3=5) and the numerator of the third fraction equals the sum of the denominators of the second and third fractions (i.e. 5+2=7). This sequence of fractions approaches √2 alternately from both sides, namely
To the left and right of 2 we find two sequences of rational numbers which both ap-proximate 2 as their limit. Since a limit is itself defined as a number (!) approximated by the terms of a sequence in such a manner that the difference between the terms of the sequence and the limit-value can be made arbitrarily small (i.e. smaller than an arbitrary rational number > 0, as it was later formulated), it is clear that the numerical character of 2 cannot be defined by means of the limit
concept, since the limit concept presupposes that whatever functions as limit must already be a number.4
It is therefore clear that each approximating value remains rational (i.e. a fraction) – and 2 is not a fraction. When the Greeks discovered irrational numbers it threatened their idea of the delimiting nature of number, for an irrational number harbours within itself the unlimited (a non-repeating decimal fraction). The effect was that they did not develop a theory of irrational numbers (in modern terms: real numbers) and in fact opted to switch to a geometrization of mathematics.
8. THE ‘RELATIONSHIP’ OF NUMBER AND SPACE TO PHYSICAL REALITY What appears to be questionable is the ‘reality’ of the infinite and the ‘existence’ of irrational numbers (such as 2). But is it necessary to start with the complexity of infinity and the real numbers? What about the fractions themselves? Given our mathematical training it is second nature to reason about halves, thirds, and so on. But where “in reality” do we find halves? It is certainly possible to count a multiplicity of entities by using natural numbers – but whereas the number of entities (regardless which kind we may have in mind) in the universe is finite, the natural numbers are (successively) infinite. Hence, if existence means physical existence, then the successive infinite sequence of natural numbers does not ‘exist’. Likewise, within physical reality most entities could be divided, but none of their parts is exactly one third or a half of the (original) whole entity. In other words, typical physical halves are not identical to spatial halves in the purely geometrical sense of the term –entailing that ( perfect’) fractions also do not ‘exist’ in ‘reality’! The same applies to‘ other original spatial figures – such as squares and circles for they are also not identical to any physical circle or physical square. Consequently, with physical existence as yardstick numerical fractions and normal spatial figures such as
squares and circles do not ‘really exist’.
9. WHAT CAN COMPUTING MACHINES ACCOMPLISH?
We may explore this issue further by reversing our perspective. We know that
physical (or: mechanical) devices may serve as an aid for numerical calculations. It
is known that already in 1642 Blaise Pascal invented a calculating machine, destined to ease the addition of numbers. Eventually more sophisticated mechanical calculating machines were invented, until the computer entered the scene with its incredible (electronic) expansion of the calculating limitations present in mechanical machines. The brilliant German mathematician, physicist and engineer Johann (later: John) von Neumann arguably made the most important contribution to what
4
Even Cauchy was still of the opinion that irrational numbers should be considered as being generated as the limits of convesrging sequences of rational numbers through this approximating process. He is therefore caught in the same circular argument since the presence of an irrational limit presupposes its existence as a number, which means that the numerical nature of irrational numbers cannot be defined in terms of or derived from convergence since the latter presupposes it.
he called a Mathematical Analyzer Numerical Integrator and Computer (acronym: MANIAC).
Before this happened the discipline of mathematics and logic witnessed the invention of an imaginary calculating machine by Alan Turing, since then known as a
Turing machine (see Turing, 1936). Such a machine is conceived as being equipped
with a control-organ that can find itself in a finite number of states (called by Turing “machine configurations”), as well as an eye and a writing apparatus. Turing envisaged a number of applications of his machines but focused his more detailed attention on the development of machines that can compute dual expansions of real numbers x (0 ≤ x ≤ 1) (see Kleene, 1952:361). The underlying aim is to show that whatever a human computer can accomplish could be analyzed into successions of
atomic acts of some Turing machine (Kleene, 1952:377).5The decisive point to be observed is that Turing machines are bound to the nature of the successive infinite (the potential infinite). The implication is that no Turing machine can make the ‘jump’ from 2 to 2 in sketch 1 above!
In spite of the spectacular improvement of the calculating power of contemporary computers the same limitation holds true for them–they cannot exceed the scope of successive (infinite) steps – which actually means that they are strictly speaking
finitistic in their operation. What Kleene has emphasized as “successions of atomic acts” reflects the nature of the successive infinite. The binary basis of computer
electronics on the one hand enables an encoding capable of expressing any numerical value (and other notational symbols) merely through a combination of zeros and ones. The length (size) of any sequence of zeros and ones could (electronically) be extended at will, indefinitely. For that reason computers can set foot on the most basic path of the infinite, infinity in the literal sense of endlessness. Yet, however far computers may pursue this path, they will always be restricted to realize only a finite part of this domain of infinite succession. If we designate the never ending progression of any given sequence (of zeros and ones) as the “structural infinite” and the realization of only a (n arbitrarily large) part of it ‘finitistic’ it is clear that not even an electronic device could be truly structurally infinite. That this state of affairs is dependent upon a proper understanding of the interconnections between the meaning of number and space will be argued in more detail below – from which it will be possible to conclude that there is no
constructive transition from the successive infinite to the at once infinite.
However, mathematical analysis and the even more general scope of set theory employ ideas that transcend the possibilities of any mechanical and any electronic device. In order to explain what is involved in these advanced mathematical domains we proceed with a question.
5
Maass distinguishes between a deterministicand anindeterministicTuring machine (in the latter case
Since 1872 modern mathematics realized that one cannot generate any real number (including an irrational number such as 2) with the aid of a converging sequence of rational numbers (as we have noted above). In order to be a limit of a converging sequence of rational numbers that limit-value already ought to be a number; consequently it cannot come into being through such a converging process.6
Broken down to the steps involved we have (see Scholz & Hasse, 1928:40-42):
6
Cantor eventually pointed out that the ideas formulated in 1872 by Heine were actually derived from him (see Cantor, 1962:186, 385).
But before we return to the implicit presence of the at once infinite in the standard notion of a limit a number of issues require our attention first.
11. MATHEMATICAL PLATONISM AND CONSTRUCTIVISM
Behind all the above considerations several assumptions are concealed. On the one hand someone like Paul Bernays – the co-worker of David Hilbert (1862-1943 who had the fame of being the leading mathematician in the world after Poincaré died in 1912) – claimed that mathematical Platonism rules the day amongst mathematicians (see Bernays, 1976:65). According to him Platonism accepts “mathematical objects” independent of the thinking mathematical subject (Bernays, 1976:63). Particularly the transfinite arithmetic of Cantor generated opposing reactions. The intuitionistic school of thought, initiated by early intuitionists like Kronecker and Le Besque, found in L.E.J. Brouwer7 and his followers competent mathematicians who rejected the assumptions of Platonism by defending the conviction that mathematical existence coincides with constructability – thereby returning to starting-points already found in Greek thought. All Western thought on infinity and continuity has been decisively influenced by Aristotle – that is, until Cantor fundamentally questioned it. Becker writes:
The decisive insight of Aristotle was that infinity just like continuity only exists potentially. They have no genuine actuality and therefore always remain uncompleted. Until Cantor opposed this thesis in the second half of the 19th century with his set theory in which actual infinite multiplicities were contemplated, the Aristotelian basic conception of infinity and continuity remained the unchallenged common legacy of all mathematicians (if not all philosophers).8
Whereas intuitionism rejects the transfinite number theory of Cantor as a phantasm (see Heyting, 1949:4), David Hilbert appreciated it as the finest product of the human intellect and proclaimed that no one will be able to drive us out the paradise created by Cantor (see Hilbert, 1925:170),
Next the intuitionism of Brouwer and his school (see also the support from Dummett, 1978) we may also mention the constructive and operational logic and mathematics of Paul Lorenzen (see Lorenzen, 1969 and 1975) and the constructive mathematics of A.A. Markov (see Kushner, 2006). All these schools of thought share the rejection of what Aristotle called the actual infinite. Strictly speaking they are therefore not really in a position to accept the ‘existence’ of real numbers (including the irrational
7
See Brouwer 1907, 1919 and 1919a.
8
“Die entscheidende Erkenntnis des Aristoteles war, dass Unendlichkeit wie Kontinuität nur in der Potenz existieren, also keine eigentliche Aktualität besitzen und daher stets unvollendet bleiben. Bis auf Georg Cantor, der in der 2.Hälfte des 19.Jahrhunderts dieser These mit seiner Mengenlehre entgegentrat, in der er aktual unendliche Mannigfalgtigkeiten betrachtete, ist die aristotelische Grundkonzeption von Unendlichkeit und Kontinuität das niemals angefochtene Gemeingut aller Mathematiker (wenn auch nicht aller Philosophen) geblieben” (Becker, 1964:69).
numbers) because at most they adhere to a position where successively infinite approximations of real numbers are constructed on the basis of rational numbers. This entails that these views still in principle adhere to the view already advanced by Kurt Geissler in 1902. He writes:
The irrational is accordingly merely as a relationship available; in itself no single irrational number exists.9
If we jump to Markov we in principle find that he still adheres to the same view: In particular, a constructive real number is a coded pair of algorithms. The first algorithm represents a sequence of rational numbers …, while the second give a Cauchy modulus for this sequence. As a syntactic object, a constructive real number contains sufficient information to effectively find rational approximations to the real number with any desired accuracy (Kushner, 2006:563).
The “desired accuracy” simply represents rational numbers approximating (in the sense of the successive infinite) the real number as close as one may wish.
12. THE UNSURPASSABLE BOUNDARY FOR ANY CONSTRUCTION10
The extreme difficulties traditionally encountered in all attempts to give a positive content to the idea of the actual infinite explains perhaps why most objections in fact used the successive infinite as measure or yardstick for mathematical existence, and sometimes even confused this yardstick with an idea of existence that connects it with physical existence. In other words, those philosophers and mathematicians who traditionally rejected the at once infinite in most instances did that by applying the concept of “existence in succession” as the norm.
13. THE NUMERICAL ORDER OF SUCCESSION
Even if we opt for the successive infinite only, its meaning remains arithmetical and never becomes “concretely physical.” In addition the numerical “possibilities” of the meaning of succession simply exceeds “physical reality” by far – the universe has about 1080atoms – and Van Dantzig wrote an article asking the question whether 10 twice elevated to the power 10 is still a finite number? Strictly speaking the successive infinite as such is ever realized – not in physical reality and not in any
9
“Das Irrationale ist demnach nur als Verhältnis vorhanden; es existiert nicht eine einzige irrationale Zahl” (Greissler, 1902:134).
10
Although within the context of arithmetic and analysis the concept constructive implies computable
Tait points out that “this is a theorem; it is not built into the notion of construction.” In other words, in general construction and computation should not be identified: “ ‘Constructive’ means that the only witnesses of existential propositions one admits are ones that can be constructed, where of course this implies some background rules of construction. From the construction of an object, a means of computing it (in cases in which this idea makes sense) may or may not be found” (Tait, 2006:213).
computer or calculating device. Its literal numerical meaning after all is that it is
without an end, i.e. endless.
What is important to realize is that there is a strict correlation between the numerical order of succession and any succession of numbers factually exhibiting this order of succession. The latter serves as the foundation for what is known as (mathematical)
induction and it safeguards mathematics from collapsing into an “enormous
tautology” (as Weyl remarked – see Weyl, 1966:86). Of course intuitionism attempts to “subjectify” this order by emphasizing the constructive nature of the successive infinite without being able to account for the given order ultimately conditioning any succession of constructions. Yet the methods of intuitionistic mathematics transcend what Hilbert intends in his finitistic Metamathematik (Beweistheorie = Proof Theory) after Gödel published his famous 1931 article on the incompleteness of axiomatic systems (see Bernays, 1976:60).
14. CANTOR'S UNDERSTANDING OF THE ACTUAL INFINITE
The original and authentic definition given by Cantor of the actual infinite (the at
once infinite) explicitly states that it is not a variable (nicht veränderlich) but firm
(fest) and determined (bestimmt) in all its parts (in allen seinen Teilen), that it is a true constant (eine richtige Konstante) (Cantor, 1966:401). Cantor's intention was to provide a purely arithmetical understanding of the actual infinite (also reflected in his claim that the “continuum” should be seen as a “perfect-coherent set” (eine
perfekt-zusammenhängende Menge – Cantor, 1966:194).
Yet, what he did not realize, is that the key elements of his definition are derived from aspects that differ from the numerical aspect. His negation of variability makes an appeal to the physical aspect, his remark that it is a constant reflects the core
kinematic meaning of uniform motion (constancy), whereas the emphasis on firm
and determined as well as on the whole and all its parts is fully dependent upon the irreducible meaning of the spatial aspect (see Strauss, 2002:27).
15. PRIMITIVE MEANING AND DISCLOSURE
By pointing this out we are actually advancing arguments for the claim that whereas the successive infinite reflects the “primitive” (not-yet-disclosed) meaning of number, the at once infinite can only be accounted for by implicitly or explicitly call upon more than merely the numerical aspect. Yet by accepting number and space as real aspects of reality (and not mere products or modes of thought) we can side-step both constructivism and Platonism. The former holds that the mathematical thinking subject constructs the meaning of number and space, whereas the latter assumes that what we see as a human disclosure of the ontic nature of these aspects actually pre-dates human contemplation since it has an eternal (a-temporal) existence independent of all human construction.
16. AN “AS IF” APPROACH
Vaihinger claims that the use of inherently antinomic constructions (designated as
fictions) may serve human (scientific) thought in surprisingly efficient ways. For
example, he characterizes mathematical constructs such as negative numbers,
fractions, irrational and imaginary numbers as “fictional constructs” with a “great
value for the advancement of science and the generalization of its results in spite of the crass contradictions which they contain.” Yet according to him these fictions are
not hypotheses (Vaihinger, 1949:57).
The co-worker of David Hilbert, Paul Bernays, indeed saw something of the nature of this regulative hypothesis – and at once he also distances himself from Vaihinger:
The position at which we have arrived in connection with the theory of the infinite may be seen as a kind of the philosophy of the `as if'. Nevertheless, it distinguishes itself from the thus named philosophy of Vaihinger fundamentally by emphasizing the consistency and trustworthiness of this formation of ideas, where Vaihinger considered the demand for consistency as a prejudice ... (Bernays, 1976:60).
Paul Lorenzen also sensed something of this approach in his remark that the actual infinite meaning attached to the “all” shows the employment of a fiction – “the fiction, as if infinitely many numbers are given” (Lorenzen, 1952:593). But what is striking in this case is that we see that the “as if” is ruled out, or at least disqualified as something fictitious, with an implicit appeal to the primitive (undisclosed) meaning of number.
17. ANTICIPATORY HYPOTHESIS
We may characterize this forward-pointing (anticipatory) hypothesis (referring from number to space) as a disclosed approach in which any successively infinite sequence of numbers may be viewed as if it is given at once, as an infinite whole or
totality. The following crucial implications should be highlighted:
(i) The multiplicity present in any at once infinity cannot escape from simulta-neously echoing succession and transcending it in their being given “at once”.
(ii) For example, the mere designation of the initial positions of the set of natural numbers – {1, 2, 3, …} – is not decisive for the kind of infinity it may capture, in other words, this representation may intend either the successive infinite or the at once infinite. Suppose we actually assume the successive infinite, then this set literally is endless. But suppose now that we start this sequence of natural numbers with 11 as the first number: {11, 12, 13, …}. Of course we can then continue restricting ourselves to the successive infinite. We may now look at the sequence of fractions
generated by substituting 11 with 1/1, 12 by 1/2, 13 by 1/3 and so on. Then we have {1/1, 1/2, 1/3, …} – still in its successive infinite sense. (iii) At this point we may now involve our spatial intuition of simultaneity (at
once). We do that by observing the one-one mapping between “all” the points of the succession {1/1, 1/2, 1/3, …} with those points on the line matching these fractional (rational) values (“magnitudes”) – as intuitively explained in the sketch below:
(iv) Through the above-illustrated mapping we can consider each one of the three successively infinite sequences [namely {1, 2, 3, …}; {11, 12, 13, …} and {1/1, 1/2, 1/3, …}] as infinite totalities, i.e. under the guidance of our spatial intuition the initial sets are interpreted as “actually infinite” (at once infinite) sets! Without the coherence between number and space the idea of the at once infinite will be intrinsically contradictory.
(v) Just look at the way in which Weyl “abuses” the standard mathematical “solution” of Zeno's paradoxes in order to disqualify the at once infinite by employing the successive infinite as yardstick. He mentions that the current solution of the paradox refers to the successive partial sums of the series 1/2 + 1/22 + 1/23+ …, 1—1/2n(n = 1, 2, 3, ...) that does not grow beyond all limits (since they converge towards the number 1) and then adds the remark that when the infinitely many partial distances are viewed as a completed totality the essence of infinity is contradicted in the claim that Achilles in the end completely passed through the ‘Unvollendbaren’ (that which cannot be completed) (Weyl, 1966:61).
(vi) This issue may be ‘simplified’ with reference to the number 1. Consider the question: is the number 1 equal to 0.999… or not? Suppose we only accept the potential infinite. Then there will always be more “fractional amounts” to be added, since however far one proceeds, there will always be more to come – in which case 0.999… is not equal to one. The other option is (in following Weiertrass, Dedekind and Cantor) to accept the “actual infinite” and straightaway to define the number 1 as the “totality” of
the decimal expansion 0.999…! (The approaches of Weierstrass and Cantor had to use the at once infinite in their account of real numbers.) 18. SIMULTANEITY AND WHOLENESS – CRUCIAL INTERCONNECTIONS
BETWEEN NUMBER AND SPACE
That simultaneity and wholeness transcend the primitive meaning of number could be argued with an appeal to the views of Bernays and Gödel. Paul Bernays holds the view that mathematical analysis deals with the conceptual clarification of
geometrical representations and that the totality-character of space obstructs a complete arithmetization of the continuum (see Bernays, 1976:VIII; 74).11 Gerhard Gentzen distinguishes three levels of the employment of the infinite in modern mathematics: (i) elementary number theory, (ii) analysis (where even individual numbers may be infinite sets) and (iii) general set theory.12
Add to these perspectives the remark made by Gödel regarding the nature of sets. Although modern (axiomatic) set theory (Zermelo, Fraenkel, Hilbert, Ackermann, Von Neumann) pretends to be a purely (atomistic) arithmetical theory, the structure of set theory actually implicitly (in the undefined term “set” or “member of”) borrows the whole-parts relation from space.13This explains why Hao Wang informs us that Kurt Gödel speaks of sets as being “quasi-spatial” – and then adds the remark that he is not sure whether Gödel would have said the “same thing of numbers” (Wang, 1988:202). The idea of wholeness or totality indeed has an original spatial meaning. Consequently, the notion of the power or cardinality of sets cannot be understood without acknowledging the interplay of succession and at once (number and space). For this reason set theory ought to be appreciated as an arithmetical theory guided, directed and deepened by the core meaning of space (continuous extension).14
11
“Bei der Analysis handelt es sich um die begriffliche Präzisierung geometrischer Vorstellungen”
Bernays, 1976:VIII); “… daß die intuitionistische Vorstellung nicht jenen Charakter der
Geschlossenheit besitzt, der zweifellos zur geometrischen Vorstellung des Kontinuums gehört. Und es ist auch dieser Charakter, der einer vollkommenen Arithmetisierung des Kontinuums entgegensteht” (Bernays, 1976:74).
12
“Admitted as objects here are not only the natural numbers and other finitely describable quantities, as the first level, as well as infinite sets of these, as at the second level, but, in addition, infinite sets of infinite sets and again sets of such sets, etc., in the utmost conceivable generality” (Gentzen, 1969:223-224).
13
In passing we may note that an intuitionistic approach such as the one found in Dummett 1978 on the one hand rejects the actual infinite (in the sense of an “infinite totality”) but then continues to use an expression like an “infinite domain” without realizing it is just a substitute for the idea of an “infinite totality” (cf. Dummett, 1978:22, 24, 57, 58, 59, 63 and so on). In fact, there is no way in which a mathematician can analyze the meaning of number without in some or other way employing terms derived from the aspect of space.
14
This view finds support in the conviction defended by Bernays, namely that the distinction between an
‘arithmetical’ and a ‘geometrical’ intuition should not be accounted for in terms of spaceandtime, but
by considering the difference between discreteness and continuity : “Es empfiehlt sich, die Unterscheidung von ‘arithmetischer’ und ‘geometrischer’ Anschauung nicht nach den Momenten des Räumlichen und Zeitlichen, sondern im Hinblick auf den Unterschied des Diskreten und Kontinuierlichen vorzunehmen” (Bernays, 1976:81)
A remarkable ambivalence in this regard is found in the thought of Abaraham Robinson. His exploration of infinitesimals is based upon the meaning of the at once infinite. A number a is called infinitesimal (or infinitely small) if its absolute value is less than m for all positive numbers m in being the set of real numbers). According to this definition 0 is infinitesimal. The fact that the infinitesimal is merely the correlate of Cantor's transfinite numbers is apparent in that r (not equal to 0) is infinitesimal if and only if r to the power minus 1 (r-1) is infinite (cf. Robinson, 1966:55ff). In 1964 he holds that “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.“ Yet he believes that mathematics should proceed as usual, “i.e., we should act as if infinite totalities
really existed” (Robinson, 1979:507).
The idea of the at once infinite is made possible by the uniqueness and mutual coherence between number and space and– cknowledging that the at once infinitea ‘borrows’ from space both its order of simultaneity and the whole-parts relationship represents a third alternative not explored in the long-standing reductionistic legacy of mathematics –either reducing space to number (starting with the Pythagoreans and re-established by Weierstrass, Cantor and Dedekind) or reducing number to space (since Parmenides and his school and revived in the last phase of the thought of Frege–see Frege, 1979:277).
19. BEYOND ANY COMPUTING DEVICE BUT WITHIN THE REACH OF HUMAN UNDERSTANDING
Because every computing device (mechanical or electronic) is bound to physical processes it is understandable why such a device can only operate on the first level of infinity distinguished by Gentzen (where the successive infinite involves natural numbers and their relationships, evinced in ratios of fractions). Only on this basis it is possible to approximate what is found within the second and third levels of the infinite according to Gentzen. What no machine can achieve is the realization of the
at once infinite something the human intellect can indeed contemplate. We may
illustrate this unique human intellectual ability by recollecting the genesis of modern set theory once the concept of denumerability surfaced in Cantor's Mengenlehre. When the elements of any set can be correlated in a one-to-one way with the natural numbers (1, 2, 3, ...) it is said that the cardinality (or power) of such a set is denumerably infinite (Cantor introduced the symbol aleph-zero: ). One can easily see that the integers are denumerably infinite, because one can count (enumerate) them by starting with 0 then move to +1, then to -1, then to +2, then to -2, and so on indefinitely. Likewise it turned out that also the rational numbers (fractions) are denumerably infinite. Count first 1/1, then 2/1, then 1/2, then 1/3, then 2/2, then 3/1, then 4/1, and so on (see Meschkowski, 1972:23):
Once this ‘triangle’ is converted into the representation below the denumerability of the rational numbers is immediately clear
However, in 1874 Cantor proved that the real numbers are not denumerable (i.e. that they are non-denumerable). In 1890 he provided his famous diagonal proof, used in the explanation below (cf. Cantor, 1962:278-281). A one-to-one correspondence could be established between all real numbers and the set of real numbers between 0 and 1. Furthermore, every real number in this interval can be represented as an infinite decimal fraction of the form xn = 0.a1 2 3 4a a a ... (numbers with two decimal representations, e.g. 0.100000... and 0.099999 ... are consistently represented in the form with nines). Suppose a denumeration x1, x2, x3, ... exists of all the real numbers between 0 and 1, i.e. of all the real numbers in the (closed) interval 0 ≤ xn≤ 1 (i.e. [0,1]), namely:
If another number can be found between 0 and 1 which differs from every xn, it would mean that every denumeration of the real numbers would leave out at least one real number, which would prove that the real numbers are non-denumerable. Such a number we can construe as follows:
It is clear that y is a real number between 0 and 1 (i.e. 0 ≤ y ≤ 1). The number y does not have two decimal representations since every decimal number in its decimal development is unequal to 0 and 9. The number is also unequal to every real number xnsince the decimal expansion of y in the first decimal place differs from the first decimal number x1, in the second differs from the second decimal number of x2 (namely x2), and in general from the nth decimal number of xn. It is clear from this that a denumeration of the real numbers will always exclude at least one real
number (“miscount” it in the denumeration), which concludes Cantor's proof that real numbers are non-denumerable.
20. THE PROOF OF NON-DENUMERABILITY IS DEPENDENT UPON A SPECIFIC IDEA OF INFINITY
Although this ‘proof’ seems to be ‘exact’ it is dependent upon the acceptance of the at once infinite. Those mathematical schools of thought that reject the at once infinite cannot conclude to non-denumerability.15 Someone who recognizes only the uncompleted infinite can never accept this conclusion, since the diagonal method then only proves that for a given constructible sequence of countable sequences (i.e. decimal expansions of real numbers) of natural numbers, yet another different countable sequence of natural numbers can be construed. Becker states this in the following way: “The diagonal method demonstrates, strictly speaking, the following: when one has a counted (law-conformative) sequence of successive numbers, a sequence of successive numbers can be calculated which differs in every place from all the previous ones” (Becker, 1973:161 footnote 2). In this interpretation non-denumerability does not feature!
A mathematical proof which apparently takes an “exact” course therefore comes to conflicting conclusions depending on the presuppositions (namely the successive infinite or the at once infinite) from which one proceeds! Fraenkel points this out emphatically:
Cantor's diagonal method does not become meaningless from this point of view, ... the continuum (i.e. the real numbers – DFMS) appears according to it as a set of which only a countable infinite subset can be indicated, and this by means of pre-determinable constructions (Fraenkel, 1928:239 footnote 1).
Most modern mathematicians do not realize that the at once infinite is different from and irreducible to the successive infinite. In terms of these two kinds of infinity there are alternative interpretations of the two key “quantifiers” of mathematical logic, namely the universal quantifier (for all: ) and the existential quantifier (for some; there exist: ) (see Gentzen, 1935:32). In addition we have to note, without being able to explain this issue in more detail here, that the familiar understanding of the symbolic expression “ ” within analysis is mistaken. If the symbolic expression indicates that a variable “approximates infinity” then it appears to specify an instance of the successive infinite. Yet the numerical value of the limit actually reveals an instance of the at once infinite. Just recollect the sketch 1 above. As long as the word /approximate/ designates successsive fractions (rational numbers) the horizontal and vertical lines of the steps respectively remain 1 when added – in which case the “step-wise” approximation of the diagonal equals 2. The moment the
15
The assumption of Cantor's diagonal proof is that all the xn's (as an infinite totality given at once) are arranged in a denumerable sequence.
symbolic expression “6 4 ” is understood in the sense of the at once infinite, we have the exact value of the diagonal at hand: 2 (which is an irrational number –and irrational numbers can only be introduced on the basis of the acceptance of the at once infinite). The ‘step’ from 2 to 2 is strictly correlated with the difference between the successive infinite and the at once infinite.
The same applies to the other example that we have used, namely where the number 1 is the limit of the “approximating sequence” n/n+1 where “n ”:
In this case one can view the finite number 1 as being ‘constituted’ by the infinite totality of all the numerical values of n/n+1–givenat once.
21. IMPLICATIONS FOR SCIENCE, ENGINEERING AND TECHNOLOGY
One of the successful features of the modern natural sciences (including the interplay of physics and mathematics) is that it opened up multiple contexts of
measurement. The training of engineers is therefore understandably intimately
connected with mathematical skills (calculations). But precisely when it comes to an account of measurement the difference between the successive infinite and the at once infinite once again surfaces. The reason for that is given in the fact that all physical measurements are bound to the rational numbers, while the theoretical structure of a metric rests on the nature of the real numbers.16And we have seen that whereas an account of the nature of the rational numbers can get away with“ ” the meaning of endlessness a truly disclosed and deepened understanding of the real numbers requires the at once infinite.
Furthermore, it should be kept in mind that a technical invention is never a
conclusion of (physical) science. Scientific analysis and argumentation are different
from a technical invention. Physical construction present in technical inventions and engineering innovations is bound to endlessness–in the sense of afinite number of steps – even though the theoretical background of modern mathematical physics inevitably has its (deepened) foundation in the employment of the at once infinite exemplified in the real numbers. The latter embodies the amazing capacity of the human intellect, manifest in its ability to disclose the primitive meaning of the successive infinite by employing the “as if” hypothesis of the at once infinite – transcending the limits of any constructive (successive infinite) approach, such as the physical practice to which science, engineering and technology is inevitably bound.
16
It is not necessary to enter here into an explanation of what this all entails - such as introducing comparative (or: topological) concepts, considering equivalence classes and groups, and so on. For a
more detailed analysis of the relation betweenmetricand measurement, see Stafleu 1972 (pp.43-49)
Although any succession of numbers could be mapped onto the natural numbers (1, 2, 3, …) and although this kind of succession can be extended indefinitely, this endlessness already exceeds physical reality (the physical universe has about 1080 atoms). The calculating power of any (mechanical or electronic) computing device is therefore irrevocably restricted to a finitistic reality that at most can explore successive infinite possibilities (i.e. its approximating power finds in the nature of the rational numbers its only operational possibility).
The meaning of the at once infinite (actual infinite) cannot be accounted for merely in numerical terms (merely in terms of successions), since both the spatial order of
simultaneity and the spatial whole-parts relation play a guiding role in the possibility
the human intellect has to view any (denumerable or non-denumerable) multiplicity as if it is given at once – and there is no constructive transition from the successive infinite to the at once infinite (see Wolff, 1971:399-400). Ultimately the irreducibility of the spatial aspect serves as the foundation for the irreducibility of the at once infinite to the successive infinite and explains why there is no constructive transition from the successive to the at once infinite. The human intellect, exploring these interrelations between number and space, is therefore more powerful than any technical device, which is always bound to a limited manifestation of the successive infinite. Consequently, the assessment of Wachter – in a slightly different context – is justified: “our most modern technology, also the ‘Super-Protonensynchrotron’, compared with our thoughts, is actually a very primitive tool” (Wachter, 1975:19). Whenever the regulative hypothesis of the at once infinite is used the human intellect explored an interconnection between number and space that exceeds the possibilities of any computing device, thereby underscoring that the possibilities of the human intellect transcends the limitations of computing devices!17
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17
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Key words: successive infinite (potential infinite), at once infinite (actual infinite), construction, limit, (non-)denumerability
