Topmenu Calendar Archive Links Faculty and staff Publications Research projects Sitemap Main page

Niels Ole Finnemann: Thought, Sign and Machine, Chapter 4 © 1999 by Niels Ole Finnemann.
| Table of Contents | Chapters: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Literature | Download pdf-file |

4. The language of logic and the logic of language

4.1 The truth of a sentence

With the invention of arithmetic, which cannot be dated, it became clear that mechanical procedures could be incorporated in human thinking. With the invention of the counting-board, which was known in such countries as ancient Babylon, now Iraq, approximately 5000 years ago, it also became clear that we are able to implement such mechanical thought procedures in physical instruments to our own advantage - but without finding it necessary because of this to refer to these or other mechanical apparatuses as capable of thought.

The reason for this may be philosophical or religious, human thought has always been seen in an eternal, spiritual light and thus superior to the finite world. But it may also be for practical and experientially conditioned reasons. Throughout the history of arithmetic people have handled mechanical procedures with the help of different, non-mechanical thinking - whether they were carried out mentally, manually or with the help of artefacts.

This is still the case. A mechanical procedure or calculation is always preceded by an analytical explanation which must at least include a definition of the components of the calculation and a motivated choice of compositional structure. Similarly, after a mechanical procedure there follows a non-mechanical handling of the result.

Confronted with this massive historical experience there would appear to be little room for the idea that all human thinking is mechanical or calculative. Nevertheless it was precisely this idea that was to become a main theme in much of 20th century science and philosophy.

In a peculiar way the breakthrough of this idea was based on mathematical physics. It is from here, on the one hand, that the idea arises that human consciousness can be regarded as a - complicated - physical system, that every mental process has a physical manifestation which should consequently be describable in mechanical terms. On the other hand, there is a manifest destabilization of the mechanical description of physical nature which also stems from here. Mathematical physics loses its ontological foundation while the question of truth is transformed from a question of nature's order to that of language.

On the face of things, these two traits would appear to contradict one another and in themselves provide no reason for regarding language - and underlying this, thinking - as a mechanical calculation procedure. However, they did actually occasion a dialectical effort to resolve the contradiction by formulating it at a higher level.

The epistemological problems in physics had arisen as a new uncertainty concerning the interpretation of mathematical formulas as models of physical phenomena. Rejecting the mathematical description of physical processes was out of the question. On the other hand it was conceivable to imagine that it might be possible to solve not only the problem of mathematical physics, but all epistemological problems by constructing a more comprehensive and independent mathematical or logical language.

With this idea in mind a number of philosophers and mathematicians including Charles S. Peirce, Gottlob Frege, Bertrand Russell, David Hilbert, the logical positivists and the young Ludwig Wittgenstein took up an idea which had been previously taken up by Leibniz, but whose roots are lost in Arabian mathematics and medieval alchemy.

Although both the German Orientalist, Wilhelm Schickard, and the French mathematician and philosopher, Blaise Pascal, anticipated Leibniz with their calculating machines, Leibniz was the first to see in the calculating machine the fundamental form of the thinking machine. Although his own computer was only able to handle the four basic arithmetical operations, his philosophical system contained the idea that human consciousness was pre-programmed - like the mechanism of a clock - and that it should be possible to formulate the logic of this mechanism in a language characterized by the precision of mathematics.[1]

Leibniz imagined that with such a reduction of logic to mathematics it would be possible to solve all conceivable problems, from the proof of God's existence and world order to the clearing up of any moral dispute. This depended, however, on the paradoxical precondition that all these and other relationships had been decided in advance, as Leibniz assumed that God had created a precisely co-ordinated synchronism between an external, physical world clock and a clock of consciousness. Thus, if we feel pain when we cut a finger this is not due to any causal connection between the external event and the inner experience, but on the contrary to the divinely instituted synchronization of the two clocks which each controls its own domain. For Leibniz there was no other connection between the external and internal than this pre-ordained synchronization.

But this idea did not originate solely with Leibniz. He refers to the Spanish mystic, Ramon Lull, also called Doctor Illuminatus, as the first to broach the idea of a universal algebra. It was the goal of Doctor Illuminatus - long before Descartes - to emancipate philosophy from theology, because reason should be founded on doubt, not faith. For this purpose he built an apparatus - Ars Magna - comprising a number of concentric circles to which were attached a series of words and ideas in accordance with a specified order. By arranging these words in different ways it was possible to form questions and, by so doing, produce another series of words which presumably expressed a more precise delimitation of the logical character of the problem.[2]

This logic - referred to disrespectfully by Descartes as "the art of Lully"[3] - which was ostensibly thought up as a defence for sceptical reason against the illusions of faith, had thus as automatic logic also the property of freeing the individual from a significant amount of difficult thinking. In this perspective the picture of the complete refinement of thought appears as the cessation of thinking.

However we balance the inherited accounts there is - for us at least - one immediately obvious difference between the ideas of Doctor Illuminatus and Leibniz on the one hand and mathematical logic, which makes its breakthrough at the beginning of the 20th century, on the other. What was once a theme for quaint alchemistic and philosophical excursions becomes - refracted through the prism of mathematical physics - the axis of a technological scientific revolution, which since World War II has assumed the character of a permanent revolution.

Regarded as a change in the history of scientific-utopian thinking this is not simply a question of a leap from utopia to reality, but also of a conceptional change-over. While the utopian dream has ancient roots, it assumes a new shape expressed in particular as new, more rigorous demands on demonstration. The result of this was not only the loss of the dream, but also the first description of the principles of a real universal calculating machine.

4.2 The logic and the life of the sign

Attempts to construct a complete mathematical-logical language constituted only one of the new symbol theories which came into being around the turn of the century. During the same period Ferdinand Saussure presented the idea of a general semiology which also laid the foundation for structuralist theories of language while Charles S. Peirce presented his ideas of a general semiotics.

All three projects have a common and general theoretical ambition, but are mutually very distinct. The mathematical-logical project differs partly in its marked constructive and innovative aim, partly in the peculiarity that it hardly concerns itself with language as its subject area. The real subject is logic, which is also the basis of reflections on common language, to the extent that this is taken up at all, as in Rudolf Carnap. The same is perhaps true of Peirce who does not, however, see his draft of a sign-theoretical relational logic as the construction of a new language, but rather as an inherent principle in all symbolic activities.

While mathematical logic became of direct significance for the development of information theories, Saussure's and Peirce's theories have only been of a less direct and later significance. As they were formulated within the same scientific-historical context and play a part in the present description of both information theory and computer technology, it will be appropriate to define the different approaches more closely.

In Peirce's general model of cognition the classical subject-object figure is replaced by a so-called triadic sign concept. The established understanding of a sign as an expression which stands for something else is replaced by a tripartite relationship between firstness, "the signal" or quality as it is, secondness, the relationship of the signal to an object and thirdness, the interpreter that defines the connection between the signal and the phenomenon. As the interpreter comprises all the ideas included in the understanding of the sign, it resembles a colossal scrap-bin with a completely arbitrary content. But not for Peirce, who claimed that it was possible to define all imaginable logical relationships between the three sign aspects, which only together comprised what Peirce called "genuine signs" as opposed to "degenerate signs". Underlying this is again the thesis that all sign relationships can be described as more or less complex combinations of triads. Corresponding to the genuine sign's general triadity there is therefore also a triad of possibilities for each of the parts. There are three possible types of signal (qualisign, sinsign and legisign), three possible referential relationships to objects (icons, indexes and symbols) and three possible types of interpreter (rhemes, dicisigns and arguments/argusigns).[4]

As all we are concerned with here is representing the general sign model it is not appropriate to run the scholastic risk of giving a more detailed account of the individual categories. For the moment it is sufficient to note the new epistemological emphasis on the interpreter function which can be understood as a reflection of - and an attempt to solve - the problem of observation in physics. The interpretation of the sign is included in the definition of the concept of observation.

Given Peirce's interest in the logical relationships of signs it is hardly surprising that he - who was characterized by Cohen as a reincarnation of Dr. Illuminatus - was also interested in the idea of a logical, thinking machine.

The secret of all reasoning machines is after all very simple. It is that whatever relation among the objects reasoned about is destined to be the hinge of a ratiocination, that the same general relation must be capable of being introduced between certain parts of the machine.[5]

Peirce understood - like all his predecessors - the logical machine as a mimetic reconstruction of the structure of logical thought, but had, on the other hand, no illusions that human thinking as such could thereby be reproduced:

Every reasoning machine, that is to say, every machine, has two inherent impotencies. In the first place, it is destitute of all originality, of all initiative; it cannot find its own problems; it cannot feed itself. It cannot direct itself between two different possible procedures... And even if we succeed [in the latter] it would still remain true that the machine would be utterly devoid of original initiative... In the second place, the capacity of the machine has absolute limitations; it has been contrived to do a certain thing, and it can do nothing else.[6]

This is indeed also true, adds Peirce, of consciousness, but in a different way, which is illustrated by our ability to continue to develop algorithmic calculations indefinitely. As will appear in the next section Gödel incidentally used very similar reasoning in his argumentation regarding the incompleteness of formal description.

As a consequence of these limitations, which were related to analogue machines, the point lay not in the possibility of replacing human thinking, but in the possibility of freeing us from boring routine work and particularly in the possibility of obtaining new knowledge of logical thinking by studying such machines. The question was, how great a part of thinking could such a machine carry out?[7]

According to Peirce, however, mechanical procedure also possesses a property which means that it cannot solely be seen as a simple, repetitive procedure. When the various parts of a machine interact, relationships also arise which have not necessarily been intended or anticipated. Peirce sees these relationships as "reasonings" which express a law which has thus been formulated by the machine. This argument goes not only for the logical machine, where laws are of a logical character, but, says Peirce, also for many physical machines where the interaction is an expression of physical or chemical laws:

In this point of view, too, every machine is a reasoning machine, in so much as there are certain relations between its parts, which relations involve other relations that were not expressly intended. A piece of apparatus for performing a physical or chemical experiment is also a reasoning machine, with this difference, that it does not depend on the laws of the human mind, but on the objective reason embodied in the laws of nature.[8]

In a narrower linguistic sense the new sign theoretical clues are laid down rather by Ferdinand Saussure's draft of a structuralistic sign theory which, in spite of its general aim to be a "a science which studies the role of signs as part of social life"[9] was constructed around an investigation of linguistic signs with the definition of the sign as an - arbitrary - unit of expression and content: "A linguistic sign is not a link between a thing and a name, but between a concept and a sound pattern."[10] This definition of the sign concept contains the entire basis of Saussure's theory of language: No matter how we look at a linguistic phenomenon it always contains two complementary facets. There is thus not only an indissoluble - but also variable - bond between a concept and a sound pattern, as well as between articulation and acoustic impressions, between the auditory-articulatory unit and the idea, between physiological and psychological, between the individual and social aspects of language, between language as ever ongoing evolutionary process (diachrony) and as institutional system (synchrony) and between language as an invariant structure or system (langue) and use (parole).[11] As language use is understood on the one hand as the sole manifestation of the invariant linguistic structure and also, on the other, as the sole manifestation of meaning-giving variation, the different interpretations of the meeting between system and meaning constitute an important dividing line between different linguistic theories.

Where Peirce introduced an interpretant as a kind of bridge builder between the expression and the content, Saussure saw the sign relationship as a series of units between the two sides. Both theories implied that any study of signification and meaning should be based on a study of the structural representation of the content. But they did so in two different ways. Where Saussure's theory directs attention towards the specific linguistic sign function, Peirce attempted - as pointed out by Linda Gorlée - to formulate a universal symbolic logic on the basis of the assumption that it is possible to identify thought content with the symbolic expressions.

Logic needs no distinction between the symbol and the thought for every thought is a symbol and the laws of logic are true of all symbols.[12]

For Peirce the linguistic form of expression therefore simply becomes in itself a less significant manifestation of an underlying, universal logic of sign relationships, while Saussure viewed linguistics as a beginning and contribution to a general theory of signs - a semiology - including different sign systems whose respective places in human consciousness would later have to be determined by psychology.[13]

To do this, however, the psychologist would have to have both a theory of signs and a sign system with which to express his analysis. In spite of all caution Saussure also paved the way - with his draft of a more precise definition of language as a distinct subject area - for an acknowledgement of the central place of the sign system in epistemology. The delimitation of the concept of language as a phenomenon with its own separate structure at the same time instates this structure as a condition for human cognition of the non-linguistic.

Saussure formulated his theory in a break with the comparative linguistics of the 19th century, because "they never took very great care to define exactly what it was they were studying"[14] and they were therefore unable to develop a systematic method. But the definition of language as a stockpile of sound patterns, signs which are composed of a connection between expression and content is not only a new, improved foundation for the same linguistic science, it also becomes the starting point for a methodological polarization which, in its general form, results in a distinction between a synchronic and diachronic description of language, each covering its own point of view with the sign concept as the only connection.

For Saussure the distinction is primarily methodological, a necessity which makes it possible to carry out a more precise description and the choice of point of view is in principle only a question of what we are interested in. If we study a slowly evolving language, we will take note of the synchronic traits, but if the language evolves rapidly, a diachronic approach will not only be more obvious, it will also be considerably more difficult to separate an area for a synchronic description:

Of two contemporary languages, one may evolve considerably and other hardly at all over the same period. In the latter case, any study will necessarily be synchronic, but in the former case diachronic. An absolute state is defined by lack of change.[15]

This simply shows how difficult operating with an isolated synchronic description is. Saussure also supports the distinction by allocating the diachronic and synchronic perspectives to two different domains of language. The synchronic is anchored primarily in grammar, while the diachronic is anchored in phonology.[16] In the later structuralist interpretation the diachronic perspective has, as a general rule, been completely abandoned or regarded as a less decisive, modifying factor.

Saussure himself set the stage for this consequence in that he assumed that diachronic changes - sound shifts, for example - primarily involve indi-vidual elements in the language system. Although such changes undoubtedly affected the entire system there was no "inner" connection between the partial, diachronic changes and the total system. The two aspects could not be studied simultaneously and language was primarily understood as a system with emphasis on the synchronic connections between its components.

The central, still unsolved problem on both sides of this polarity, however, turns out to be the same, namely the relationship between a language as a system and the manifested use of language. The synchronic language system must not only be separated from the diachronic time axis of the phonological expression variation, it must also be separated from the semantic variation which is manifested in the same usage as the language system. Unlike phonological variation, semantic variation cannot be described as a particular, delimited variation of a single link in a language system. Semantic variation is not only manifested as individual variations, but is also expressed as a trait of genre and style. Hence, it is difficult to see why semantic variation should not occupy a place at the level Saussure delimited as language system or language construction. In other words, there is no basis for the concept of an invariant language system.

Saussure also concedes that the diachronic perspective cannot be reduced to phonological or other subordinate changes regarding the synchronic description.

When the phonetic factor has been given its due, there still remains a residue which appears to justify the notion that there is a "history of grammar". That is where the real difficulty lies. The distinction - which must be upheld - between diachronic and synchronic calls for detailed explanations which cannot be given here.[17]

A more detailed explanation along these lines has still not appeared and we could therefore ask whether this indeterminacy in the relationship between language system and language use, which appears to be valid for all languages sometimes referred to as "natural" languages, is not a - perhaps even very central - property which at the same time separates them from mathematical, algorithmic and logical languages?

This question will be taken up for further discussion in chapters 6-9, where it will be claimed that the concept of an invariant language system cannot be valid for languages that allow an indeterminate semantic variation of the utilized notation system's smallest units of expression because in this case, such a semantic variation would also be able to include the established rules of the system.

The relationship between the diachronic and synchronic description was neither included as a problem in mathematical logic nor in Peirce's semiotics. But according to Saussure the problem is manifested in disciplines which, like linguistics and economy are concerned with "a system of equivalence between things belonging to different orders. In one case, work and wages in the other signification and signal". Such systems are characterized by simultaneously manifested, different`values' which together comprise the system at a given stage, while the individual components vary individually over time.[18]

The three sign theories were in many ways extremely different mutually. Mathematical-logical symbol theory is concerned with the relationship between mathematical and logical representation, each of which is understood as a well-established and consistent symbol system that, in one or another combination can comprise a general sign theoretical foundation. Peirce's theory contains a draft of a new sign theoretical logic, while in this connection Saussure sets his sights - rather uncharacteristically - on a more modest goal, a narrower definition of a new starting point for a later general sign theory.

Although these theories emerged independently of one another and are mutually incompatible in their original form, their direction was the same: The attempt to describe language as an independent system with immanent laws which were either independent of or actually controlled the semantic reference. Sign production came to be regarded as a result of sign relationships contained in the language system. Semantics was regarded either as invariant in relationship to the language system, or as a function of language use and language system.

Herein lies the implicit and - it will be claimed in a later chapter - also dubious postulate: that the semantic dimension of language is completely manifested in the symbolic expression system - wholly contained or expressed in language use and surrounded by the language system which, conversely, is postulated as inaccessible to semantically motivated variation. The common goal was thus to describe an independent, closed system of rules for the articulation of meaning.

Saussure's theory was formulated against the background of the "pre-scientific foundation" of historical linguistics where although language had been regarded as a form, the form was seen as an external vehicle for the articulation of meaning. Instead, he claimed, that thought, before it is expressed in the distinctions of language, must be understood as an amorphous mass, "chaotic by nature". The semantic content can therefore only exist through the linguistic oppositions, "the contact between [sound and thought] gives rise to a form, not a substance" which again implies that the semantic level is allocated to the individual signs.[19]

For Peirce and the formal, mathematical-logical symbol theory, the background was a destabilization of the representational validity of the mechanical description of nature.

There were apparently two equally disagreeable alternatives. If the aim was to maintain a systematic or mechanical model, a loss of referential validity would have to be accepted. If the aim was to maintain the demand for referential validity, the method of systematic or mechanical description would have to be abandoned.

The main currents of 20th century linguistics thus take on the appearance of an emancipation from an understanding of language as submerged in the pre-ordained meaning content of history and nature. Hume's philosophical critique of the concept of causality had, so to speak, caught up with linguistics, the semantic bond to the described world was broken.

When a fixed correspondence between the concept and the conceived is no longer regarded as given, conceptualization, linguistic representation, emerges as a separate substance and as the place where the question of truth is decided. With the transformation of ontology to epistemology the reference of language to the world outside language becomes woven into the reference of language to itself. Referentiality no longer exists as an assumed or obvious possibility, instead it becomes an object for linguistic reflection, while at the same time language emerges more clearly as an independent, autonomous system of pure forms.

In spite of the mutual divergences, which will appear again in later chapters, together they represent the first marked - as yet only theoretical - expressions of a secularization of the relationship of science to language. This secularization has traits in common with older, nominalistic assumptions which similarly doubted that language refers to an order outside language, but it is distinguished by the objectivizing view of language as a self-reliant phenomenon that can be described.

While Saussure aimed at a systematic description of the mechanisms of common language, the logical and mathematical symbol theories attempt to respond to the lost referentiality by formulating a new, formal and consistent symbolic language. Hereby the idea itself of an abstract and formally defined symbolic system became firmly anchored in many other disciplines beside mathematical physics.

4.3 The idea of a mechanical decision procedure

It might be thought that logically oriented philosophers would be the first to cast doubt on the idea of a mechanical logic which makes the logician superfluous, robs his previous efforts of any connection with the more elevated mental occupations and relegates philosophical logic to civilization's prehistoric archive for happily done with, now superfluous business. But this is far from being the case. Nobody else has proposed - let alone attempted to develop - the idea of a mechanical logic with the same disinterested fervour and perseverance as can be found in the history of logical philosophy. In addition, the most ambitious and powerful expression of these efforts is to be found precisely at the point where the mechanical paradigm, with its background in physics, really got into difficulties. There can be little doubt that inspiration was to a great degree derived from the speculative daring which in its time led to the successful formulation of the basis of mathematical physics. Galileo's "measure what can be measured", Descartes' analytical geometry and formulation of mathematics as the critical, sceptical weapon of reason against ignorance and Newton's largely successful application of a relatively simple system of axioms to a description of the planetary system, which was in itself an expression of the fact that it was possible to develop general methods of description which must clearly take precedence over questions of empirical evidence.[20] The problems of mechanical physics had to be regarded in this perspective and the central theme of philosophical logic therefore became the relationship between logic and mathematics itself. The overall goal was to provide a proof theory, that is, a general, formal proof that it was possible to decide whether a procedure carried out in a formal, symbolic logical language was correct or incorrect.

The meta-mathematical programme - of David Hilbert - resulted in a number of precisely formulated questions proposed at two international mathematical congresses in 1928 and 1930. Among the questions raised there were three in particular which came to occupy people: Can mathematics be regarded as complete in the sense that every mathematical statement can either be proved or disproved? Can mathematics be regarded as consistent in the sense that a valid operational sequence will never lead to incorrect statements (such as, for example, that it will never be possible on arithmetical lines to arrive at results such as 2 + 2 = 5)? And can mathematics be regarded as decidable or provable, i.e. is there a finite method which in principle can be used on every assumption with the guarantee of a correct decision as to the truth of the assumption? The last problem is the so-called ]Entscheidungsproblem.[21]

Hilbert himself was convinced that all these questions could be answered in such a way as to make it possible to declare that it had been proved that mathematical logic was a complete and consistent descriptive language. At the 1930 congress he rounded off his lecture by declaring that ein unlösbares Problem überhaupt nicht gibt.[22]

But far from everyone shared Hilbert's optimistic expectations regarding formal mathematical description. Mathematician E. L. Post had thus as early as the 20's been on the trail of formal problems which could not be decided with the help of any finite method. Others - such as John von Neumann - had, similarly in the 20's, argued that there was not only no known proof that all mathematical problems in principle had a finite solution, there was no reason at all to believe that such could be found. On the contrary, the lack of such proof was the raison d'être of mathematical thinking:

...the contemporary practice of mathematics, using as it does heuristic methods, only makes sense because of this undecidability. When the undecidability fails then mathematics, as we now understand it, will cease to exist; in its place there will be a mechanical prescription for deciding whether a given sentence is provable or not.[23]

Even before Hilbert had been able to present his whole programme at the 1930 congress, the two first of the three questions mentioned had found a clear and equally surprising answer.

On the previous day, mathematician Kurt Gödel presented one of the 20th century's most epoch-making mathematical proofs, Gödel's theorem, which in a nutshell states that arithmetic is either inconsistent or incomplete, as he showed that there are "relatively simple" arithmetical sets which contain at least one assumption the validity of which cannot be decided within the premises of the given formal system.[24]

With Gödel's proof culminated the idea of a mathematical-logical epistemology and logical positivism lost its philosophical foundation. Instead of a general truth function there was now a formal proof that there was a problem of description and decision which could not be solved within the framework of formal logic - as Gödel wrote:

The true reason for the incompleteness which attaches to all formal systems of mathematics lies... in the fact that the formation of higher and higher types can be continued into the transfinite... while, in every formal system, only countably many are available.[25]

While Gödel closed one door with this conclusion, he opened another with his method, the formal treatment of formal systems.

As such this method had already been developed in the various attempts to connect the mathematical and logical descriptions. The assumption here, however, was that there was a logical relationship: that mathematics is logic (Frege-Russell) or that mathematical problems could be handled with a meta-mathematical logic (Hilbert). In both cases the method was deductive, an attempt to reach the mathematical expression through analytical reduction. Gödel took a different path as he simply numbered the individual sentences in a formal system: "we now set up a one-to-one correspondence of natural numbers to the primitive symbols of the system"[26] and then proceeded to handle the demonstration process on the basis of number theory. Gödel thus demonstrated how it was possible, with a simple and arbitrary coding or addressing procedure - "Gödel-numbering" to handle a logical-symbolic system in a numeric system, where the first system was represented only by an address. As a whole the demonstration was extraordinarily complicated, but the coding procedure itself was almost hair-raisingly simple.

The methodical principle, arbitrary re-coding, contains at least in germ a number of formal procedures which have since found widespread use. In the present connection it is particularly interesting that the method contains a general model for the algorithmic handling of formal procedures, among them also other algorithms - second-order handling - and that it exploits a "scanning principle" as a coding procedure.[27]

The method, however has - still - not been emancipated from that limitation which was the first result of its use. It is impossible to formulate a general, formal demonstration procedure for the completeness of a formal description.

In this way Gödel's theorem is part of 20th century mathematical logic in the same paradoxical way as quantum physics is of 20th century physics, as a method of description which extends the descriptive potential by limiting the validity of the description.

Gödel, however, answered Hilbert's two first questions with his method. The third remained. Gödel had introduced a distinction between formally correct, demonstrable sentences on the one hand and non-demonstrable sentences on the other and presented a new method of demonstration which extended the area of use for formal demonstrations, which depended on the performance of a finite number of formal, step-by-step operations with fully deterministic rules of arithmetic. But he had only shown that any known formal system was incomplete because it must contain at least one sentence which could not be demonstrated within the system's own framework. He had not invalidated the possibility that there could be a general, finite method to decide ]whether an arbitrary mathematical sentence could be demonstrated or not. But this problem too - Hilbert's Entscheidungsproblem - was now coming close to solution.

If it were possible to confirm Hilbert's thesis, it must be assumed that it should be possible to perform any logical procedure along mathematical-algorithmic lines with the help of mechanical procedures. "It is well known", wrote Gödel in 1931,

... that the development of mathematics in the direction of greater precision has led to the formalization of extensive mathematical domains, in the sense that proofs can be carried out according to a few mechanical rules.[28]

But, it appeared, a confirmation of Hilbert's thesis required not only the description of a mechanical demonstration procedure, it must also be shown that this procedure could be performed with a finite number of operations. It must be possible to decide when it could conceivably be stated that there never would be a decision. This again required a precise definition of what was understood by a finite mechanical procedure.

In the middle of the 30's no fewer than three different suggestions for such a definition emerged. It could quickly be shown that the three suggestions were equivalent even though they had been worked out in different ways and that they implied that it was not possible to formulate a general method to decide whether an arbitrary sentence could be proved. One of these definitions distinguished itself, however, by being formulated on the basis of an arbitrary theory of numbers.

It was with this definition of a finite mechanical procedure based on the theory of numbers - developed in an attempt to answer Hilbert's third question - that, in 1936, the then 24 year-old English mathematician Alan Turing supplied the first theoretical formulation of the principles of the "universal" computer.[29]

Go to top

Notes, chapter 4

  1. C.f. Bolter, 1984: 143 and Davis, 1988: 150. Leibniz was one of the first European philosophers to take the binary system into consideration. Among other things, he believed that it provided an almost exemplary proof of the existence of God as it was a system which allowed the Almighty to create everything out of nothing. Augarten, 1984: 34. The idea of using the binary system in a calculating machine should perhaps have occurred to him, but would hardly have been possible to utilize at the time. The binary system is almost impossible to handle without the help of extremely complex mechanical or electronic apparatuses, its use as notation in a universal calculating machine also demands - as will be shown in chapters 5 and 7 - a conceptual break with the understanding of the binary form as a system of notation with fixed relationships between the notation unit and its value (as number or rule).
  2. Cohen, 1966: 33.
  3. "But on further examination I observed with regard to logic that syllogisms and most of its other techniques are of less use for learning things than for explaining to others things one already knows or even, as in the art of Lully, for speaking without judgement about matters of which one is ignorant." Descartes (1637) 1985: 119.
  4. Both in Peirce and later, changing terminology is used in referring to the individual components of the so-called sign trichotomy. The terminology here is from a number of manuscript fragments printed in Peirce: Collected Papers, volume 2. The fragments were collected by Justus Buchler under the title Logic as Semiotic: The Theory of Signs and printed in his selection of Peirce texts in Buchler, 1940: 98-119. C.f. also Gorlée, 1990, who describes the development of terminology and meaning in Peirce's view of monadic and dyadic ("degenerate") signs which include such things as biological and bodily signs and mental signs in which the triadic element is missing.
  5. Peirce, 1887: 168. Cohen's characterization is from Cohen 1966: 112.
  6. Peirce, 1887: 168-169.
  7. Peirce, 1887: 169 and 165.
  8. Peirce, 1887: 168.
  9. Saussure (1916) 1983: 15 (33).
  10. Saussure (1916) 1983: 66 (98).
  11. After Saussure (1916) 1983: 8-8 (23-24). The most common translation of parole seems to be "speech", which accords with Saussure's concept of language as spoken, not written. But since it is sometimes necessary to include or refer to the manifestation of written language, it will also be translated - in agreement with Hjelmslev's terminology - as language use or usage.
  12. Peirce (1865, Writings. 1.166) quoted here after Gorlée, 1990: 72.
  13. Saussure (1916) 1983: 16 (33).
  14. Saussure (1916) 1983: 3 (16).
  15. Saussure (1916) 1983: 99-100 (142).
  16. Saussure (1916) 1983: 99 (141), 133-142 (185-197). It should also be mentioned that Saussure also expanded and re-interpreted grammar by adding lexicology to morphology and syntax and by describing these - mutually overlapping - aspects on the basis of what he believed was a more basic distinction between syntagmatic and associative relations. Syntagmatic relations exist in relationship to the surrounding signs in the speech chain. Associative relations are concerned with the relationship to other possible candidates for the same place.
  17. Saussure (1916) 1983: 141 (196-197).
  18. Saussure (1916) 1983: 79-86 (114-123).
  19. Saussure (1916) 1983: 110 ff. (155 ff.). Here there is a parallel with the movement of physics from the idea of matter defined by form to the idea of amorphous matter and pure form.
  20. Empiricism too assumed (and developed on the basis of) such a theoretical-constructive epistemological foundation, necessitated by the attempt to penetrate the prejudices of immediate experience.
  21. C.f. Hodges 1983: 90-92.
  22. Quoted after Robin Gandy, 1988: 63.
  23. Von Neumann, 1961: 265-266. Quoted here from Robin Gandy's translation, 1988: 66. The original quotation is as follows: ...die Unentscheidbarkeit ist sogar die Conditio sine qua non dafür, da[[beta]] es überhaupt einen Sinn habe, mit den heutigen heuristischen Methoden Mathematik zu treiben. An dem Tage, an dem die Unentscheidbarkeit aufhörte, würde die Mathematik im heutigen Sinne aufhören zu existieren an ihre Stelle würde eine absolut mechanische Vorschrift treten, mit deren Hilfe jedermann von jeder gegebene Aussage entscheiden könnte, ob diese beweisen werden kann oder nicht.
  24. Gödel, (1931) 1965: 6 and Hodges, 1983: 91 ff. Gandy, 1988: 68. Gödel's proof was printed in 1931 and expanded in 1935. Quoted here from a reprint in Davis, 1965.
  25. Gödel, (1931) 1965: 28-29.
  26. Ibid: 13.
  27. By scanning I mean here that a phenomenon is represented or handled via an arbitrary symbol - just as a wardrobe number represents a coat. The scanning concept has since spread together with the information concept and is used both of cognitive and visual processes and as a term for certain investigation procedures (such as search procedures in computer science, the eye's search procedure in perceptual psychology, by among others, Gregory and Gibson, each with his own interpretation, and ultra-sound scanning in medicine). It could be said that the scanning perspective, which is a principle of fragmentary representation, today plays a role as a cognitive and visual paradigm, which resembles the role of system perspective in the 19th century and that of central perspective in the 17th and 18th centuries. C.f. Finnemann, 1989: 163-172 and 1991: 170-172.
  28. Gödel (1931) 1965: 5.
  29. The three definitions are Church-Kleene's, Gödel-Herbrand's and Alan Turing's. C.f. Kleene, 1988: 34-36. Gandy, 1988: 69-88 and other contributions in Herken, 1988. The coincidence of time has given rise to much speculation regarding the possible direct and indirect lines of inspiration. It has, however, been reasonably well established that Turing formulated his proof without knowledge of the others. Turing came last, but his article was published because his method was so different to Church's.