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Niels Ole Finnemann: Thought, Sign and Machine, Chapter 2 © 1999 by Niels Ole Finnemann.
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2. The origin of a new concept of information

2.1 Missing information - the thermodynamic demon

Modern information theories are in agreement concerning their origin in thermodynamics in the last part of the 19th century, more precisely the statistical thermodynamics of the Austrian physicist, Ludwig Boltzmann in 1872.[1]

In spite of many references, however, most are more than sparing, often restricted to quoting Warren Weawer's remark that:

Boltzmann's observation in some of his work on statistical physics (1894) that entropy is related to "missing information" inasmuch as it is related to the number of alternatives which remain possible to a physical system after all the macroscopically observable information concerning it has been recorded.[2]

A laconic, but apposite clue. Information theory takes its point of departure in considerations of the way in which it is possible to calculate the indeterminate by describing indeterminacy as a quantity of a finite number of alternative, not yet decided possibilities. The hunt for the thus more narrowly defined, missing information became a central theme within both physics and the later information theory.

What is missing, however, and what more precisely takes its starting point here, is in dispute. The dispute is not simply concerned with the localization of a definite body of missing knowledge, but also with the interpretation of the epistemological implications. This is the reason why one and the same problem has given rise to different interpretations within physics and created the starting point for the paradigm of information theory. It is the latter clue which is central in this connection, but it cannot be pursued without a glance at physics, because the paradigm of information theory is not only derived from the concept of missing information in physics, but also appropriates other concepts from the thermodynamic reformulation of mechanical theory.

This chapter contains an account of Boltzmann's contribution to this reformulation, as he not only identified missing information with his foundation of statistical thermodynamics, but also established the conceptual framework which is the starting point for the paradigm of information theory. This is first and foremost true of his description of the physical problem of observation which is connected with the discrepancy between macro-physical order and micro-physical (molecular) "disorder", his use and interpretation of statistical methods and his contribution to the development of the concept of finite space as an abstract and arbitrary system.

Together, these elements contain a basic renewal of the mechanical paradigm, because the mathematical description is developed here as an abstract model which can be applied to both physical and non-physical phenomena and processes. Thermodynamics hereby breaks with the understanding of the relationship between matter and form in classical mechanics and at the same time opens the way for the emancipation of mechanical theory from physics.

In classical mechanics physical matter is defined on the basis of form and extent, the form is understood as the defining property of matter and difference in form is interpreted as material difference. In statistical thermodynamics, on the other hand, form is defined as an independent structure which can organize an arbitrary material and the material is regarded - in Boltzmann still only latently - as amorphous and without structure. The same forms and structures can thus also be imagined as being incorporated in different domains/substances. Mechanical theory can now be thought of as a purely formal system of mathematical relationships which can be applied to an arbitrary physical, biological or mental substance. It is true that the demand of classical physics for a mathematical abstraction which corresponds to physical reality is not abandoned, but the demand is manifested as a descriptive ideal which cannot be fulfilled within the framework of classical physics.

The abandonment of a materially bound form concept, which opens the way for the development of mechanical theories in a number of new domains, also gives rise to another difficult problem, however, because the concept of amorphous matter removes the justification for a distinction between different domains - such as between the physical, the biological and the mental.

Given these far-reaching innovations, it is hardly surprising that several mutually different interpretations and answers are given. There are also two different paths which lead from Boltzmann's thermodynamics to the later information theories. One takes its point of departure in the concepts of missing information and entropy and leads - as Weawer pointed out - directly to Shannon's mathematical theory of communication. The concept of finite space is here interpreted as physical space. The second path takes its point of departure in the concept of formally defined, finite space and passes, via mathematical logic, to Alan Turing's theoretical description of a universal computer. The concept of finite space is interpreted here as logical space.

The two different paths from the problem of observation in physics would later meet again, as both arrived at the idea that it must be possible to describe both biological, perceptual and conscious processes, as well as the content of consciousness as local, finite physically-mechanically performed processes which take place in time and space. The two paths together thus comprise a significant - although not the only - precondition for post-war information theories, cybernetics, theories of artificial intelligence, cognitive science and artificial life. As Boltzmann's theoretical deliberations took the same direction, his work also provides an early account of the epistemological problems which arise in the later efforts to use one and the same mechanical paradigm in the description of physical and biological processes, of the processes of the brain and consciousness and of the content of consciousness.

Boltzmann's presentation of the problem

Boltzmann's work as a physicist took its point of departure in thermodynamic theory as it had been formulated in the middle of the last century. According to the first law of thermodynamics the amount of energy in the world is constant and, according to the second law - the law of increasing entropy - nature is subject to a law of irreversible development which will gradually lead to the so-called "heat death", where all differences in energy - and hence all kinds of organization - have been neutralized.[3]

The theory, however, could only be confirmed at a macro-physical level through a measurement of temperature and pressure conditions, it was impossible to account for the order of the individual particles in the thermodynamic system. Here, spectroscopic analyses had on the contrary provided evidence of a complicated micro-physical structure that could not be described on the basis of ordinary theoretical assumptions.

It was James Clerk Maxwell who had formulated the problem. Although it was not possible to observe the micro-physical processes, it was possible, noted Maxwell, to imagine an ideal observer, a "demon", equipped with such refined, but scientifically describable means of observation that this, unlike the physicist, could observe each, individual molecule which moved within a closed physical system. He then demonstrated how such a demon could work as a kind of perpetual motion machine as it could move energy from a colder to a warmer place without performing any work, thereby undermining the second law of thermodynamics - on the degradation of energy. This does not, however, undermine the law of the constancy of energy, which means that there can be no question of a perpetual motion machine which is capable of producing energy from nothing.[4]

The talent of the hypothetical demon not only raised the question of micro-physical order, but also of how the micro-physical system, comprising a very large number of molecules moving in mutually uncoordinated paths and unceasingly colliding with one another, can still, at the observable, macro-physical level, show unchanged, constant properties.

Molecular thermal motions are most probably such that a given state of motion is not shared by a large group of neighbouring molecules, but that in spite of constant mutual influence each molecule pursues its own independent path, appearing as it were as an autonomously acting individual. One might therefore think that this autonomy of the parts would at once have to show itself in the external properties of bodies for example that in a horizontal metal bar the right and now the left end must become spontaneously hotter according as the molecules happen to vibrate more intensely at one or the other place, or that if in a gas a large number of molecules happen to be moving towards the same point at the same time, a sudden increase in density must occur there. However, we observe none of this, and the reason why this is so is nothing other than the so-called law of large numbers.[5]

In another, more recent formulation, the question is posed as follows: how can a system made up of particles which obey mechanical laws that are invariant with regard to the direction of time nevertheless develop in a certain direction?[6]

Boltzmann's thesis was that it was necessary to give up traditional methods of description for the benefit of statistical methods designed to calculate the probability of a molecular system being in one or other of its possible states and then explain why a system could not move from a probable state to another, equally probable, state, but only to a more probable state.

Boltzmann assumed that there was a corresponding number of different combinations (imagined micro-physical states) for every macro-physical state. He hereby describes a given macro-physical state as a closed - spatial - system which is subdivided into an - arbitrary - number of smaller "phase spaces" so that it would be possible to describe the micro-physical (molecular) state on the basis of the distribution of the molecules in these phase spaces.

In the micro-physical system the most improbable state is characterized by the highest degree of order. This state corresponds to a situation where all molecules are concentrated in a single phase space. Entropy here is zero and the number of possible combinations assumes the minimum value 1.[7] The most probable state is the opposite and characterized by maximum disorder, the energy has degraded to heat energy, the molecules are "spread throughout the system" and the number of possible combinations here assumes the maximum value for a given system depending on the number of molecules in the system.

Boltzmann thus abandons a classic, fully deterministic prediction. But, he claims, entropy can be regarded as proportional to the number of possible combinatory states, expressed in the formula:

S = k log W

where S is entropy, k a mathematical constant (later designated Boltzmann's constant) and W the thermodynamic probability which expresses the possible number of molecular combinations with the same macro-physical properties (same distribution structure).[8]

With this description of the system "as though" it comprised a large number of independent particles, each behaving individually in accordance with mechanical principles, Boltzmann had indicated a method for predicting the total state of the system with exactly the degree of statistical precision required, even though it was not possible to describe the behaviour of the individual particle.

The system was characterized by increasing molecular disorder (equal distribution of molecules throughout the system) or "progressive elimination of all original asymmetry"[9] and it behaved, in spite of the molecular chaos, in accordance with the law of increasing entropy.

Hereby, Boltzmann believed, the problem of missing information in thermodynamics had been solved. The statistical and probabilistic description of the state of the thermodynamic system was a complete, exact description which could also reconcile the law of entropy with a classical, mechanistic and deterministic theory of motion.

The means to this was a new method where a number of - atomic - particles was described in relation to a finite physical space which was divided into smaller units and included all possible spatial positions. The idea of the finite space itself was presumably quite obvious as it is the epitome of the containers which were used to store various gases. Similarly, the purely formal or arbitrary subdivision of the space resembles a simple reference to the three-dimensional, spatial system of co-ordinates. But it nevertheless implies a break with the classical conception of nature as one - infinite - cohesive mechanical universe in favour of a conception of nature as a number of locally limited, finite systems. The break occurred as Boltzmann abandoned the description of the individual molecule's individual state in favour of the probable distribution state of the total system relative to a formal lattice structure in a closed space with a finite number of possible positions.

With this break the method of analytical subdivision becomes a completely arbitrary method,as the analytical procedure is no longer a means to dissolve a phenomenon into its component parts, but on the contrary, a means to structure a formal reference system for describing - statistical properties of - phenomena independently of individual variations.

But the solution had its price. The statistical description of molecular "chaos" could not be understood as a phenomenological description of molecular nature. Boltzmann's answer to the information that was missing on the individual particles meant that the door opened up on another area of missing information of a more fundamental, epistemological character. Namely the missing information which manifests itself as a difference between a deterministic, physical description and a statistical and probabilistic description.

There were two areas in particular which - notwithstanding the interpretation - were difficult to handle on the basis of the classical Newtonian paradigm in which force was described as an - in itself immaterial - function of discrete particles' mass and speed (or distance). One was the discovery and description of the many different types of energy. It was impossible to reconcile the Newtonian laws of motion, that described the meeting of particles as a collision, with the experimental examples of wave interference and transformations between the different energy forms. The second was the transition from a description of physically visible or perceptible phenomena to the description of physical micro-processes which could not become the object of directly perceived observations, but only be studied indirectly through macro-physical, recordable effects.

Both of these problems were generally acknowledged, the dispute was about - and is about - their implications. The most obvious - and at the time most common - starting point would have been a reformulation of physics based on wave theory, because thermodynamic theory pointed to energy as the basic physical substance. But Boltzmann, starting with a statistical description of the micro-physical system on an atomic basis, chose a different path.

His solution to the specific problem of description therefore necessarily brought about a re-interpretation of scientific epistemology and during the 1880's he turned increasingly away from the work on thermodynamics to questions connected with the epistemology of physics in general. This change of direction meant that he absented himself from the history of physics for a very long period. During this period his deliberations are seldom referred to and when they are, they are treated rather as an expression of a more outgoing personal and philosophical interest which had little relevance to physics.[10] It is hardly possible to decide whether this was also the reason why he became tired of life, but a certain bitterness in his latest work indicates something of the sort. Boltzmann committed suicide in 1906.[11]

2.2 The price of information - Boltzmann's dilemma

If Boltzmann experienced problems in gaining a deserved hearing for his theoretical deliberations, this was not least due to the fact that he was unable to accept the inability to solve a problem which - still unsolved - would become central to 20th century physics, namely the relationship between the descriptions of inorganic, micro-physical nature based on the wave and particle theories respectively and the relationship between the micro-physical and macro-physical levels.[12]

In his attempts to solve this problem Boltzmann started with the successful statistical description of molecular systems. This description was based on classical atomistic premises which now, however, had to be formulated as a statistical and probabilistic description. The mechanical procedures which were part of the description could not be understood as mechanisms which existed in nature:

If the molecules and atoms of the old theory [Newton's] were not to be conceived of as exact mathematical points in the abstract sense, then their true nature and form must be regarded as absolutely unknown, and their groupings and motions, required by theory, looked upon as simply a process having more or less resemblance to the workings of nature, and representing more or less exactly certain aspects incidental to them. With this in mind, Maxwell propounded certain physical theories which were purely mechanical so far as they proceeded from a conception of purely mechanical processes. But he explicitly stated that he did not believe in the existence in nature of mechanical agents so constituted, and that he regarded them merely as means by which phenomena could be reproduced, bearing a certain similarity to those actually existing... Maxwell himself and his followers [continued Boltzmann, thinking not least of himself] devised many kinematic models, designed to afford a representation of the mechanical construction of the ether as a whole as well as of the separate mechanisms at work in it: these resemble the old wave mechanisms, so far as they represent the movements of a purely hypothetical mechanism. But while it was formerly believed that it was allowable to assume with a great show of probability the actual existence of such mechanisms in nature, yet nowadays philosophers postulate no more than a partial resemblance between the phenomena visible in such mechanisms and those which appear in nature.[13]

By looking at mechanical theory as a mental model which always and in principle only expressed an approximation "bearing a certain similarity", according to Boltzmann a number of old questions disappeared of their own accord.

As we know that both material "points" and "forces" are simple mental images, we no longer need to speculate about how it is possible for a force to be emitted from a point which is simply a mental construction, or how points can be united and become extended. As it was also possible to refine the description of points and forces "as closely as we please" to an image of the spatial world, this re-interpretation could be depicted as a practical expansion - rather than a theoretical loss - of possible cognitions. This bore a similarity to the old dispute on the relationship between matter and energy. By looking at the theoretical concepts as mental images it was possible to avoid falling back on the old metaphysical debates as to whether matter or energy is "truly existent".[14]

Correspondingly, from another article:

All our ideas and concepts are only internal pictures, or if spoken, combinations of sounds. The task of our thinking is so to use and combine them that by their means we always most readily hit upon the correct actions and guide others likewise. In this, metaphysics follows the most down-to-earth and practical point of view, so that extremes meet. The conceptual signs that we form thus exist only within us, we cannot measure external phenomena by the standard of our ideas. We can therefore pose such formal questions as whether only matter exists and force is a property of it, or whether force exists independently of matter or conversely whether matter is a product of force but none of these questions are significant since all these concepts are only mental pictures whose purpose is to represent phenomena correctly.[15]

The theory thus allowed the Newtonian model to be preserved by understanding the concepts of both energy and matter as mental pictures which, in principle, were only capable of providing an approximative expression of certain traits in nature's organization.

The idea of the mental, pictorial character of mathematical physics itself is reminiscent of Descartes' theory of consciousness, but with the difference that the mathematical picture is now understood as a hypothetical approximation, the legitimacy of which depends on its appropriateness. Mathematical consistency is no longer a secure basis for a correspondence between concept and the conceived.

It can be mentioned in passing here that Descartes saw this correspondence as divinely given and certain. Boltzmann was also very much aware of the religious foundation of the epistemology of science, but attempted to eliminate its significance by pointing to the common - and inadequate - anthropocentric basis of all the different ideas of god inherent in epistemological considerations.

Here, too, belongs the question of the existence of God. It is certainly true that only a madman will deny God's existence, but it is equally the case that all our ideas of God are mere inadequate anthropomorphisms, so that what we thus imagine as God does not exist in the way we imagine it. If therefore one person says that he is convinced that God exists and another that he does not believe in God, in so saying both may well think the same thoughts without even suspecting it. We must not ask whether God exists unless we can imagine something definite in saying so rather we must ask by what ideas we can come closer to the highest concept which encompasses everything.[16]

There is no direct connection between the truth and the mental representation - human consciousness - but it is possible to draw them closer together with the help of an increasingly sophisticated and complex model construction and the experimental experience of "facts".

This was true both of classical mechanics and statistical thermodynamics and Boltzmann took an emphatic stance as one of the few advocates of a classical atomistic physical theory of his period, but re-interpreted as a suitable mental model which could not be abandoned for the present. At the same time it should also be mentioned that he often emphasized the great probability that the mechanical model might have to be rejected - or radically changed - at some future date. The postulate was simply that at the time there was no basis for doing so.

2.3 The sign and the designatum

It is still a matter of debate whether Boltzmann represented a realistic/materialistic theory or broke with the idea of a mimetic/realistic correspondence between the description and the described.[17] A decision regarding this discussion can hardly be made because he expressed himself as an adherent of both positions. We will probably not be much mistaken if we assume that his original point of departure was within the realistic tradition, but it is equally clear that he was in favour of retaining the atomistic model even though it could no longer be seen as a realistic theory. This is the schism which is at the heart of his theoretical work and it increasingly forced him to exchange the choice between two different theoretical ideas of nature for a transition from a classical idea of direct representation (mimetic reflection or Cartesian correspondence) in the relationship between the world and the scientific description to a new idea of the approximative description as a principle and unavoidable epistemological condition.

... it cannot be our task to find an absolutely correct theory but rather a picture that is as simple as possible and that represents phenomena as accurately as possible. One might even conceive of two quite different theories both equally simple and equally congruent with phenomena, which therefore in spite of their difference are equally correct.[18]

Furthermore, Boltzmann is almost prepared to desert the idea of analogical representation in favour of a semiotic description where scientific concepts are seen as an independent sign system (where the greatest possible mathematical determinism must be attempted) which does not represent, but refers to another - natural - system.

Hertz makes physicists properly aware of something philosophers had no doubt long since stated, namely that no theory can be objective, actually coinciding with nature, but rather that each theory is only a mental picture of phenomena, related to them as sign is to designatum.[19]

He had, however, at best only a vague idea that there was a great deal of slippery material in the matter he had taken up. Even though, for example, in introducing his lectures on mechanical principles he carefully corrected himself and suggested the sign concept rather than the concept of mental pictures, he failed to go into further detail regarding the implications of sign theory.

Nobody surely every doubted what Hertz emphasizes... namely that our thoughts are mere pictures of objects (or better, signs for them), which at most have some sort of affinity with them but never coincide with them but are related to them as letters to spoken sounds or written notes to musical sounds.[20]

He was not aware that the relationship between the sign and the designatum, far from being given, would on the contrary become a main theme of the 20th century. He was by inclination a realist and appears, in spite of his emphasis on the figurativeness of mental representation, to have regarded the character of figurative and verbal representation as an already clarified, obvious matter.

The comparison of the relationship between the spoken sound and written letter was not simply intended as a pedagogical reference to an assumedly well-known matter from a different area of experience. It was a direct expression of the fundamental idea in his understanding of the character of mental construction of all theoretical and particularly mathematical-physical thinking. That he did not - like Charles Peirce, similarly prompted by thermodynamics - take the further step towards a general sign theory is connected with the fact that he had a different view of the way in which it might be possible to negotiate the gulf which opened up between the sign and the designatum. The mental picture which pointed the way to physical nature was not simply a sign, it was also itself physically manifested in time and space.

2.4 The physics of thought

It appears as though this idea grew out of his deliberations on mechanical principles, which did not simply exist as conceptual, mental pictures, but in the form of independent mechanical apparatuses. Boltzmann saw these apparatuses (machines, instruments etc.) as materializations of our mental representations which not only - as for Descartes - included the inner representations, but also the physical models and tools we surround ourselves with and use:

When therefore we endeavour to assist our conceptions of space by figures, by the methods of descriptive geometry, and by various thread and object models our topography by plans, charts and globes and mechanical and physical ideas by kinematic models - we are simply extending and continuing the principle by means of which we comprehend objects in thought and represent them in language or writing. In precisely the same way the microscope or telescope forms a continuation and multiplication of the lenses of the eye and the notebook represents an external expansion of the same process which the memory brings about by purely internal means.[21]

This is remarkable because here Boltzmann expresses the opinion that there is no path from the one "great machine", nature, to the small machines, except through human consciousness and sign creating competence. This also explains why a universal mechanical paradigm must regard consciousness as a function of the "great machine" in order to allow the existence of the small, while a "local" mechanical paradigm for well-defined finite spaces both provides the space for a finite consciousness and for a finite machine. On the other hand, this point of view provides no answer to the question of how a finite mechanical system can create an extension of itself which is at the same time an independent, closed system.

Boltzmann insisted, however, on the mechanical idea and would not himself describe it as a semiotic understanding of technology either. Rather, he regarded the possibility of externalizing mental pictures as experimental confirmation of the utility of the mental picture and - something that certainly had a far-reaching perspective - as an expression of continuity behind the distinction between the concept and the conceived. The technological materialization of mental ideas was a kind of confirmation of the validity of these ideas. That the familiar mechanical technologies had no obvious similarity to other visible phenomena in the surroundings produced no further deliberations.

In regarding models and mechanical apparatuses as implemented consciousness Boltzmann exceeded the bounds of the Cartesian distinction between the inner, which has no extension, and the outer, which has, precisely in keeping with his repeated insistence that behind the apparent jumps in nature there are gradual or continuous transitions, "nature knows no jumps".[22] This had, he claimed, been confirmed experimentally time after time both in physics and chemistry. Now, however, the question arose as to whether the same also held true of the relationship between physics, biology and consciousness?

Descartes' idea that human consciousness, although it belongs to the natural world, but floats freely in a separate substance, had not only been contested by Thomas Hobbes in the 17th century, but also early on in the 19th. Hobbes believed that thinking was a purely mechanical system like all other phenomena in the world. He therefore saw no problem in the relationship between the physical and mental aspects of thought processes. This theme was raised, however, in the 19th century. While both Goethe, Romantic philosophy and scientifically-oriented sensory physiology, as well as physicists such as Müller and Helmholtz attempted to describe the physics of sense perception[23], Boltzmann, inspired by Darwin's biological theory of development, goes a considerable step further and claims that not only sensation, but also consciousness and the thinking process can be understood as physical processes:

The intimate connection of the mental with the physical is in the end given to us by experience. By means of this connection it is very likely that to every mental process there corresponds a physical process in the brain, that is, there is an unambiguous correlation and that the brain processes are all genuinely material, that is, are representable by the same pictures and laws as processes in inanimate nature. In that event, however, it would have to be possible to predict all mental processes from the pictures that serve to represent brain processes. Thus all mental processes must be predictable from the pictures used for representing inanimate nature without change of the laws that govern it... All these circumstances make it extremely likely that an (objective) world picture is possible in which the processes in inanimate nature play not only the same but even a much more comprehensive role than mental processes, which latter are then related to the former only as special cases to general ones. Our aim will not be to establish the truth or falsehood of one or the other world picture, but we shall ask whether either is appropriate for this or that purpose while we allow both pictures to continue alongside each other...

The brain we view as the apparatus or organ for producing word pictures, an organ which because of the pictures' great utility for the preservation of the species has, conformably with Darwin's theory, developed in man to a degree of particular perfection, just as the neck in the giraffe and the bill in the stork have developed to an unusual length. By means of the pictures by which we have represented matter (no matter whether the most suitable pictures will turn out to be those of current atomism or some others) we now try to represent material brain processes and so to obtain at the same time a better view of the mental and a representation of the mechanism that has here developed in the human head, making it possible to represent such complicated and apposite pictures.[24]

The idea of technology as an externalized materialization of mental pictures is thus closely connected with the idea that mental processes are not only physically materialized, but can also be described with the concepts of physics. That he ventured upon such deliberations was not least due to the fact that the thermodynamic description of physical systems had reached a new stage of higher complexity which better allowed ideas on complex or hierarchic physical systems in which the higher levels possessed other - more well organized - properties than the simpler, but less organized systems. If such a description of biological phenomena could be given it also implied that it would be possible in principle to construct artificial organisms, including organisms which think like people.

Imagine there could be a machine[25]that looked like a human body and also behaved and moved like one. Inside it let there be a component that receives impressions of light, sound and so on, by means of organs that are exactly built like our sense organs and the nerves linked with them. This component is further to have the ability of storing pictures of these impressions and by means of the pictures so to stimulate the nerve fibres that they produce movements that are totally similar to those of the human body. Unconscious reflex movements would then naturally be those whose innervation did not penetrate so deeply into the central organ as to generate memory pictures there. It is said to be a priori clear that this machine behaves externally like a man but does not sense. It would indeed retract the burnt hand just as quickly as we do, but without feeling pain...In our fictitious machine every sensation would exist as something separate. Similar sensations would have much in common and dissimilar ones less. Their course in time would be that given by experience. Of course no sensation would be simple, each would be identical with a complicated material process, but for one who does not know how the machine is built, sensations would again not be measurable by length and measures, he could no more represent them by spatial and mechanical pictures than we can our own sensations. However nothing more is given by experience. Thus everything we are empirically given of the mental would be realised by our machine. The rest we arbitrarily add in thought or so it seems to me. Like any other person, our machine would say that it is aware of every existence (that is, it had thought pictures for the fact of its existence). Nobody could prove that it was less aware of itself than a human. Indeed, one could not define consciousness in some manner such that it applied less to a machine than to men...[26]

With this - hypothetical - idea of a reconstruction of a human being based on the complex laws of physical nature Boltzmann not only anticipated the theoretical and practical efforts of the 20th century to construct intelligent machines, he also formulated three central criteria for this project. First, that this would have to be built on the basis of the concepts we use in describing physical nature, because all mental and biological phenomena have a basic physical realization. Second, that such a reconstruction would necessarily include a reconstruction of the human sensory apparatus, because sensory and experimental experience are conditions for knowledge and thinking, and third, that the possibility of such a project is entirely dependent on the definition of consciousness.

The point that Boltzmann saw, however, was also that definitive arguments can never be produced against this possibility because any definitive argument would contain such a specific definition of consciousness that consciousness could be reconstructed with the help of the identical, testable specifications. We can never say never. As will appear in chapter 5, this is exactly the same consideration that Alan Turing uses as an argument for the future possibility of being able to refer to the modern computer as a thinking machine without expecting contradiction.

Whether it is possible to manufacture such a copy is still a question of belief, but it is quite legitimate to discuss the basis for doing it. Boltzmann's human machine implies the precondition that we are able to reconstruct human organs on the basis of their micro-physical parts and functions. However, not only do we not possess that knowledge, we do not possess the means to handle the necessary amount of knowledge either. On the other hand we know that we can only obtain these means if it is possible to carry out some kind of - mathematical, for example - synthesis of the necessary knowledge. This conflicts with two of Boltzmann's central premises. First, a mathematical synthesis would only be an approximation which expressed certain, limited aspects of what is described. Second, the very idea of a mathematical synthesis is incompatible with his demand for a precise correspondence between the thought and its physical manifestation. A mathematical synthesis could not have the same extent as the unsynthesized expression.

The condition for carrying out Boltzmann's project is thus that his precondition, the unambiguous correspondence between the physical and mental aspects of thought processes, is not valid. The project dissolves into a paradox, it is only possible if it is impossible and this is due to the fact that Boltzmann, in a circular fashion, ignores his own starting point: mental representation, even in the most rigorous mathematical synthesis, is only an approximation which in principle cannot reproduce all relevant physical properties. This understanding of approximation is incompatible with a deterministic theory.

In the final analysis, the paradoxical in Boltzmann's project was connected with the attempt to avoid the threatening indeterminism of the physical theories with the help of a deterministic theory of consciousness. Even if the physical restrictions are ignored, the idea of a deterministic theory of consciousness appears to be extremely difficult to reconcile with the idea that thinking has a content. As it can only refer to its own, previously known, determined preconditions the thought would not be able to produce anything. Prediction is reduced to a simple, meaningless articulation, a sign which is only capable of referring to its own preconditions - or in Boltzmann's case: its physical manifestation - is not a sign of anything at all.

Boltzmann's paradox, however, cannot simply be explained away as the result of a one-off blunder. It must rather be understood as an early formulation of an epistemological field of tension which has retained its paradoxicality in the 20th century. Behind the paradox lies also an extension of the scientific field of reflection which is still far from being thoroughly worked out. This is true both of his idea of a physics of thought, the idea of an artificial reconstruction of biological and mental systems, and a preliminary semiotic understanding of technological artefacts. It is also true of his draft for a dynamic ecology.

2.5 Thermodynamic biology?

Boltzmann built up his idea of a hypothetical machine-human on two central theoretical preconditions. One was the new, more complex picture of physical - especially micro-physical - nature and the thermodynamic theory of development in particular. The second was Darwin's theory of biological evolution. Also propounded in this hybrid is the idea of biological evolution as a function in and of the thermodynamic.

The basic idea is that, seen from a thermodynamic perspective, biological organisms can be described as highly-organized physical systems. The fact that this - from a cosmic point of view - extremely improbable situation actually exists and even, according to Darwin's theories, continues to develop towards still higher forms of organization, can be explained partly on the basis of the assumption that the universe as a whole is enormous and can therefore contain local, more highly developed systems, partly on the basis of the assumption that a continuing degradation of energy takes place within such locally existing systems. The energy which comprises the conditions of life for biological organisms is released through this process:

The general struggle for existence of animate beings is therefore not a struggle for raw materials - these, for organisms, are air, water and soil, all abundantly available - nor for energy which exists in plenty in any body in the form of heat (albeit unfortunately, not transformable), but a struggle for entropy, which becomes available through the transition of energy from the hot sun to the cold earth. In order to exploit this transition as much as possible, plants spread their immense surface of leaves and force the sun's energy, before it falls to the earth's temperature, to perform in ways as yet unexplored certain chemical syntheses of which no one in our laboratories has so far the least idea. The products of this chemical kitchen constitute the object of struggle of the animal world.[27]

The idea that human intervention in the natural process of energy transformation could be of great significance for conditions of life must have been rather remote to Boltzmann. It is more remarkable that three-quarters of a century would elapse before the entropy-ecological clue he hints at here as a possibility would attract broader scientific interest, particularly because physics, during the course of this century, has produced a dramatic expansion of energy releasing technological potential.

Even without this development, which includes both a quantitative expansion of 19th century macro-physical energy releasing techniques and a qualitative expansion with the 20th century's micro-physical, Boltzmann's theory would have been far from adequate, however, and his premises are also doubtful. The inadequacy lies, among other things, in the physical approach. Universal heat death is irrelevant by comparison with the - much narrower - biological boundary conditions which are neither within the scope of Darwin's nor Boltzmann's ideas. The doubtful premises lie in the idea of development.

The idea of development is beset with two uncertainties - both in Darwin and Boltzmann. One is the lack of a possibility for pointing out or defining certain natural initial conditions. The other is lack of an explanation of how a cell, or a more complex organism, creates itself as a biological entity and how this type of physical condition can again produce mental phenomena. For example Darwin assumed in On the Origin of Species[28] that there was an original (divine) creation of a few biological species, (or perhaps only a single) while Boltzmann believed that it was sufficient to assume that very complex atomistic processes could reduplicate themselves

...by forming similar ones around them. Of the larger masses so arising the most viable were those that could multiply by division, and those that had a tendency to move towards places where favourable conditions for life prevailed.[29]

This is the same idea of an already existing, mechanically executed "tendency" ("instinct" or "intentionality"), i.e. a force which motivates a complex of atoms to reproduce themselves as a whole which can then search for "the favourable conditions for life" - which is the prerequisite in Boltzmann's hypothesis on the origin of consciousness or intentionality:

Sensitivity led to the development of sensory nerves, mobility to motor nerves sensations that through inheritance led to constant strong compelling messages to the central agency to escape from them we call pain. Quite rough signs for external objects were left behind in the individual, they developed into complicated signs for complex situations and, if required, even to quite rough genuine internal imitations of the external, just as the algebraist can use arbitrary letters for magnitudes but usually prefers to choose the first letters of the corresponding words. If there is such a developed memory sign for the individual self, we define it as consciousness.[30]

The lack of a possibility for defining the initial conditions, however, not only implies what Boltzmann is quite aware of, that the theory is hypothetical, but also that the hypothesis takes as its point of departure that - biological or cognitive - phenomena which it will explain as the result of a physical development must already exist before this development takes place. Darwin assumed the existence of a few divinely created biological cells and simple organisms, while Boltzmann assumed that molecular systems of a certain physical complexity would receive the properties we describe as biological and mental, including a purpose-oriented autonomy. The only question the hypothesis fails to answer is that which motivated it, namely what can make a mechanical-physical system produce biological and mental properties which allow the system, among other things, to "escape" or avoid what we call "pain"?

This objection tells not only against Darwin's and Boltzmann's attempts to reformulate and expand a deterministic theory of nature, it also reveals the weakness which appears in any deterministic theory in the unavoidable meeting with consciousness. Not because any specific criticism can be made of determinism, but because any deterministic theory of consciousness is a contradiction in terms. If consciousness is determined by physical or congenital, hereditary or mutated biological processes, there is no sense in talking of human perception of the world's organization because no statement in such a case can be related to anything else, it is exclusively a passive function which is in rapport with the physical or biological system in which it is realized at a given - already disappeared - time in a certain place.

Haugeland makes a similar criticism of Thomas Hobbes' and David Hume's philosophical models of the mechanical structure of thought in pointing out that, each in his own way, they end precisely by not being able to explain the trait which makes thinking thinking, unlike the mechanical processes which are not.[31]

Haugeland calls this "the paradox of mechanical reason", but apparently believes that it either is, or soon will be, capable of solution.

Perhaps the idea of automatic symbol manipulation is at last the key to unlocking the mind...

he says in his conclusion, but does admit that another possibility can be imagined

Perhaps the programmable computer is as shallow an analogy as the trainable pigeon - the conditional branch as psychologically sterile as the conditioned reflex... after thirty years, the hard questions remain open.[32]

When Haugeland, who in all honesty acknowledges his dislike of what he calls the "intellectual anaemia" of scepticism, finds himself forced to come to such a sceptical conclusion, it is well founded. A theoretical determinism concerning consciousness cannot be saved by replacing the mechanical motor with a formal automaton, however many built-in conditional clauses it contains, because the formal procedure, unlike human consciousness, cannot describe the rule structure of the formal procedure or interpret its significance.

The cognitive void which manifests itself as a logical circularity - even an automatic circuit - at the same time makes deterministic theories of consciousness immune to criticism.

It is perhaps slightly more paradoxical in Boltzmann's case than in others because of the degree to which he directed his physical research precisely towards the questions which in particular undermine the classical deterministic assumptions of natural science and because, philosophically, he placed such emphasis on the freedom and fallibility of thought - also with regard to the perceptions of natural science.

Boltzmann takes great pains to emphasize that it is not only the senses, but also human thinking that is fallible, but claims that this can also be explained on mechanistic premises on the basis of Darwin's theory.[33] This is true both of fallibility and the surmounting of its limitation. The idea appears to be that this development, also within the realm of thinking, has the happy logic that the fittest will win. In mechanical theory, however, there is no room for any criterion of fitness. Another problem with Darwin's theory is that it is purely retrospective, it derives the prehistory of the later development which itself can only be explained on the basis of an even later development. As it thus contains no indicators for the future, it provides no means to decide what is valid in the present.

It is not easy to understand how it is possible to reconcile the idea of consciousness as an unambiguous function of physical laws with the idea of correct and incorrect scientific theories. But it is also remarkable because Boltzmann, as a corollary to Maxwell, so strongly emphasizes mathematical physics' character of mental pictures, which in the best case can represent an approximation of certain selective traits in the physical world. It was here more than anywhere that Boltzmann saw missing information of a more fundamental and unavoidable character than that missing information which is concerned with the number of alternative, micro-physical possible states.

2.6 Mathematics as an approximative model

Boltzmann gave several reasons for this schism. One lay in physical theory. Although the formulation of a number of mechanical models of physical energy processes had succeeded, it was clear that they should be interpreted as conceptual analogies, they did not express nature's actual "inner structure".[34] It is possible to express laws for both interference and collision in mechanical models, but not join them together to a theoretical whole. This perhaps initially affected only the understanding of energy, but subsequently also the realistic understanding of mechanistic particle theory and the idea of nature as one great machine. The understanding of theory as no more than a model was a consequence of the thermodynamic description.

Another reason lay in the understanding of mathematical representation itself. Paradoxically, this limitation emerges from the success of mathematical description. It became evident that it was possible to describe many, quite different physical processes using the same mathematical formulas:

It often happens that a series of natural processes - such as motion in liquids, internal frictions of gases, and the conduction of heat and electricity in metals - may be expressed by the same differential equations and it is frequently possible to follow by means of measurements one of the processes in question - e.g. the conduction of electricity just mentioned. If then there be shown in a model a particular case of electrical conduction in which the same conditions at the boundary hold as in a problem of the internal friction of gases, we are able by measuring the electrical conduction in the model to determine at once the numerical data which obtain for the analogous case of internal friction.[35]

Physics was later able to confirm, to a great extent, that this applicability also held true of the equations which in particular ensured Boltzmann a place in the history of physics.[36] But the advantage is connected with the fact that the mathematical procedures have no reference to what separates the represented systems. The same mathematical expression can only describe different physical processes because it does not describe all the significant physical aspects of any individual process. Connected therefore to each individual, specific use is a detailed explanation of the specific conditions for use. Here, Maxwell expressed himself more distinctly as - to a greater degree than Boltzmann - he emphasized that the analogies based on the mathematical equivalence between different physical processes] were specific and that any extended generalization of this equivalence must be based on step-by-step experimental instances.[37] The choice of which mathematical expression that expresses general laws of nature lies outside the scope of the mathematical description.

The use of the same mathematical expression in describing different physical processes thus does not indicate that these processes can be subsumed under one, general law. This multiple application is only possible because it is a question of different physical processes which, under certain conditions, can be regarded from the same reductive point of view. Conversely, a generalization of the mathematical description must represent the physical differences.

This is not only true of qualitatively different physical phenomena, but also of purely quantitative differences:

...a mere alteration in dimensions is often sufficient to cause a material alteration in the action, since various capabilities depend in various ways on the linear dimensions. Thus the weight varies as the cube of the linear dimensions, the surface of any single part and the phenomena that depend on such surfaces are proportionate to the square, while other effects - such as friction, expansion and condition of heat etc. vary according to other laws.[38]

Mathematical precision for Boltzmann is neither a certain nor sufficient basis for physical knowledge. The mathematical formulas are rather excellent schemes or models for handling things because they are independent of both the conceptual ideas from which they are derived and of the specific physical processes which are handled with these models and with the calculative possibilities which are connected with them.[39]

Boltzmann's deliberations on the approximative character of mathematical description - notwithstanding their preliminary and tentative form - are not far removed from the views of newer mathematics on the arbitrary, symbolic character of mathematical idealizations and it is possible to discern in them the beginnings of a thematization of the epistemological problems which lie in the application of mathematics to physics.

2.7 Summing up

When we read Boltzmann today it is almost impossible not to be struck by the consistency with which he formulated and pursued the epistemological questions which are still under discussion in relationship to the interpretation of the physics of thermodynamics. It is not only Boltzmann's still discussed, specific results and answers, but also the methodical procedure and a broad range of the problems discussed that have retained their topicality.

If we join Warren Weawer in claiming that Boltzmann was the first to point out a specified relationship between missing information and physical entropy, we can also add that he thereby not only laid the foundation for an understanding of the indefinite as a number of alternative possibilities which could be calculated with the help of statistical probability methods;[40] with his concept of the formal, arbitrary variable and finite reference system, he also laid the foundation for a new mechanistic model of description and raised many of the epistemological problems which were attendant on this method, although not all.

Statistical description meant that he had to refrain from describing the individual particles. He thereby left unsolved the main question, the relationship between micro and macro-physical order, but also added a new, namely that of the epistemological status of statistical description. Boltzmann attempted to answer this question in several ways. First, by claiming that statistical description was just as precise and deterministic as classical atomistic description, but the price was that neither of them could be considered as phenomenologically valid. He continued to accept mathematical description as explicatively valid, but still maintained that it could only be approximative. He hereby conferred a new form on the epistemological problem, namely that of a general problem of observation and description.

It was as an answer to this that he formulated his draft of a physics of thought. Although he formulated this idea on the basis of deterministic and physical thinking and therefore ran into the problem of deterministic description in the form of mutually conflicting premises, he still paved the way for the scientific use of the philosophical idea of consciousness as a dynamic system which would later give rise to distinctive theoretical innovations.

What he thus attempted to make cohesive can be summed up in the following themes, each of which has attained central significance as an area of discussion in the 20th century, but seldom in a similar, collected form:

In the clue pointed out by Warren Weawer, the connection between thermodynamics and information theory lies first and foremost in the connection between the concepts of entropy and information. This connection, however, is not quite as simple as Weawer appeared to assume and for Boltzmann it already implied a number of more far-reaching considerations on the epistemology of physics and the possibility of describing biological and mental processes (including the content of the latter), as well as the physical execution of these processes on a physical basis.

Although both Boltzmann and Weawer believed that the problem of missing information had been solved, they had completely different ideas both of the problem and its solution. For Boltzmann this involved finding a mathematical expression for an (invisible) physical process, which could not be described exhaustively. He sought a method for extracting knowledge on the way in which such physically indefinite micro-processes could manifest themselves as a physical whole with well-defined physical properties and believed that he had found this with the statistical and probabilistic description.

For Weawer, who interpreted Claude Shannon's mathematical information theory, it involved on the contrary defining a general mathematical goal for the frequency of occurrence of physically well-defined informational entities. Here the problem was not the extraction of knowledge either, but the stable transport of knowledge.

Both Shannon and Weawer appeared - as will be shown in chapter 6 - to understand mathematical information theory as a generalization of Boltzmann's thermodynamic definition of entropy, as Shannon "simply" eliminated the physical constant (Boltzmann's constant) in the formula. It was precisely this constant, however, that contained the connection to the entropy concept. The mathematical formula which the two theories had in common has nothing to do with the entropy concept, but is exclusively a - relative - yardstick for a statistical (im)probability, the calculative result of which incidentally rests entirely on and varies with the chosen area of application. The mathematical yardstick itself has no content.[41]

Where "missing information" in thermodynamics stems from the lack of a possibility for establishing the initial conditions for a physical system - and thereby also the lack of possibility for predicting the movements of the individual molecules - mathematical theory is interested solely in physically well-defined quantities. These quantities, however, are not concerned with the organization of nature, but with - physically defined - symbolic notation forms. The question of physical-phenomenological precision and the question of the validity of knowledge, are not included in the mathematical information theory.

The path from Boltzmann's to Shannon's theory thus traverses a gulf which involves both the missing information and the problem of observation in physics. That Shannon did not become involved with these questions can first and foremost be explained by the fact that his own path back to Boltzmann went via the Hungarian physicist, Leo Szilard, who at the end of the 1920's had proposed a theoretical analysis which was intended to explain how it would be possible to maintain thermodynamic principles if the analysis included a measurement of the energy used to transfer information from the system to the observer.

Szilard proposed the theory with a - debatable - postulate to the effect that it contained a solution to the theoretical problem of observation. But it also contained a more precisely - and narrowly - defined concept of physically defined information, as information is defined as a measurable amount of energy.

As the information concept with this definition becomes a synonym for a physical phenomenon, it has no place as an independent concept in physics. On the other hand the definition contains the conceptual basis for a description of physical and informational systems as parallel mechanical systems because the physical and informational entities are joined like two sides of a coin.

As Szilard's theory thereby becomes an important link in the transformation of the mechanical paradigm from a physically to a mechanically founded informational paradigm, it will be returned to in chapter 3, while Shannon's theory will be gone into in greater detail, but in chapter 6, as the intervening chapters concern the - simultaneous - theoretical development of the idea of the finite, mechanical symbol procedure towards the symbiosis of mechanical and symbolic thinking in modern information theory.

Prior to this, the discussion of Boltzmann will be rounded off with a glance at a couple of the other clues which, during the course of this century, have secured him continued - and for the past 25 years growing - attention both in and outside physics.

While the English physicist, J.D. Bernal, completely ignored Boltzmann in his voluminous history of science from the 1950's,[42] there is widespread agreement today that the honour of having introduced statistical-mechanical description into the history of physics should be ascribed to him. Since then, statistical thermodynamics has pursued another path, laid out almost at the same time, independently of Boltzmann, by the American physicist, J.W. Gibbs, who instead of describing a single system with many interacting molecules, described a number of such systems with the whole system as an entity.[43]

But a lasting significance is ascribed to Boltzmann for his two more specific contributions to thermodynamic theory, namely his description of the entropy concept as a mathematically well-defined yardstick for the disorder, or probable state, of a molecular system and his formulation of the so-called Boltzmann equation, a mathematical expression of the state of equilibrium of a system comprising a great number of particles.[44] Certain mathematical equations accompany both of these more durable results, which have since acquired - and are still acquiring - many new areas of use. Shannon's theory is an example of such use.

There can be little doubt that these - rather tardily recognized - results will secure a more visible place in the history of physics for Boltzmann, but his rehabilitation is less interesting than the discussion and disagreement that have emerged in the discussion of the implications of these results, which concern both the understanding of thermodynamic irreversibility and the relationship between mechanistic and statistical methods of description. The thermodynamic entropy theorem as it was formulated by Boltzmann is still an object of discussion, as he showed how macro-physical irreversibility could appear as a - statistical - result of a mechanical (and thereby reversible) description of molecular processes.

Boltzmann thereby synthesized, says Ilja Prigogine, three forms of description which had arisen separately, namely the dynamic description based on the laws of mechanics, the probabilistic and the thermodynamic descriptions.[45] But that synthesis which was the answer to the problem for Boltzmann was rather the new question itself.

Boltzmann's influence can be traced not only in statistical mechanics in physics and in Shannon's and Weawer's mathematical communication theory, but also in Ilja Prigogine's thermodynamic theory, which is a new attempt to surmount the conflict between the reversible chronological symmetry of mechanics and the asymmetry of thermodynamic chronology.

According to Prigogine the concept of dissipative structures makes it possible to eliminate the stochastic, probabilistic element in Boltzmann's theory.[46] Thermodynamic entropy, irreversibility, is explained here on a causal-dynamic, mechanistic basis which presumably contains an asymmetric ]time concept. Mechanical reversibility is hereafter a borderline case.

Another, far-reaching actualization of Boltzmann's work can be found outside the realm of physics and information theory - in a comprehensive treatise, Filosofi, by Danish philosopher Johs. Witt-Hansen. Boltzmann's statistical thermodynamics is described here as a radical break with classical dynamics, as the statistical point of view - used as an explanatory principle - implies a new understanding of the concept of order. According to Witt-Hansen, order is understood here as a combinatory phenomenon which can be described on the basis of statistical probability, while classical Newtonian mechanics was founded on a causal deterministic idea of order where every phenomenon can be localized in a well-defined time-space connection.

Witt-Hansen sees Boltzmann's statistical description of the thermodynamic principle of irreversibility - today often called "the arrow of time", after Arthur Eddington[47] - as a first and prototypical example of what he calls "mathematical generalization", or logically based extension of the conceptual framework. He sees herein a general principle for transcending the explanatory limitations of a given scientific paradigm. In Boltzmann's case the limitations of classical, mechanical description.

Although Witt-Hansen refers directly to Prigogine as a precondition, he draws a different - less realistic - conclusion in that he sees the value of Boltzmann's efforts in another area, namely in his contribution to the development of the principle of mathematical generalization, which according to Witt-Hansen constitutes the only stable foundation for science and today is becoming "fruitful in biology, sociology and futurology".[48]

That he thereby presents a less critical interpretation of Boltzmann than we otherwise find in present-day discussions, does not necessarily affect his argumentation for regarding the principle of mathematical generalization as the proper answer to explanatory limitations of a scientific theory, but it does indicate a problem. Boltzmann's statistical explanation of thermodynamic irreversibility was later rejected as inadequate, the status of statistical description is still an object of discussion and the law of thermodynamic entropy cannot be regarded as definitively proven. Like Boltzmann, Witt-Hansen also glides from purely epistemological to ontological reasons because, among other things, he places so much emphasis on the concept of natural constants.

When considered together these theories first and foremost indicate that even more recent research has failed to lead to any final evaluation either of Boltzmann's results, or of his own interpretations. Boltzmann's topicality lies not only in some of the answers he provided, but also in the questions and new points of view he formulated. The reach and character of these questions explain why he himself increasingly moved from scientific to epistemological and theoretical problems.

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Notes, chapter 2

  1. Boltzmann, 1872. Both Kronig, Clausius and Maxwell had anticipated the statistical point of view in thermodynamics during the 1850's, but it was Boltzmann who was the first to give it a precise mathematical form. Cf. P. & T. Ehrenfest (1912) 1959: 1-2, Prigogine, 1983: 405, Cohen & Thirring, 1983: V.
  2. Warren Weawer, (1949) 1969: 3. Despite a certain amount of work, I have not succeeded in verifying the term "missing information" in Boltzmann.
  3. The entropy concept ("transformation content" Greek: en + tropein) was formulated in 1865 by Rudolf Clausius, roughly at the same time as the energy concept ("work content" Greek: en + ergon). Both concepts were created as part of the recognition of the connection between the material forces of nature: mechanical, electrical, magnetic force and heat. Entropy is defined as a reduction of available energy as it degrades into unavailable (heat) energy. The relationship between the different types of energy was described as a relationship between higher and lower forms of energy. The idea was first formulated by Carnot who established a utility principle on the criterion of accessibility or inaccessibility, but this was gradually reformulated to a distinction between forms of energy of different ranks and quality by Lord Kelvin - alias William Thomson - and Clausius. Mechanical energy can be transformed into heat energy, while heat energy cannot be fully transformed into mechanical energy, which has a higher rank. The same is true of electricity, while chemical energy (e.g. combustion) occupies an intermediate position. The first formulation of the law of entropy as "heat death" stems from physicist Helmholtz in 1854.
  4. Maxwell illustrated his argument with a suggested experiment: The starting point is the experimental knowledge that gas molecules in a closed container at a uniform temperature move at unequal speeds. The container is divided into two parts, A and B with a divider in which there is a small hole and it is assumed there is an observer (the demon) who can see the molecules and open and close the hole. If the observer opens and closes the hole so that the faster molecules are able to pass from A to B and the slower molecules from B to A, the temperature in B will rise, while it will fall in A. He has moved energy from a colder to a warmer place without performing any work, which is at variance with the second law of thermodynamics on the irreversible growth of entropy (in that his own operations are not seen as part of the system): Maxwell, (1871) 1970: 308-309. Cf. Goldmann, 1983: 122 ff. and Klein, 1973: 74-77.
  5. Boltzmann, (1886) 1905: 33-34. English translation: Boltzmann, 1974: 19-20.
  6. S.R. Groot, Boltzmann, 1974: 3.
  7. The salient point here is that this description does not emphasize the question as to where molecules are gathered in the space - in which of the imagined phase spaces - but only as to the combinatory or structural distribution of the molecules relative to the phase space. It is also a precondition that the molecules move in discrete and instantaneous transitions between states, so that each particle at all times is in a certain phase space.
  8. The - final - notation given here is from Max Planck. Thermodynamic probability is calculated as the possible distribution of molecules in a three-dimensional, geometric model of a given system which is imagined as divided into cells in which a given number of molecules can be placed. In the improbable state the molecules are gathered in one cell (thereby giving only one possible state). In the most probable state they are equally distributed in all cells (which would be the case for the maximum number of possible states of the system). Cf. Witt-Hansen, 1985: 49-50. Flamm, 1983: 265.
  9. Witt-Hansen, 1985: 50.
  10. A large part of this work was published under the not particularly apposite title Populäre Schriften in 1905. According to Klein, 1973, Boltzmann's difficulties in making himself understood were due to the fact that he reformulated his position several times without explanation. Another of Boltzmann's problems, however, was that many physicists - including Maxwell, who was otherwise concerned with similar questions - considered him prolix and quasi-metaphysical.
  11. Flamm 1973: 13 and 1983: 274.
  12. The problem had already manifested itself in the divergence between Newton's and Huygen's understanding of light as respectively a particle or wave phenomenon. It became more urgent, however, because no progress had been made in clearing it up in spite of an increase in knowledge. It was no longer enough to "simply" differentiate between what was known and what was not yet known. The problem now lay in the relationship between the acknowledged laws of physics.
  13. L. Boltzmann, (1902) 1974: 217-218.
  14. L. Boltzmann (1899a), 1905: 216, 219. English translation, Boltzmann 1974: 91, 93.
  15. L. Boltzmann (1899b), 1905: 257- 258. English translation, Boltzmann 1974: 104.
  16. L. Boltzmann (1897b), 1905: 187. English translation, Boltzmann 1974: 75.
  17. The two points of view are represented by respectively Broda's and Klein's contributions in Cohen and Thirring, 1973.
  18. L. Boltzmann (1899a), 1905: 216. English translation, Boltzmann, 1974: 91.
  19. Ibid: 215-216. Respectively: 90-91.
  20. L. Boltzmann, (1897a), 1974: 225.
  21. L. Boltzmann, (1902) 1974: 214. My emphasis.
  22. L. Boltzmann, (1886) 1905: 47. English translation. Boltzmann 1974: 29.
  23. Cf. Jonathan Crary, 1988.
  24. L. Boltzmann (1897b) 1905: 178-179, English translation. Boltzmann, 1974: 68-69. The English translation, probably correctly, regards the German edition's "automistik" as a printer's error for "atomistik".
  25. Boltzmann adds in a footnote: By a machine I naturally mean merely a system built up from the same constituents according to the same laws of nature as inanimate nature, but not one that must be representable by the laws of current analytical mechanics; for we are by no means sure that the whole of inanimate nature can be represented by these latter. (1974: 76, note 12).
  26. Ibid: 183-184. English translation. Ibid: 72f.
  27. L. Boltzmann (1886) 1905: 40. English translation, Boltzmann, 1974: 24.
  28. Charles Darwin, (1859) 1964: 484. Facsimile of the first edition.
  29. Boltzmann(1886) 1905: 49. English translation, ibid: 31.
  30. Ibid: 49. English translation, ibid: 31.
  31. Haugeland, (1985) 1987: 23-44. According to Haugeland the problem for Hobbes, who regards consciousness as part of the corpuscular world, is that in order to explain the mechanics of thought processes he must differentiate between the individual thought units (parcels) and the laws of motion which regulate the movement. These, however, have the character of thought themselves (rules of calculation) and their activity must therefore also be explained, which again demands an underlying motor whose activity must be explained so that this results in an endless recurrence or in an unexplained assumption of an inner homunculus which makes the system move. Hume attempts, says Haugeland, to avoid Hobbes' problem by denying that thinking refers to the surrounding world. He "simply" sees the mechanics of thought as an analogy to mechanical physics and thus ignores the question of what separates this mechanical system from other mechanical systems, but is still far from providing an answer to what gives this mechanics its content of thought. Where Hobbes has a homunculus, Hume has nothing.
  32. Haugeland, (1985) 1987: 253-254.
  33. Boltzmann understood Darwin's theory as an - exhaustive - mechanical theory even though Darwin's evolutionary logic is filled with intentional and instinctual forces (both individual and with regard to species) and assumes a divine creation long after the origin of the physical universe.
  34. Paul Feyerabend draws a direct parallel between Boltzmann's and Maxwell's understanding of form analogies and Bohr's understanding of the "figurativeness" of wave and particle concepts. Feyerabend 1981, I: 12 (and note 29). It is perhaps more surprising that Einstein, in spite of his fundamental determinism, also expressed great scepticism of the "precision" of applied mathematics. The differences between pure and applied mathematics are very great, indeed, wrote James H. Fetzer and adds, As Einstein remarked, insofar as the laws of mathematics refer to reality, they are not certain, and insofar as they are certain, they do not refer to reality. Fetzer 1990: 259.
  35. L. Boltzmann, (1902) 1974: 220.
  36. Cf. Cohen and Thirring, 1973 and Groot in Boltzmann, 1974.
  37. According to Feyerabend, Maxwell distinguished mathematical formulas from physical hypotheses and from form analogies. He believed that the mathematical formulas lacked heuristic potential. They can help to trace the consequence of given laws, but at the expense of the "visibility" and "context" (connections) of phenomena. Hypotheses are useful as guides, they keep sight of the physical subject, but also confuse because they are generalizations expressed in a conceptual, theoretical medium. The concept of form analogies serves to emphasize the need to test the constituent parts of all hypotheses, step by step. Feyerabend 1981, 1: 12.
  38. L. Boltzmann (1902) 1974: 220.
  39. Cf. Boltzmann (1897c) 1905: 158-162 and (1890) 1905: 80. English translation, Boltzmann, 1974, 54-56 and 1974, 36.
  40. Warren Weawer (1949) 1969. See chapter 6.
  41. Cf. Donald McKay, 1983: 489.
  42. J.D. Bernal, (1954). Norwegian edition, 1978.
  43. Cf. P. and T. Ehrenfest (1912) 1959, which contains a detailed discussion of Boltzmann's and Gibbs' works, John von Neumann (1932) 1955: 360.
  44. The formulation given here summarizes S.R. Groot's characteristic in Boltzmann, 1974: IX-X. In E.D.G. Cohen and W. Thirring, 1973, the editors emphasize in the introduction that the Boltzmann equation provided the first-ever precise, mathematical basis for a discussion of the conditions for a state of equilibrium.
  45. Prigogine, 1973: 407.
  46. Prigogine, 1973: 443-445. Prigogine’s theory is also statistical, as the molecular processes are described as »ensembles« and not as the movements of singular molecules. The phenomenological or realistic interpretation of the statistical foundation is also disputed. Cf. Danish physicist Torben Smith Sørensen, who points out that it is impossible to ignore the lower limit of description in quantum mechanics. Sørensen, 1987: 48.
  47. Arthur Eddington (1928) 1930.
  48. Johs. Witt-Hansen, 1985: 44-52. Witt-Hansen's theory stems from a discussion of thermodynamics which intensified within physics and in the area between physics and biology, among other things due to the fact that many physicists, especially after World War II, began to take an interest in biology. Cf. S.B. Dev, 1990. Flamm, 1983, believes he can demonstrate that there is a direct line from Boltzmann to quantum physicist Erwin Schrödinger who published his famous What is Life? in 1943, which provided inspiration in such areas as the development of information-theoretical molecular biology.