Reductionism

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Jacques de Vaucanson : Mechanical Duck (1738)

Reductionism is a philosophical or scientific doctrine, depending on the area of ​​application , according to which a system is completely determined by its individual components ('elements'). This includes the complete traceability of theories to observational sentences , of concepts to things or of lawful connections to causal - deterministic events. The theory assumes that a cause is followed by exactly one effect, which in turn is the cause of another effect (see cause-effect diagram ) . Several effects of a cause, different causes of an effect and repercussions on causes are not considered. The reductionist basic assumption assumes that every phenomenon can nevertheless be described in full if enough data on the object of investigation are known.

Reductionism can be represented as a general scientific program or restricted to a certain scope. Reductionism in the first sense is committed to the ideal of unified science, according to which all phenomena in the world can in principle be explained by the most fundamental science that is seen in microphysics. A reductionism in the second sense can be represented between different areas of science, for example between psychology and neurobiology , between chemistry and physics or ethics and the descriptions of behavior , but also e.g. B. between politics and economics .

The opposite position is the philosophical concept of holism , in which a holistic view is required in order to be able to predict (or at least assess) interactions, side effects and repercussions of causes, chaotic developments and effects on other systems (“ The whole is more than the sum of its parts ”). Since holistic approaches are methodologically much more difficult to grasp and often do not allow generally valid conclusions, they have so far rarely been found in most scientific disciplines.

Unified science and reductionism

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Elementary particles
Scheme by Oppenheim and Putnam, 1958. The upper layer should be composed of the lower layer and should be able to be reduced to this.

The idea of ​​a unified science based on the static top-down scheme of the entities of Oppenheim and Putnam requires a general reductionism . The ideal of science here is the return of all individual sciences to a fundamental science . The dynamic bottom-up model of emergent self-organized processes and systems supports a uniform hierarchical development in nature and society. Both ideas strive for the unity of science, and an all-encompassing and unified view of reality or reality .

A general reductionism requires a number of individual reductive theses: It is assumed that chemistry can in principle be reduced to physics , biology to chemistry, psychology to (neuro) biology and social contexts to (social) psychology . It is also assumed that reductive relationships are transitive : if a science A has been reduced to B and B to C, then A has also been reduced to C. These convictions together result in the thesis that even social relationships can in principle be reduced to physics.

The unified scientific reductionism got its classic formulation in the 1958 essay The Unity of Science as a Working Hypothesis by Paul Oppenheim and Hilary Putnam . Oppenheim and Putnam assumed that unified science was a realistic goal of scientific research. Today's reductionists, however, see unified science as an ideal that could be theoretically realized, but can practically never be achieved by human research.

Emergence, materialism and dualism

It would be wrong to equate materialism with general reductionism and dualism with anti- reductionism . Although dualism is certainly fixed on an anti-reductionism - an immaterial entity cannot be reduced to a material one - many philosophers try to formulate a non-reductive materialism. The popularity of positions that seek to combine anti-reductionism with materialism has increased enormously in the last few decades due to the anti-reductionist arguments described below. There are also philosophical positions that go beyond the alternative materialism vs. Want to position dualism. Examples are neutral monism and conceptual pluralism, such as Nelson Goodmans .

A term that has received increasing attention in the debates about non-reductive materialism is “ emergence ”. It should be noted, however, that this term is used with two different meanings in today's debates. In a weak sense, a property is emergent if and only if it has arisen from a complex configuration. In this sense, the ability of a robot to recognize complicated visual patterns is emergent . For the reductionism debate, this emergence term is of no interest because nothing speaks against the principle of reducibility of the weakly emergent property.

On the other hand, we speak of “emergence” in a strong sense when a system has a property that in principle can not be derived from the properties of the system components. A term of emergence understood in this way, as v. a. was developed by CD Broad , seems to rule out the possibility of a reduction in principle. If the system property A can not in principle be explained by the system properties X, Y and Z, then there does not seem to be any possibility of reducing A to these system properties.

A controversial question is to what extent the emergence theoretical concept enables a non-reductive materialism to be formulated. For a non-reductive materialist the following statement seems to be appropriate: A is a material property, but cannot be reduced because A is emergent. Against such emergence-theoretical materialism , however, the objection is often that it is incomprehensible how one could call A a material property if A can not in principle be derived from the fundamental material properties. This question is systematically examined and answered, for example, by M. Bunge and M. Mahner.

How reductions work

The emergent formation of H 2 O from H 2 and O 2 is a classic example of a lack of reducibility: It is an autocatalytic, branched chain reaction of approx. 6 × 10 22 particles per 18 g of gas. The complex structure and properties of the water molecules cannot be predicted from the properties of the hydrogen and oxygen molecules.

The main topic of the epistemological debates is the methodology of reducing theories. If a true theory A has been successfully reduced to a theory B, the phenomena described in A have been explained reductively . The classic model of theory reduction was formulated by Ernest Nagel in the book The Structure of Science (1961). Nagel suggested making the following requirements for a successful reduction:

Nail reduction: A theory A is reduced to a theory B if and only if all laws of A can be derived from the laws of B.

There should be popular examples from the history of science for such derivation of law. For example, the Galilean law of fall can be derived from the laws of Newtonian mechanics . However, this is only a special application of a formula of mechanics that takes into account gravity as acceleration. As soon as a large number of entities or complex interactions play a role, e.g. B. in the oxyhydrogen reaction , the reduction of the laws of ideal gases to the dynamics of the gas molecules, or the reduction of the chemical bond to the atomic forces, the reduction fails. If, for example, one wants to reduce all the phenomena that can be observed when dealing with water to the chemical theory of H 2 O, then, according to Nagel, one must be able to derive all known laws about water from chemical laws, e.g. the experience that Water boils at sea level at 100 ° C. However, the term “cooking” does not appear in chemical theory, which is why a derivation of the law from chemical theory seems impossible. According to Nagel, bridging principles are needed for such cases , which combine the vocabulary of theory with that of experience. For example, cooking would have to be identified with certain molecular movements.

Too much criticism has been expressed about the approach of Nagel. Strict deductions, such as those demanded by Nagel, seem rarely to exist, so that there is an increasing search for liberal definitions of reduction. It is also controversial whether there might not also be formulations that do justice to the phenomenon of multiple realization (see below). After all, with regard to the status of the bridging principles, Nagel has always remained unclear as to which theory is formulating this.

Emergent self-organized processes

Alternatively, there is a scientific approach to a thorough understanding of processes and systems that explains the development of nature and society: the ontological , process-oriented model of emergent self-organization . It can be seen as an extension of biological evolution. In this model there is a uniform basic process for the processes of the world, which works from the big bang through the development of life, the functioning of the brain to the processes of human society: elements that interact with each other arise naturally and mostly spontaneous systems with new structures, properties and capabilities. The processes are influenced by the conditions in their environment.

Since emergent systems can again be elements of further emergent processes, a hierarchy of increasingly complex systems has developed naturally and recursively in the course of the development of the world . The emergent structures, properties and capabilities cannot be predicted from the properties of the elements and must be determined empirically through observations, measurements, etc. Emergent processes are mostly fed back and therefore non-linear; their sequence is determined by the deterministic chaos . Structures and systems are formed due to the non-linearity of the processes.

Arguments for Reductionism

History of science

Reductionism is mostly motivated by the fact that people are impressed by the explanatory success of modern natural sciences. With reference to this successful explanation, one can formulate an inductive argument for reductionism: Since so many theories have shown that reduction is in principle possible, one should assume that reductions are also possible in areas that have not yet been explained. From a reductionist perspective, one can also point out that the history of science has shown that areas of theory that have fundamentally eluded reduction have largely been banned from science. Classic examples are the witches believe or astrology .

Various objects can be argued against a justification of reductionism from the history of science. On the one hand, it is possible to doubt that it has really been shown for many theories that a reduction is possible. This is because reductions have so far only been carried out for very limited areas. In addition, one can doubt that induction is convincing: If a theory A can be reduced, this does not necessarily mean that B can also be reduced. After all, theories about different phenomena are often structured very differently, so that the induction argument ultimately acts like a very uncertain conclusion by analogy.

causality

In the philosophy of science, therefore, most of the arguments in favor of reductionism refer not to the history of science but to considerations of causality. The classic argument points out that there are causes for an event on different levels. An example: If a person takes a headache pill, various causes can be given for this event, for example: 1) the sensation of headache - that would be a mental explanation, 2) biological processes that trigger certain muscle contractions or 3) microphysical processes, which cause other microphysical processes that make the tablet swallow.

Reductionists now argue that this variety of causes is problematic: It is highly implausible that swallowing the tablet has three independent causes. After all, there is such a multitude of causes in every action and it would be a miracle if all of these actions consistently had several independent causes. It is much more plausible it is that we ultimately have to do here with a Cause: The headaches are nothing more than a biological process and the biological process again nothing but a microphysical process. But if one accepts this solution to the problem of multiple causes, one must also accept reductionism here, since the headache is ultimately identified with a microphysical process.

Supervenience

Even if the term supervenience was originally used for an anti-reductionist motivation, today it is often used for reductionist argumentation. The idea of ​​supervenience is as follows:

A supervises B if A cannot change without B changing. Supervenience relationships can be discussed using simple examples: The fact that a person has hair on their head cannot change without something changing on the microphysical level at the same time. Hence this fact supervises over microphysics - but not the other way around. Now, reductionists agree with some anti-reductionists that everything supervises over microphysics: political, biological or psychological facts cannot change without changing microphysical facts at the same time.

It is now argued that these supervenience relationships can only be made understandable within the framework of a reductionist theory. Without reductionism it would be completely puzzling that everything supervises over the microphysical facts. If, however, one recognizes reductionism, there is a very simple explanation for the fact that A supervises over B: A is nothing other than B.

Arguments against reductionism

While reductionism was the orthodox position in philosophy of science for much of the 20th century , anti-reductionist positions have been becoming more and more popular for around 30 years. On the one hand, this is due to the fact that the sciences did not unify in the 20th century, but rather diversified. On the other hand, the (strong) reductionism in the natural sciences is only applicable to a limited extent in our familiar macroscopic world, but not in the world of atoms, molecules and living beings and in cosmic processes. Third, the new anti-reductionism is closely related to the development of the philosophy of mind and the problems encountered in the reductive explanation of consciousness .

Multiple realizations

Illustration of Fodor's anti-reductionist argument: The macro properties (M1 and M2) are realized through different physical properties (P1 – P6). There is no uniform physical micro-law for the uniform macro-law. Rather, the physical states are linked by various causal relations.

The essay Special Sciences - The Disunity of Science as a Working Hypothesis by the cognitive scientist and philosopher Jerry Fodor from 1974 is probably one of the most influential texts critical of reductionism. According to Fodor's thesis, individual sciences such as psychology or economics cannot in principle be reduced to microphysics, since the laws and properties that are described by the individual sciences cannot be represented by the laws and properties of physics.

Fodor argues that quite different objects have the property of being a means of payment - gold, dollars, shells, etc. Even if these objects have one economic property in common that distinguishes them from all other objects, it is unlikely that gold , Dollars and shells have a physical property that sets them apart from all other objects. But this means that this economic property cannot be reduced to a physical property. In philosophy one also speaks of a multiple realization of the property.

Fodor also argues that the individual sciences describe laws that cannot be traced back to physical laws. Fodor's example is “Gresham's Law”: If two currencies exist at the same time, one of which is more valuable than the other, then the more valuable currency is forced out of payment transactions and saved. Since currencies are now realized in multiple ways, it seems plausible that - depending on the realization - very different laws are involved on the microphysical level. This means, however, that the laws of individual sciences cannot be reduced either and that the individual sciences are therefore irreducible.

Puzzling phenomena

In addition to the multiple realization, there is another argumentative strand that is directed against reductionism. Many anti-reductionists refer to phenomena that, in principle, cannot be described from the perspective of the natural sciences. The existence of such phenomena would pose even greater problems for the reductionists than the multiple realizations. Objects that have been realized in different ways, such as currencies, can at least be described by a science and do not represent a challenge for a naturalistic position. In contrast, the phenomena mentioned here should fundamentally be withdrawn from scientific access.

The question of the reductive explainability of mental states was already controversial in the early 18th century

Some examples of phenomena that anti-reductionists believe to be scientifically undetectable:

  1. Consciousness : Consciousness or the mind is often viewed as a phenomenon that is fundamentally beyond a purely scientific description. One reason for this is that mental states have the property of being experienced in a certain way. If you prick your finger with a needle, for example, not only are complex biological processes going on, but it is also painful. But the biological processes do not seem to make it understandable in any way how someone experiences pain .
  2. Moral properties: Actions have moral properties. Now the fact that an action is legitimate, reprehensible or good or ethical does not seem to result in any way from a scientific description. The reason is that moral terms are normative while scientific descriptions are generally viewed as descriptive . The direct transition or the equation of normative to descriptive statements is rejected as a naturalistic fallacy .
  3. Aesthetic properties: The argumentation for aesthetic properties is similar to that for moral ones. The scientific description has no aesthetic vocabulary, which is why a reduction here seems implausible.

In particular, the thesis that consciousness cannot be explained reductively often leads to a general rejection of materialism. The concrete formulations of the anti-materialist positions are diverse. On the one hand, classic substance dualisms are represented. On the other hand, there are also various anti-materialist positions that also set themselves apart from substance dualism. These include various forms of pluralism , neutral monism , aspect or property dualism and relativism .

Pluralist criticism

While the two previous objections described individual phenomena that are supposed to be irreducible, the pluralistic criticism is generally oriented. A pluralist explains that people have very different approaches to the world and that there is no reason to assume that these approaches can all be reduced to one another. Pluralists also admit that there are reductions, but they argue that reductionism is based on a one-sided preference or absolutization of the physical description of the world. Two currents can be distinguished within pluralism . On the one hand, there is an anti - realistic tendency that declares that it is hopeless to look for a description-independent “way of being” of the world behind the various descriptions of the world. Nelson Goodman can be regarded as its most important representative . On the other hand, there are also realistic pluralists, such as John Dupré , who combine a pluralistic ontology with their position .

Epistemological alternatives

In contrast to the unordered pluralism and the simple double aspect theory, higher standing, multi-digit relation terms (meta-relations) were developed ( explication ). The combination of certain reference systems in physics or in perceptual psychology are simple examples. Strategies of contextual and relational thinking, which combine statements in a mutually complementary manner, even summarize statements that appear contradictory and categorically completely different, are more complicated .

The principle of complementarity coined by Niels Bohr and the differentiation between perspectives ( perspectivism ), which goes back to Gottfried Wilhelm Leibniz , can be understood as meta-relations. What is required is the ability and willingness to change the perspective (the frame of reference) in a phenomenally adequate manner. These strategies show that beyond the enduring and seemingly insoluble controversy between the belief systems of the monists and the dualists, new forms of non-reductionist thinking can be developed.

Lack of reflection on categories and category mistakes

The more recent discussion of reductionism is mainly shaped by Anglo-American analytical philosophy , while the fundamental definition of categories and frames of reference is largely excluded. This other position is characterized by a quote from Nicolai Hartmann , whose comprehensive theory of categories also emphasizes the errors of categorical boundaries. "Every special area of ​​being has its own categories that belong to it only, which in no way can be replaced by other categories and which in turn can never be easily transferred to other areas of being".

In principle, the question remains whether the philosophical contributions to the discussion on reductionism should not be more closely linked to the philosophy of science and methodology of the empirical disciplines. The often very simplistic “didactic examples” widespread in philosophical literature show how far the gap is from current research strategies. The specialist knowledge of the relevant disciplines and the relevant criteria there are required to justify definitions and reductions and to better identify category errors.

Special position of psychology and other human sciences

If a moderate reduction or a micro-reduction in the smallest steps is strived for in biology, then it is a matter of tracing back biological laws to laws of chemistry and physics. In a corresponding attempt in the field of psychology, there is the fundamental difficulty that empirical psychology does not have any strict laws (in the sense of causal explanation , nomology ). What is meant here by the “liberal definition of reduction” mentioned remains unclear. When repeating a psychological experiment, due to the experience side and the subjective attitude of the participants (test person behavior), no complete constancy of conditions ( ceteris paribus principle) can be asserted and, due to frequent individual exceptions (singular relationships), not even regularities; Instead of predictions, it is widely believed that only statistical relationships and statistically founded expectations are possible.

According to the conception of physicalism (naturalism) in the form of reductive or eliminative materialism, the neurophysiological functions ultimately form a system that works according to the laws of physics. Reduction means here: Sentences about mental brain functions are replaced by sentences about neuronal brain functions or they are derived from them ("naturalization of consciousness"). Can the language of the content of the experience be translated into a brain physiological language without any loss of information? (See Norbert Bischof's qualitative reductionism .)

Reductionist tendencies of the opposite kind

Usually, reductions are only described as “leading down” from the top down towards basic physical laws; scientific sentences about “higher” functions should be replaced by less structured and categorically simpler sentences.

From a reverse perspective, reductionist tendencies in the simplification (abstraction) of empirical facts, research controversies and theories can often be noticed in philosophical representations. There are theoretical concepts of physiology and biology that form very complex structures of morphological and functional relationships, while these conceptual and categorical differentiations seem to have disappeared in some philosophical and psychological statements. The overall psycho-neuro-physiological system of an emotion is so complex - and far exceeds today's attempts at comprehensive theories - that an adequate description cannot yet be foreseen. How simple, on the other hand, are the vocabulary and principles of most philosophical-anthropological and experiential-psychological presentations on "emotion" . The language-analytically derived comparison of the first-person perspective and the third-person perspective also reduces the psychologically and neurophysiologically complex process of interactive self-perception and perception of others, both of which also include emotional and motivational aspects, to the methodologically more accessible cognitive functions and thus to a comparatively simple philosophical-cognitive scheme .

Unsolvable problems in theoretical physics

At the end of 2015 it was proven that a fundamental mathematical problem in quantum physics is unsolvable . It is therefore impossible to determine the macrostate of a material from them, even with a theoretically complete knowledge of all microstates .

Debates on reductionism in public

The term reductionism does not only play a role in the epistemological debates. It is also often used in public disputes, but has a rather vague meaning there. Usually the term here has a negative connotation and is not differentiated from the term scientism . When using the term in this way, the accusation is usually in the foreground that scientific descriptions are given preference in an illegitimate way over artistic, humanities or social science descriptions.

This critical use of the concept of reductionism is often in the tradition of a culture-critical philosophy, such as the Frankfurt School . With Max Weber's concept of the disenchantment of the world , it is argued that the advancing natural sciences covered more and more areas of the human world and thus devalued non-scientific ways of describing. Reductionism is understood as the dogma of an “imperialist” science.

Klaus Holzkamp (1972) saw in conventional psychological research an individualistic narrowing and at the same time a reductionistic elimination of the “specifically human, ie. H. social level of life activity ”. The path of "modern psychology" isolates people from their socio-historical living conditions. In a anthropological reductionism leads Gerd Jüttemann (1991), the enduring crisis of psychology back. The alleged lack of presuppositions often ends with the reflection of one's own image of man and its implications instead of striving for the appropriate application of methods and the reflexivity of the procedure.

Even if the concept of reductionism has mostly negative connotations in public, various scientists and philosophers are increasingly trying to interpret it as positive. Prominent examples are the biologist Richard Dawkins and the philosopher Daniel Dennett . They argue that anti-reductionist intuitions are motivated by outdated metaphysical and theological prejudices. Within a materialistic worldview one can only welcome reductions.

See also

literature

Classic texts

  • Paul Oppenheim , Hilary Putnam : The Unity of Science as a Working Hypothesis. In: Minnesota Studies in the Philosophy of Science, 1958. The classic formulation of the unified science program
  • Ernest Nagel : The Structure of Science. Harcourt, Brace & World, New York 1961, ISBN 0-915144-71-9 . Comprehensive epistemological work, contains the classic formulation of reductions
  • Clifford Alan Hooker: Towards a General Theory of Reduction. In: Dialogue. 1981. Alternative model of reductive explanations

Criticism of reductionism:

  • Jerry Fodor : Special Sciences. In: Synthesis. 28, 1974, pp. 97-115. Classic essay on the justification of the autonomy of the individual sciences
  • John Dupré : The Disorder of Things. Metaphysical Foundations of the Disunity of Science. Harvard University Press, Cambridge 1993, ISBN 0-674-21260-6 . Formulation of a pluralistic metaphysics. Emphasis on biology
  • Jürgen Habermas : Freedom and Determinism. In: German magazine for philosophy. 52/6, 2004, pp. 871–890 and in: Between Naturalism and Religion. Philosophical essays. Suhrkamp, ​​Frankfurt am Main 2005, ISBN 3-518-58447-2 . Habermas' commitment to anti-reductionist aspect dualism in the context of the free will debate
  • Paul Feyerabend : Against the method constraint. Suhrkamp, ​​Frankfurt am Main 1986, ISBN 3-518-28197-6 . Relativistically based anti-reductionism

Current controversies:

  • David Charles, Kathleen Lennon (Eds.): Reduction, Explanation, and Realism. Oxford University Press, Oxford 1992, ISBN 0-19-875131-1 . Collection of mostly reductionist essays. Contains texts on individual topics such as social sciences or morals.
  • Jochen Fahrenberg : On the theory of categories in psychology. Complementarity principle. Perspectives and change of perspective. Pabst Science Publishers, Lengerich 2013, ISBN 978-3-89967-891-8 PDF file; 5.5 MB, 573 pages Overview of category theory and relational terms in psychology and biology, contains arguments against a discussion that is different from current empirical research and category theory.
  • Magda Hengle: Behavioral reductionism: on the reception of behaviorism in sociology , 1978, DNB 780772563 (dissertation University of Hamburg, Department of Philosophy and Social Knowledge 1978, 331 pages, 21 cm).
  • Frank Jackson : From Metaphysics to Ethics. Oxford University Press, Oxford 1998, ISBN 0-19-825061-4 . Defense of general reductionism by a former dualist
  • Wolfgang Deppert : The reductionism problem and how to overcome it. In: W. Deppert, H. Kliemt, B. Lohff, J. Schaefer (eds.): Theories of Science in Medicine. A symposium . Berlin 1992, ISBN 3-11-012849-7 , pp. 275-325.
  • Jaegwon Kim : Physicalism, or something near enough. Princeton University Press, Princeton 2005, ISBN 0-691-11375-0 . Overview of Kim's theory, contains arguments against anti-reductionist positions such as dualism or non-reductive materialism
  • Achim Stephan : Emergence. Mentis, Paderborn 2005, ISBN 3-89785-439-2 . Most comprehensive German-language presentation of the concept of emergence.
  • Manfred Stöckler: Reductionism. In: Joachim Ritter ua (Hrsg.): Historical dictionary of philosophy . Volume 8. Wissenschaftliche Buchgesellschaft, Darmstadt 1992, ISBN 3-7965-0115-X , pp. 378-383.

Web links

Wiktionary: Reductionism  - explanations of meanings, word origins, synonyms, translations

Individual evidence

  1. ↑ a b Reinhard Wagner: Communication of basic systems science concepts. Diploma thesis, Karl Franzens University Graz , Berlin 2002, pdf version , pp. 3–4. accessed on July 21, 2019.
  2. ^ Raphael van Riel and Robert van Gulick: Scientific Reduction. In: Stanford Encyclopedia of Philosophy. Retrieved January 30, 2018 (Section 1. Historical background ).
  3. ↑ Mario Bunge, Martin Mahner: About the nature of things, Hirzel 2004
  4. ↑ a b c Günter Dedie. The power of natural laws - emergence and collective skills from elementary particles to human society, 2nd edition, 2015 tredition
  5. ↑ RB Laughlin: Farewell to the world formula, Piper 2009
  6. ↑ Philip Clayton: Emergence and consciousness, Vandenhoeck & Ruprecht 2008, http://digi20.digitale-sammlungen.de/de/fs1/object/display/bsb00083882_00001.html
  7. ↑ G. Jetschke: Mathematics of Self-Organization, 2nd edition, Harri Deutsch 2009
  8. ^ Karl-Helmut Reich: Developing the horizons of the mind: Relational and contextual reasoning and the resolution of cognitive conflict . Cambridge Univ. Press, Cambridge, 2002, ISBN 0-521-81795-1 .
  9. ↑ Nicolai Hartmann: The structure of the real world. Outline of the general theory of categories. 2nd Edition. de Gruyter, Berlin 1949, p. 92.
  10. ↑ Martin Mahner, Mario Bunge: Philosophical foundations of biology. Springer, Berlin 2000, ISBN 3-540-67649-X .
  11. ^ Rainer Westermann: Philosophy of Science and Experimental Methodology. Hogrefe, Göttingen 2000, ISBN 3-8017-1090-4 .
  12. ↑ Norbert Bischof: Psychology. A basic course for the discerning. Kohlhammer, Stuttgart 2008, ISBN 978-3-17-020909-1 , p. 230.
  13. ↑ Dirk Eidemüller u. a. (TUM / DE): Unpredictable solids. Even complete knowledge of the quantum properties does not guarantee the calculability of the macrostate. In: pro-physik.de - the physics portal. Wiley-VCH , December 10, 2015 .
This version was added to the list of articles worth reading on December 16, 2005 .