II. Abstraction, idealization, sign

 

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The treatment of material according to mathematical principles

Polygones as symbolic models of the cosmos

Idealization

Abstraction

Signs need to be decoded

Charles Sanders Peirce: Theory of signs

Max Weber: Ideal types

Ideal towns and buildings

Bali villages mirror the cosmological total system

 

 

 

The treatment of material according to mathematical principles

 

The increasing differentiation of the building forms and decorations since the Early Renaissance made necessary an expansion of geometry: perspective, stereometry and construction of regular polygones were developed.

Already around 1490 one of the first mediators between official science and mundane practice, Luca Pacioli (Leonardo Olschki 1918, I, 157), dealt with this need. His „Summa" (1494) is the first popular text in which the five regular polygones are described. They have a symbolic meaning revealed by the scientific investigation oft humanism and the study of Plato (I, 172, 223).

 

Leonardo Olschki points out, that the technique to produce the  regular poygones devised by Pacioli directly goes back to the treatment of materials for decoration purposes. „The treatment of the material, as stone and marble, for structural as well as sculptural purposes according to the mathematical principles of the time required precise determination of volume and reliable information for the transformation of a body into another. The times of treating materials by eye were definitly gone“ (I, 218).

 

We find the model of a dodecahedron in the size of a hand on a picture which Jacopo de Barbari painted in 1495 of Luca Pacioli. For Pacioli’s book „De divina proportione“ (written 1497, published 1509) Leonardo da Vinci drew numerous wooden models of geometrical figures, amongst them for the first time an ikosidodekaeder.

 

 

Polygones as symbolic models of the cosmos

 

Plato speaks in „Timaios" suggestively of a heliocentric world order and visualizes its harmony by stereometric objects and their interrelations (Olschki, I, 222).  Since Pacioli’s peculiar econjunction of mysticism and practice  the mystic component of the Platonic solids became a linchpin in scientific research (I, 220-227).

It was crowned in the cosmography of Johannes Kepler. Here the regular polygones are symbolic models of the cosmos (see also Friedrich Wagner 1970, 58f.). The familiar picture  of 1596 in J. D. Bemal (1970, 397) is one of the first visualizations of a theory (= explanation).

 

 

Idealization

 

In his „Discorsi“ (1638) the Italian Galileo Galilei describes the idealization of the inclined plane. To this diverging as well as interesting thoughts and experiments are given by Viktor A. Stoff (1969, 52-53), I. Bernard Cohen (1977), Ernan McMullin (1985), Leszek Nowak (1992), Martii Kuokkanen (1994), Andreas Hüttemann (1995), Vasilis Raisis (1999) and Jan M. Zytkow (1999).

Other important idealizations in the technical field are:

·        ideal conditions,

·        machines without friction loss, etc.

Economic idealizations are:

·        “fair dice”,

·        efficient capital market, self-regulating market,

·        "homo oeconomicus", etc.

 

 

Abstraction

 

Abstraction means to work out and use "pure cases", e. g. ideal gas, black body, ideal circle, mathematical pendulum or point, incompressible fluid, perpetuum mobile, etc.

 

Peter Achinstein (1965, 102-120) deals with „theoretical models“ as the

·        Bohr model of the atom

·        Billiard ball model of gases

·        Corpuscular model of light

·        Shell model (and liquid-drop model) of the atomic nucleus

·        Free-electron model of metals

And he argues „these models are quite distinct from other conceptions sometimes called models“.

 

Manfred Stöckler (1994, 49-50) treats some idealized representations of physical systems or simplifying theories of them (“Konkrete Theoretische Modelle”), e. g. the first three „theoretical models“ of Achinstein as well as

·        Solid-state models, , e. g. free electron model of metals, e. g. potential pot model of metals

·        Core models

·        Droplet model

·        Skin model

·        Core cluster model.

 

One of the first to deal with idealization was the Polish philosopher Leszek Nowak (1972, 1980, 1992). Since 1990 he, Jerzy Brzezinski et al. edited more than a dozen volumes of the ”Poznan Studies in the Philosophy of the Sciences and the Humanities” on “idealization”. Niall Shanks (1998) published an omnibus on idealization im modern physics.

 

For abstractions in the social and economic field Max Weber proposed in his interpretative sociology the „ideal type“. It is an abstraction of a set or real beahvior and serves to understand social and economic phenomena. Weber („Economy and Society“, German 1922; Engl. 1968, II, §2) differentiates four categories of "ideal types" of behavior:

·        zweckrational (rational means to rational ends)

·        wertrational (rational means to irrational ends)

·        affektuell (guided by emotion) and

·        traditional (guided by custom or habit).

 

 

Signs need to be decoded

 

The use of signs goes back to early mankind. Alexander Marshack (1972; 1976) analyzed regular scores on 500,000 year-old ox ribs and interpreted them as notation of a calendar and representation of lunar cycles. Other scores may have been used for calculations. Since the early cultures, signs span such extremes as secret marks and marks of stonemasons to mathematical, logical and algebraic signs. In contrast to symbols which are mostly pictorial and self-explanatory, signs need a key in order to be decoded. But often signs and symbols are left undifferentiated or used in very different ways in different contexts.

 

 

Charles Sanders Peirce: Theory of signs

 

In 1868 the founder of pragmatism, Charles Sanders Peirce, formulated: "We have no ability to think without signs" (1967, 186). One can see his theory of signs, which he later dismounted substantially, also as model theory.

 

Already in his early papers of 1868 Charles Sanders Peirce developed the rudiments of his theory of signs (Virgil C. Aldrich 1932; Arthur Walter Burks 1949; Elisabeth Walther 1962, 1989; Douglas Greenlee 1964). In 1893 he introduced the trinity of icon, index and symbol. In 1902/03 he refined his theory. But since 1893 he published only some minor hints on his approach It did not get attention before the „collected papers“ were published in the beginning 1930s. Unfortunately Mechtild Keiner in her thesis (1978) could not rely on all microfilm records of Peirce’s manuscripts.

 

 

Max Weber: Ideal types

 

Around 1900 the German sociologist Max Weber conceived for his science, especially comparative sociology, the „Idealtypus“ or ideal type. It is an abstraction of empirical phenomena, not an ideal or a perfect thing nor a statistical average.

 

He described (1904) the construction of an ideal type in the following manner:

„An ideal type is formed by the one-sided accentuation of one or more points of view and by the synthesis of a great many diffuse, discrete, more or less present, and occasionally absent concrete individual phenomena, which are arranged according to those one-sidedly emphasized viewpoints into a unified analytical construct (Gedankenbild). In its conceptual purity, this mental construct cannot be found empirically anywhere in reality. It is a utopia.

Historical research faces the task of determinig in each individual case, the extent to which this ideal-construct approximates to or diverges from reality“ (English 1949, 90).

 

 

Ideal towns and buildings

 

There are many real (not: ideal) towns, villages and buildings following a construction principle, e .g.

·        Neunkirch (1260)

·        Pienza (1459-62)

·        Klagenfurt (after 1518)

·        Jülich (ca. 1550-1600)

·        Sabbioneta (1554-71)

·        Valletta (1566-71)

·        Palvanova (after 1593)

·        Freudenstadt (after 1599)

·        Mannheim (after 1600)

·        Glückstadt (after 1617)

·        Saarlouis (after 1680)

·        Neuf-Brisach (1699-1703)

·        Karlsruhe (after 1715; the „fan city“)

·        Arc-et-Senans (Claude-Nicolas Ledoux, 1775-1777)

·        Washington (1792-1800)

·        La Chaux-de-Fonds (after 1794)

·        New York City according to the „Commissioner’s Plan of 1811“

·        Le Locle (after 1833)

·        Parts of Paris; e. g. the Boulevards built by Baron Georges-Eugène Haussmann after 1850

 

Borobudur (Java), the biggest temple of Buddhishm  in he world, looks from the air as a mandala. Seen from the side the whole bulding resembles a gigantic stupa, 110 meters long and 43 meters high. The ten layers symbolize

1.      the daily reality, the realm of desire

2.-7.  the medium sphere, the realm of form

8.-10. the nirvana, the realm of formless, crowned by a 11 meter high stupa, the „satori“.

 

 

Bali villages mirror the cosmological total system

 

All viillages on the isle of Bali follow strong principles of construction. They mirror the cosmological total system: The axe ist he line between the sea (the realm of the bad) and the mountains (the seat of the gods).

Corresponding to the cosmic order the village is divided in three parts, each with a temple representing the meaning of the zone. Equally the three zones represent the life cicle of man from the mountains to the sea: birth, life and death.

·        In the mountain-oriented zone stands he temple of origin, consecrated to Brahma, the creator, and the forebears of origin.

·        Around the center, the crossing of the main street with a smaller street, there is located the temple of Vishnu, the preserver, the congregation hall and often a pavillon for the music and the palace of the local aristocrat.

·        Besides the village down to the sea there is the temple of death, home of the goddess of death. Nearby are the places for burning, burying, and funeral as well as the cementery.

 

 

Bibliography

model: special topics - Abstraktion/ Idealisierung - Idealstadt - Idealtypus/ ideal type (nach Max Weber, 1904)