Mueller Science - concept of model - definition of model

The Concept of Model: Definitions and Types

(If you wonder at some wording, please consult the tentative translation key)

Difficulties with the definition of model

Each definition of „model” is insufficient: It covers only a small range of the reach of use. In particular all definitions with the terms „original”, „reality” and „representation” cannot convince.

The German “Brockhaus” (1971) offers a convenient entrance (translated):

"model <ital.> pattern, ideal, reproduction or draft of things (increased, reduced or in actual size). Apart from real things models can also be mental constructions.”

We have to accept the fact that the definition of a term always requires several other terms, which mostly are also ambiguous or used differently. Thus the definition of “Brockhaus” relies on: thing, pattern, ideal, reproduction and draft, real and mental construction.

The otherwise most popular ambiguous descriptions are: function, system and structure, object, process and behavior, phenomenon, ens and entity, hypostasis and construct, power and ability, amount and size.

A concise overview of all fields of meanings in the range of “model” is given by:

12 main meanings of mould, moulding, modulation, module, model and modulus

The multiple development of the concept “modulus”

see: Nachschlagewerke für Begriffsgeschichte

Latin „modulus“ (derived from “modus”, meaning: little measure) has been absorbed at different times by the European languages (Walther von Wartburg, 1966):

first from the 11th -14th century for pattern and mold as „Modul“ und „Model“ by German, as „modle“, "molle", “mole” and „moule“ by French, as „mòdano“ by Italian and as „mould“ by English
in a second attempt in the sense of the Vitruvian measure on the way of the Italian „mòdulo“ (13th cent.) in the middle of the 16th century as „module“ by French and by English
and finally for architectural model and figure on the way of the Italian „modello“ (1355; first: model or drawing, only since 1417 solely for small-scale device) likewise in the middle of the 16th century as “modelle” and „modèle“ by French, as „Modell“ by German and as „model" and “modell” by English.

Furthermore we have to note that Latin „modulus“ (as well as e. g. „exemplar“) from 23 BC to 1750 was used by scholars (but only since 1450 for architectural models and for other small scale images of real objects).

In English „modulus“ is used until today in physics and mathematics, in German in mathematics.

Some dictionaries mention also „modulation“ and the respective verb descending from „modulus“. This happend in music in the 14^th century meaning singing or making music harmonic. Later the meaning was expanded to forming something according to due mesure and proportion.

(Walther von Wartburg in his etymologic dictionary, 1966, separates the descent from „modulation“ from the descent from „modulus“. Indeed already Vitruvius used „modulatio“.)

The respective verbs are:

German: modeln; modulieren; modellieren
English: mold or mould; modulate or modulize; modelize or model
French: mouler; moduler or moduliser; modeler
Italian: modanare; modulare; modellare.

1611 - 1717 - 1771/81 - 1828: Early definitions

In Randle Cotgraves Dictionary French-English (1611) we read:

“Modeler: To modell, forme, fashion, plot, cast in a mould.

Modelle (f.): A modell, patterne, mould, plot, forme, frame.

Modulation (f.): Modulation, harmonie, musicall proportion, pleasant tuning.

Module (m.): A modell, or module; that whereby a whole worke is measured, proportioned or squared;
also, the measure, bignesse, or quantitie of a thing;
also, a certaine measure in conduits or conveyances of water;

also, modulation, melodie, or measure in Musicke.

[Moulage (m): Grist, or a ginding;

Also, Multure, the fee, or toll thats due for grinding.]

Moule (m.): A mould (wherein a thing is cast, formed, or forged;)
also, a high shore, or strand by the sea side; or as Mole, a Peere, &c.
Bois de moule. Bellets, or logs, of a certaine size; or which have beene assized by the Mouleur.
Chandelles de moule. Candles made in moulds; (great) Christmas candles.

Moule de bois. A Wood-stacke, or pile of wood.

Moulé: m.ée. (f.) Moulded; cast, or framed in a mould.

Mouler. To mould, or cast in a mould; to frame, or forge by mould;
also, to appoint a mould for, prescribe a size unto.

Moulerie (f): A moulding; a forging by mould, a casting in a mould.

Moulle. as Moule.“

Note that the early definitions of model mostly do not mean architectural models, but abstract models. The philosopher Blaise Pascal (1657) for instance saw the model as „ouvrage d'esprit ou action morale, dont on peut s'inspirer“.

One of the earliest definitions in German is with Johann Huebner (1717, column 1078):

„Modele (inserted in the ed. 1741: ,Modell, Exemplar’), eine cörperliche Abbildung eines Dinges ins kleine (a small scale representation), oder nach dem verjüngten Maas-Stabe, sonderlich die Abbildung einer Vestung in Holtz, Gips, Thon, oder auf der Erde selbst.

Die Mahler und Bildhauer nennen alles, was sie nachzumachen (for imitating) sich vorsetzen, ein Modell, und also nennet man auf der Mahler- und Bildhauer-Academie denjenigen ein Modell, welcher sich gantz nackend vor die Schüler darstellet oder hinleget, damit man nach ihm zeichnen möge.

lnsgemein werden Modelle genannt, die von Holtz, Gips, Wachs, oder Thon gemachten kleinen Figuren (little figures intended to be reproduced in big scale) von Bildern, Häusern, Palatiis oder Machinen, nach welcher hernach das grosse soll verfertiget werden, daher an vielen Höfen, sonderlich wo grosse Schlösser erbauet werden, die so genannten Modell-Tischer und Wachs-posirer seyn, welche vorher ein cörperliches Modell nach dem auf dem Papier vorgezeichneten Aufriss, nach dem verjüngten Maass-Stabe verfertigen müssen, damit sich der Bau-Herr eine so viel bessere ideam von dem aufzurichtenden Gebäu vorstellen, und so lange es noch ins kleine ist, die Fehler so viel besser daran können corrigiret werden.

Ein Modell heist man auch, die in den Parterren oder Lust-Gärten angebrachten zierlichen Blumen-Betten-Figuren (flower beds) (1741: ‚Beeten’), bestehende entweder in schönen, und auf das Wappen alludirenden Figuren, oder künstlich geschlungenen Zügen und Gängen."

The text in the first edition of the „Encyclopaedia Britannica“ (1771) is poor. It is a simple translation of three sentences from the French „Encyclopédie“ (1765), only regarding the architectural model:

“Model, in a general sense, an original pattern, proposed for any one to copy or imitate.

This word is particularly used, in building, for an artificial pattern, made in wood, stone, plaster, or other matter, with all its parts and proportions, in order for the better conducting and executing some great work, and to give an idea of the effect it will have in large.

In all great buildings, it is much the surest way to make a model in relievo, and not to trust to a bare design or draught.”

The English „Cyclopaedia“ (1781) has picked up this wording literally, but has added substantial translations from the “Encyclopédie”, e .g. at the beginning:

“St. Paul’s church is said to be built after the model of St. Peter’s at Rome.”

And afterwards:

“There are also models for the building of ships &c. and for ordinary stair-cases, &c.

Model, in Painting and Sculpture, is anything proposed to be imitated. And

Hence in the academies, they give the term model to a naked man, disposed in several postures, to give an opportunity to the scholars to design him in various views and attitudes.

The sculptors have little models of clay or wax to assist them in their designs of others that are larger, in marble &c. and to judge of the attitude and correctness of a figure.

Statuaries likewise give the name model to certain figures of clay or wax, which are but just fashioned, to serve by way of guide for the making of larger, whether of marble or other matter.”

Wilhelm Traugott Krug (1827-29) specified in his dictionary of philosophical sciences:

„Modell ist das Muster (pattern), nach welchem man sich in irgend einer Beziehung (in wissenschaftlicher, künstlerischer oder sittlicher Hinsicht) richtet, wodurch also eine gewisse Handlungsweise (modus agendi) bestimmt ist.

Das Modelle kann demnach entweder schon gegeben sein (wie wenn jemand nach einer natürliche Gestalt oder lebenden Figur zeichnet) oder erst von dem hervorgebracht werden, der sich künftig danach richten will. Letzteres thun besonders die bildenden Künstler, um ihren Werken die höchstmögliche Vollendung zu geben; sie modellieren erst das Werk, bevor sie es ausführen.

Aber auch derjenige modelliert, welcher einen Entwurf (draft) zu einer Rede, Abhandlung, Schrift oder zu einem wissenschaftlichen Systeme macht. Denn wenn er diesen Entwurf nachher ausführt, so richtet er sich nach demselben; und ebendeswegen machte er den Entwurf.“

Again 100 years later (1934-36) the logicians Rudolf Carnap, Morris Raphael Cohen and Ernest Nagel as well as Alfred Tarski saw the model as „fulfilment” of axiomatic systems and formalized theories. A classical definition of Alfred Tarski reads: „A possible realization in which all valid sentences of a theory T are satisfied is called a model of T.”

Is all cognition „in models”?

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.

In his famous book „The Logic of modern Physics” physicist Percy W. Bridgman wrote 1927:„I believe that the model is a useful and indeed inescapable tool of thought, in that it enables us ton think about the unfamiliar in term of the familiar.“

1949 the social and political scientist Karl W. Deutsch formulated at a symposium: „Men think in term of models.“

The mathematician and philosopher Herbert Stachowiak (1973) went further. He postulates that all „cognition is cognition in models and by models“. It means that any contact wit the world, „being out – passive or active – for recognizing of somewhat”, is „relative to certain subjects, intentional selecting and centering and in temporal limitation of its relation to the original”.

In 1989 Christian Wissel maintained: „The human spirit is unable to think otherways than in models.“

Other cognitive and emotive relations

But it could be that in ordinary life there are other than cognitive relations to things, e. g. by means of

intuition

meditation

mystical absorption

empathy

affection

touch

introspection

insight

“Wesensschau”, etc.

see also: Intuition

An abundance of meanings and examples

The abundance of things called „model” is not easy to reconcile.

Everything can be a model, but it is not necessary

Everything existing or image able can be a model, but must not be a model. For example each human can be a model for other humans – but he must not. He can be a model for the artist, or he is imitated, without intention or without noticing it. And he can be a good or bad model. Think of the relation parents-child. Children learn, Albert Bandura says (1961), by models.

On the other hand each human in Jewish-Christian view is an image, i.e. the image of God. Analogous the child can be regarded as reduced image or approximate copy of father or mother.

For analogies the assortment is large: A ball or an apple can be an analogy for the universe or the earth, an onion can be an analogy for the soul (Albert Wellek 1950), a clock or a (idealized) machine can be an analogy for the state (Hobbes, Justi), for humans (La Mettrie) or for the universe (Newton-Leibniz), etc.

Disturbance by the abundance

William of Humboldt and Ludwig Wittgenstein again and again emphasized that the language lives in its use! Therefore it is of no use to complain about that there are so many other terms also used for model, and that so many things can called “model”.

The 20 volume „Oxford English Dictionary” (1933, 2. ed.1989) and the „Grande Dizionario della Lingua Italiana” (1961ff) register each approximately 40 various basic meanings (with subsections) of „ model”.

Wolfgang Brezinka (in an study “'Modelle' in den Erziehungstheorien" 1984), professor of pedagogy at the University of Konstanz, worked out 15 different meanings of model on 20 pages (see also below), but allows for pedagogy only one meaning: the teaching aid „ for the visualization or elucidation of an original” (354). In his eyes the other 14 uses are „useless”.

In 1902 the „Encyclopaedia Britannica” had a beautiful contribution of Ludwig Boltzmann on „model” (five columns), nine years later were added „artists' models” and „model-yachting”. In the 1990 edition - 29 volumes! - there is no more article on „model” neither on „modelling”. Instead we read 65 pages on „motion pictures” Also in „Collier's Encyclopedia” (20 volumes; 1993) we do not find „model” as key word.

On the other side the French „Encyclopaedia Universalis” has reprinted in a new edition (Noel Mouloud et al. 1992) twenty years old texts unchanged.

Therefore it is advisable to look as curiously and impartial as possible for the real use of „model” in science and in everyday life.

Modern Dictionaries: What do they tell us?

The „Duden for foreign words” (roughly translated in English)

Although the terms Model and Modell are registered in the “Deutsches Wörterbuch” of Jacob and Wilhelm Grimm (1885), they often were regarded as „foreign words” (e. g. by Hans Schulz 1942).

In the “Duden for foreign words” (1990) we read under the heading „Modell (das)”:

1. Pattern, ideal

2. Draft or reproduction in smaller scale (e. g. of a building)

3. Form (in wood) for the establishment of the mold

4. Article of clothing, which is unique

5. Humans or objects as ideal for a work of the forming art

6. Type, type of a product

7. Simplified representation of the function of an object or of a process, which facilitates or enables research

8. Mannequin (viz. Model 5)

9. Callgirl.

(Linguists take loving care of their objects: In the 1997 edition of this “Duden” under point 9 it was attached „veiling” in parentheses.)

Under „Model (der) und Modul (der)” we read:

1. Radius of the lower part of a antique column

2. Hollow form for the establishment of pastry or for forming butter

3. Raised printing form for printing textiles or wallpapers

4. Pattern for embroideries and knitwear

5. (das Model =) Mannequin, photo model

Under „Modul (der)” we read:

1. see Model 1-4

2. Mathematics:

a) underlying relation, underlying proportionality factor

b) divisor (natural number), in respect to which two whole numbers are congruent, i. e. by division give the same reminder

c) absolute amount of a complex number

3. Physics, technology:

a) material constant (e. g. Young's modulus)

b) measure for the calculation of the tooth size with gearwheels

Under „Modul (das)” we read:

1. interchangeable, complex part of a device or a machine, which forms a closed function unit (particularly in electro-technology)

2. a unit consisting of several components within an overall system, which can be exchanged at any time (computer science).

Under „Modulation (die)“ we read:

1. Alteration of a frequency for transmitting information on wires or wireless

2. change for one key in another (music)

3. adjustment of volume and timbre in vocal music (e. g. in singing; music).

Under “Moulage” (der – also: die) we read:

Imprint, cast, especially colored anatomic wax model (of organes).

What do English dictionaries say?

model n.

1 a representation in three dimensions of an existing person or thing or of a proposed structure, esp. on a smaller scale (often attrib.: a model train).

2 a simplified (often mathematical) description of a system etc., to assist calculations and predictions.

3 a figure in clay, wax, etc., to be reproduced in another material.

4 a particular design or style of a structure or commodity, esp. of a car.

5 a (foll. by of) an exemplary person or thing (a model of self-discipline).

b (attrib.) ideal, exemplary (a model student).

6 (often foll. by for) a Person or thing used, or for use, as an example to copy or imitate (an unlikely model for emulation).

7 a a Person employed to pose for an artist, photographer, etc.

b a Person employed to display clothes etc. by wearing them.

8 a garment etc. by a well-known designer, or a copy of this.

model v. (modelled, modelling; US modeled, modeling)

1 tr. a fashion or shape (a figure) in clay, wax, etc.

b (foll. by after, on, etc.) form (a thing in Imitation of).

2 a intr. act or pose as a model.

b tr. (of a Person acting as a model) display (a garment).

3 tr. devise a (usu. mathematical) model of (a phenomenon, system, etc.).

4 tr. Art cause to appear three-dimensional.

(H. W. Fowler, F. G. Fowler (Ed.): The Concise Oxford Dictionary of Current English. Ninth Ed. 1995)

model n.

copy of an object a copy of an object, especially one made on a smaller scale than the original (often used before a noun)
particular version of manufactured article a particular version of a manufactured article (had traded in her car for the latest model)
MANUF something copied something that is copied or used as the basis for a related idea, process, or system
FASHION somebody paid to wear clothes somebody who is paid to wear clothes and demonstrate merchandise as a profession, e. g. in fashion shows and photographs for magazines and catalogues
simplified version a simplified version of something complex used, e. g. to analyse and solve problems or make predictions (a financial model)
perfect example an excellent example that deserves to be imitated
PAINTING artist’s subject somebody who poses for a painter, sculptor, photographer, or other artist
SCULPTURE small version of sculpture a small version of a sculpture, from which a finished work is copied [this entry was later dropped]
ZOOL animal copied by another animal an animal species repellent to predators which another animal mimics for protection
LOGIC interpretation an interpretation of a theory arrived at by assigning referents in such a way as to make the theory true
FASHION exclusive garment the first sewn example of a couturier's or clothing manufacturer's design, from which a new line of garments is produced.

model v. (-els, -elling, -elled)

vti. work as fashion model to work as a fashion model, wearing clothes, make-up, and other items in order to display them to others
vi. be an artists model to sit as a modal for somebody such as a painter or photographer
vt. base something on something else to base something, especially somebody's appearance or behaviour, on somebody or something else (She modelled herself on her older sister)
vt. shape something to make something by shaping a substance or material, e. g. clay or wood
vti. make simplified version of process to make a model of a process or system as a way of analysing and solving problems or making predictions [this entry was later dropped].

modelling n. (US: modeling)

fashion model’s work the work of a fashion model
making models the activity or hobby of making models
PSYCHOL demonstration of behaviour the demonstration of a way of behaving to somebody, especially a child, in order for that behaviour to be imitated

modulation n.

adjustment of sound adjustment of the tone, pitch, or volume of sound, or of something that produces sound, for example, a musical instrument or the human voice
slight alteration slight alteration that makes something less strong, forceful, or severe
PHYSICS process of modulating carrier wave the process of changing the amplitude or frequency of a wave, used in radio broadcasting to superimpose a sound signal on a continuously transmitted carrier wave

module n.

self-contained interchangeable unit: an independent unit that can be combined with others and easily rearranged, replaced, or interchanged to form different structures or systems
education short course of study: a short course of study that forms part of a larger academic course or training program, e.g. any of the elements that form part of a degree program
aerospace part of space vehicle: one of the self-contained units or craft that make up a space vehicle
Architecture unit of measurement: a unit of measurement or a standard, used especially in measuring architectural elements

modulus n.

physics coefficient: a coefficient expressing the degree to which a substance exhibits a particular property
mathematics division number: a number by which two other numbers can be divided so that both give the same remainder
mathematics absolute value: the absolute value of a complex number
mathematics logarithm factor: the factor by which a logarithm of one base must be multiplied to become the logarithm of another base

moulage n.

making of a cast of something as evidence the process of making of a mould or cast of something, e. g. a footprint, in the course of a criminal investigation
mould or cast a mould or cast made in the course of a criminal investigation

mould n. (US: mold)

container for making a shape a container that gives a shape to a molten or liquid substance poured into it to harden
frame a frame on which something is formed or built
object made in a mould an object that was formed using a mould
distinctive type a particular type that has a distinctive character or nature (a leader in the heroic mould)
set of assumptions a fixed pattern or framework of assumptions, especially when regarded as restricting (negotiators who break out of the traditional diplomatic mould)
= moulding n. 1
[later added] shape of mould the shape or form of a mould
[later added:] general shape the general shape or form of something

mould v. (moulds, moulding, moulded)

vt. make something in mould to shape or form something in a mould
vt. give something shape to shape or give form to something
vt. influence somebody’s character to guide or influence the growth or development of somebody or something
vti. fit the contour of something [later: make something cling] to fit closely by following the contours or acquiring the shape of something
vt. METALL make a mould from something to make a material into a mould to be used in casting metal
vt. ARCHIT put moulding on something to decorate something with a moulding

moulding n.

ARCHIT, WOODWORK Decorative strip a strip of wood or some other material that is used to decorate or finish a surface of a wall or apiece of furniture
Sth made in a mould sth that is produced using a mould

(Encarta World English Dictionary. Bloomsbury 1999)

For „modulus“ the “Oxford English Dictionary” has in addition:

“A constant multiplier, coefficient, or parameter involved in a given function, transformation, etc.”, with three specifications.

For „modulo“, prep. (a and b) and adj. (attrib.; B.), the „Oxford English Dictionary“ has:

a. With respect to a modulus of (a specified value).

b. In extended use. (a) With respect to an equivalence defined by (some feature), disregarding differences indicated by (some unimportant feature); (b) taking into account (a particular consideration, aspect, assumption, etc.).

B. Modular; involving calculations with respect to a modulus.

moulding n. (US molding)

1.a an ornamentally shaped outline as an architectural feature, esp. In a corniche.

1.b a strip of material in wood or stone, etc. for use as moulding.

2. similar material in wood or plastic etc. used for other decorative purposes, e. g. in picture-framing.

(Concise Oxford Dictionary, 1990)

module n.

A standard or unit of measurement.
Archit. A measure of proportion among the parts of a classical order, the size of the diameter or semidiameter of the base of a column shaft usually being taken as a unit.
A standard structural component repeatedly used, as in a building, computer, etc.; cubic modules used in the design of a table
A preassembled, self-contained unit, often a component or subassembly of a larger structure: a housing module,; a lunar module
Obs. A mere image.

modulate v. t.

1. To vary the tone, inflection, or pitch of

2. To regulate or adjust; temper, soften

3. Music To change or cause to change to a different key

4. To intonate of sing

5. Electronics To alter the frequency or amplitude of (a radio carrier wave)

6. v. i. Electronics To alter the frequency or amplitude of a carrier wave

7. v. i. Music To change from one key to another by using a transitional chord common to both.

modulus n. pl. –li.

Physics A number, coefficient, or quantity that measures a force, function, or effect: modulus of elasticity
Math. The logarithm of e to the base 10.

mold v. t.

1 To work into a particular shape or form; model; shape.

2 To shape or cast in or as in a mold; make an a mold.

3 In founding, to form a mold of or from.

4 To ornament with molding.

mold n.

1 A matrix for shaping anything in a fluid or plastic condition: distinguished from cast.

2 Hence, that after which something else is patterned, or the thing that is molded.

3 Form; nature; also, kind; character.

4 The physical form; shape: now applied to the human form.

5 A molding, or number of moldings. See synonyms under MODEL.

(Funk & Wagnalls: New International Dictionary of the English Language. Comprehensive Edition. World Publishers 1987)

module n.

1. One of the inherent cognitive or perceptual powers of the mind.

2. Detachable compartment of a spacecraft.

3. Computer circuit consisting of an assembly of electronic components (as of computer hardware).

4. A self-contained component (unit or item) that is used in combination with other components.

(Webster’s Online Dictionary)

Comprehensive classifications of models

Each author has another classification of models.

Models in architecture, painting and sculpture as well as in casting

In the famous French „Encyclopédie“ (1765, Vol. 10) we find the following kinds of models:

„tout ce qu’on regarde comme original, & dont on se propose d’exécuter la copie. Ce mot se prend au simple & au figuré, au physique & au moral“. Example: une femme, “modèle précieux pour un peintre“
“en Architecture: original qu’on propose pour l’imiter, ou pour le copier.. Modèle est en particulier en usage dans les bâtimens, & il signifie un patron artificiel…“.
“en Peinture tout ce que les Dessinateurs, les Peintres, les Sculpteurs se proposent d’imiter
                           un homme, qu’on met tout nud à l’académie
                           des figures que les Sculpteurs modelent d’après le modèle à l’académie
                           des figures de terre ou d’argile, de plâtre, de cire, qu’ils embauchent pour leur servir de dessein, & en exécuter de plus grandes, soit de marbre soit d’une autre matière“
“dans les ouvrages de fonte, le modèle est en quelque façon l’ouvrage même, dont le métal prend la forme; la matière seule en fait la différence“
“une couche de ciment & de terre, de la forme de la cloche qu’on veut fondre, & de la même épaisseur que la cloche doit avoir“

Models for instruction and to illustrate theories in physics

The physicist Ludwig Boltzmann differentiated 1892 (90f, 97) three types of models (translated):

1. Models for instruction in mathematics and physics like geometrical „forms in gypsum, models with fixed and mobile cords, bars and links of all kinds ... mechanical models, visual wave planes, thermodynamic planes in gypsum, wave machines of all kinds, apparatus to make visible the laws of the refraction of light and other laws of nature”;

2. calculating machines, which „take over from humans the real operations of calculating, beginning with the for basic operations and ending wit integrating”;

3. the models of theoretical physics: “For the time being all these mechanical models existed only in mind. They were dynamical illustrations in fantasy and could not be practically realised. Because of their great relevance they provoked realisation at least of their basic types.“

Also: draft and ideal

In his article in the “Encyclopaedia Britannica” (1902) Boltzmann took into consideration also non-scientific models:

1. „a tangible representation, whether the size be equal, or greater, or smaller, of an object which is more either into actual existence, or has to be constructed in fact or in thought”;

2. “a thing, whether actually existing or only mentally conceived of, whose properties are to be copied”

a) in foundries the object of which a cast is to be taken

b) the animate figures which serve the painter as types. The sculptor first makes a model oft the object he wishes to chisel in some plastic material such as wax

3. "In anatomy and physiology, models are specially employed as aids in teaching and study, and the method of moulage or chromoplastic yields excellent impressions of living organisms.“

Material and formal models

Arturo Rosenblueth and Norbert Wiener (1945) differentiated between material and formal or intellectual models. Karl Wolfgang Deutsch (1951) and many others followed them.

Also the Russian philosopher Viktor A. Stoff (in “Modellierung und Philosophie” 1969, 36-50, 315-323) differentiates material from mental models (fig. 1) whereby he sees drafts of houses and towns, location plans and spatial models of molecules as material models, but geographical maps and chemical structural formulas as mental models. The various degrees of illustrativeness are to be discussed.

Mechanical or Mechanistic, organismic and other models

Following the physicist Joseph Larmor physicist Georg William de Tunzelmann distingushed in 1910 three „models“ of the universe:

the mechanical,
the molecular and
the mental.

In 1927 Joshua Craven Gregory separated „mechanical“ from „animate“ models. It was only in the 17th century when – by René Descartes and Robert Boyle - mechanical models replaced the animistic models of „primitive men“ to Van Helmont.

In 1949 Karl Wolfgang Deutsch introduced the distinction of mechanistic (since Newton and Hobbes) and organismic (since Rousseau and Burke) models.

In 1951 Herman Meyer differentiated scientific models in mechanistic ones (e. g. Newton’s „Principia”), arithmetic (e. g. game of dice) and axiomatic ones (e. g. for the kinetic theory of gases).

Iconic, analogue and symbolic models

In their “Introduction to Operations Research” (1957, 155ff) C. West Churchman, Russell L. Ackoff and E. Leonard Arnoff differentiate between three kind of scientific models:

iconic
analogue
symbolic.

Of the latter they describe in individual chapters the following models:

Inventory
Allocation
Waiting-time
Replacement
Competitive.

The initial breakdown into iconic, analogue and symbolic models has been followed by many other scientists and teachers. Roger Minshull (1975, 33-59) is very critical to this classification.

Later the categories of the symbolic models have been modified with the assistance of Patrick Rivett and Maurice W. Sasieni (see Patrick Rivett 1972, 34-50).

Scale, analogue, mathematical and theoretical models

In a lecture in December 1958 the linguist Max Black (1962, 219-243) differentiated „scale models” from „analogue models”. The first are mostly reduced images, eventually chemical or social experiments. Black relates them with Charles Sander Peirce’s “icon”.

Analogue models consist of another material than the original; they describe only its structure. They are based on the principle of “isomorphism“. For Black „mathematical models” are nothing else than mathematical „treatments” without explaining power.

For „theoretical models” Black refers to Maxwell and Thomson and states, they are only verbal descriptions, “half understood metaphors“. They lead to mystifications and confusion of concepts. Black considers the use of such models as a detour. Finally he introduces the term “archetype” for „root metaphor”, „basic analogy” or „ultimate frames of reference”.

Malcolm Baker (2004, 32) differentiates four kinds of 3D-models:

Proposals for a large-scale work or a machine still to be made
Reproducing on a small scale something that already exists
Making visible a structural or mechanical principle not evident in the completed large-scale work
Representing what is only described in texts.

Symbolic and iconic models

At the Utrecht colloquium in January 1960 Gerhard Frey introduced the distinction of symbolic (e. g. a set of equations), primarily iconic (e. g. a wiring diagram) and secondarily iconic models (e. g. the theories of the classical mechanics). A symbolic model, to which there is a completely copying iconic model, is an iconic-symbolic model. In contrast the Heisenberg matrix mechanics and the Schroedinger wave mechanics are not-iconic-symbolic models.

In 1974 Danielle and George Arthur Mihram tried a new “taxonomy of models”:

physical (replicas, quasi-replicas and analogue)
symbolic (literal) (descriptions, simulations and formalizations)
hybrid (material-symbolic) (analogue-simular, replica-descriptive, replica-simular and others).

Models of data

„Models of data” were introduced by Patrick Suppes in 1960 at the international congress in Stanford.

Abraham Kaplan’s methodology for the behavioral sciences

In his methodology for the behavioral sciences the philosopher Abraham Kaplan (1964, 267-268, 273-275) distinguishes five kinds of models:

„(1) any theory more strictly formulated than is characteristic of the literary, academic, or eristic cognitive style, one presented with some degree of mathematical exactness and logical rigor;

(2) a semantical model, presenting a conceptual analogue to some subject-matter;

(3) a physical model, a nonlinguistic system analogous to some other being studied;

(4) a formal model, a model of a theory, which presents the tatter purely as a structure of uninterpreted symbols;

(5) an interpretive model, providing an interpretation for a formal theory.“

Elsewhere (1964, 258) Kaplan sees models as „something eminently worthy of imitation, an exemplar or ideal“ and he thinks that this sense of the term is by no means irrelevant to its use in contemporary methodology. Furthermore (1964, 332-346) he distinugishes two models of explanation: pattern model and deductive model.

Kaplan sees several dangers or failings of the use of models as

Overemphasis on symbols
Overemphasis on form
Oversimplification (what was neglected is something important for the purposes of that very model)
Overemphasis on exactness or rigor
Map reading (not all its features correspond to some characteristic of its subject-matter)
Pictorial realism (forgetting that the similarity exists only in an given perspective).

Scientific models

Around 1965 geographer Richard John Chorley distinguished

Mathematical models

deterministic
stochastic

experimental models

scale models
analog models

natural models

those which compare the phenomenon being studied with something better known
those which compare the phenomenon being studied with something which is more accessible for study or experiment.

The „Handwörterbuch der Organisation“ (1969) differentiates seven very wide used meanings of “model” in science (translated):

· Theories in general or subclasses of theories

· Potential theories (un-interpreted calculuses)

· not-linguistic descriptions, i. e. simplified real reproductions – iconic or material models -, e. g. planetarium, hydraulic model of the economy, globe, map

· Terms and theories, which refer to so-called “pure cases”, e. g. ideal gas, perfect market, perpetuum mobile, „homo oeconomicus”

· Realization (interpretation) of an axiom system (calculus); considering logical-mathematical things one speaks of a formal, considering material (empirical) things of a material model of the axiom system

· A class of structures (sets of equations), which differ only in the numerical specification

· Theoretical models in the range of the theories of isomorphism and homomorphism.

In his extensive review of models in general and of models in geography Roger Minshull (1975, 24-25) lists 36 different definitions of model, rough divided in

Hypothesis and theory
Description, representation, abstraction
Something on which to work
Framework or ideal
Synthesis, analogue
Psychological crutch.

Graphic, technical and semantic models

In 1957 Herbert Stachowiak differentiated causal, conditional and structural explanatory models. Later he gave this up and finally found the concise presentation of models (fig. 2) in his comprehensive „Allgemeine Modelltheorie” (1973), where he differentiates between three main groups (translated):

graphic models (predominantly two-dimensional), e. g. photos and movies, drawings and paintings, maps and medical atlases, diagrams

technical models (predominantly three-dimensional) like relief, historic record, planetarium, auto test track, flight simulator, prostheses and computer models, in addition, animal experiments, polls, group dynamics and business games

semantic models (predominant in the head), in particular perception systems (= internal models of the external world) and cogitative models, e. g. the public prosecutors plea, Hoelderlin’s „Hyperion”, myths, scientific theories and forecasts. External sign-models are colloquial language, writing and Braille - all models of other models and from each other.

Descriptive and non-descriptive models

Sociologist Raymond Boudon differentiated 1970:

A) descriptive models, which classify, order or measure, e. g. in psychometrics or dimensional analysis

B) non-descriptive models, namely:

a) theoretical, which analyse a term or a set of axioms to explain a given reality

general, e. g. game theory, theories of economic cycles, formal grammar theory
specific, e. g. in group dynamics

b) inductive, which explain observed reality; they are verifiable theories; they are either more predictive or more explicative and are used

in experimental situations (e. g. mathematical learning theory)
by the observing sciences (e. g. econometry and opinion surveys in sociology)
to analyze the relation between the elements of a system („structural“ models).

Systematic grid

In the „Handwörterbuch der Betriebswirtschaft“ (1975) Richard Koehler uses a grid to differentiate 16 kinds of models, e. g.:

· Purely mental analogy considerations

· Subject-internal interpretations of a written calculus

· Representation of the formal structure of normative ways of thinking

· Un-interpreted calculuses

· Linguistic representation of ideal assumptions

· Semantic interpretation of an axiom system

· Semantically interpreted systems of statements

· Iconic models, as reduced representations, sculptures, circuits.

In the eyes of the architect and city planner

In his thesis (1976) the architect Juergen Oestereich formed for a special field, namely town planning, four classes of models:

· Substantial models (topographic images)

· Conditional models (e. g. calculuses of use, traffic models, models of local centers)

· System models (simulation models, e. g. the population distribution model by Ira S. Lowry, 1964, or „Urban Dynamics” of Jay W. Forrester 1971)

· Reflection models (e. g. models of the system action or collective acting).

And he writes: „If we regard the function of models, namely to organize and keep available knowledge and to make binding a selection of it for planning, we can judge also false use. Such is the case, if substantial models, which stem themselves from norms, are to be justifying non-normative acting - or in reverse, if we understand conditional system or reflection models as founding norms.“

Models in pedagogical texts

Wolfgang Brezinka (1984, 836-839, 352) differentiates the following general kinds of models:

Three-dimensional image of a material object
original
archetype, ideal image
means to visualize a theory
means to construct, develop and apply a theory
(isomorph or structural similar) theory
not-linguisitc entities, in which is realized a linguistic given axiomatic-deductive system of clauses
formalized representation of an empirical theory

Especially in pedagogical texts Brezinka (1984, 340-352) finds the following 15 kinds of models, which he describes in varying detail:

teaching aids (to visualize or elucidate an original): reproduction models and symbol models
object of imitation: „example“, „model person“
prototype: „experiment school“ and „example school“
plan: „lessons model“ („period image model“ and working aids) and „reform model“
experiment
category: conceptions or approaches and forms
guideline
example
world in miniature
mental image
paradigma: „point of view“
theory: scientific as well as practical theories
empirical theory, insofar she is a pattern for an empirical theory of another field
mathematical theory, insofar she is a pattern for proposing hypotheses of an empirical theory
partial rough draft of a theory: frame of refenence and hypothesis.

Various models

Hermann Fertig (1977, 27-28) distinguishes four kinds of models in physics:

Isomorph models (physical systems)
Abstract models (equation schemata)
Approximative models (physical theories)
Formal models (physical theories fulfilling as given system of axioms).

Further Fertig collects about two dozens other kinds of general models (28-35), discusses the relation of theories and theoretical models (35-42) and deals with „mathematical model systems“ (42-57) and „models for measurement“ (95-126).

With respect to empirical theories Wolfgang Balzer (1982, 142) distinguishes two kinds of models:

„Potential models are structures resulting from the condensation of various concepts relevant to the theory.“
„Models on the other hand are structures in which also textual axioms are valid, i. e. laws, in which the various concepts are related.“

„If we ‚forget’ the textual axioms of a model then the components of the model make up a potenial model. On the other hand if we have a potential model, whose components fulfil in addition the textual axioms, then this potential model is already a model.“

In addition Balzer knows a lot of other models, e. g. theoretical (8), partial (44, 173, 283) and real (152, 193), mathematical (248) and non-mathematical (250), measure models (114, 282, 312) and space-time models (165).

Note the direction of model building. It is not the question if a theory is a model - and if yes of what it is a model - but otherwise: observations are interpreted as models according to a theory already there.

In 1992 defined K. Preston White, Jr. in the “McGraw-Hill Encyclopedia of Science and Technology” (310): “A model is one entity used to represent some other entity for some well-defined purpose. Examples of models include:

(1) An idea (mental model), such as the internalized model of a person's relationships with the environment, used to guide behavior.

(2) A picture or drawing (iconic model), such as a map used to record geological data, or a solids model used to design a machine component.

(3) A verbal or written description (linguistic model), such as the protocol for a biological experiment or the transcript of a medical operation, used to guide and improve procedures.

(4) A physical object (scale model, analog model, or prototype), such as a model airfoil used in the wind-tunnel testing of a new aircraft design, or an electronic circuit used to simulate the neural activity of the brain.

(5) A system of equations and logical expressions (mathematical model or computer simulation), such as the mass- and energy-balance equations that predict the end products of a chemical reaction, or a computer program that simulates the flight of a space vehicle.”

In the “Wiley Encyclopedia of Electrical and Electronics Engineering” (1994) K. Preston White distinguishes the same five kinds of models, but in other words.

With great effort economist Dietrich Zschocke (1995, 221-226) lists 65 definitions of model mostly in the realm of representation. He studies them thoroughly and finds out (226-261) that mathematics and real sciences differ strongly in the use of „model“. In mathematics model is a interpretation of a given system of axioms, in the real sciences model is an abstraction or generalization of an archetype (object, thought, bleary idea, non transparent problem).

For economics Zschocke (1995, 269-306) differentiates:

Real and ideal models
Formal (= syntactic), semantic and mathematical (numeric, parametric or not-numeric) models
State and decision models.

In his voluminous work "Informationstechnologie und Gesellschaft" (1993, 184) William Steinmüller differentiates three kinds of model:

"static and dynamic models: A static model for instance is a structural drawing (a state is represented); dynamic models can be of two kinds:
(a) the original changes (not the timetable, but the trains); the timetable dynamically represents the temporal/ local movements of the trains;
(b) the model is dynamic, moves (e. g. the dynamic systems of mathematics);
material/ external and idealistic/ internal:
(a) material or external models (dummies for crashtests, wooden car bodies for the wind tunnel, atom models for physics instruction, model railwy) and
(b) idealistic (e. g. mathematical formulas, texts in books, CAD diagram in computers): Only idealistic models are information or are processed by information systems. It is better to call them
(c) internal models, because it is only the system internal interpretation that differentiates them from material models;
hierarchical models of models of... (model subject has model over addressee1, which has model over addressee2 ...)".

Ronald N. Giere (1999) differentiates in an „unified interpretation of the nature and role of models in science” the following kinds of models:

· Instantial models and analogy

· Representational models

· Other material models

· Abstract models

· Hypotheses

· Theoretical models.

Oliver Thomas (2005, 29-33) lists 35 definitions or descriptions of „model“ in the range of business informatics of the years 1990-2005 and adds one of 1974 and 1981. He differentiates by means of careful analyses of the literature and based on Franz Lehner (1995) four concepts of „model“:

· general (on the basis of Herbert Stachowiak, 1973)

· axiomatic or mathematic (after Alfred Tarski, 1935, 1944, 1954)

· representation-oriented or representation-theoretic (after Erich Kosiol, 1961; Adolf Angermann, 1963; Werner Kern, 1964)

· construction-oriented or constructivist (Hans-Georg Knapp, 1978; Michael Gaitanides, 1979; Wolf-Rüdiger Bretzke, 1980; Reinhard Schütte, 1998).

Models in anthropology

Anthropologist Bradd Shore (1996, 44-71, 311-373) proposes an ethnographic conception of mind and distinguishes two kinds of models:

public artifacts „in the world“ (instituted models)
cognitive constructs „in the mind“ (mental models)
a) personal, e .g. mnemonic strategies
b) conventional, e. g. handshakes.

Especially for anthropology Mariella Combi (2004, 168-172) describes some prevailing models:

1) Patterns of culture (Ruth Benedict)

2) Basic or systemic models (Alfred L. Kroeber)

3) Conscious and unconsicous, mechanical and statistical models (Claude Lévi-Strauss)

4) Segmentary models (Edward Evan Evans-Pritchard)

5) Explicit and implicit models (Edmund R. Leach)

6) Operational, representational and explanatory models (Peter Caws)

„All the above models aim to solve the same fundamental problem about the relationship existing between reality, representation and imaginary.“ Combi herself proposes „a sensory model“.

Simulation and constructs

Eric Winsberg (1999, 279-284) distinguishes five kinds of models with respect to computer simulation:

Mechanical
Dynamical
Computational
Creative ad hoc, and
The model of phenomena.

Giovanni Boniolo (2004, 73-84) gives a taxonomy of models as fictive constructs:

. Principle models („they offer only a simulacrum account of the hypothetical representation“)

. Focusing models („deductively connected to a theory“)

. Replacing models („structured as a theory“)

. Phenomenological models („constructed to save the phenomena“)

. Object models („they enable the theories ... to have the consequences empirically checkable“).

Boniolo differentiates them from „mathematical schemes“ which he sees not as models. Another difference is between mathematised representations and mathematised models.

Some functions of models

Marxist philosophers

Georg Klaus (1969), philosopher in Eastern Germany, differentiated, among others, three main groups of models (translated):

Structural model: „Model, whose analogy with the original consists in the correspondence of characteristics of the structure and which imitates the thereby founded function and certain behaviors”
Function model: „Model, whose analogy refers to the function of the original”
Behavioral model: „Model, whose analogy to the original refers to the equality of behavior, but not necessarily to the equality of structure, function or substrate (i. e. the material and energetic state).“

Furthermore Georg Klaus sees four groups of purposes of the use of models:

Models serve

the substitution of function (or: expansion, reinforcement)
the gaining (or improvement) of knowledge and afterwards
the imparting of knowledge or
the control of behavior (from training over project engineering up to the generation of the original itself).

In the „Wörterbuch der Philosophie“ (Leipzig 1969, Reinbek 1972, 731-732), edited by Georg Klaus and Manfred Buhr, eight model functions are differentiated by Klaus-dieter Wüstneck:

Cognition (enzyme models, structured model of the protein, cybernetic turtle, animal experiments in medicine – also: atom model)
Explaining and demonstration (Maxwell’s model of the electric current, educational displays, case studies as training situation)
Indication (visual strain models, heart simulator)
Variation and optimisation (simulated network of energy distribution, operations research models)
Verification (experimental model)
Project engineering or construction (technical construction drawings, project engineering models for plants or towns)
Control (stored project models, e. g. models in the copy milling lathe, learning models, general staff map with information on positions, sand box model)
Substitution (artificial limbs or organs of humans, heart lung machine).

In his comprehensive book „Modellierung und Philosophie” (1969) Viktor A. Stoff dedicates - with much Marxist theory and literature – a whole chapter to each function of models

· as means of experimental research

· as specific kind of reflection

· as abstraction of a special kind

· as means for interpretation and scientific explanation

· as means for thought experiments.

Heuristic, prognostic and various other functions of models

In his classic „The Nerves of Government” (1953; German 1969, 44) Karl Wolfgang Deutsch differentiates four functions of models:

· organizing

· heuristic

· forecast

· measuring.

Walter Popp (1970) differentiates similarly six functions of models:

heuristic
prognostic
instrumental
technological
innovative
critical of ideologies

Christian Salzmann (1972/1974) and Wilfried Buddensiek, Franz Josef Kaiser, Hans Kaminski (1980) broadened these functions by:

structuring, description, explication
make up theories
training/ imparting
stimulation or impulse
regulation/ control
substitution
valuation
inspection and evaluation

Other functions

Roy Lachman (1960) dtisinguishes four functions of models:

Models Providing Modes of Representation
Models Functioning as Rules of Inference
Interpretational Function of Models
Models Providing Pictorial Visualization.

Afterwards he describes the „kinetic theory of gases“ as well the „statistical learning theory“ and adds some remarks on analogy.

S. E. Elmaghraby (1968) recognized at least five legitimate and common uses of models:

1 an aid to thought

2 an aid to communication

3 purposes of training and instruction

4 a tool of prediction

5 an aid to experimentation.

For scientific models Stachowiak (1973) differentiates:

Demonstrating models (didactical models) for illustrating relations
Experimental models for finding (heuristic models) and examination of hypotheses
Theoretical models for the imparting of knowledge
Operational models of possible “external ranges of goals” to make available aids for decision and planning.

Geographer Roger Minshull (1975, 60-106) distinguishes 9 uses of models:

1. in research

device to represent systems
stage in theory construction
device to separate laws from generalizations
simulation device
experimental medium
tool and procedure

2 in teaching and exposition (aid demonstration)

simplification
compendium
norm (ideal) and system of reference.

Stephan Hartmann (1999) sees six functions of models in contemporary physics:

a) Apply a theory

b) Test a theory

c) Develop a theory

d) Replace a theory

e) Explore the features of a theory

f) Gain understanding.

With Daniela Bailer-Jones Stephan Hartmann differentiates in the „Enzyklopädie der Philosophie” (1999, 856) the following functions of models in science. They (translated):

· allow the use and test of theories (Bunge 1973; Redhead 1980)

· are a tool for construct theories (Hartmann 1995)

· support the understanding of abstract theories and formalisms (Duhem 1906)

· mediate between a theory and the world (Morrison, Morgan 1998)

· serve as pragmatic substitute for theories (Hartmann 1995)

· allow the description and processing of data (Suppes 1962)

· are a module of computer simulations (Hartmann 1996)

· help to establish a causal relation between events (McMullin 1984; Cartwright 1983 and 1989)

· allow the understanding of a concrete object or system (Harré 1970; Hartmann 1999)

· are components of scientific explanations (Achinstein 1968; Cartwright 1983)

· are used as educational aids in teaching

· help constructing and evaluating of experiments (Hartmann 1996).

Further classifications of models

An odd differentiation stems fro the Bernese graphologist Barbara Saegesser (1975). If models of the natural sciences can refer to visible and perceptible objects (as in classical physics), then we have restriction models, if the objects are too „small” and can only be supposed, as in atomic physics, then we have fiction models.

Inspired of Klaus and Stachowiak we can also make classifications of models according to the grid „type of existence / type of emergence” (Roland Mueller 1977; fig. 3) or „type of existence / type of function” (Roland Mueller 1988; fig. 4). The quadruple occurrence of the architectural model suggests that the function of the same object can have a great variety. In addition the kind of existence can vary: I can mere imagine an architectural model, I can describe it in words, I can draw it, paint it or photograph it.

Characteristics of scientific models

The Belgian philosopher Robert Franck (2002, 5-8) gives an overview on ten characteristics of scientific models:

Models provide a simplified representation of the phenomenon.
Scientific models are testable.
It is the model itself tat, in the scientific approach, becomes the object of the study.
Scientific models may represent that which is essential to the object.
A scientific model is conceptual.
A scientific model allows the possibility of measurement and calculation.
A scientific model allows explanation of the phenomenon.
A model is a fictive representation of reality.
Scientific models represent systems.
A model is isomorphic to the system that it represents.

12 basic meanings of Modell, Model, Modul

Outlining a very general and nevertheless comprehensive summary of the meanings of “Modell”, “Model” and “Modul” we can use fig. 5. It does not show real relations, it is simply a sketch and is based on the four most important functions of models, i. e. image, ideal, draft and substitute.

We can differentiate twelve basic meanings:

First:

8 relational kind of models, thus models of things, model for a thing, models for persons, e. g. proposals on the organization or reform of education (in Germany: Honneffer model, comprehensive school model), patterns for dresses, the miniature reproduction of a crocodile locomotive, the draft of a new shopping center etc.

Second:

4 correlative kinds of models. That sounds similar as “relational”. The difference is: Either we have an unity which can stand alone (e. g. a module for operations in space, a model dress, a stamp or mold) or we have a lot of accurately the same items, mostly mass-produced (“Prêt à porter” dresses, shoes, furniture, devices and apparatus). Of the legendary Ford „Model T” (1908-27) there were built 15 million pieces.

In German baking tins, printing forms, stamps and molds are partly named “Model”, partly “Modell” (in Switzerland also „Foermli”, i. e. little forms). The embossed or formed things however are seldom called “Modell” (rare cases are: artistic castings) or “Model” (e. g. in Switzerland “model bacon” and “es Moedeli Anke”, a piece of butter in form of on an ingot).

All classifications and allocations are discussable

All classifications and allocations are arbitrary. For example one could separate the “prevailing opinion” (herrschende Meinung; “belief system”) from the numerous alternatives as understanding, interpretation, instruction. Also one could separate three-dimensional illustration and interpretation of an axiom system. And the sample, the specimen („Muesterchen”), the textile pattern and the guinea pig could each be a group for itself.

On the other hand one could combine „formulated ideas” (formulierte Ideen) and „draft”, perhaps even “ideal”.

Thirdly one can add contours and templates to the forms for stamping, minting, embossing. In turn stamps and seals can be added to the ideals.

Finally it can be asked whether for instance notations of ballet steps or sketches of military operations are really „schematic” reproductions and not attempts to mere visualize movements and arrangements.

Basic linguistic observations and basic questions

Basic observations are: An original can be a model (e. g. a nude), but also a copy is called model (Ford's „Model T”). The prototype is mostly a draft and therefore of experimental character, on the other hand it is an exemplary ideal. A pattern is not a sample is not a „Muesterli”. Type is so ambiguous that type and kind offer only a small cut-out from the meaning spectrum.

Sometimes the form and the formed have the same designation, e g. stamp, cast, contour, seal, type.

Today the “model joiner” does no longer make tables, but „ wooden or synthetic models for the mold for metallic machines and implements or parts of them”. The “model locksmith” shapes such forming devices not only in metal, but also from gypsum.

Basic questions of content are:

Are hypotheses or data models or even theories either “images of the reality” or better “drafts” (fictions?), perhaps “formulated ideas” or “imaginations”, eventually “ideals”, in the sense of “pattern”?

Or: What is a Madonna of Raffael: image of a woman, illustration of an idea, draft of an ideal, ideal itself, standard for other women or other painters?

A play can be nearly everything: imagination of the dramaturge, reproduction of the „Comédie humaine”, moral institute (thus ideal), draft of a better world, illustration of a lifelong lie, analogy to the everyday behavior of humans ("we all play theatre”).

Models of models

And more: There are models of models of models, etc.

Let us take the figure of God as the creator. We have the subsequent models:

The human behavior serves as model for the divine behavior. Hundred years of psychoanalysis say, men projects himself, with its desired strengths and with all his weaknesses in the stars.

God establishes the world in exemplary way. Man takes an image oft it. From this he makes an artistic representation. Of it we can make a photograph, which is reproduced in a book, after which I made a transparency for the overhead projector. If we shoot a video from this lecture, that would be a further image (model), etc.

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Christian Salzmann: Gedanken zur Bedeutung des Modellbegriffs in Unterrichtsforschung und Unterrichtsplanung. Pädagogische Rundschau 26, 1972, 468-485; verändert nachgedruckt in Leo Roth, Gerhardt Petrat (ed.): Unterrichtsanalysen in der Diskussion. Hannover: Schroedel 1974, 171-205.

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