Keywords: models, analogies, idealization, truth, molecular spring-and-ball models.
Chemistry, therefore, shows how far analogical thinking is entrenched
in science. Analogy in general is not a simple notion; in this case, however,
its meaning is clear: the geometrical and mechanical properties of the
macroscopic model are assumed to match closely (though not completely)
the properties of the corresponding real molecule.
The SB model of chemistry thus provides a concrete example of thinking
by analogy in science, and hence a good starting point in a general study
In this perspective, a reflection on models in science is a chapter
of any study on the relation between sensible reality and the procedures
by which science derives knowledge about it. Let us admit that models are
the tools of scientific thinking: physical models are tools of descriptive
thinking, mathematical models at large are the tools of
analogical thinking. But what exactly does the term
‘tool’ mean in this case? This is no easy question, because, as the SB
model shows, at least physical models have different functions depending
on whether one is studying the world of our ordinary experience or the
submicroscopic world. To find an answer we have to examine the nature of
scientific truth taking the ontological structure of sensible reality into
account. In plain words: if the statements of science concern models, how
can they be expected to disclose the truth about what lies ‘out there’,
what appears to be independent of the observer’s will and wishes?
Let us thus grant that, in order to be true, a statement need not match reality in all its aspects, relations, and/or details, but has to match it only as far as it goes. This consideration applies both to scientific concepts and to relations between them.
The difficulties begin at this point. When a zoologist thinks of the term ‘animal’, does that evoke any picture in his imagination? This question should be turned to the psychologists, but it will suffice here to answer that when thinking of an animal at least some people do imagine a lion, a bee, an amoeba, so as to visualize those properties which make them animals, ignoring at the same time those which are specific of their class, order or minor subdivisions. The ‘ignoring’ here can be an extremely fast and subconscious process, but even ordinary people know by observation on their children that this combination of a concrete image with an abstraction process is learned in tender age.
The fact that physical models provide images is a reasonable explanation for the extensive use of models in science. If this is granted, why should we waste more time in working towards a precise definition of models, in contrast with the current opinion that definitions are not necessary? Why not try to understand what models are just by means of examples, in accordance with the practice of modern education? The answer can be given in two points.
First, ‘understanding by examples’ can only be considered sufficient by people not familiar with introductory mathematics, where that view is shown to be limited to negative assertions.
Secondly, there has been a decline of the belief that science is the only way to knowledge of a reliable kind. This trend is good in itself, but it is liable to go too far, and make scientists lose confidence in their work. It would seem that the only defensive measure is to bring to light the scope and limits of scientific ‘truths’, by making explicit a number of tacit beliefs till now implicit in scientific thought. Now, an important unexpressed belief is that models are ‘tricks’ used during the preliminary stages of research, and that science should reach beyond them. If this belief is mistaken, and models are indispensable tools, or indeed built-in features of the scientific enterprise, then it becomes indispensable that we should know how to use them.
It thus appears that a clearer understanding of models is worthwhile.
In order to proceed in this direction, we must consider how science arrives
at ‘truth’, and in what sense the scientists qualify their statements as
In practice – though seldom at a philosophical level – the statements falling within the realm of science are said to be true if they are:
(b) faithful to reality.
Horses and other objects or beings of the same size belong to the world accessible to our direct experience. However, science is also concerned with objects and processes that are not directly perceived by our senses. In what sense, if any, are they reduced by science to models or phantoms of what they really are? We have already an idea of the answer from the SB model (Sect. 1.1). To make that idea more explicit we have to examine briefly the ‘ontological structure of sensible reality’. This rather formal expression applies to the result of an attempt to describe reality as an ordered collection of objects and beings – entities – of different sizes, degrees of complexity, accessibility to observation.
Let us consider an example discussed by Heidegger: a tree in full bloom standing before us on a meadow. Is the tree real? Or is it only a collection of nuclei and electrons suspended as it was in emptiness? The latter question is answered positively by physicalistic reductionism, but on further reflection, it appears to be untenable. As we have tried to show in a previous paper, a tree is a collection of particles, but it is not just that; because it is the actualization of one system out of the huge number of systems those particles can make up. The tree is also a collection of cells and dead material; it is a collection of leaves and branches and roots, and it is a tree, indeed a blooming tree, with its colors and its pleasant smell.
Examples of this sort suggest a general view of what ‘to exist’ means that is unconceivable in a physicalistic frame of mind. According to this view, the physical world consists of objects (physical systems) more or less independent but clearly distinguishable from one another. Concrete examples of those objects, roughly in descending size order, are:
Careful examination of the list above will give the reader a feeling for the nature of the ‘size ladder’ of material reality; that ladder, which seems at first sight to extend from infinity to infinity, was the object of Pascal’s famous reflection on ‘the two infinities’.
Before the DA objects, large objects are listed whose entire reality cannot be perceived by all our senses, if at all. Even in the case of a familiar object such as a mountain one can easily realize that for us to see it as a whole it must be so remote that we cannot touch it, and vice versa. A star is only perceived as a shining point, or not at all.
As to the objects listed after those which belong to the DA level, a mite or a cell can only be examined visually with a microscope, and cannot be detected by touch. A molecule cannot even be seen by simple magnification, because it is smaller than the wavelength of visible light; so that what we call an ‘image’ obtained, say, by scanning tunnel microscopy is a theoretical construction based on the formal similarity between matter waves and light waves. Even when an image can be obtained with light, it is actually based on an analogy, albeit a very close one. The varying theory-dependence of our knowledge of material entities too large or too small for man’s direct access – and the resulting problems with the nature of our ‘perception’ of them – suggest that we should attribute to material reality a layer structure from the point of view of intelligibility. This level structure may or may not have an ontological import, but it is essential for the ‘existence judgments’ which science is expected to pronounce.
The size levels (or layers) should not be confused with another aspect
of the structure of reality, namely complexity. Each object or being in
the universe only belongs to one size level, whereas different complexity
levels are simultaneously present in any given object. Moreover, objects
such as a microscopic mite and a whale belong to the same level of reality
on the complexity scale and are by far more complex than a star: despite
their difference in size they are studied by the same discipline – zoology.
In short, complexity is another feature of reality according to which entities
should be classified and disciplines distinguished. We have discussed it
elsewhere in connection with chemistry, and shall
not pause on it here, because all that is needed about levels of reality
in connection with models has been said above. The major inference to be
drawn is that, since man has no direct perception of a large majority of
the objects which science has detected in the physical universe, his knowledge
is based on analogies with ad hoc objects, possibly artificial,
as is the case with the SB model of molecules.
These examples show, if need be, that, although the etymological roots of words are seldom present in our minds, even speculative thinking has its roots in the everyday sensible experience.
According to historians, even mathematics began with concrete, visualizable problems; and there are grounds for believing that formulas are visualized by the mathematicians for their minds to work on, even in transfinite mathematics. However that may be, in the sciences of nature visualization (and sometimes association with other sensations, like sounds) plays a fundamental role, not only as regards creativity but as regards understanding.
If – and to the extent to which – this is so, it implies that the categories
of our direct-access level of reality are those which we use for understanding
objects and processes belonging to size and (perhaps) complexity levels
far remote from that level. This is why those models which are representations
at man’s DA level of material objects not directly accessible are indispensable
tools of science.
Let us denote by R the real size level of an object, by DA man’s direct-access level, by M-DA a physical model belonging to the DA level. The procedure under consideration can then be schematized as follows:
We recalled at the beginning that the word ‘model’ can be used for a mathematical model, already discussed by other authors, and for a physical model, which appeared to require further reflection. A brief examination of the SB model of molecules confirmed the claim that science studies an idealized copy of reality, and suggested that we should therefore explore the role of idealizations and analogies in our endeavor to apprehend and comprehend reality. Idealizations (reduction to standard types) provide analogies used in science to cope with difficulties arising from those details which make each perceived object unique. Reality beyond the thresholds of direct perception is apprehended by means of analogies constructed ad hoc, namely with very special, often artificial systems, real or imaginary, belonging to our direct access level of reality. This led us to focus our attention on these physical or ‘analogical’ models and their relation to reality.
Physical models thus serve a double purpose: they allow scientists to
ignore inessential details of entities belonging to the ordinary environment
of man without dismissing their collocation on space and time; and they
provide counterparts in that same ordinary environment of entities not
immediately accessible to the five senses of man. To grasp the ontological
significance of models of objects lying beyond our direct perception, from
galaxies to electrons, layers or ‘levels’ of reality corresponding to size
and to complexity should be considered. There is a size level of reality
which can be perceived by man with his five senses; it may be called ‘man’s
direct-access’ (DA) level. It turns out that our knowledge, including abstract
thinking, is based on analogies with material objects or processes at the
DA level. Ad hoc objects (‘models’) at the DA level supporting analogies
for objects not directly accessible to our senses are essential tools of
 M. Born, J. R. Oppenheimer: ‘Zur Quantentheorie der Moleküle’, Annalen der Physik, 84 (1927), 457-84.
 Of course, the results of a calculation of the vibrational properties of a molecule will usually match the observed vibrational frequencies only if the parameters are chosen in the proper way. This is consistent with the fact that the theory is based on an analogy.
 For example, when he established Galilean relativity, by considering flies in a ship in perfect uniform motion.
 For mathematical models, cf. the survey by Salvo D’Agostino, Sandro Petruccioli (eds.): Mathematical Models and Physical Theories, published as a reprint from: Rendiconti dell’Accademia Nazionale delle Scienze detta dei XL(Rome), serie V, vol. IX, parte II, 1985, pp. 65-198. The general question of mathematics vs. description in terms of visualizations was treated by Henri Poincaré: La science et l’hypothèse (1902), Paris, Champs-Flammarion, 1968; cf. G. Del Re: ‘Poincaré et le mécanisme’, Philosophia Scientiae (Nancy), 1 (1996), spec. issue 1, 55-69.
 Although for our present purposes this intuitive description of realism is sufficient, there would be much to add in the frame of a general theory of thought and perception. The Kantian critique to essentialism and the difficulties which it raised in connection with realism is well known to everybody since secondary school.
 Il Saggiatore (1623), Ed. Naz. Op. Galileo, Firenze, Barbèra, 1896, vol. VI, p. 235.
 This is a cursory mention of the view on science adopted for this study, and is not intended to dispose of the numberless studies on scientific knowledge published in this century.
 Henri Poincaré, op.cit., p. 148f.
 A. S. Eddington, The Nature of the Physical World (1935), London, J.M. Dent & Sons, Everyman’s Library, 1947, pp. 5-10 and passim. Note that Eddington was writing when science was assumed to coincide with physics.
 Martin Heidegger, Was heißt Denken? – Vorlesungen des Wintersemesters 1951-52, Stuttgart, Reclam, 1992, pp. 25-29. Cf. A. S. Eddington, loc.cit.
 Del Re, ‘On the ontological status ¼ ’, op.cit.
 Mario Bunge: ‘Is Chemistry a Branch of Physics?’, Zeitschrift für Allgemeine Wissenschaftstheorie, 13 (1982), 209-223.
 Strictly speaking, only the objects that man can handle belong to the DA level; but the separation we are trying to establish has a somewhat fuzzy border, and, as the reader certainly sees, it is not worthwhile to try to be more precise.
 Blaise Pascal, Pensées (ca. 1660), ed. by Louis Lafuma, Paris, Éd. du Seuil 1963, pp. 199-72.
 G. Del Re: ‘Chemistry and Complexity’, in: G. Costa, G. Calucci, M. Giorgi (eds.), Proceedings of the Third International Symposium on Conceptual Tools for Understanding Nature, Singapore, World Scientific 1997, pp. 153-164; cf. also works by the same author already cited. The complexity levels are very close to what N. Hartmann called Seinsschichten (layers of reality). This was pointed out in connection with chemistry by J. Schummer: Realismus und Chemie, Würzburg, Königshausen & Neumann, 1996. Independently from Hartmann (and perhaps in a less systematic form), also S. Alexander proposed around 1920 a layer structure of reality.
 A great merit of Heidegger was precisely that he brought this point to the forefront of philosophical thought. It is unfortunate that the recent dramatic decline of classical studies has caused a loss of feeling for the original values of most philosophical words, which is slowly but steadily spreading from the Anglo-Saxon world to the Latin and the German countries.