The relation between a type and its tokens can be said to be a one-over-many
relation, which is also the relation between universals and particulars. This relation usually
establishes a hierarchical order of things, for the two things so related belong to different
levels. On the other hand a one-to-one relation brings two things of the same level into an
order. Such an order is usually referred to as ordered pairs. Classification of things is a good
example of one-over-many relation, while functional relations provide excellent examples for one-to-one relation.
The distinction between the two types of relations is rather well known, and hence hardly requires any further illustrations. However, we find it insightful to view the matter in a different manner. Since one-over-many relation introduces difference in the level of things
related, we tend to depict such a relation by placing the things one above another in avertical
manner. Let us therefore call the systematization that is established by the employment of
one-over-many relation avertical systematization. An axiomatic system is a vertical system-
atization because the axioms and the theorems have a one-over-many relation. The structure of a hypothetico-deductive model is also vertical for the same reason. Taxonomic system- atization is another example of vertical systematization, because the classes are arranged in several levels—one higher class including the subclasses in a nested manner.
Likewise, since a one-to-one relation is obtained between things of the same level,
we can visualize them side by side in a single horizontal plane because the things thus re-
lated belong to the same level. Therefore, let us call the systematization that is obtained
by a one-to-one relation ahorizontal systematization. The various parameters that are func-
tionally related in a physical system can be viewed as an excellent example of horizontal
systematization, because all the parameters thus related are located in a system together,
hand-in-hand, to form a sort of a horizontal ‘plane’. For the same reason models also are horizontally systematized. However, the relation between a model and a physical system, as stated in a scientific assertions, is not horizontal but vertical. Because one model can have
several physical systems as its instances, and as stated in the above chapter (§5.2 page 127),
the proper relation that obtains between them is of the one-over-many kind. There can be several levels of models that can be placed one upon the other, depending on their gener-
172 Chapter 6. Inversion
Stegm¨uller are structures related in a vertical order, though each kind of model is horizontally
systematized. We shall shortly return to this point below.
Another kind of systematization should also be noted for the sake of giving a more or less complete picture regarding the kinds of systematizations obtainable in science. This third
kind isevolutionary systematization. This kind of systematization is achieved by mapping the
taxonomic order of systems on one hand, with a temporalantecedent and consequent relation
of them on the other hand. The manner in which different organisms or systems have been
located in a temporal evolutionary scale on the basis of theantecedent and consequent relation
constitutes an excellent example of this third kind.17
These three kinds of systematizations can be viewed as the x, y, andz coordinates
of scientific knowledge. The map of scientific knowledge that we are going to draw will make use of this manner of visualizing the different ways of scientific systematization. In this thesis, however, we will not deal with the vertical and evolutionary systematizations, for the objective of our study is to highlight the role of inversion in the generation of scientific objects such as definitions, models, and systems that come under the horizontal kind of systematization.
This distinction between the horizontal and vertical systems can now throw more light on the nature of the distinction between what is and what is not a statement. We will regard a statement (or an assertion) as an instance of a vertical system, because types and tokens belong to logically distinguishable categories of intension and extension respectively.
We think that nothing is a statement if it is not a relation between an intension (type) and
extension (tokens). This notion of statement can be applied to both individual statements and general statements, because in case of the general statements the place of extension will be a class of tokens, while in the case of individual statements it will be a token. Meta-level statements can also be interpreted in the above manner. Even for a relational statement this specification is sufficient, because in such a case the extension will be a pair, or a triad, etc., depending on whether the predicate is diadic, or triadic, etc. In fact all scientific assertions
must be relational statements, in this sense, because—as stated above (§5.2 page 130)—the
model and physical systems are stated to be related by isomorphism, over and above the type and token relation. Thus we think relational statements are special kinds of subject-predicate statements. A complete argument for this claim cannot be worked out here. We will however presuppose this, and develop the rest of the thesis on this basis.
We will regard any structure or system as a nonstatement iff the relation between
17
Though we originally intended to illustrate this kind of systematization as a separate case-study in the thesis, we could not ultimately incorporate due to time constraint.
the terms is horizontal, i.e., the terms belong to the same level. The terms may all belong to the extensional category—as in the case of a physical systems, or the terms may all belong to the intensional category—as in the case of models and scientific definitions.
It is our claim that scientific knowledge is a product of both horizontal an vertical
systems.18
We further propose that horizontal structures of science, namely models, scien- tific definitions, and physical systems, are structures constructible on the basis of inversion, while vertical systems of science, such as taxonomic and hypothetico-deductive systems, are constructible on the basis of negation.
Horizontal structures are necessarilynonstatemental in nature. Stegm¨uller’s models
are horizontal structures, though each of them is related to the next level of models in a vertical order. It is this possible relation that enables the metalevel statements. In order to see the relevance of this observation let us consider the nature of the work of a physicist, theoretical physicist, and a pure mathematician. The levels which a theoretical physicist, for example, would mostly be dealing with are always above physical systems. His concern is usually studying the properties of the objects of a theoretically modeled world (possible world). The statement of a scientist when engaged in this sort of work can be stated to be applicable to the simulated world alone. A mathematician can be said to be working at other levels higher than a theoretical physicist. The statements made at this higher level would be descriptive of the models that become instances of the abstract algebraic structures. More higher levels of abstract engagements can further be identified where foundational attempts
resembling those of Felix Klein’sErlanger program, or of the French structuralists’ program
of Bourbaki. At the lower level when a scientist states a relation between a model and the
realizable physical systems that are believed to belong to this world, we get what can be
called the scientific assertions which are either true or false. All other statements possible at higher than this level can be said to be providing the semantics (conditions of truth and
falsity) to the possible applications of the various levels of conceptions.19
The difference in the levels of the various possible statements that scientists could (and do) make is another reason why a large structure such as classical particle mechanics
should not be considered a single nonstatement (structure) as Stegm¨uller suggests. Therefore,
we think that the defenders of the nonstatement view should make room for the various possible statements scientists make, by following our suggestion of vertically ordering the
18
Since we are not elaborating the third evolutionary system at present, and we cannot not visualize at the moment what kind of picture might emerge after incorporating the third ‘dimension’, the above statement may be regarded tentative.
19
It may be less confusing if we could conventionally name the different levels of statements possible by identifying them with the level involved.
174 Chapter 6. Inversion
various possible models. The number of levels of models need not be just three as Stegm¨uller
suggests and we see no a priori reason for such definite specification. The defenders of the
semantic approach should allow for enrichment of the view by distinguishing the different levels of models. We think that the conditions suggested for distinguishing the statement and nonstatement components on the basis of vertical and horizontal systematization should be acceptable to the defenders of the semantic approach and nonstatement view. With these suggested alterations, we see the possibility of lessening the problems of adjustments between the two structuralist positions.
Though the scientists’ ultimate objective can be perceived as dealing withthis world,
their engagements higher up in the world of abstractions can in no way be regarded as non- epistemological on the ground that they do not deal with ‘hard’ truth. Most conceptions, i.e., nonstatements, of science have taken birth in these fertile contexts. Though several examples can be cited, the example of the abstract construction of the group of invariant transformations of velocity by Lorentz and its subsequent application by Einstein to the case of light can be regarded a paradigmatic one. To distinguish the nature of their achievement
we can say that Lorentz invented the concept of velocity invariance, and Einstein discovered
an application for the former’s invention. This is one of the areas where a generativist has to search for the possible patterns of discovery (or invention), the other area being the genesis of physical systems from phenomena, as suggested in the last chapter, which will be dealt with in greater detail in the case-studies.
Most of the attention of philosophers of science has been paid to studying the logical properties of vertical structures or systems, which any way are definite constituents of scientific knowledge. However, the study of horizontal structures has not attracted many thinkers in mainstream philosophy of science. The study of horizontal systems, models, definitions, and physical systems, as important constituents of the anatomy of science has only recently, after the failure of the positivist’s attempts, attracted the attention of the followers of the semantic approach. We think very strongly that these new non-traditional categories of understanding scientific knowledge will provide a significantly richer framework for future studies on this subject.