It has been suggested above that the operations involved in the inverse-definite- descriptions, though broadly classifiable as either additive or multiplicative, need to be re-
garded as distinct operations. The ampliative nature emerges out of the multiplicity—in the
sense of being many—of the operations. In this section we will provide further justification
for such a multi-operations-view. One part of the argument that is already suggested above
is that each of the operations generate or construct different structures representing different state-of-affairs.
When the functional relation specified in the definitions is intended to be a gener- alization of relationship between the dimensions, how could one think of several operations
making up one relation? One might say that this appears quite counterintuitive as well as
unnecessary. In order to make our position appear plausible we will introduce a distinction between ampliative and non-ampliative operations.
Consider the case of operation on numbers. Given an equation x ×y = z, the
operation × is generally considered to take any two numbers, x and y, from a given set
S, and yield a number z belonging to the same set S. It would not be considered a ‘well-
behaved’ operation if it yields a znot belonging to the setS. But what about the cases when
we consider adding or multiplying things other than numbers?
Could we multiply 10 apples with 20 pebbles? Do we get 200 of ‘appebles’ ? How
aboutadding10 apples with 20 pebbles? We get an aggregate of 30 objects that includes both
apples and pebbles. Since we do obtain an aggregate of 30 countable objects, the operation makes sense. But there seem to be no meaningful product obtainable from multiplying apples
with pebbles. There seems to be some cases where some operations do not make sense. That
is some operations do not generate any meaning, while others do.
Let us take another case. When a line of length 20 meters is added to another of
10 meters, we get a line of length 30 meters. What we get can still be called a line. What
happens when we multiply the two lines? Do we get a line of 200 meters? It depends on the
context. If we are measuring the length of some thing, such as a road, with a tape of length 10 meters, having performed the operation 20 times we compute the length of the road as 200 meters. However, in this case while the 10 stands for the length, the 20 does not stand for the length, but for the number of times an operation is performed. Whether we measure
188 Chapter 6. Inversion
with a tape of 20 meters 10 times, or vice versa we get the same length, and so the operation of multiplication has some sense in this case.
In another situation, we might multiply apparently the same kind of values, but we
would report that the result is 200 square meters. Here also it makes sense because we are
measuring the area of a surface. Are we performing the same operation in both the cases? Definitely not. In the former case, though we are using the multiplication table, actually what we are doing is nothing but addition, as many times as the operation is repeated. This is a trivial application of multiplication. In the latter case, however, we are engaged in a
non-trivial operation. The result is no longer a length, but area. We will call such non-trivial
operations ampliative operations. Insofar as 10 and 20 are numbers, and 200 also a number,
our operation continues to be of an arithmetical kind. But when the numbers represent magnitudes of certain dimensions, they cease to be mere numbers. One might suppose that this is in no sense a startling fact that needs to be talked about in a doctoral dissertation.
The non-trivial matter is the epistemological significance of theamplification involved in the
process. When we start talking about area as a function of the sides of a rectangle, we have not made any inductive generalization, rather we have constructed (defined) a concept. This
concept of area can also be identified in terms of inverse-definite-descriptions.30
In certain situations we may have to follow a different procedure of addition. Spe- cially when we are operating with a vector magnitude, such as displacement. Here the direction of displacement along with the magnitude matter. Thus vector addition is another operation that follows a different, but definite logic.
Though it is possible in a higher order theory to have a very general definition for obtaining the areas of any shape, the manner in which we make sense of that higher order operations follows a definite path of generative history. A large number of differentiable operations, each with a specifiable and non-trivial meaning have entered into that process of obtaining a general theory of space. That the general theory is not just a single concept becomes clearer to us when we begin applying the ‘global’ formula to generate ‘local’ formulae. Possession of a general formula is not a sufficient condition for the mathematician’s ability to solve a specific problem. His abilities depend on how good he is in generating the specific formula for the specific purpose. Generalizability of a large class of operations conceals the fact that insofar as they are general such theories remain ‘deaf-and-dumb’. The context of learning as well as the context of application can demonstrate the amplifiability of such
30
Piaget’s investigations on the development of concepts such as ‘length’ ‘area’ ‘volume’ etc., as disussed in The Child’s Conception of Geometry 1960, demonstrate that the nature of the operations involved in learning each of the above mentioned concepts are independently closed under distinct group of operations. Invertibility (reversibility) of the operations is one of the necessary conditions of acquiring these concepts.
general theories.
We therefore think that there are as many operations as there are different contexts of application. Every operation cannot be thought of being applicable to every situation
promiscuously.31
In this connection Karl Popper’s views are supportive of our standpoint. He is of the opinion that the arithmetic of natural or real numbers is helpful in describing
certain kinds of facts, but not other kinds.
[W]e may note that the calculus of natural numbers is used in order to count billiard balls, or pennies, or crocodiles, while the calculus of real numbers pro- vides a framework for measurement of continuous magnitudes such as geometrical
distances or velocities. . . .We should not say that we have, for instance, 3.6, orπ,
crocodiles in our zoo. In order to count crocodiles, we make use of the calculus of natural numbers. But in order to determine the latitude of our zoo, or its distance
from Greenwich, we may have to make use of π. The belief that any one of the
calculi of arithmetic is applicable to any reality . . .is therefore hardly tenable.32
He is arguing there that each calculus should be viewed as a distinct semantical system. Since each calculus presupposes certain kind of operation/s, we may say that each semantical system differs from the other on the basis of what kind of operations are constitutive of the systems.
Popper argues that a proposition such as ‘2 + 2 = 4’ can be thought of in two different senses. One of them is to consider it as a logical truth. In the second sense it may be taken to mean a physical manipulation (operation), such as, say, counting apples or pebbles. In the second sense the universality of the proposition becomes doubtful. Popper’s examples are interesting and very useful, for they bring home several points that we are presently pursuing.
[I]f you wonder what a world would look like in which ‘2+2 = 4’ is not applicable, it is easy to satisfy your curiosity. A Couple of rabbits of different sexes or a few drops of water may serve as a model for such a world. If you answer that these examples are not fair because something has happened to the rabbits and to the drops, and because the equation ‘2+2 = 4 only applies to objects to which nothing happens, then my answer is that, if you interpret it in this way, then it does not hold for ‘reality’ (for in ‘reality’ something happens all the time) but only for an
abstract world of distinct objects in which nothing happens.33
Two contexts become demarcated here. One is a context where nothing happens, and the
other is a context where something happens. We think that the ‘operation’ that happens
31
This thesis can be viewed as opposing Quine’s famous thesis on natural kinds.
32
Karl Popper 1962, Conjectures and Refutations, p. 211.
33
Ibid,p. 212. Note that here the term ‘model’ is used to refer to an example of a world (possible). We would prefer ‘model’ for the possible world, and ‘system’ for the actual world where we find the model instantiated.
190 Chapter 6. Inversion
when rabbits or drops meet cannot be captured by the ‘poor’ sense of the operations supplied by arithmetic. Arithmetical operations do not have generative potential in the non-trivial sense. In order to bring out the qualitative differences between various operations in various contexts, we need to differentiate each such operation by a proper methodological procedure. We think that the above method of constructing inverse-definite-descriptions could bring it out. Since every operation has an identity, which can be uniquely described by the inverse- pairs, we could talk about the semantic differences that exist in the various operations that are possible in the ‘rich reality’.
Another very crucial point is with regard to what Popper says about the transfor- mation of semantical systems into scientific theories. He says that
in so far as a calculus is applied to reality, it loses the character of alogical calculus
and becomes a descriptive theory, which may be empirically refutable; and in so
far as it is treated as irrefutable, i.e. as a system oflogically true formulae, rather
than a descriptive scientific theory, it is not applied to reality.34
This passage serves us a double purpose. First, it brings out the point that a semantical system gets transformed into a descriptive refutable and empirical scientific theory when applied to reality. Since a semantical system is irrefutable, it is neither true nor false—it is immune to refutations. Second, it clarifies Popper’s notion of scientific theory: it is an
application of a semantical system to reality. For him nothing is a scientific theory if it
has no potential application. For us nothing is a scientific concept if it has no potential
application and nothing is a scientific assertion if is not a statement applying a scientific concept. The nature of the difference is that Popper demarcates statements into scientific or not on the basis of falsifiability, while we are suggesting a demarcation of concepts into scientific or not on the basis of whether a concept specifies an invariant identity expressible in terms of inverse-definite-descriptions or not. In consequence we would pass semantic systems as legitimate objects of scientific knowledge because it is sufficient for them to have potential application. Growth in the number of distinguishable semantic systems is already a partial growth in scientific knowledge.
Lorentz is indeed a scientist, for he did construct a meaningful semantic system. Einstein is also a scientist, for he found a truthful application for the meaningful semantic
system that Lorentz invented. Should we say, Einstein discovered and Lorentz invented?
Sometimes it is less confusing if we do differentiate the activities of different scientists by different terms. We would say that both activities have epistemological relevance, unlike
34
Popper, who found greater relevance in Einstein’s activity. Since the construction of seman- tic systems are based on a knowledge of proportionalities that are available from either direct or experimental knowledge, empirical constraints necessarily enter in the process of gener- ating models. Thus though there is an element of theoretical (mental) construction in the generation of semantic systems, it is constrained by relations that obtain in ‘reality’. There- fore, none of the structures thus formed can be devoid of empirical content, and in most cases unanticipated counter-inductive knowledge gets generated. Einstein’s relativity theory is a good example of a non-inductive, inversion based construction, which surprised many due to its distance from ordinary understanding.