CARACTERIZACIÓN DE INDICADORES CINEMATICOS DE FALLAS

Galileo’s contribution, as observed above, was in convincing people that physical nature can be quantified, and in making the mathematization of science possible. In that process he argued for the need of idealization and experimentation for understanding and validating scientific knowledge. The counterfactual nature of scientific conceptions and the need of not only physical experiments, but also thought experiments has been brought to light in his deliberations. Descartes too was not only convinced that physical nature can be

quantified, but actually identified mathematical (geometrical) dimensions with the physical.

[I]t is not merely the case that length, breadth, and depth are dimensions, but weight also is a dimension in terms of which the heaviness of objects is estimated. So, too, velocity is a dimension of motion, and there are an infinite number of

similar instances.”21

However, Descartes allowed some distinctions in relating them to actuality and possibility— physics is to actuality and mathematics is to possibility.

The difference consists just in this, that physics considers its object not only as a true and real being, but as actually existing as such, while mathematics considers it merely as possible, and as something which does not actually exist in space,

but could do so.22

Physics, then, becomes applied (actualized) mathematics. This development has far reach- ing implications for the advancement of modern science. In ancient times multiplication of

dimensions other than geometric or arithmetic are thought to be impossible.23

Unless the di- mension of, say, mass is multiplied with the dimension of motion (velocity) no quantification

21

Quoted in Mason 1956,Main Currents of Scientific Thought: A History of the Sciences p. 132.

22

Conversation with Burman(V 160, C p.23), quoted in Bernard Williams 1978,Descartes: The Project of Pure Inquiry p. 259.

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of motion could be achieved in terms other than merely saying that something moves faster

than some other thing. Development of physics without allowing the functional correlation

or covariation (read multiplication) of geometrical dimensions and physical dimensions can be stated to be impossible. Thus the subject matter of physics and mathematics have found a common ground, such that they could develop, henceforth, dialectically, if not hand in hand. Anyone familiar with the development of both mathematics and modern physics after the 17th century, would not deny that neither mathematics nor physics could have developed independent of each other. The foundational contribution of Descartes is extremely relevant for enforcing such a development of both the fields. Since the study of such a development is a subject in itself, we shall not divert our attention to that here. It is sufficient to observe here that Descartes’ contributions in working out a common framework for mathematical physics have been more fundamental than that of Galileo. However, when one looks at the comparative abilities of finding applications of mathematical knowledge in solving concrete problems Galileo’s success is more commendable than Descartes. Modern science could not afford to miss either of them.

Descartes also proposes a joint method of Analysis and Synthesis, which is clearly

conceived as a method of discovering and ordering knowledge. In Regulae he proposes rules

for the direction of the mind. His rules IV, V and VI are as follows: Rule IV: There is a need of method for finding out the truth.

Rule V: Method consists entirely in the order and disposition of the objects to- wards which our mental vision must be directed if we would find out any truth.

We shall comply with it exactly if we reduce involved and obscure propositions

step by step to those that are simpler, and then starting with the intuitive appre- hension of all those that are absolutely simple, attempt to ascend to the knowledge of all others by precisely similar steps.24

Rule VI: In order to separate out what is quite simple from what is complex, and to arrange these matters methodically, we ought, in the case of every series in which we have deduced certain facts the one from the other, to notice which fact is simple, and to mark the interval, greater, less, or equal, which separates all the others from this.

Rule V is a clear statement of the joint method of analysis and synthesis. However, we see that relational knowledge of things is what is sought, and not Aristotelian essences. The

ultimate goal or aim of the analytic regression, as is clear from Rule VI, is not the simplequa

simple but the simple ‘relatively’ to the other terms of the series. Also notice that the ‘series’ does not imply that we are to consider that things or facts can be arranged in a conceptual

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56 Chapter 2. The Marriage of Mathematics and Natural Science

classification similar to that adopted by the Aristotelians. The series is not a static ontological classification based on genus and specific difference but an implicatory sequence of antecedent and consequent in which the important and decisive factor is the logical relation of one to

the other.25

Also to be noted is the use of the term ‘propositions’, and not classes.

Rule VI says that in order to know what is simple and complex, we should arrange terms in relative and absolute order. Descartes defines an absolute term as one which con- tains within itself the pure and simple of which we are in quest. Examples of such terms are independence, cause, simple, universal, one, equal, straight and so on. Relative terms on the other hand are those which are ‘related’ to the absolute and deducing them involves something other than the absolute concepts. Examples of such terms are what ever is con- sidered as dependent, effect, composite, particular, and so on. Note that the terms in the independent category includes basically primary mathematical terms, and in the dependent category includes the secondary non-mathematical terms. Thus the method, couched in terms of analysis and synthesis, tends toward mathematical objects of knowledge, which is about divisions, shapes and motions.

The method of analysis ultimately reduces the problem by a regressive and gradual

division until we reach a term which ismaxime absolutum. From the discovery of themaxime

absolutum the method of synthesis can begin, which is the arrangement of the facts discovered by analysis, in such an order that they will be successively relative and more concrete terms

of the implicatory series will issue as the solution of the problem.26_.

Thus Descartes’ program is to interpret nature in the form of an axiomatic structure of the whole system, by establishing indubitable foundations and the deducing from them the rest of the phenomena. Following such a maxim he tried to construct a system, which is purely mechanical in character, i.e. it employs no principle other than the concepts employed in mechanics, such as shape, size, quantity, motion etc.

Gradually Descartes realized how difficult was the program he visualized. Later he not only diluted the rigid architectonic approach of deducing everything from first principles, he allowed room for hypothetical premisses that are compatible with the first principles in his system. This point comes out vividly in the study of Larry Laudan (1981), who writes that:

After trying to deduce the particular characteristic of chemical change from his first principles (i.e., matter and motion), he concedes failure. His program for the

derivation of the phenomena of chemistry and physics froma priori truths remains

25

Beck, L.J. 1952, p. 161.

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uncompleted. His first principles are, he admits, simply too general to permit him to deduce statements from them about the specific way particular chunks of mat- ter behave. ... Not content to leave anything unexplained, Descartes departed from his usual devotion to clear and distinct ideas and advocated the use of inter- mediate theories (less general than the first principles, but more general than the phenomena), which were sufficiently explicit to permit the explanation of individ-

ual events and which were, at the same time, compatible with, but not deducible

from, the first principles. Descartes recognized that all such intermediary theories were inevitably hypothetical. Because their constituent elements were not clearly and distinctly perceived, it was conceivable that they were false. After all, nature is describable in a wide variety of ways and the fact the an explanation worked was no proof that it was true. Like any good logician, Descartes realized that “one may deduce some very true and certain conclusions from suppositions that

are false or uncertain”.27

This development in Descartes turns out to be highly significant for understanding the role of the method of hypothesis in the later developments of science. This moderately modified stand also brings Galileo and Descartes closer than before. In the earlier section we have noted why Marsenne in his letter to Descartes was critical of Galileo. Whatever be the significance of this later realization in the context of the development of the hypothetico- deductive methodology, as Laudan tries to stress, the significance of this in the development of problem oriented (paradigmatic) science, as opposed to architectonic science, should also be noted.