Galileo’s contribution, we think, lies in synthesizing the Aristotelian empirical method and the mathematical or geometrical method of the Greeks. Galileo’s program is to translate scientific experience into experience that can be expressed in mathematical terms. Anyone who attempts to accomplish this must be able to face a problem long known in history. The problem has been posed by the historians of science as one between the Platonists and the Aristotelians. The former thought that nature is mathematical in character and the latter thought that mathematical descriptions are neither true nor false, while physics tells us the truth about the world by following the empirical method.1
Unlike the Platonists, Galileo was not trying to apply mathematics or geometry to describe the Platonic world of ideas, but was attempting to apply geometry to the real physical world, which is believed to be hidden behind the phenomenal world.2 For traditional Platonists the problem of application does not arise because the perfection of Beings can not be applied to or matched with the imperfection of Becomings. The Book of Nature which Galileo was intending to read, which he believed is written in the language of mathematics, is not a book of the Platonic kind. If that had been the case, truly speaking, Galileo would have had no problem to solve. Whatever Galileo had contributed is acknowledged as a remarkable achievement mainly because he tried to apply mathematical order also in a domain which had been traditionally conceived as non-mathematizable.
In what follows we shall observe that Galileo took some significant steps, first to enable the application of mathematics to physical phenomena; second to suggest that the material hindrances be eliminated in order to find the objects that are independent of sensory experience and convention; and third to suggest a mathematico-experimental method, Galileo’s version of the joint method.
In the Dialogues Concerning the Two Chief World Systems (1632) Simplicio, like a true Aristotelian, expresses doubts about Galileo’s project.
... [T]hese mathematical subtleties do very well in the abstract, but they do not work out when applied to sensible and physical matters. For instance, mathematicians may prove well enough in theory that spheara tangit planum in puncto ... ; but when it comes to matter, things happen otherwise. What I mean about these angles of contact and ratios is that they all go by the board for material and sensible things. 3
Now Galileo should either show how a physical (real) plane touches a physical sphere at a point or show how an ideal plane can touch an ideal sphere over many points over a surface. In fact Galileo’s answer consists in realising that both are geometrically possible. Salviati, who speaks for Galileo, responds to Simplicio’s objection after long deliberations.
Salviati: Are you not saying that because of the imperfection of matter, a body which ought to be perfectly spherical and a plane which ought to be perfectly flat do not achieve concretely what one imagines of them in the abstract?
Simplicio: That is what I say.
Salviati: Then whenever you apply a material sphere to a material plane in the concrete, you apply a sphere which is not perfect to a plane which is not perfect, and you say that these do not touch each other in one point. But I tell you that even in the abstract, an immaterial sphere which is not a perfect sphere can touch an immaterial plane which is not perfectly flat in not one point, but over a part of its surface, so that what happens in the concrete up to this point happens the same way in the abstract ... 4
This argument contains one of the most central thesis of Galileo, in his attempt to show that mathematics can be the language of the book of nature. Butts reformulates the central point as follows:
For any x, y and t, if x is a perfect material sphere and y is a perfect material plane, and t is a definite interval of time, and x and y remain perfect through t, then x and y touch one another in a single point when y is struck as a tangent of x.5
It is indeed a token statement of applied geometry, which describes a particular condition or situation of the world. Given that the antecedent can never be satisfied by actual solid objects, (he agrees with his predecessors on the point) the statement will always be a true counterfactual, because the ‘fact’ to which it is applied is not directly given in experience.6
What happens in the world,therefore, is what happens in geometry. This is Galileo’s first move in the direction of achieving his target. Given this, what should one do in order to see mathematical order in the world? His answer is that one must deduct the material hindrances, or defalking the impediments of matter. This, according to Butts, is the second central theses of Galileo.7 This consists in choosing only those characters that can be mathematically expressed and eliminating those characters that fall outside mathematical description.
Just as the computer who wants his calculations to deal with sugar, silk, and wool must discount the boxes, bales, and other packings, so the mathematical scientist (filosofo geometra), when he wants to recognize in the concrete the effects which he has proved in the abstract, must deduct the material hindrances, and if he is able to do so, I assure you that things are in no less agreement than arithmetical computations.8
This is the condition of mathematization. In other words deducting material hindrances would mean creating a set of ideal conditions such that abstract effects can be actualized in the concrete world. This must be the real reason for experiment in science. We shall return to this a little later.
This point can be appreciated in relation to the other very important distinction Galileo introduced, which eventually became a very popular theme of philosophical speculation, namely primary and secondary qualities. One might think that this distinction is necessary for accomplishing Galileo’s program. But we will show below that this distinction does not play the said role of finding out mathematizable properties, and therefore has no methodological significance. The two famous passages from the The Assayer clearly indicating the distinction are as follows:
Now I say that whenever I conceive any material or corporeal substance, I immediately feel the need to think of it as bounded, and as having this or that shape; as being large or small in relation to other things, and in some specific place at any given time; as being in motion or at rest; as touching or not touching some other body; and as being one in number, or few, or many. From these conditions I cannot separate such a substance by any stretch of my imagination. But that it must be white or red, bitter or sweet, noisy or silent, and of sweet or foul odour, my mind does not feel compelled to bring in as necessary accompaniments. Without the senses as our guides, reason or imagination unaided would probably never arrive at qualities like these. Hence I think that tastes, odours, colors, and so on are no more than mere names as far as the object in which we place them is concerned, and that they reside only in the consciousness. Hence if the living creature were removed, all these qualities would be wiped away and annihilated. But since we have imposed upon them special names, distinct from those of the other and real qualities mentioned previously, we wish to believe that they really exist as actually different from those.9
To excite in us tastes, odours, and sounds I believe that nothing is required in external bodies except shapes, numbers, and slow or rapid movements. I think that if ears, tongues, and noses were removed, shapes and numbers and motions would remain, but not odours or tastes or sounds. The latter, I believe, are nothing more than names when separated from living beings, just as tickling and titillation are nothing but names in the absence of such things as noses and armpits.10
Of the two lists Galileo gives the former is the list of primary qualities, while the latter is of the secondary qualities. The qualities included under the head of primary qualities is very revealing of the non-Platonic position of Galileo. To be in space and time, and being in motion are considered primary qualities and the objects of scientific knowledge. Let us recall that for Plato the objects of episteme are Forms, which are not located in any space, are eternal, and since they are Beings, they do not become, so no change and no motion can be attributed to them. It should also be noted that the primary qualities are about corporeal and not incorporeal ‘substance’, and therefore there is no doubt that Galileo is quite unlike Plato.
Secondary qualities (like tastes, odours etc.) are not in bodies which do have certain other qualities called primary qualities (like shapes, number, motion etc.) to excite in us the experience of the former. Primary qualities are considered to be some sort of causes impinging in us the sensations. What is given to us in our consciousness is therefore considered as effects due to the senses and what is not immediately (directly) given to us are the independent things of the world, because Galileo says shapes, numbers and motions would remain even if our senses were removed.
As far as Galileo’s program of mathematizing the real world is concerned, primary qualities are indispensable, and secondary qualities dispensable. However, this relationship between primary qualities and mathematizable qualities is unwarranted. When Galileo talks of deducting the material hindrances, one may say, he certainly has in mind the secondary qualities. Let us look at Galileo’s analogy. (See the quotation above.) For the purpose of determining the amount of sugar in a warehouse a clerk neglects (deducts) the contingent facts, such as the sugar is in bags, or in boxes, or in open containers and so on. If he wants to measure the weight, size, shape etc. of the container, though primary, are to be deducted. Is this ‘deduction’ based on any water tight compartmentalization of primary and secondary qualities or does it depend on any other factor? If one wants to measure some thing by volume, some other factors should be eliminated than those mentioned above, and if one wants to consider the geometrical forms, both volume, weight, along with others become eliminable. Therefore in the process of applying mathematics to the world, the principle of deducting material hindrances can be employed only as a way of approaching the measurable, and what gets deducted depends on what quality one desires to measure. Galileo cannot be right if he says that only primary qualities are measurable. Butts also criticizes him for grouping all sensory qualities as secondary, and therefore not measurable and also for holding that secondary qualities are not in the object. He argues that Galileo can be right that motions are the cause of heat, and still be wrong that the heat in no sense exists in the object, e.g., the boiling water. Certainly the thermometer measures something, and it is not a something that exists merely as a potentiality to produce a sensation of heat in a perceiver.11
We therefore think that the distinction between primary and secondary qualities is not necessary for finding measurable qualities, and therefore for mathematization. It is incorrect, therefore, to confuse material hindrances with secondary or sensory qualities. The above example shows that even relational qualities like mass, volume etc., which are clearly primary qualities according to Galilean criteria, can also become hindrances if what we want to measure is say shape or number or something else. Therefore we conclude that this distinction has no methodological significance. What is significant for the program of idealization is deducting hindrances, not necessarily material hindrances. What counts as a hindrance cannot be stated in certain terms.
In this connection it is important to consider another distinction that Galileo makes between extensive and intensive modes of knowing.
[H]uman understanding can be taken in two modes, the intensive or the extensive. Extensively, that is with regard to the multitude of intelligibles, which are infinite, the human understanding is as nothing even if it understands a thousand propositions; for a thousand in relation to infinity is zero. But taking man’s understanding intensively, in so far as this term denotes understanding some propositions perfectly, I say that the human intellect does understand some of them perfectly, and thus in these it has as much absolute certainty as Nature itself has. Of such are the mathematical sciences alone; that is, geometry and arithmetic, in which the Divine intellect indeed knows infinitely more propositions, since it knows all. But with regard to those few which the human intellect does understand, I believe that its knowledge equals the Divine in objective certainty, for here it succeeds in understanding necessity, beyond which there can be no greater sureness.12
Undoubtedly such pronouncements must have played a very fundamental role in Galilean days, when humanism was on the rise. The message is clear: Human beings can know Nature as perfectly as God. This would make a clearly different kind of response to the Sophists’ challenge that we have discussed above, and is quite non-Platonic. Extensively we may never be able to exhaust all the variety of nature. Since extensive knowledge is based on non-mathematical qualities, and if each such quality refers to some essence of a thing then there are as many essences as there are qualities. Since there is no limit to kinds of things, complete knowledge of them is impossible. To this extent the Sophists should have agreed with him. But the intensive knowledge of mathematical objects is possible. Plato’s response looks similar to Galileo in the sense that both of them thought that true knowledge is about mathematical objects. However, as already mentioned above, Galileo’s mathematical objects are different from Plato’s. But it should be remembered that Galileo is not here responding directly to the Sophists; he is arguing against the Aristotelians who believed that episteme is about these innumerable essences, and that the knowledge of them is possible. This is a clear departure from Aristotelianism. If these observations are correct they should indicate sufficiently that Galileo is neither Platonic nor Aristotelian, but is original in many ways.
Galileo’s opposition to Aristotelian essences becomes more clear in his letter to the Jesuit mathematician, who denied that the sun-spots could be on the sun itself, for as the most luminous of bodies the sun could not generate its opposite, darkness. Galileo bursts out at him - as though things and essences existed for the sake of the name, not the names for the sake of the things. He writes that he does not find any advantage in understanding the essences of substances.
If I ask about the substance of the clouds, I am answered, they consist of a damp mist; if I wish to know further what this mist is, so I am taught perchange that it is water rarefied through the force of warmth. If I remain in my doubt and wish to know what water really is, in all my investigations I will only learn in the end that it is that fluid which runs in streams and which we continually touch and taste: a knowledge which to be sure enriches our sense perception, but leads us no further into the interior of things than the notion I had of clouds to begin with.13
Our knowledge of nearby objects is not more than that of distant objects like the moon and the sun. But with respect to intensive knowledge, our knowledge of the celestial objects is better than that of nearby objects.
For do we not know the periods of the planets’ revolutions better than the different tides of the sea? Have we not grasped the spherical form of the moon much sooner and more easily than that of the earth?14
Having denied importance to the extensive mode of knowing Galileo chose the intensive mode of knowing.
The objects of knowledge of this intensive mode are relational forms of things, “their position, their motion, their form and size” etc., and are therefore mathematizable or measurable. Characterization based on certain relational qualities has “absolute certainty as Nature itself has”. We are reminded of Aristotle’s desire to know things as clearly as they are known by nature. However, as we just observed, they differ on the issue of what are the objects that are known by nature. Aristotle thought we can know the essence of things, which is his object of knowledge, i.e., by extensive knowledge, while Galileo thought that we can understand Nature’s language better by intensive knowledge. The objects of scientific knowledge have clearly undergone a transformation. Succinctly we may say that the nature of transformation with respect to Aristotle is from qualitative episteme to quantitative episteme, and with regard to Plato it is from absolute Forms to relational Forms, which includes dynamic and static relational forms.
According to Galileo’s version of the joint method of analysis and synthesis, the scientist begins with a hypothetical assumption. The hypothesis does not come immediately from observation and the measurement of facts, but rather from an analysis of the mathematical relations involved in a given problem. Only after the mathematical relations involved in the initial hypothesis have been demonstrated by the method of composition, does it possess a quantitative meaning and implication that it can be compared and measured with observations and experiments.15 What is involved in mathematical analysis? Galileo illustrates the method of mathematical analysis, thus:
When ... I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of velocity, why should I not believe that such increase takes place in a manner which is exceedingly simple and rather obvious to every one? If now we examine the matter carefully we find no addition or increment more simple than that which repeats itself always in the same manner. This we readily understand when we consider the intimate relationship between time and motion; for just as uniformity of motion is defined and conceived through equal times and equal spaces (thus we call motion uniform when equal distances are traversed during equal time-intervals), so also we may, in a similar manner, through equal time-intervals, conceive additions of velocity as taking place without complication, thus we may picture to our mind a motion as uniformly and continuously accelerated when during any equal intervals of time whatever, equal increments of velocity are given to it.16
The mathematical analysis of the problem first consists in understanding “the intimate relationship between time and motion”. Then, motion is “defined and conceived through equal times and equal spaces” arriving at the definition of uniform velocity: a motion is uniform when equal distances are traversed during equal time intervals. One might ask ‘Why define uniform motion?’. It could not have been because Galileo thought motion is always uniform. But because uniform motion is a simple kind of motion, which can be defined and experimentally realized for empirical study. Similarly, i.e., in the same simple manner, he defines uniform acceleration. Thus acceleration and velocity have a specific definition, and as a result a specific meaning. Then having obtained the definitions, he postulates hypothetically the law of free fall, which is a statement that asserts that the distance increases proportionally to the square of time. For this he gives a plausibility argument that “we find no addition or increment more simple than that which repeats itself always in the same manner”. Why square of time, why not simple proportionality? Galileo did not arrive at this without false starts. In the initial stages he never analyzed the matter in terms of acceleration. Since the details are presented in the case study, we shall postpone further discussion till the substantial details are also available. It is sufficient to observe here that there were many false starts before he could finally arrive at the law.
Thus the most important phase in the context of discovery is first to have clear and precise definitions of measurable (mathematical) parameters of a phenomenon, such as velocity and acceleration, in terms of certain other measurable parameters, such as space and time. This phase is the initial mathematical analysis, followed by an hypothetical assumption. Based on definitions and assumptions certain theorems (consequences) are proved to demonstrate the internal coherence. Galileo spends a lot of time in his later works proving a number of theorems, explicating the semantic content of the assumptions and definitions. This is the complementary mathematical synthesis. Then Galileo proposes that the hypothesis be verified by experimental observations.
After completing the method of mathematical resolution and composition, which is based on definitions, the mathematically demonstrated hypothesis can now be compared and measured with observations and experiments.
If experience shows that such properties as we have deduced find confirmation in the free fall of natural bodies, we can without danger of error assert that the concrete motion of falling is identical with that which we have defined and assumed; if this is not the case, our proofs still lose nothing of their power and conclusiveness, as they were intended to hold only for our assumptions - just as little as the propositions of Archimedes about spirals are affected by the fact that no body is to be found in nature that possesses a spiral motion.17
Here Galileo is more than clear that the theoretical analysis based on definitions and assumptions (hypotheses) has its own value, whether we actually find them in reality. Here lies the significance of mathematical physics per se. If we can find the mathematical objects, properties of which are well known in the concrete world, then and only then we “can without danger of error assert” that the world is as we have defined and assumed in the definitions and assumptions. Even if we cannot find the counterparts of such theoretical objects in the actual concrete world, the knowledge of the defined object would not be entirely worthless. The demonstration, therefore, will be valid whether or not an application is found. However, Galileo is not for complete theoretical research without caring to verify it empirically.
Why is Galileo introducing a new experimental method, apart from the mathematical method? Why is it that only experimental observation and not mere observation can demonstrate mathematical hypotheses? This is because it is only in an experimental situation, which tries to mimic ideal conditions, that idealized mathematical propositions can hold, however approximately. This is to create an ‘environment’ where the material hindrances are deducted. The affine manner in which the mathematical objects are constructed and demonstrated cannot be obtained in the world of ‘open’ experience. Therefore we need to construct a ‘closed’ experimental world free from material hindrances. Thus Galileo felt the simultaneous need of both mathematical and experimental methods of scientific investigations.
This picture of Galileo’s methodology appears to have affinities with the hypothetico-deductive methodology proposed in the beginning of the twentieth century by Popper, Hempel etc. However in Galileo’s method definitions are given more fundamental status than hypotheses, for the latter are formed on the basis of pre-constructed definitions. Thus the origin of hypotheses has a clear basis, unlike in Popper’s view where any basis is denied. The problem however still persists, because it is not clear how one would construct definitions. We discuss the role of inversion in constructing definitions in Chapter 6. Though Galileo is not entirely explicit, he gives clear clues about how he constructs them, after which he postulates hypotheses. Here he would make use of the idea of inversion, and therefore we will postpone the details to a latter part of the thesis. We are content here to state that Galileo too believed in the methodological theme of analysis and synthesis.
We have observed in the beginning of this section that Galileo is linking two methods together: the Euclidian method of analysis and synthesis and the Aristotelian method of resolution and composition. Randall Jr. (1940) argued that Galileo is influenced by the Aristotelians of the school of Padua. Gilbert (1963) argued in response to Randall’s thesis that he is influenced by the Greek mathematicians. We think that both these claims are true. We have observed above that he differs with both Plato and Aristotle in a significant manner. The Greek mathematicians on the other hand clearly influenced Galileo, but they were not concerned with the philosophical problems for supporting either mathematical or experimental physics. The Aristotelian influence is also clear in his desire to solve specific problems. We should therefore understand him as a great blending character. Peter Machamer (1978) correctly observes that Galileo belongs to a tradition of mixed sciences.
The tradition is that of the mixed sciences, which is itself a tradition blending mathematics and physics (or natural philosophy), blending Platonic (or neo-Platonic) and Aristotelian elements, blending reason and observation.18
Galileo’s method has another feature that requires special mention, and this is also a major point of difference between Galileo and Descartes. Galileo is not only interested in pure mathematical mechanics per se. He is interested in those principles that are exemplified in nature. For that he fixes his subject matter by defining the natural phenomena to be studied.
And first of all it seems desirable to find and explain a definition best fitting natural phenomena. For anyone may invent an arbitrary type of motion, and discuss its properties; thus for instance some have imagined helices and conchoids, as described by certain motions which are not met with in nature, and have very commendably established the properties which those curves possess in virtue of their definitions; but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions.19
The first remark clearly suggest that he is not inclined to do pure mathematics like the Greek mathematicians. Here Galileo is clearly referring to Archimedes, whose work on helices and conchoids in geometry is well known. This should not be taken to mean that Galileo is against pure mathematics, but he is appealing to complement definitional knowledge by applying it to the actually occurring phenomena. Galileo’s ultimate interest is to define natural phenomena in analogous terms with mathematical objects. His attempt is to apply mathematics in the world of natural phenomena. These remarks suggest that though Galileo is following the Greek mathematicians, he followed them with a difference. And this difference, we think, consists in Galileo’s interest in local problems. Let us recollect that Aristotle also thought that fixing the subject matter is a crucial feature of natural science, contrary to the Platonic idea of Universal science. Descartes is a Platonist on this issue, while Galileo is not. Both Descartes and Plato, it well known, are great system builders. They believed and attempted to systematize science in a architectonic manner. Descartes thought that Galileo’s approach was piecemeal; he wanted to construct science not from merely plausible hypotheses but from indubitable clear and distinct first principles as foundations. He accuses Galileo of having built mechanics without foundation.
I find that in general he philosophizes much better than the usual lot for he leaves as much as possible the errors of the School and strives to examine physical matters with mathematical reasons. In this I am completely in agreement with him and I hold that there is no other way of finding the truth. But I see a serious deficiency in his constant digressions and his failure to stop and explain a question fully. This shows that he has not examined them in order and that, without considering the first causes of nature, he has merely looked for the causes of some particular effects, and so has built without any foundation.20
Descartes appreciates Galileo’s inclination to mathematics, but demands greater rigor, for Galileo did not, as Descartes thought, “stop and explain a question fully.” While it is true that Galileo did not stay forever in the mathematical world, he cannot be accused for not having answered or explained a question fully. Insofar as specific contributions towards the science of motion are concerned, Galileo succeeded better as compared with Descartes. Descartes’ contributions to mathematical analysis are undoubtedly more sophisticated than Galileo’s, but it is not legitimate to accuse Galileo for his inclination to solve specific problems. One of the characteristic features of modern science, that comes out clearly in the studies of T. Kuhn also, is that it develops by attempting to solve local problems. Descartes, we think, has failed to see the significance of solving ‘petty’ problems.
Towards the end of the above passage, Descartes criticizes Galileo because the latter looked always “for the causes of some particular effects” without any foundation. Here also Descartes’ understanding of Galileo has to be questioned. Because for Galileo, the cause-effect relation is not central, as it is in Aristotle’s physics. He is interested in the mathematical relationship between the relevant measurable parameters of the phenomena under study. Traditionally there has been too much emphasis on the cause and effect relation in the philosophical accounts of science, as is evident from the writings of Aristotelians. We will see in Part-III that the distinction between cause and effect is not central to the Galilean approach. It is rather well known that Galileo did not so much look for causes of motion, but emphasized mathematical (functional) relationships between different measurable parameters. Though, later Newton returns to the question of causes of motion, his notion is functionally defined, unlike Aristotle’s notion of cause and effect, which none in the 17th century accepted. In a functionally defined causal relation, the cause and effect can be interchanged, or reversed. We will see below that this reversibility is due to the invertible relation or symmetry of most mathematical relations. It is well known that the theoretical knowledge of science has a meta-theoretical property called symmetry. If theoretical knowledge had been grounded on Aristotle’s notion of cause and effect which is necessarily asymmetrical, mathematical physics would not have been possible.
Before we look at Descartes joint method, we shall summarize the above discussion. Galileo defined new objects of scientific knowledge as relational properties of measurable dimensions--another possible response to the Sophists’ challenge--and accordingly devised a new joint method, which we have characterized as mixed, for it contained both mathematical and experimental components. He differed significantly from both Platonic and Aristotelian thought, and also in a subtle manner from the Greek mathematicians--his thought is unique and original. His main problem was to find applications of mathematical knowledge to natural phenomena.
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 reaching 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 dimension of, say, mass is multiplied with the dimension of motion (velocity) no quantification 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 towards 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 apprehension 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 simple qua 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 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 contains 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 considered 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 is maxime absolutum. From the discovery of the maxime 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 from a priori truths remains 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 matter behave. ... Not content to leave anything unexplained, Descartes departed from his usual devotion to clear and distinct ideas and advocated the use of intermediate theories (less general than the first principles, but more general than the phenomena), which were sufficiently explicit to permit the explanation of individual 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.
Galileo’s second important successor Newton was closer to him in the sense that he is also a member of the mixed tradition. He tried to keep a proper balance between an unlimited confidence in mathematics unchecked by experience, and mere experimenting unaccompanied by mathematical analysis and demonstration.28 His statements on method, therefore, sounded much like Galileo. He gave his method more experimental coloring than Galileo had done, for the latter did not feel the need to check by observation mathematically deduced consequences. For Newton the logical inclusion of a proposition within a deductive system was not a sufficient proof of its ‘truth’. As rightly pointed out by Randall, the experimental analysis of instances in nature forms a part not only of the method of discovery but also of the verification.29
In the Opticks appears Newton’s classic statement of the joint method of analysis and synthesis, with its experimental fervor.
As in mathematics, so in natural philosophy, the investigation of difficult things by the method of analysis, ought ever to precede the method of composition. This analysis consists in making experiments and observations, and in drawing several conclusions from them by induction, and admitting of no objections against the conclusions, but such as are taken from experiments, or other certain truths, for hypotheses are not to be regarded in experimental philosophy. And although the arguing from experiments and observations by induction be no demonstration of general conclusions; yet it is the best way of arguing which the nature of things admits of, and may be looked upon as so much stronger, by how much the induction is more general. And if no exception occur from phenomena, the conclusion may be pronounced generally. But if at any time afterwards any exception shall occur from experiments, it may then begin to be pronounced from compounds to ingredients, and from motions to the forces producing them; and in general, from effects to their causes, and from particular causes to more general ones, till the argument end in the most general. This is the method of analysis: and the synthesis consists in assuming the causes discovered, and established as principles, and by them explaining the phenomena proceeding from them, and proving the explanations.30
It may be noted that the term ‘analysis’ is used to refer to the experimental and empirical context, unlike the modern usage of the term to the logical and deductive context. Accordingly the term ‘synthesis’ refers to deductive proof. The terms are used to refer to the same contexts as in the Aristotelians of the School of Padua at Italy, as elaborated above. This terminological inversion, as indicated above, must be due to the later linguistic orientation of philosophers, specially after Kant. It is typical, for Aristotelians, to consider the effects or phenomena as complex, therefore to be analyzed until they reach the causes, which are regarded as simple. The later modern philosophers use the term ‘analysis’ mostly to denote the logical movement from the more general statements to the more specific statements, while inductive movement from specific to general statements is regarded as synthetic. This inversion of terms demands historico-philosophical explanation. Again, we are afraid, we cannot meet the demand here, but must remain content with the observation.
The events mentioned in the method of synthesis, though include induction, are not mere simple unidirectional inductive movements. But it is characterized as dialectical, i.e., checking errors and collecting instances, ultimately arriving at the general. It is the well known view of Newton that in this context hypotheses should not be brought in. So much has been written, which is ridden with confusion regarding Newton’s cryptic views on the role of hypotheses, we shall not add anymore to it. However, it should be noted, that it is typical of the scholars of that period to believe in only those postulates that are ‘deducible’ from given experience. If Descartes allowed in the last resort some room for hypotheses, it is not because it is desirable to have them, but because we have nothing better than them. However the difference between Newton and Descartes should be noted. Newton wanted that the principles be ‘induced’ experimentally, while Descartes’ earlier program was to deduce them from the clear and distinct principles. Thus the nature of the kind of reason they have envisaged is qualitatively different. Now for Galileo, as elaborated above, the first step was to construct the definitions, and then the hypotheses. Considering the deficiencies of both inductive and hypothetico-deductive methodologies that developed, it is Galileo’s position that needs to be reconsidered. In the view that we are going to defend, constructing definitions will be considered the first step in the context of discovery.