The paper will appear in:
(Ed.) Kuokkanen, M., Idealization, Structuralism and Approximation, 
(Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. 42 [1995])
Amsterdam/Atlanta, GA.: Rodopi.

Leszek Nowak
Department of Philosophy, Adam Mickiewicz University

REMARKS ON THE NATURE OF GALILEO'S METHODOLOGICAL REVOLUTION

The present writer has already had an opportunity to conjecture that the methodological breakthrough made by Galileo consisted in introducing the method of idealization and concretization to physics (1971, 1980, pp.35-36). In this paper, I would like to transform this loose conjecture into a hypothesis, that is, to substantiate it more thoroughly and to analyze its explanatory power.

 1.

     There is a consensus among philosophers of science that the methodological breakthrough in the natural sciences dates back to Galileo. According to an old stereotype, this breakthrough is taken to have consisted in Galileo's rejection of the a priori dogmas of Aristotle's physics thus allowing the observation of the world unprejudiced by any assumptions. "It is by relying on observation that Galileo became the founder of modern physics" states Ernest Mach (1976, p.101). The legend has it that Galileo has used the tower of Pisa to drop various heavy objects to the ground and proved - contrary to Aristotle - that they fall with an equal speed [i].

    Today, it is easy to correct the legend. As the historians of science made evident, Pisa's tower had been used not by Galileo but by Coresio, an Aristotelian, in order to prove that the heavier the objects the quicker they fall[ii]. What is more important, Galileo's breakthrough did not consist in returning to the observation of nature. This would have been superfluous because the Aristotelian theory was always very closely connected with everyday common-sense experience. The basic type of argumentation in Aristotle's Physics was a reference to everyday observation. It is not a systematic theory but a totality of explanations that take their origin in observations available to everybody (Cohen (I.B.) 1960, pp. 22ff, Crombie 1959, p.101):  

Aristotle's physics originates from common sense data; it builds an extraordinarily coherent unity from them. Those data or facts which constitute its foundation are extremely simple, and so obvious that they could create the basis for physics based on naive realism in every epoch (Lesniak 1968, p.xviii).

And Galileo was fully aware of this claiming that "Aristotle... preferred sensory experience to any considerations" (Galileo 1962, p.225). Indeed, there was no reason for Galileo to bring Aristotle closer to the facts. He was quite close to them. According to Aristotle, there are two types of motion: natural and forced. All bodies move toward the center of the world (identified with the center of the Earth) naturally. Each movement in any other direction is unnatural and requires an explanation by revealing a force responsible for it. A body moves as long as the moving force operates; when the force ceases to act, the body falls down to the Earth or finds itself in the state of rest. Observation of moving objects teaches us that two factors influence the movement of bodies: external force and resistance of the environment. In order to make objects move, external force has to be greater than resistance. If the resistance of the environment is constant, then the velocity of a body will be directly proportional to the moving force. The question of how objects behave when there is no resistance was never answered by Aristotle, for first it should be made clear what is a vacuum. And vacuum has never been noticed within this world (Aristotle 1968, 216b).

Galileo knew very well, too, that vacuum does not exist in the physical world. This, however, did not prevent him from asking a question which was senseless in Aristotle's physics: how will "the perfectly round ball move on the plain which is smoothly leveled in order to eliminate all external and accidental obstacles" upon an assumption that the "resistance arising when the ball makes its way and all other obstacles that could arise" are not considered at all? (Galileo 1962, p.155).

In order to raise this question several assumptions had to be adopted:

(1)   (a) the rolling ball is perfectly round,

(b) the plane is ideally smooth

(c) the plane is perfectly spherical

(d) the resistance of environment equals zero.

These are obviously idealizing assumptions - i.e. ones consciously adopted as deforming the empirical reality for simplification goals - and therefore all the considerations conducted under them lead to idealizational statements, i.e. conditionals necessarily possessing some idealizing conditions in their antecedents (more about these notions cf. my 1972, 1980, Chap. VIII)[iii]. Galileo believed that four conditions (a-d) are equivalent to the claim that no external forces affect a moving ball. Therefore, he was in a position to claim that if all external obstacles were removed, the movement would last as long as the plane extended, going neither downwards nor upwards. Therefore, if space was infinite, the movement would also be limitless and hence infinite (Galileo 1962, p.158).

His answer to the above quoted question was then his law of inertia: (G) "Imagine any particle projected along a horizontal plane without friction; ... this particle will move along this same plane with a motion which is uniform and perpetual, provided the plane has no limits" (Galileo 1963, p.234)    which may be read as a conditional: (2) if x is a rolling ball which is perfectly round, projected along an ideally smooth and perfectly spherical unlimited plane and the resistance of the environment exerted upon x equals zero, then x moves along this plane with uniform perpetual motion.    Galileo was aware of the idealizational nature of thesis (2) as he commented further on:

Even horizontal motion which, if no impediments were present, would be uniform and constant is altered by the resistance of the air and finally ceases (1963, p.242).

Why then formulate such a statement? For Galileo the reasons for formulating (2) are twofold.

The first way of using the law of inertia is based on the observation that although the movement described in (2) ceases due to the resistance of the air, "the less dense the body, the quicker the process" (ibid.). And conversely, the more dense the body, the slower does the movement disappear. Therefore, there is a possibility to approximate (2) to the movement of sufficiently dense bodies. In particular, the movement of celestial bodies is, as Galileo stresses, similar to the movement described by the law of inertia (ibid., p.251). Another circumstances in which the law of inertia is approximated is the state of balance among the forces. Galileo considers the case of a body falling toward the Earth from a great height which originally moves with a uniformly accelerated motion but after some time the resistance of the air balances "the natural acceleration downwards common to all bodies". Then the body is supposed to move with a approximately uniform motion. This case does not strictly satisfy but merely approximates the law of inertia as there are still numerous forces affecting the body under consideration. For

if we consider only the resistance which the air offers to the motions studied by us, we shall see that it disturbs them all and disturbs them in an infinite variety of ways corresponding to the infinite variety of form, weight, and velocity of the projectiles (Galileo 1963, p.242).

 That is why in the case of balance of forces one can at most expect an approximation to the law of inertia.

The second way of using the law of inertia is based on the presupposition that the formula (2) can be corrected for some types of motions that occur in the conditions that are far away from the ideal circumstances postulated in (2). As Galileo goes on to say immediately after the above quoted formulation of the law of inertia:

But if the plane is limited and elevated, then the moving particle, which we imagine to be a very heavy one, will on passing over the edge of the plane acquire, in addition to its previous uniform and perpetual motion, a downward propensity due to its own weight; so that the resulting motion which I call projection, is compounded of one which is uniform and horizontal and of another which is vertical and naturally accelerated (1963, p.234).

And he proves (ibid., p.235ff) a theorem stating that such a movement may be described as semi-parabolic. The proof is based on the "superposition of two different states", namely, the inertial force "which if acting alone would carry the body at a uniform rate to infinity, and the velocity which results from a natural acceleration downwards common to all bodies" (ibid., p.207). This theorem may be read as the statement (ibid., p.235): (3) if x is a rolling ball which is perfectly round, projected along an ideally smooth and perfectly spherical but limited and elevated plane so that x is carried with a naturally accelerated vertical motion and the resistance of the environment exerted upon x equals zero, then the path of motion of x is semi- parabola.    (3) may be claimed to be a concretization of (2): some of the idealizing assumptions are waived -- they are replaced with their realistic negations and appropriate corrections to the consequent of the conditional are introduced (see more about this notion in my 1980, Ch. VIII). Then, according to Galileo, the idealizational law of inertia may be applied to actual cases either by approximation or by concretization[iv].

Let us add that, as is well known, Galileo's formulation of the law of inertia is not, in the light of Newton's mechanics, entirely correct (e.g. Krajewski 1974, Such 1978). Galileo's law of inertia was still based on some elements of Aristotelian conceptual background (shared also by another scientific revolutionary, Copernicus - see Dingle 1959, pp.24/25). First and foremost the notion of movement is a bit burdened by the notion of natural movement: Every body - states Galileo - which finds itself in the state of rest but is left to itself and able to move - begins to move, if the Nature offers it an inclination to reach a definite place (Galileo 1962, p.19). Hence the movement counts not from a place on but toward a place. For this reason Galileo rejects the notion of infinite movement:

Motion along the straight line is in its very nature infinite since the straight line is infinite...It is then inconceivable to claim that something moves along the straight line, that is, to the goal which is unreachable (ibid., p.18).

Galileo postulating the movement of an ideal object rolling infinitely on an ideally smooth surface presupposed at the same time that this surface is at each point equidistant from the center of the Earth. His law of inertia states that a body moving with a constant velocity around a circle does not stop until external forces begin to operate. This is the reason why Galileo regarded the motion around a circle as natural. A paradigmatic example was for him the case of a ship that could move around the Earth without stopping, if no external obstacles appear. Briefly, for Galileo the movement around the circle could be an inertial motion, for Newton, it could be not; for Newton the movement around the circle is instead an accelerated motion which is preserved due to a certain force. These are, however, substantive limitations of Galileo's discovery. As far as their methodological background is concerned one may say that the very method with which he invented his law of inertia was a real novelty in comparison to the then widespread methods of investigation. He gave up Aristotelian inductivism to adopt a method of making assumptions "even if they [are] not strictly true" (Galileo 1963, p.241). The Galilean breakthrough consisted then in systematically imagining what a given phenomenon would be like if the factors considered to be secondary did not act upon this phenomenon at all. And that is what was typical for the innovation Galileo brought into the body of methods applied in the natural sciences[v]. Galileo systematically applied the method of idealization. That was the real sense of the revolution in the natural sciences which bears his name[vi]. Let us look at his method a little more carefully.  

2.

 This is how Galileo presents his theoretical goal in the analysis of the phenomenon of free fall:

 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 motion (Galileo 1963, p.154).

The point is which features pertaining to the "observed falling bodies" are claimed to be essential and which secondary. "[T]he intimate relationship between time and motion" belongs to the former and due to it "we may picture to our mind a motion as uniformly and continuously accelerated when, during any equal intervals of time whatever, equal increments of speed are given to it" (ibid., p.155). More strictly,

The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances (ibid., p.167).

One of the secondary factors is the air resistance. It is a negligible influence because

it disturbs all [the movements] and disturbs them in an infinite variety of ways, corresponding to the infinite variety in the form, weight and velocity of the projectiles ... Of these properties of weight, of velocity, and also of form, infinite in number, it is not possible to give any exact description; hence, in order to handle this matter in a scientific way, it is necessary to cut loose from these difficulties; and having discovered and demonstrated the theorems, in the case of no resistance, to use them and apply them with such limitations as experience will teach (Galileo 1963, p.242).

Thus, Galileo adopts the idealizing assumption that the body falls in perfect vacuum, i.e. in the conditions in which the forces of friction equal zero. Apart from that Galileo adopted an assumption to the effect that the Earth's gravity g is constant. For him, it was simply a constatation of fact, that is a realistic condition. Galileo's formulation of the law of free fall may be reconstructed as follows:

(4)    if fb(x, e) & v⁰(x) = 0 & r(x) = 0 & g(e) = const, then s(x) = 1/2 gt²(x),

where 'fb(x, e)' means 'x is a body falling in the direction of the Earth (e)', 'v⁰' stands for the initial velocity, 'r' -- the forces of resistance of the medium, 's' -- a distance covered by the falling body, 'g' -- the constant of the Earth's gravitation, 't' -- the time of fall. Of these assumptions, two are realistic. One delineates the universe of discourse (fb(x, e)) and the other postulates that the Earth gravitation is constant. The remaining two postulate that the initial velocity equals zero (v⁰(x) = 0) and there is the lack of any resistance of the medium, i.e. the perfect vacuum (r(x) = 0)). They may be taken to be idealizing assumptions.

Taking into account the falling bodies whose initial velocity differs from zero one must waive the condition v⁰(x) = 0 and correct the Galilean equation[vii]:

(5)    if fb(x, e) & v⁰(x) > 0 & r(x) = 0 & g(e) = const, then s(x) = 1/2 gt²(x) + v⁰(x)t(x) .

The procedure of concretization of the law (3) with respect to the assumption postulating the free fall in the perfect vacuum is marked by Galileo only intuitively. He refers to the law of Archimedes (a body submerged in fluid loses as much weight as is the weight of fluid pushed aside by this body) and takes it that a body falling in air loses as much weight as is the weight of the air pushed aside by this body. As he claims,

"Assuming...that all falling bodies acquire equal speeds in a medium which...offers no resistance to the speed of the motion, we shall be able accordingly to determine the ratios of the speeds of ...bodies moving...through different space-filling, and therefore resistant media. This result we may obtain by observing how much the weight of the medium detracts from the weight of the moving body, which weight is the means employed by the falling body to open a path for itself and to push aside the parts of the medium... And since it is known that the effect of the medium is to diminish the weight of the body by the weight of the medium displaced, we may accomplish our purpose by diminishing in just this proportion the speeds of the falling bodies, which in a non-resisting medium we have assumed to be equal" (Galileo 1963, p.72).

 It is quite clear that the operation performed on the law of free fall (3) is one of concretization. It is, however, equally clear that it was done on an intuitive level.

Later Boyle succeeded to create the "pneumatical engine" as he called his air pump and to empirically determine that in the physical vacuum bodies with different shapes fall down with the same velocity, exactly as Galileo's law predicted (Newton 1962, vol.2, p.543). This was a(n approximate) confirmation of Galileo's law (Conant 1953, pp.52ff)[viii].

3.

The interpretation outlined above shares some traits with the two main interpretations of Galileo's method, viz. the Platonist and hypothetico-deductive. Both interpretations stand against the inductivist stereotype mentioned at the beginning of this paper but in quite opposite ways. According to the Platonist interpretation, the breakthrough made by Galileo consisted in the fact that he opened the "book of nature written in the language of mathematics" for the first time. This presupposes that "the world of thoughts and the world of phenomena correspond to one another[,]... that the laws...which as such embrace freely general notions and connections between notions, still have reality and validity in the Nature; in other words, that the reasonable is also real" (Snell 1858, p.41). Today, this view is held by Koyre (e.g. 1968) who claims that for Galileo the empirical reality is merely manifestation of eternal Platonic ideas and his alleged experiments were thought experiments making his abstract reasoning easier to get along. According to the hypothetico-deductive interpretation (Drake 1973, Shapere 1972), the role of actual experiments was absolutely crucial for Galileo's way of making science - putting forward hypotheses, deducing the observational consequences and testing the hypotheses with reference to the data: "At each stage of inquiry, sense experience must be combined with reasoning and with mathematics to afford a sound basis of deduction" (Drake 1972, p.266).

 The interpretation outlined above attempts to give justice to both the idea of discovering abstracts traits of reality in the language of mathematics - this is how the basic idealizational laws are found, and the idea of empirically testing the correctness of these discoveries - this is why the concretization of idealizational laws is indispensable. Still, it differs from both of them. On the one hand, idealizational laws are hypotheses about ideal worlds which may easily prove to be false in them. If magnitudes H, m and n influence F, then an idealizational law which states, say, an actual dependence of F on H but abstracts from n and fails to abstract from m would be false in a world characterized by F-facts and H-facts but lacking both m- and n-facts. Already this fact that idealizational statements can be false in the ideal domains determined by their idealizing antecedents suffices to distinguish them (e.g. Galileo's law of free fall) from mathematical theorems which are supposed to hold in the domains determined by their assumptions. On the other hand, the hypothetico-deductive method does not embrace the method of idealization. I have already had an opportunity to support this opinion (cf. 1980, Chap.V)., so I shall restrict myself here to one new argument. As is well known, Popper's scheme of testing a theory is that of modus tollens (1959, para.18). If law t implies basic (observational) statement b and if non-b, then non-t. Assume, however, that t is an idealizational statement of the form [n = 0  F = f(H)]. What it implies then is not b but an idealizational conditional [n = 0  b] which cannot be tested by direct observation; hence b cannot be possibly rejected on this basis. That is why, the operation of concretization is necessary. So, from [n = 0 F = f(H)] its concretization [n /= 0 F = f'(H, n)] is derived. If it implies [n /= 0  b], then, provided that n /= 0 is realistic, b can be obtained. A possible rejection of b can therefore ceteris paribus testify to the falsity of the initial premiss [n = 0  F = f(H)].

4.

 Obviously, the interpretation of Galileo's method outlined above is an interpretation, nothing more. How then could its possible superiority over the two rivals be proved? There is evidently no such way. There are, however, some ways in which such a superiority could be argued for. One of them is this. Let all the three readings of Galileo establish some methodological facts about Galileo's method. It is likely that all of them would agree that the three methods are characteristic of Galileo's way of doing research: (a) mathematical language, (b) experiments, (c) idealization. And they differ not in stating these historical facts but in the significance they attach to them. The question is now which of these three ways of reading Galileo's methodology has a greater explanatory power. It seems to me that from (a) one could perhaps derive (c), but not (b).[ix] As I tried to argue a moment ago, (b) is unable to explain (c) and I do not see how (b) could explain (a).

In contrast to the two alternatives, the idealizational interpretation can, I conjecture, explain both (a) and (b) in terms of (c). It will be seen that the idealizational statement [n = 0  F = f(H)] presupposes in its consequent a substantive instantiation of the mathematical formula X = f(X); in other words, the formula F = f(H) of this statement obtains by a substantive interpretation of the variables Y (as F ) and X (as H ) in Y = f (X). Similarly, the consequent of its concretization obtains by the substantive interpretation of the mathematical formula Y = f '(X, Z) which in a special case transforms into Y = f(X). In a way, idealizing conditions are necessary to deform the phenomena so that the idealized counterparts of those phenomena fall under the mathematical functions. Our world is not written in the language of mathematics but its idealized deformations are. That is why idealization has a logical priority before the application of mathematics in a given domain.

The significance of the experimental method can also be explained in terms of idealization. Consider the same idealizational law as above. If it is not known how n influences F, or if the mathematical formalism allowing to build f-expressions of two variables is not yet known at all, the concretization of this law is excluded. The best way of testing the law is then a rough approximation. It may happen, however, and often does, that in actual conditions the influence of n upon F is not negligible. But if conditions where it is negligible can be technically created, then this statement could be approximated: [E & n ~ 0  F ~ f(H)], where E is a condition limiting application of this statement to the experimentally created circumstances. One of significant functions of the experiment is then to (approximately) test idealizational laws.

In sum, if we hypothetically assume that it is idealization which is the core of the Galilean method, we become able to explain the other important components of his methodology, viz. the mathematization of his theory and the experimentalization of his research practice. And if I am correct that it would be difficult to explain the other two elements of Galileo's method from the point of view of either of the remaining alternatives, then it appears that the proposed interpretation has some advantage in comparison to them.

5.

The Galilean revolution consisted in making evident the misleading nature of the world image which senses produce. We only see phenomena which are the joint effect of all the relevant influences. As a result, senses do not contribute in the slightest to the understanding of the facts. In order to understand phenomena the work of reason is necessary which selects some features of the objects through idealization and in their idealized models recognizes some other features of the empirical originals. These models differ a great deal from their sensory prototypes, what is more, they present images of hidden relationships which could not be grasped with the aid of experience at all. Science idealizing phenomena opposes common sense:

experiences which clearly state against the annual movement - says Galileo - are seemingly so contrary to the theory that...I cannot find the words to express my admiration for Aristarches and Copernicus who managed to put reason into a frame which forced the senses to withdraw their trust in the apparent meaning of sensory data...[This proves how great is] the elevation of these minds which accepted these views and took them as true ones overcoming the testimony of their own senses with the quickness of minds and preferring that which reason dictated to what senses and experiments seemed to offer (Galileo 1962, pp.353-54).

[That is why] The philosopher-geometrician who wants to investigate in reality what has abstractly been proved should exclude the interfering influences of matter from the calculations (Galileo 1962, p.225).

This gap between the abstract world of laws and the world of senses can be filled with the aid of concretization which takes into account what has been previously abstracted from. Due to this, abstract laws become more and more realistic and the distance between them and the actual facts diminishes. Idealization and concretization constitute the essence of the method whose adoption in physics Galileo had initiated. This method had been systematically applied by Newton. Also the understanding of it had been deepened in Newton's Principia. But that is a separate story.

REFERENCES

Aristotle (1968).  Fizyka (Physics).  Warszawa: PWN.

Barr, W.O.  (1971).  A Syntactic and Semantic Analysis of Idealizations in Science.  Philosophy of Science, 38, 258 - 272.

Brzezinski, J.  (1985).  Consciousness: Methodological and Psychological Approaches (Poznan Studies in the Philosophy of the Sciences and the Humanities, 8).  Amsterdam: Rodopi.

Brzezinski, J., Fr.  Coniglione, T.A.F.  Kuipers and L.  Nowak (Eds.) (1990).  Idealization-I: General Problems (Poznan Studies in the Philosophy of the Sciences and the Humanities, 16). Amsterdam/Atlanta: Rodopi

Butterfield, H.  (1958).  The Origins of Modern Science 1300-1800. London: Bell&Sons (quoted after Polish translation, Warszawa 1968).

Cartwright, N.  (1983).  How the Laws of Physics Lie? Oxford: Oxford University Press.

Cartwright, N.  (1989).  Capacities and Their Measurement.  Oxford: Oxford University Press.

Cohen, I.B.  (1960).  The Birth of a New Physics. London/Melbourne/Toronto: Heinemann.

Cohen, R.S.  and M.  Wartofsky (1974) (Eds.).  Methodological and Historical Essays in the Natural and Social Sciences (Boston Studies in the Philosophy of Science, XIV).  Dordrecht/Boston: Reidel.

Conant, J.B.  (1953).  On Understanding Science.  New York: The New American Library.

Coniglione, Fr., R.  Poli and J.  Wolenski (Eds.) (1993).  Polish Scientific Philosophy: The Lvov-Warsaw School (Poznan Studies in the Philosophy of the Sciences and the Humanities, 28). Amsterdam/Atlanta: Rodopi.

Crombie, A.C.  (1959).  Medieval and Early Modern Science, vol.I. Cambridge: Harvard University Press (quoted after Polish translation, Warszawa 1960).

Dingle, H.  (1959).  Copernicus and the Planets.  In: Lindsay (1959), pp.18-27.

Drake, S.  (1972).  Galileo Galilei.  In: Edwards (1972), vol.3, pp.262-66.

Drake, S.  (1973).  Galileo's Discovery of the Law of Free Fall. Scientific American, 228, 84-93.

Edwards, P.  (Ed.), The Encyclopedia of Philosophy.  London/New York: Macmillan & the Free Press. 

Galileo, G.  (1962).  Dialog o dwoch wielkich systemach swiata, Ptolemeuszowym i Kopernikanskim (Dialogue on Two Principal Systems of the World, Ptolemeic and Copernican).  Warszawa: PWN. 

Galileo, G.  (1963).  Dialogues Concerning Two New Sciences (trans.  H.  Crew and A.  de Salvio). New York/Toronto/London: McGraw-Hill.

Harre, R.  (1981).  Great Scientific Experiments.  20 Experiments that Changed Our View of the World.  London: Phaidon Press.

Hartkaemper, A.  and H.-J.  Schmidt (Eds.) (1981).  Structure and Approximation in Physical Theories.  New York/London: Plenum Press.

Koyre, A.  (1955).  A Documentary History of the Problem of Fall from Kepler to Newton (Transactions of the American Philosophical Society, vol.45, part 4).  Philadelphia: The American Philosophical Society.

Koyre, A.  (1968).  Metaphysics and Measurement.  Cambridge, Mass.: Harvard University Press.

Kuokkanen, M.  (1988).  The Poznan School Methodology of Idealization and Concretization from the Point of View of a Revised Structuralist Theory Conception.  Erkenntnis, 28, 97-115.

Kuokkanen, M.  and T.  Tuomivaara (1992).  On the Structure of Idealizations.  In: Brzezinski and Nowak (1992), pp.67-101.

Kupracz, A.  (1990).  O dwoch ujeciach idealizacji w naukach empirycznych.  Proba analizy porownawczej (On Two Approaches to Idealization: A Comparative Analysis).  Poznan: Polish Academy of Science Press.

Kwiatkowski, T.  (1993).  Classification of Reasonings in Contemporary Polish Philosophy.  In: Coniglione et al.  (1993), pp.117-67.

Lesniak, K.  (1968).  Aristotle's Philosophy of Nature (Filozofia przyrody Arystotelesa).  In: Aristotle (1968), pp.IX-XXXI.

Lindsay, J.  (1959) (Ed.).  A Short History of Science: Origins and Results of the Scientific Revolution.  A Symposium.  Garden City.  N.Y.: Doubleday.

Ludwig, L.  (1981).  Imprecision in Physics.  In: Hartkaemper, A. and H.-J.  Schmidt (1981), pp.  7-19.

Mach, E.  (1976).  Knowledge and Error.  Dordrecht/Boston: Reidel.

Magala, S., L.  Nowak (1985).  The Problem of Historicity of Cognition in the Idealizational Concept of Science.  In: Brzezinski (1985), pp.18-35.

Newton, I.  (1962).  Principia (translated A.  Motte, revised by F. Cajori).  Berkeley: University of California Press.

Niiniluoto, I.  (1986).  Theories, Approximations and Idealizations.  In: R.B.  Markus, G.J.W.  Dorn, and P.  Weingartner (Eds.), Logic, Methodology and Philosophy of Science VII. Amsterdam: North-Holland, pp.  255-89.

Niiniluoto, I.  (1990).  Theories, Approximations and Idealizations.  In: J.  Brzezinski, F.  Coniglione, et al.(1990), pp.  9-57.

Nowak, L.  (1970).  O zasadzie abstrakcji i stopniowej konkretyzacji (On the Principle of Abstraction and Gradual Concretization).  In: Metodologiczne zalozenia 'Kapitalu' Karola Marksa (Methodological Assumptions of Karl Marx's 'Capital'). Warszawa: KiW, pp.  117-213.

Nowak, L.  (1971).  Galileusz nauk spolecznych (Galileo of the Social Sciences).  Nurt, 1, 39-41.

Nowak, L.  (1972).  Theories, Idealization and Measurement. Philosophy of Science, 39, 533-47.

Nowak, L.  (1980).  The Structure of Idealization.  Towards a Systematic Interpretation of the Marxian Idea of Science, Dordrecht/Boston/London: Reidel.

Piekara, A.  (1964).  Mechanika ogolna (The General Mechanics). Warszawa: PWN.

Pitt, J.C.  1991, Galileo, Copernicus and the Tides, Theoria et Historia Scientiarum, 1, 83-94)

Popper, K.R.  (1959).  The Logic of Scientific Discovery.  London: Hutchinson.

Ostwald, W.  (1908).  Grundriss der Naturphilosophie.  Leipzig.

Shapere, D.  (1972).  Newtonian Mechanics and Mechanical Explanation.  In: Edwards (1972), vol.5, pp.491-96.

Snell, K.  (1858).  Newton und die mechanische Naturwissenschaft. Leipzig: Arnold.

Such, J.  (1978).  Idealization and Concretization in the Natural Sciences (Poznan Studies in the Philosophy of the Sciences and the Humanities, 4).  Amsterdam: Gruener, 49-73 (translation of the Polish paper published in Studia Filozoficzne, 11/1972).

Suppe, F.  (1972).  What's wrong with the Received View on the structure of scientific theories?.  Philosophy of Science, 39, 1-19.

Wallace, W.A.  (1974).  Galileo and Reasoning ex suppositione.  The Methodology of the Two New Sciences.  In: Cohen/Wartofsky (1974), pp.79-104.

Copyright © 1995 by
Leszek Nowak
epistemo@main.amu.edu.pl