PART TWO: ENLARGING DEMONSTRATION
It would take us too far afield to comment on the medieval discussions of the nature of demonstration in the detail they deserve. Our main goal in this essay is to analyze the three primary patterns that have been proposed over the years for explanation/ inference in natural science. We have already seen one of these, and must rather summarily move on to the other two. But it is worth asking first about the extent to which the transition to other inference-patterns was already under way among medieval writers on the nature of science. To what extent did medieval commentators on the Posterior Analytics show an awareness of the difficulties involved in finding “commensu- /25/ rately universal” relationships, the only kind yielding demonstration proper? Did they, for example, relax the demand for premisses that would be seen to be not only true but necessarily true, once properly understood? Did they take steps to systematize the process involved in epagoge in order to ensure that the regularities discovered in nature would be genuinely universal ones? We shall see that on the whole the requirements of strict demonstration were not relaxed, but that significant efforts were made to grapple with the problems that these requirements imposed. We shall also see that some contemporary attempts to construe medieval enlargements of the doctrine of demonstration as actively preparing the way for modern accounts of scientific method are, though laudable in the generosity of their intention, considerably overstated.
Looking at the Middle Ages as a whole, one would obviously have to separate two quite diverse methodological traditions, the Aristotelian and the nominalist. And one would have to bear in mind the great diversity within both of those traditions themselves. The Aristotelians remained faithful on the whole, to the ideal of demonstration se down in the Posterior Analytics, while developing some aspects of that doctrine, the distinction between demonstrations propter quid and quia, for example, much more fully than Aristotle had /26/ done. The nominalists began to shape the notion of inductive generalization, entirely rejecting the notion of necessary connection between essence and property on which the older notion of demonstration had been based. The idea of causal explanation as tentative and “consequential,” resting in large part, that is, on the verified observational consequences drawn from it, only sporadically made its appearance and, among Aristotelians, never as any other than a stage on the way to demonstration proper. In lieu of a detailed historical treatment,
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we shall pick out for brief comment three of the leading representatives of the tradition of the Posterior Analytics, Grosseteste, Aquinas, and Zabarella, each of whom has a special interest for our story.
Grosseteste. Forty years ago, Alistair Crombie in a widely discussed book declared that:
As a result of their attempts to answer the Greek question: How is it possible to reach with the greatest possible certainty true premisses for demonstrated knowledge of the world of experience? the (sic) natural philosophers of Latin Christendom in the thirteenth and fourteenth centuries created the experimental science characteristic of modern times.34 /27/
This was the “continuity thesis,” announced earlier by the histo- rian/philosopher of science. Pierre Duhem, now proposed in its strongest possible form. More specifically, Crombie claimed that the appearance of Robert Grosseteste’s commentary on the Posterior Analytics (c. 1225), which made that difficult work accessible for the first time to Latin readers,35 in conjunction with Grosseteste’s own works on optics and other scientific topics, initiated a new approach to natural science. Grosseteste’s distinctive contribution was “to emphasize the importance of falsification in the search for true causes and to develop the method of verification and falsification into a systematic method of experimental procedure.”36
Later historians of the period have, on the whole, been unsympathetic to this reading of Grosseteste. In a recent study of that philosopher’s work, James McEvoy writes:
If a broadly adequate methodology be regarded as a necessary and quasi-sufficient basis for scientific advance, then it becomes at once essential and embarrassingly difficult to account for the relative scientific sterility of the late medieval and early Renaissance period, if one grants – and this is the force of Crombie’s main thesis – that the requisite methodology was already available from around 1240 onwards.37 /28/
To assess the force of this objection to Crombie’s thesis, one would have to distinguish between methodology, understood in terms of practical prescriptions regarding working procedure, and conception of science, taken to specify the sort of knowledge-claim that counts as science proper. Crombie was, of course, claiming originality for Grosseteste and his successors on both scores. It is only the second, his conception of science that concerns us here. Being right about one’ conception of science does not necessarily translate into practical success in one’s scientific work. Likewise, failure in the latter regard does not necessarily connote failure in the former, so we have to look at the matter more carefully. Did Grosseteste transform the Aristotelian notion of demonstration into something akin to the modern idea of experiment-based theoretical inference?
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To begin with, he certainly did stress the importance of experience as the basis for scientific knowledge, even more perhaps than Aristotle had done.38 His motive for this, it should be noted, was in large part theological. Because of sin, man’s higher powers were corrupted but the humbler power of perception, by standing firm in the rout, enabled the higher powers to recover their proper function of apprehending the essences of things.39 In practice, the features of the Posterior Analytics that he chooses to dwell on are often the empir- /29/ ical examples that Aristotle uses to such good effect. For example, when discussing the nontwinkling of the planets, he produces his own explanation for the twinkling of starlight: the greater the distance of any object from us, he suggests, the smaller the angle it subtends at the eye and thus the greater the strain on vision. This strain and the tremor it induces in the virtus visiva that goes out from the eye to the object causes the appearance of twinkling.40 When he discusses, at considerable length, the connections between lack of incisors and possession of multiple stomachs in horned animals, he simply brings together what Aristotle and some of his later commentators, like Themistius, have to say in different contexts about this topic.41 There is no new observation involved, and Grosseteste almost certainly had not himself seen all of the animals mentioned. He does, however, draw attention to a consequence that Aristotle had left unstated: though all horned animals lack upper incisors, not all those that lack upper incisors possess horns (camels and hinds, for example). Thus the properties are not mutually implicative, or to put this in the traditional way, the major premiss in any attempted demonstration of a causal relation between -the properties would not be convertible. In cases like that of the camel, Grosseteste adds (again following Aristotle), the animals have other means of defense so that horns /30/ are not needed. (This leaves the camel’s lack of upper incisors unexplained; Aristotle and Grosseteste try to make do with a teleological explanation for this by noting that the camel’s hard palate substitutes for the lack of incisors.)
Had Grosseteste been of a mind to challenge or revise Aristotle’s account of demonstration, this would have been an obvious context in which to do so. Another context would have been where he comments on a particularly difficult passage (II, 16 and 17) in the Posterior Analytics, and asks whether the same effect might not be known from experience to be explainable in terms of several different possible causes.42 Again, he forbears. The most illuminating contexts are those in which he discusses different “opinions” regarding the explanation of specific phenomena. Here one does find him suggesting that the “opinions” may be tested by seeing whether inferences drawn from them conflict with observation or general principle. Crombie draws attention to two such examples, in particular.
In one, Grosseteste reviews four “opinions” regarding the nature of comets and rejects all four of them on the grounds that their consequences are at odds with observation, as well as with “the principles of the special
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sciences.”43 He advances a fifth view on his own account: a comet is a fire of a transformed nonterrestrial sort linked to its /31/ present star or planet by an attraction akin to that between magnet and iron. Though the theory seems fanciful from our perspective, it is governed by the insight that comets share in the revolutions of the heavenly bodies and hence cannot be of a simply terrestrial nature. In another passage, he gives an example to illustrate how the universal is reached by the mind through “the ministry of the senses.” The taking of scammony (a purgative widely used in the ancient Greek world) is commonly accompanied by the discharge of red bile.44 Is the former the cause of the latter? We cannot see such a causal relation directly. But “frequent observation” of the two events as co-occurring leads us to suspect (estimare) a third factor which is itself not observable, that is, a causal relation between the two. The power of reason (ratio) then comes into play and suggests, by way of test, that scammony should be administered when all other (known) causes of red bile have been excluded from operating. If there is still a discharge of red bile in such cases, “a universal is formed in the reason” affirming the causal connection between the two sorts of occurrence, thus giving rise to a “universal experiential principle.”
Though Crombie may have been overenthusiastic in his original formulation of the continuity thesis, he was surely right to take passages such as these to presage a manner of reasoning in /32/ science that differs fundamentally from demonstration. But the extent of the change is by no means clear, as the last example shows. Scientific inquiry still terminates in the reason’s directly grasping a universal causal connection between the administration of scammony and the discharge of red bile. Does this claim to understanding continue to rest on the adequacy of the inductive procedures followed, the exclusion of other relevant causal factors and the testing of the invariability of the alleged correlation? In this case, how are we to know that the connection between the two kinds of occurrence is of an essential nature? Or, on the other hand, are we to suppose that the grasp of the causal connection becomes at this point a direct intuitive one, given that one has now an adequate familiarity with the concepts involved? These are the alternatives that faced Grosseteste and his successors in one guise or another. He did not himself, so far as one can discover from the texts, resolve in favor of what we may call modernity. But the fact that the issue itself can be so clearly documented in his work and in that of other later natural philosophers, is already a major contribution in its own right.
Aquinas. Thomas Aquinas was assuredly one of the most perceptive contributors to discussions of the nature of science in this period. His interests in the issue were, of course, primarily theo- /33/ logical and not physical. Unlike his mentor, Albertus Magnus, be devoted little of his abundant energies to empirical inquiries into the natural
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world. Nonetheless, he was well-versed in the natural science of his day, particularly in astronomy.45 His own most considerable “natural” work was his commentary on the Physics of Aristotle. The Physics, it should be remembered, is for the most part devoted to careful conceptual analysis. It begins from the ways in which certain very general terms are used in common speech and it systematically analyzes the consequences of these legomena.46 It ranks itself, therefore, G.E.L. Owen remarks, “not with physics in our sense of the word, but with philosophy. Its data are for the most part the materials not of natural history but of dialectic, and its problems are accordingly not questions of empirical fact but conceptual puzzles.”47 In terms of the kind of evidence it relies on, it belongs rather more to the tradition of Parmenides than to that of the “physicists,” as Aristotle calls them, despite its frequent references to the latter. Physics I , for example, seeks the necessary and sufficient conditions for the correct application of the term, “change,” and rests, not on a series of observations of different sorts of natural change, but upon the simple ability to use the term, “change,” correctly, an ability dependent /34/ on experience, of course, but only in the most general way.
Analysis of this kind falls naturally into a deductive pattern once the dialectical preparations are made. Aristotle sets out to prove his claims: that an actual infinite cannot exist, that motion cannot have had a beginning, and so on. But there are no demonstrations in the strict sense, since he is not dealing here with essences or natural kinds. The problems we have seen regarding twinkling stars and horned animals do not arise in “philosophy of nature,” as this genre later came to be called.” (An analogue of these problems may appear when causes are postulated, as for example in the case of continuing “violent” motion.) The most general terms in which we articulate our descriptions of the world around us (“motion,” “place,” “time”) are assumed to refer in a straightforward way, and are given precising definitions, if necessary. No need, then, to test hypothesis against specific observations, or the like. As long as one stays with “philosophy of nature” in this sense, the problems we have been discussing in regard to demonstration in natural science do not appear. But they are apt to emerge, in different guises elsewhere, as indeed they do for Aquinas.
There is a tension in his thinking, it has often been suggested, in regard to our knowledge of sensible things.48 On the one hand, he is emphatic /35/ in claiming that knowledge of the quiddities of natural things constitutes the appropriate object of the intellect: “The proper object of the human intellect is the quiddity of a material thing, [when that thing has been] apprehended by the senses and the imagination.”49 On the other hand, he also maintains that the essences of physical things are hidden from us: “Our power of knowing (cognitio) is so weak that no philosopher can ever fully discover the nature of a single fly.”50 “The essences of things are not known to us . . . .”51 Commenting on the text of Aquinas, Jacques Maritain
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expresses the contrast in dramatic fashion: Is it not scandalous that though our intelligence has as its connatural object
the essences of corporeal things, in face of them it meets such serious impediments that it has to be content, in a vast sector of our knowledge of nature, with the imperfect intellection we call “perinoetic?”52
The notion of an imperfect (“perinoetic”) understanding is prompted by texts like these: “In sensible things, the essential differences are unknown to us, so they are signified by accidental differences which originate from the essential ones, as a cause is signified by its effect.”53 “Substantial forms are of themselves unknown to us; we learn about them from their proper acci- /36/ dents.”54 “Since we do not know essential differences, sometimes . . . we use accidents or effects in their place, and name a thing accordingly.”55
“Since substantial differences are unknown to us, or at least unnamed by us, it is sometimes necessary to use accidental differences in their place . . . for proper accidents are the effects of substantial forms and make them known to us.”56
Aquinas was obviously far less optimistic than was Aristotle about the ability of the human intellect to extract a knowledge of essence from the regularities noted in perception. He recognizes that the features of sensible bodies that are accessible to human powers of perception are not, in general, part of essence; the forms abstracted by the mind in consequence of such perception are not substantial forms. We may say that fire is a simple, hot and dry body, but this is not to specify essence directly; rather it is to designate in terms of phenomenal qualities which are the effect of essence.57 Can one infer from these effects back to their cause, essence? How do the proper accidents make the substantial forms known? Which of the two strands in his thinking are we to emphasize here, the pessimistic one (the essences of natural things are hidden from us) or the more optimistic strand (we can discover essences through the clues afforded by the proper accidents)? Or ought we combine these two: /37/ essences of physical things are indeed initially hidden (i.e., we do not have immediate access to them in the intellectual processes involved in perception) but they can be progressively discovered by means of indirect modes of inference?
In support of this last suggestion:
The human intellect must of necessity understand by composition and division. For since the intellect passes from potentiality to act, it has a likeness to generable things, which do not attain to perfection all at once but acquire it by degrees. In the same way, the human intellect does not acquire perfect knowledge of a thing by the first apprehension; but it first apprehends something of the thing, such as its quiddity, which is the first and proper object of the intellect; and then it understands the properties, accidents, and various dispositions affecting the essence. Thus it necessarily relates one thing with another by composition
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and division; and from one composition and division it necessarily proceeds to another, and this is reasoning .58
But how exactly does this reasoning work? “Composition” is a linking of attributes, “division” a separating. How is one to discover the /38/ necessary linkage he mentions? Are we back to postulating an ability of mind simply to see the necessity of a causal linkage once the attributes themselves are fully understood, the kind of ability that Aristotle had ultimately to invoke in epagoge? Aquinas gives us very little to go on here, and one reason is not far to seek. There were few, if any, plausible candidates for inference from accidents to essence in the natural science of his day. (Discussions of the nature of the rainbow might have afforded the best clue.) In one text, he suggests that the senses must be the arbiter in such a process:
Sometimes the properties and accidents of a thing revealed by the sense adequately manifest its nature, and then the intellect’s judgement of the thing’s nature must conform to what the sense reveals about it. All natural things, limited to sensible matter, are of this sort. So the terminus of knowledge in natural science must be in the sense, so that we judge of natural things as the sense reveals them, as is clear in the De Caelo.59
What are the consequences of this initiative for the doctrine of demonstration? In his Commentary on the Posterior Analytics, Aquinas makes much of the distinction between demonstration /39/ propter quid and demonstration quia, to which only a single passage in the Posterior Analytics itself was devoted. A major reason for this shift in emphasis was undoubtedly the importance of the distinction for Aquinas’ theology: we can arrive at demonstrative knowledge of God’s existence, he argues, but the demonstration can never be better than quia since we lack the knowledge of essence, the appropriate definition, that a demonstration propter quid would require as middle term:
From every effect the existence of its proper cause can be demonstrated [by a demonstration quia] so long as its effects are better known to us . . . . Hence the existence of God, since it is not self-evident to us, can be demonstrated from those of His effects which are known to us . . . . When the existence of a cause is demonstrated from an effect, this effect takes the place of the definition of the cause in proving the cause’s existence. This is especially the case in regard to God, because in order to prove the existence of anything, it is necessary to accept as a middle term the meaning of the name, and not its essence, for the question of its essence follows on the question of its existence.60 /40/
The problem here, of course, as with any demonstration quia, would be (as we have seen) to prove convertibility, that is, to show that God is the only possible explanation of the effects in question, and to show this assertion itself to be necessary, to be something more than plausible hypothesis.
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Aside from the elaboration of the quia-propter quid distinction, however, the discussion of demonstration in Aquinas’ Commentary stays remarkably close to the original. The examples of the nontwinkling planets and the phases of the moon are presented without comment. First principles are said to be arrived at by means of induction, understood as an abstraction of universals from sensible particulars; many times experienced.61 There is no hint as to how exactly the assertion of causal relationship is to be arrived at. If we have to infer from perceived phenomenal qualities back to the unperceived essence that is their cause, how is the universal corresponding to this essence to be formed? Epagoge may work to relate perceivable accidents, but how does it link those in turn with the substantial forms Aquinas holds to be causally responsible for them?62 If the essences of sensible things are hidden, as Aquinas says they are, is not the role of strict demonstration in natural science so circumscribed as to be almost nonexistent? /41/
It is puzzling, on the face of it, that someone who had so clearly realized the seriousness of the barriers facing inquiry into natural essence would not have allowed some hint of this challenge to appear in his exposition of the canonical doctrine of demonstration in the Posterior Analytics. In a recent Aquinas Lecture, Alasdair MacIntyre argued that demonstration for Aristotle and Aquinas, as an “achieved and perfected knowledge,” is an ideal constituting the goal of inquiry, one rarely perhaps reached.63
Rational justification can thus take two quite different forms:
Within the demonstrations of a perfected science, afforded by finally adequate formulations of first principles, justification proceeds by way of showing of any judgement either that it itself states such a first principle or that it is deducible from [one] . . . . But when we are engaged in an inquiry which has not yet achieved this perfected end state, that is, in the activities of almost every, perhaps every, science with which we are in fact acquainted, rational justification is of another kind.64
This second kind belongs first and foremost to dialectics. Principles will be formulated provisionally; “apodictic theses” will be tested against “empirical phenomena,” and reformu- /42/ lated, if necessary, in the light of such tests.65 Even the very telos of the inquiry itself, the conception of the sort of science that ought be aimed for, is open to modification in the light of results achieved along the way. Progress will thus “often be tortuous, uneven, move inquiry in more than one direction, and result in periods of regress and frustration. The outcome may even be large- scale defeat . . . .”66 “What had been taken to be a set of necessary apodictic judgements, functioning as first principles, may always turn out to be false.” Hence:
No one could ever finally know whether the telos/finis of some particular natural science had been achieved or not. For it might well appear that all the conditions for
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the achievement of a finally-perfected science concerning some particular subject- matter had indeed been satisfied, and yet the fact that further investigation may always lead to the revision or rejection of what had previously been taken to be adequate formulations of first principles suggests that we could never be fully entitled to make this assertion.67
Such principles are “necessary,” then, only in the weak sense that they are stages on the way to a true, but quite possibly unreachable, judgment /43/ that “presents to us actually how things are and cannot but be.”68
What are we to make of this account? It affords a perceptive description of what might fairly be said to be the “received view” in contemporary philosophy of science of the status of theory in natural science. But can it claim a warrant in the text of the Posterior Analytics or of Aquinas’ commentary on that work? MacIntyre allows that the Aristotelian- Thomistic tradition has to be supplemented here by the insights of such contemporaries as C. S. Peirce and Karl Popper in order to arrive at this highly fallibilist conception of science, one which construes “first principles” as tentative hypotheses open to continuing modification in the light of new observational evidence. But surely this is rather radical “supplementation?” What entitles us to call the resultant view “Aristotelian- Thomistic?” The transformation that has come about in the conception of natural science in modern times has been largely due to developments in empirical inquiry itself, to an internal dynamic working within the history of science; it is to be judged in the first place, then, by reference to the history of science. MacIntyre remarks that it is in the spirit of the Aristotelian-Thomist tradition to test a conception of inquiry against the actual history of that form of inquiry. This may be so, but the conception of inquiry that /44/ emerges from this testing may well diverge from the Aristotelian tradition sufficiently to make the claim that it is a natural extension of the doctrine of the Posterior Analytics a rather forced one.
There are two sticking points in the way of such a continuity thesis. The warrant for an Aristotelian demonstration lies ultimately in the recognition on the part of the intellect that the premisses, properly understood, are “self- evident,” i.e., carry their own internal warrant. MacIntyre himself seems to say as much. Argument to first principles, he notes:
cannot be a matter of dialectic and nothing more, since the strongest conclusions of dialectic remain a matter only of belief, not of knowledge. What more is involved? The answer is an act of the understanding which begins from but goes beyond what dialectic and induction provide, in formulating a judgement as to what is necessarily the case in respect of whatever is informed by some essence . . . . Insight, not inference, is involved here . . . .69
This catches the rationalist aspect of epagoge, to be sure. But then he imposes a significant qualification: the judgment of the intellect in regard to essence still has to contend with “constraints” /45/ imposed
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by dialectical and inductive considerations, and the insight it affords requires “further indication,” namely, a check as to whether the proposed premisses/principles do, in fact, provide a “causal explanation of the known empirical facts.”
Insight into “what is necessarily the case” is therefore, apparently not of itself sufficient to warrant the first principle, the premiss of the demonstration. Quite complex-sounding forms of inference, continuing tests of the proposed principle against the empirical facts, are also needed. It is difficult, however, to find a justification for this restriction in the text of the Posterior Analytics itself. The claims made there for the nous that is consequent upon epagoge show no such hesitation, as we have seen. But perhaps it is to the text of Aquinas (though not, it would seem, to his commentary on the Posterior Analytics) that we ought to be looking. One relevant passage:
The ultimate end which the investigation of reason ought to reach is the understanding of principles, in which we resolve our judgements. And when this takes place, it is not called a rational procedure or proof but a demonstration. Sometimes, however, the investigation of reason cannot arrive at the ultimate end, but stops in the investigation itself, that is when two possible solutions /46/ still remain open to the investigator. And this happens when we proceed by means of probable arguments, which are suited to produce opinion or belief, but not science. In this sense, rational method is contradistinguished to demonstrative method, and we can proceed rationally in all the sciences in this way, preparing the way for necessary proofs by probable arguments.70
This is much more promising. But it raises new questions. These “probable arguments” can produce only “opinion.” How does the transition to a first principle actually come about, then? How do the probable arguments prepare the way for demonstration proper? Does a demonstration arrived at in this way rest in any respect on the probable arguments? It would seem not, for if it did, it would remain provisional. And its warrant would no longer be self-evidence. But if it does not rest on them, why were they needed? There is an unresolved difficulty here, one that will surface again much later in Descartes’ Discourse on Method . The divide between science and opinion, legacy of Greek ways of thinking, represents a sharp dichotomy, not a continuum. MacIntyre suggests, as we have seen, that demonstrative science is to be regarded as an ideal that lies at the horizon of inquiry. This has the merit of deflecting the troublesome problem of how the transition can /47/ ever be made from opinion to science: in practice, it is never made or at least we can never know that it has been. What the scientists of today would call “science” would, therefore, have to be labeled “opinion,” or at least something other than science. And the traditional Aristotelian claim that the human intellect can, on the basis of sense-experience, directly grasp the relation between a particular kind of
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effect and its proper cause in a definitive way, would have to be either set aside or at the very least forcefully reinterpreted.
There is a further sticking-point in the way of those who seek to establish a strong continuity between the Aristotelian and the contemporary conceptions of science. The phenomenalist cast of Aristotle’s account of epagoge has already been noted. It imposes a severe restriction on the concepts available for deployment in a demonstration. To demonstrate in natural science is to discover a causal relationship between perceptible features of sensible bodies, features that are regularly found together. Aquinas comments:
If universals, from which demonstration proceeds, could be grasped apart from induction, it would follow that someone could acquire scientia of things which he could not sense. But it is impossible for universals to be grasped apart from induction.71 /48/
Universals can come to be known only through induction (epagoge). Or in the idiom of abstraction, the concepts in terms of which a science of sensible bodies is to be constructed can only originate in perception: the form can come to exist in the mind only if it be abstracted from sensible instances of that form. An oft-quoted scholastic maxim made this restriction quite explicit: nihil est in intellectu quod non prius fuerit in sensu.
But now let us return one further time to the nontwinkling planets. As we have seen, Aristotle himself attempted a theoretical explanation of the proposed causal link between distance and twinkling in terms of the attenuation of visual rays emitted from the eye of the observer. These visual rays are clearly not themselves observable, nor is their attenuation. And their attenuation may itself require the introduction of further theoretical entities to explain it. Explanations of this sort are dotted throughout Aristotle’s work, particularly the Meteorology. The all-important spheres on which the circular motions of the celestial bodies are said to depend would furnish the most striking example.
When Aristotle is faced with the need to show that the link between two kinds of feature is a genuinely causal one, he quite often in practice postulates an underlying structure or process not itself observable, instead of just relying on the /49/ assertion that the link can be intuitively seen to be a necessary one. The necessity is thus mediated by the postulated entity; the causal link is held to be necessary in virtue of this entity. But where is the requisite universal to come from? There are universals for A and B (twinkling and distance) but not, it would seem, for C (the visual ray). C has to be somehow constructed in imagination, relying on elements drawn from perception in other contexts, no doubt. But the nature of C is constructed mentally in response to the request for explanation; it is not the result of abstraction. The test of this constructed concept lies in its ability to explain, not in its being properly abstracted from sensible particulars. The shift from
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abstraction to construction means that the resultant form of inference cannot be demonstration, unless the requirements for demonstration be greatly weakened, not to say transformed.
Neither Aristotle nor Aquinas addresses this problem. Aquinas’ detailed account of the abstraction characteristic of natural science (the first degree of abstraction from matter) gives no hint that something other than straightforward abstraction may be needed at the crucial moment in constructing causal explanations.72 Nevertheless, if one looks at what he has to say, not about natural science, but about “divine science,” a possible response suggests itself. We can come to /50/ know things that transcend sense and abstraction-based imagination, he says, by beginning from the level of sense and imagination and then arguing from this level as from an effect to a cause which surpasses it.73 We cannot be said to know the nature of such a cause (answering to quid est?), only that it exists (answering to an est?). Still, in order to know that a thing is, something must be known of what it is. Thus, in order to know that God and other immaterial beings exist, we have to be able to postulate something, at least, of what they are or, at least, of what they are not.74 When the cause so transcends the effect as God does the physical universe, “we take the effect only as the starting point to prove the existence of the cause and some of its conditions [e.g., the power to create], although the quiddity of the cause is always unknown.”75
The barrier here to knowability is difference of nature, and ultimately transcendence of nature. And the response is to construct an incomplete and imperfect definition of a cause that would be sufficient and (more problematically) necessary to account for significant general features of the world of sense. This is, in striking ways, similar to the sort of construction that a retroduction from effect to cause in natural science might also require. The visual ray does not transcend the sensible order as God does, but it does differ enough /51/ from it in regard to its accessibility to human modes of perception that a not-entirely-dissimilar constructive form of inference has to be employed to reach it. One could say, then, that the sort of retroduction that Aquinas employs to enable him to affirm the existence of God and the angels might have suggested how to proceed in natural science when causes inaccessible to sense appear to be required. Admittedly, Aquinas specifically excludes this parallel, insisting that natural science and divine science differ precisely on this point.76 Nevertheless, a way out has been, if not opened, then at least indicated, and the tight requirements of demonstration propter quid have been found incapable of satisfaction in least one domain of science, and a weaker alternative has been allowed.
It would be wrong, of course, to suggest that this relaxation is what actually led to the later acceptance of theoretical and nondemonstrative forms of inference in natural science. The change took a very long time,
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and the main inspiration for the change came from progress in the natural sciences themselves. It gradually came to be realized that the causal agencies underlying explanation in the natural sciences, if not as remote from abstractive terms anchored in perception as Aquinas had declared God to be, still required a /52/ new and nonabstractive approach to the concepts required to define them.
Galileo and the Paduan Tradition. One other philosopher should be mentioned in that context, Jacopo Zabarella, with whom yet another continuity thesis has been linked. There is space here only to summarize the argument; doing it justice would require a full-length study. J. H. Randall suggested in a wide-ranging work, The School of Padua and the Emergence of Modem Science (1961), that Galileo’s notions of scientific method were heavily dependent on the traditions of the school where he spent much of his teaching career, Padua, and particularly on the logical works of Zabarella:
The logic and methodology taken over and expressed by Galileo and destined to become the scientific method of the seventeenth century physicists . . . was even more clearly the result of a fruitful critical reconstruction of the Aristotelian theory of science, undertaken at Padua in particular . . . . [In] its completed statement in the logical controversies of Zabarella . . . it reaches the form familiar in Galileo . . . .77 /53/
The claim gave rise to a lively controversy. Two objections, in particular, were raised. One was that the connections between Galileo and Zabarella had not been clearly enough established, that although there were some resonances between the logical terms used by Galileo here and there in his scientific works, there was no real evidence of influence and no obvious medium for it. The second objection was that Randall had conflated logic and methodology. It was one thing to say that Galileo’s logic (more exactly, his conception of science, his view of what kind of knowledge-claim scienza makes) had some affinities with that of the Paduan tradition. But it was quite another to claim that his methodology, the methodology that laid such a distinctive stamp on the natural science of those who followed him, also derived from Padua. This seemed far less plausible; indeed, it found (and would still find) few defenders. Galileo gradually evolved a complex methodology involving controlled experiment, repeated measurement, mathematical idealization, and much more, which was strongly opposed by his Paduan Aristotelian colleagues and certainly finds few resonances in their tradition. Since it was his methodology and, of course, his actual discoveries in mechanics and elsewhere, and not merely his concept of science, that shaped what /54/ came after, the Randall thesis was generally thought to fail, or at least to be greatly overstated.
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In the last ten years, it has been restated and strengthened by William Wallace. Zabarella still retains his role, but the thesis has been broadened. The “canons” of Galileo’s new science, and hence of science in the Galilean tradition, Wallace suggests, “were those of Aristotle’s Posterior Analytics read with the eyes of Aquinas, and appropriated by him from the Jesuits of the Collegia Romano.”78 Wallace has shown, to most people’s satisfaction, that two short commentaries on logical topics, dismissed by the editor of the National Edition of Galileo’s works as juvenile school-pieces, were written by Galileo in his mid-twenties when he was just beginning his career as a teacher of mathematics and natural philosophy.79 Further, by a remarkably painstaking piece of detective work, Wallace has also shown that these two pieces are almost entirely derivative from lecture-notes composed by some contemporary Jesuit teachers of natural philosophy at the Collegia Romano, notably the notes of a certain Paolo Valla. And Valla drew heavily on Zabarella as well as on the Thomist tradition. What this establishes is that Galileo was cognizant of the Jesuit, and indirectly of the Paduan, tradition of commentary on the Posterior Analytics, notably on the topic of demonstration, to which one of the /55/ two sets of his notes is devoted. Whether (as Wallace supposes) the notes represented Galileo’s own views, then or later, is a quite different matter. That Galileo should have carefully summarized and paraphrased portions of the lecture notes of senior colleagues, who were teaching the courses he himself might be called on to teach, does not give strong reason to describe these notes as conveying his own “logical doctrine.”
Several features of this doctrine are said to mark the scientific work of Galileo’s maturity. On this the continuity claim rests. First and foremost is the alleged appearance of the regressus form of argument characteristic of the Paduan Aristotelian tradition, and especially of Zabarella’s logic. The regressus was the combination of the quia and propter quid inferences we have already seen, the first establishing the existence and something of the nature of the purported cause and the second demonstrating the effect from this cause. Zabarella and his colleagues had developed an elaborate analysis of this back-and-forward mode of explanation. Jardine describes Zabarella’s theory as one of “vast and obsessive complexity”80 Crucial to this analysis was the occurrence of a period of consideration or reflection (negotiatio or investigatio) between the back-to-cause and forward-to-effect phases. The first of these phases yields an indistinct (confusa) notion of the cause. /56/ The intellect then wrestles with this notion, somehow clarifies it, and finally comprehends the cause sufficiently to allow the propter quid demonstration to be completed with assurance. In particular, the intellect is said to have the ability to “see” that the crucial convertibility condition holds (equivalently ruling out the possibility of alternative causes).
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Though this intermediate phase cannot simply reduce to the epagoge of the Posterior Analytics (which is required before the first phase can even get under way), the attribution to the intellect of the ability to discern the nature of the required cause is reminiscent of the traditional doctrine of nous. There can be no doubt that the intention of the Paduan Aristotelians was to show how the regressus could provide strictly demonstrative explanations. It would be tempting to take the intermediate phase as a forerunner of later hypothetical modes of explanation, but Wallace and Jardine are surely correct in excluding this reading.81 Hence, if Galileo is to be seen as a proponent of the regressus notion of proof, he has also to be construed as a defender of Aristotelian demonstration as the appropriate mode of proof in natural science.
Jardine, to the contrary, argues that far from promoting regressus, Galileo was actively critical of its use as a model of proof: /57/
Galileo was well aware of the contemporary Aristotelian theory of scientific demonstration, had a sure insight into its weaknesses, rejected it outright, and set up in its place as a crucial part of his propaganda for the union of mathematics and natural philosophy a method of inquiry modelled on a classical account of the quest for proofs in geometry.82
The first thing to say here is that in the works of his scientific maturity Galileo never alludes to the method of regressus one way or the other, either to affirm it or reject it. True, he frequently uses the term “demonstration,” but it carries with it the connotation of “convincing proof, no more.” And he links it quite often with mathematics phrases like “the rigor of geometrical demonstration,” “the purest mathematical demonstration” support Jardine’s argument that Galileo’s notion of demonstration is associated by him much more explicitly with geometry than with the syllogism.83 To the extent that the properties of necessity and of convertibility appear, it is because of the mathematical form in which his arguments in mechanics are conveyed. Since he sets aside causal explanation in terms of gravity, and confines himself to kinematical measures of space and time only, the issue central to regressus (arguing from effect to efficient cause) simply does not arise in /58/ his mechanics. Furthermore, he returns again and again to the Platonic-Archimedean theme of “impediments,” the various obstacles that arise when one tries to apply an idealized mathematical system to the complexity of the material world.84 Such a system applies only approximately, and approximation is something that has no place in the classic conception of demonstration propter quid . A necessary truth about nature cannot be just approximately true.
Galileo’s law of falling bodies affords a clear illustration. It is the Aristotelian in the dialogues, Simplicio, who keeps objecting that the necessity one finds in purely mathematical inference cannot readily be transferred to claims about the material world. As long as Galileo’s account
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of uniformly accelerated motion be taken simply as a mathematical definition, the issue of demonstration does not arise. It is the claim that this is, in fact, the sort of motion that occurs were a body to fall in vacuo at the earth’s surface that raises the problem. Galileo gives two sorts of arguments in support of his claim. One is that uniform acceleration is the simplest mathematical form that this motion could take and hence is the one that Nature would employ. The other is that the assumption that in vacuo fall is, in fact, uniformly accelerated is supported by the “very powerful reason” that it “corresponds to that which physical experiments show /59/ to the senses.”85 The first is reminiscent of strict demonstration: Galileo is asking us to see that motion must take place in this way. (But, of course, we know from Newton’s vantage-point that the law is not exact. The acceleration of in vacuo fall gradually increases. Nature does not always act in the simplest way.) Galileo relies also on a second line of argument, which is that consequences drawn from the assumption of uniformly accelerated motion can be experimentally verified. But, of course, this is no longer a demonstrative form of argument: it rests on the extent to which, and the precision with which, the consequences of the supposition have been observationally verified. There is no suggestion that falling motion must necessarily follow this law.
Wallace responds to this objection:
Galileo’s way of presenting and justifying this definition [of the motion of fall in vacuo] has elicited criticism from some, who see him as employing a hypothetico- deductive method such as characterizes modern scientific investigations, and thus as falling into the fallacy of affirmatio consequentis when using the implied consequences of his definition to support it as the antecedent. It is true that the definition can be regarded as a suppositio, and therefore that the demonstrations to follow are made /60/ ex suppositione . . . . From a formal point of view, moreover, a suppositio has the character of an ipotesi [hypothesis], and thus its value might be judged by its ability to save the appearances, regardless of whether or not it describes a situation that is actually verified in the order of nature. It is for this reason that Galileo repeatedly makes the distinction between supposizioni that are true and absolute in nature, and those that are false and made purely for the sake of computation . . . . Galileo’s principles, the definition of naturally accelerated motion included, must stand or fall on their own merits, and not merely on the basis of one or two consequences drawn from them.86
This long quotation is instructive. It helps us understand why Wallace is so averse to allowing Galileo’s science to be called H-D and why he labors so hard to drape it in the mantle of the Aristotelian-Thomist tradition. Unless Galilean mechanics can be construed as demonstrative, it is reduced to simply “saving the appearances,” thus separating it from what “is actually verified in the order of nature.” There are echoes here of the distrust among Aristotelian natural philosophers of the late Middle Ages for the epicycles of the Ptolemaic astronomers, and the philosophers’ way of dismissing as “fictive” /61/ (the alternative to “demonstrative” that Wallace often
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employs) anything which simply rests on “saving the appearances.” But Galileo’s constant appeal to “the very powerful reason” that one of his suppositions is verified by the consequences drawn from it is appealing to the “appearances,” i.e., the experimental results. Those who see Galileo as employing hypothetical forms of inference on occasion do not (as Wallace suggests) take this to imply that this makes his reasoning fallacious. They would reject the over-simple dichotomy between demonstrative and fictive, and maintain that a hypothetical argument, one sup- ported by the consequences drawn from it, can have any degree of likelihood up to practical (not, however, absolute) certainty. Insofar as Galileo’s argument for his law of fall did carry force, it was because of the fact that it did so successfully “save the appearances,” i.e., fit the phenomena of experiment. If there was fallacy, it might be said that it was in Galileo’s appeal to the simplicity of Nature, in his vain attempt to present the appearance of strict demonstration.
Galileo’s telescopic discoveries opened up for discussion a host of questions about the natures of the beings now coming into view: sunspots, comets, and the like. There were surprises too in regard to the lunar surface, the variable illumination of Venus, and four small points of light that /62/ seemed to accompany Jupiter. Galileo’s method in dealing with these phenomena was to postulate a cause which might explain the observed phenomena, and try to find as much evidence as possible in support of his hypothesis. There was nothing particularly mysterious about this method; it is, when all is said and done, little more than common sense. Would it have been inhibited or favored by an appeal to the regressus tradition? One could, perhaps, argue either way. But the negative side is surely the stronger. The regressus tradition did insist on convertibility, on finding a cause to which one could infer with necessity from the effect to be explained, and on an interval of intellectual reflection on the nature of the proposed cause. The Paduans had assuredly never taken negotiatio to designate a period when alternatives are systematically explored, anomalies dissolved, and further positive evidence accumulated.
In some favorable cases, like the inference that mountains are the cause of certain variable shadows on the lunar surface, Galileo could infer almost directly from effect to cause, that is, exclude alternative possible explanations with a high degree of assurance. This does not, however, make his proof demonstrative in the Aristotelian sense. What made his supposition amount to certainty in his own mind was that it accounted so neatly for the appearances. But it could not be /63/ deduced from them; several (highly plausible) assumptions had to be made which in turn would require support from the consequences drawn from them. Commenting, Wallace allows that such assumptions are in fact present and remarks that until Galileo could be assured of the truth of such auxiliary assumptions, all he had was “opinion” and not “science.” (Once again, this is too sharp
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a dichotomy; instead, there is a spectrum of likelihood ascending as far as practical certainty.87) Wallace goes on to note that Galileo was aware of an objection to his claim about mountains on the moon. If there are lunar mountains, how can the edge of the lunar disk be seen as quite smooth in the telescope? He attempted to dissolve the objection, but it was not until 1664 that telescopes were sufficiently powerful to show that the moon’s outline is in fact slightly irregular. Does this mean that until 1664, all that astronomers had was opinion in regard to the lunar mountains? And that it became science with the 1664 observation? And, in any event, does not the dependence of the case for lunar mountains on this observational consequence show how artificial it is to force Galileo’s argument into the mold of Aristotelian demonstration, one purporting to yield “certain knowledge based on true causes?”88 /64/
Nothing has been said about Galileo’s use of the phrase “ex suppositione,” another supposed link to earlier logical doctrines and specifically to the Thomist tradition of commentary on the Posterior Analytics.89 Nor has an adequate distinction been drawn between what Galileo believed himself to be doing and what, from our perspective, he was actually doing. When tracing his links with earlier logical traditions, it is the former that is the more important; in assessing his influence on his successors in regard to this issue, it might be the latter that one would stress. Galileo was quick, often too quick in our estimate, to claim certainty for his conclusions. And in mechanics, at least, he sought principles which would, as far as possible, carry conviction in their own right. On the issue that meant most to him, that of the double motion of the earth, he sought for proof in such supposed consequences as the ebb and flow of tides, and the curved paths of the sunspots. The more acute kinematical arguments for the earth’s motions, showing how much more “natural” it is to attribute motion to the earth than to attach various sorts of motions to the heavenly bodies, he called “plausible reasons,” and remarked: “I do not pretend to draw a necessary proof from them, merely a greater probability.”90 /65/
In short, Galileo aimed when he could at demonstration, in the sense of conclusive proof. But when this was not available, he would settle for as high a degree of probability as the evidence would warrant, showing no inclination to regard the resultant merely as “opinion.” He used consequential modes of argument all the time, but never formulated a “method of hypothesis” and would probably have been reluctant to regard such a method as “the” method of science. All in all, then, even though Galileo was fond of using the term “demonstration,” there is little to warrant the claim that he was influenced in a significant way by the elaborations of the notion of demonstration in the older tradition of the Posterior Analytics.
As one looks at later seventeenth-century natural science, one imme- diately notices a significant difference between mechanics and the other
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sciences. In the mechanics of Descartes as of Newton, there was an emphasis on demonstration, on certainty, a suspicion of hypothesis. Perhaps this might be seen as an echo of the Aristotelian requirements for episteme, though in a mathematicized context remote from that of the Posterior Analytics. On the other hand, in other parts of natural philosophy, in optics, in chemistry, there was a growing realization that hypothesis is not only unavoidable, but even respectable, and efforts were made to formulate criteria in terms of /66/ which it should be judged. Both of these strands are already found in different parts of Galileo’s work; there is thus no single “Galilean” heritage in that regard.91