Plato's "Homework Problem" - saving the appearances (retrograde motion) while providing a mathematical account of the motions underlying the appearances.

Instrumentalism and Realism in Renaissance Astronomy

Ockham's razor (the principle of parsimony) - and the problem of circularity in the preference for the simple

The Religious (Neoplatonist/neopythagorean) as central to the development of Copernicus's system

Empirical grounds against the Copernican system

Additional religious grounds supporting the Copernican system

Political (equalizing/anti-hierarchical) consequences of Medieval/Neoplatonic/Copernican "recentering" of the universe from the Earth to the Sun

Kepler as Protestant/Neoplatonic/Neopythagorean (so Thomas Kuhn) - with additional comments from Morris Kline; the planetary laws of motion - including the third law as the foundation of Kepler's Harmonia Mundi ("Music of the Spheres") - as fulfilling the Pythagorean faith that underneath the complexity of appearances lies a simple, elegant, indeed musical mathematical order

[A NASA education module offers wonderful computer simulations illustrating the three laws. Thanks to Amy Johnston for finding these for us!]

Pine's summary: the Copernican Revolution as triumph of religious/philosophical/mathematical views

First Writing Assignment

Plato's "Homework Problem" rests on two basic assumptions of Greek philosophy/science:

a) that the universe is ordered, a kosmos whose underlying principles of order are ultimately mathematical - indeed,

they "should" prove to be aesthetically pleasing as well - where this aesthetic dimension includes a preference for "simplicity" over complexity (an assumption first articulated by Aristotle - and later at work in the famous Ockham's razor, of Medieval fame...); and

b) that this commitment to a mathematical underpinning at the same time must "save the appearances," - i.e., what appears to us cannot be rejected without further ado as "mere" illusion (the Parmenidean development rejected by most subsequent philosopher-scientists).

This means, more broadly:

• only a strong faith in the order of the cosmos could sustain the students of Plato in tackling this problem [of retrograde motion]. Many popular treatments of science prior to the seventeenth century imply that ancient civilizations failed to understand the real universe because they were dominated by a religious and philosophical dogmatism. On the contrary, the religious and philosophical ideas of the ancient Greeks encouraged precise observation of the eccentric motion of the planets and sustained the belief that the use of reason would eventually result in an explanation. (134: emphasis added, CE)
  
  

If the astronomers and other philosopher/scientists of the 1500's had difficulty with accepting Copernicus's heliocentric hypothesis, then, it was not because of religious dogmatism - but, at least to some degree, because of straightforward empiricism: given the data and prevailing understanding of physics, there was plenty of reason to reject the heliocentric hypothesis.

[As E.A. Burtt puts it:

Contemporary empiricists, had they lived in the sixteenth century, would have been first to scoff out of court the new philosophy of the universe.

- from:The Metaphysical Foundations of Natural Science, quoted in Pine, 149.

This reference to experience, indeed, is explicit in the Medieval whom Pine quotes:

• No experience whatsoever could prove that the heavens rotate daily and not the earth.
  
  -- Nicholas Oresme, Bishop of Lisieux, 1377: in Pine, 130
  

[This same Bishop, he will later point out, argues against a literal interpretation of the Bible as part of an effort to challenge Galileo.]

Pine points out that

• ...by the time of Plato and the Academy, all three ideas usually associated with our modern view of the universe had been proposed: a universe in which the Earth is not unique, a revolving Earth, and a rotating Earth. All of these views, however, originated from metaphysical and cosmological concerns. (135)
  
  

These included the 5th ct. atomist Democritus, who proposed a universe of infinite space, inhabited by an infinite number of suns and earths, in which there is no center. This modern sounding view was based simply on a deduction from the atomists' metaphysics - their belief

• that reality consisted of an infinite number of atoms moving in an infinite space. As with the origin of many ideas in science, it remained in the background waiting to be plucked if needed. (135)
  

Pine reviews the initial responses to Plato's "homework problem," including Aristarchus of Samos, the "Copernicus of antiquity," - because he proposed a heliocentric system.

Pine points to Kuhn's thorough resume of the reasons against such a system, 137f. In sum, Kuhn notes:

• The Greeks could only rely on observation and reason, and neither produced evidence for the earth's motion. Without the aid of telescopes or of elaborate mathematical arguments that have no apparent relation to astronomy, no effective evidence for a moving planetary earth can be produced. The observations available to the naked eye fit the two-sphere universe very well (remember the universe of the practical navigator and surveyor), and there is no more natural explanation of them. It is not hard to realize why the ancients believed in the two-sphere universe. The problem is to discover why the conception was given up. (from The Copernican Revolution, 42-44, in Pine, 138).
  

Four theories emerged by the end of the 3rd ct. B.C., each with specific strengths and weaknesses:

System	Strength	Deficit
Eudoxus (geocentric)	accorded with common-sense observations of the Earth's apparent stationary position; used perfect circles; explained planetary regression.	Could not explain why planets appear brighter when they retrogress
Apollonius/Hipparchus (modified geocentric - epicycle/deferent)	Explained brightness of planets in retrogression	Violated Pythagorean assumption that all bodies must move uniformly about a central point.
Heraclides (partial heliocentric)	Explained brightness of planets in retrogression - especially Venus and Mercury	How can the Earth rotate without flying apart? No explanation of how the Sun's orbit could pass through the orbits of Venus, Mercury, and the Moon.
Aristarchus (complete heliocentric)	Explained brightness of retrogressing planets; mathematically accurate - but...	no more accurate than the epicycle-deferent system. Open to Kuhn's list of objections.

(from Pine, 138)

This leads to the discussion of instrumentalism and realism:

• Several systems to choose from, but which one is right? What is the real physical universe like? For a true Platonist it did not matter! The question was irrelevant....Plato developed a metaphysics and epistemology that essentially ruled out the possibility of ever answering this question. Plato recognized the limitations of empirical knowledge and, because of the influence of Protagoras, he concluded that a complete knowledge of the physcial world was impossible. No matter how much evidence one has for a generalization about the physical world, that generalization may still be shown to be false some day. Moreover, false empirical generalizations can "work." Accurate predictions can be made that are observed in the world of our experience....True conclusions can be deduced from false premises. (139)
  
  

(Pine further uses the example of potentially infinite number of mathematical models which will fit the same data points, 139-140.)

The situation of several competing theories - each of which "covers" or explains the facts with equal degrees of accuracy - in fact confronts us with two questions:

1) how do we know which of these best "works" as a description of reality (which is what the realist expects of a scientific theory)?

a) again, the empirical data is not decisive here - each theory explains the facts, the empirical data, equally well.

b) given the lack of decisive empirical evidence - we are then forced to turn to other methodological guidelines to decide between competing theories. As the Copernican case makes clear, these guidelines include:

• i) Ockham's razor - which has us prefer the "simpler" theory.
  
  Problems:
  
  
  • a) why should we assume that the simpler theory is more likely to provide us with an accurate - i.e., "realistic" - account of Nature?
    
    As it turns out, this assumption is precisely that - one grounded in an earlier Greek (specifically, Aristotelian) assumption that nature is simple.
    
    But this means:
    
    To prefer the simpler theory as the "better" account of Nature/Reality (Ockham's razor), as resting on
    
    the assumption that Nature/Reality is simple
    
    -- is to beg the question or fall into the logical trap of circular reasoning.
    
    
    Logical tangent: In begging the question/circular reasoning, we appear to be arguing from a premise/s to a conclusion supported by but different from the evidence and claims provided in the premise/s: but, upon closer examination of the premises and the conclusion, we find that the conclusion merely restates what is already established in the premise/s.
    
    
    The astute will notice here that the structure of deductive argument and the structure of explanation are similar in an important respect:
    
    

Structure of Explanation	Deductive Argument Structure
explanans ("explainer") // --> explanandum ("what is explained")	premise/s // --> conclusion

In both cases, we do not expect or find satisfying simply the repetition in the explanandum / conclusion what is already expressed in the explanans / premise/s, e.g.:

Structure of Explanation	Deductive Argument Structure
The Sky is Blue // --> The Sky is Blue	The Sky is Blue // --> The Sky is Blue

To say that "The sky is blue because the sky is blue" is both a logical circle and a circular "explanation."

By the same token, if the only reason we have for prefering the simpler hypothesis is Ockham's razor - which in turn rests on the assumption that Nature/Reality must be simple - we land in the same sort of circularity:

Hypothesis choice in the face of competing theories

Deductive Argument Structure

Greek/Aristotelian assumption: Nature is simple // --> Ockham's razor: prefer the simpler hypothesis // --> [applied to the conflict between Copernican, Ptolemaic theory] // --> prefer Copernican theory On the realist assumption: since science provides us with an accurate account of Nature/Reality as it really is // --> Copernican theory shows Nature/Reality to be simple (at least: simpler than Ptolemaic theory) // --> Nature/Reality is simple

Premise 1: Nature is simple // --> Conclusion 1: prefer the simpler hypothesis // --> Premise 2: Copernican theory is simpler than Ptolemaic theory // --> Conclusion 2: prefer Copernican theory Premise 3: Realism: science shows us Nature/Reality as it really is // --> Conclusion 4: Nature/Reality - as portrayed by Copernican theory - is simple // --> Conclusion 5: Nature/Reality is simple

ii) Beyond the problem of circularity - the criterion of "simplicity" is also problematic. What, precisely, made the Copernican system clearly "simpler" than the Ptolemaic? [See Pine's comparison between the two systems on this point, p. 147. Ultimately, the Copernican system is more mathematically elegant - and calculations were easier. But this leaves the question: was it real?]

2) even if we can somehow decide on a single theory among its many competitors - this does not preclude the possibility of discovering a new theory tomorrow. But if theory is supposed to represent Nature/Reality - given that theory changes, how can we ever believe that a given theory accurately represents Nature/Reality?

In the face of these difficulties, Plato and his followers, as Pine notes, concluded that "...the best we can do is have models of the physical world that work," - i.e., that allow us to make accurate predictions of future behavior. (141)

This positions is called instrumentalism or operationalism:

• This view states that scientific theories, especially those that involve abstract mathematical devices, are tools, instruments, or calculation devices and should not be interpreted as real. (141)
  
  

Pine goes on to use the example of the quadratic equation to predict the motion of a projectile. The solution to the equation describes two motions: one curving up and one curving down and through the earth.

• Because we have never witnessed projectiles going backward through the solid Earth, the instrumentalist asserts that the equation is a device that enables us to predict where the cannonball will land, but that it should not be interpreted literally. We should not think of the mathematics as describing the actual motion of the cannonball. (141)
  
  

[Further example from quantum physics of the electron as a "smear" of energy that spreads to infinity, according to the mathematical description.]

So - if methodological guidelines such as Ockham's razor involve us in circularity, and if there is no empirical data to force us to prefer one theory over the other (since, in the case of the Ptolematic vs. Copernican systems, both predicted the motions of the planets with equal degrees of accuracy) - what can make us choose between theories?

Pine points out that in Copernicus's case, the answer at least partly involves religion - and not simply Medieval Christian beleifs, but also Neoplatonist belief (itself a mix of Platonism and Christianity). On this view,

• ...a vital figure such as God, although eternal and nonmaterial, would have a "materialized copy" of Itself. Just as God was a creative force of immense potency responsible for sustaining all life, the the Sun, responsible for light, warmth, and fertility, could be the only appropriate material manifestation of God. According to Copernicus, highly influenced by this Neoplatonic belief, "in this most beautiful temple" of a universe, there is no better place but the center to place this "luminary...from which He can illuminate the whole at once." In other words, Neoplatonism demanded that man be replaced with God as the central concern. (146)
  
  

You will recognize here the powerful impact of Plato's allegory of the cave from The Republic, in which the Sun stands as the emblem of the Good and thus the final goal of knowledge. On Pine's showing - relying primarily on Kuhn - it is this essentially philosophical/religious belief concerning the primacy of the Sun that tips Copernicus in the direction of a heliocentric possibility.

In addition, empirical considerations argue in favor of treating the Copernican system as an instrumental device for calculation, not a realist account of the universe:

• Copernicus requires that the Earth's revolution around the sun really involves circular motion around two invisible points, "one for the circle of the Earth and another for the center of the center of the circle of the Earth." (148)
  
  If the Earth rotates, why is there not a constant - and incredible - East wind?
  
  When stones are thrown into the air - why are they not blown away, or land in a different spot?
  
  Why is the Earth not ripped apart by centrifugal forces?
  
  Finally - why is there no parallax or parallactic motion among the stars (i.e., different relative positions apparent to observers on the Earth as we occupy different locations in space over the course of a year's orbit around the Sun)?
  
  

Pine concludes:

• The best observers of the time could detect no such parallactic motion. The serious astronomer of the sixteenth century had difficulty reconciling these "facts" with a moving Earth. The most intelligent thing to do was to accept the new Copernican system as a calculation device but not something that was literally real. (148)
  
  

One of these best observers, in fact, was Tycho Brahe (1546-1601), who rejected the Copernican system in part because of the "wasted space" it seemed to involve - i.e., in violation of his religious conception of a created order reflecting God's sense of purpose and planning. As a result, Brahe attempted to further develop the Heraclidean system - and in fact succeeded in developing a mathematical equivalent to the Copernican system, but one which did not violate Scriptural references supporting a geocentric view and the common/empirical sense of the day. (148f.)

So far, then, empirical data and mathematical elegance do not solve the problem - we are left to religious, and, it will turn out, political considerations...

Pine points out that additional philosophical and religious reasons can be found to support the Copernican hypothesis. In particular, one can answer the problem of parallax by assuming that the universe is very large, perhaps even infinite:

• Thus, both Nicholas de Cusa (1401-1464), before Copernicus, and Giordano Bruno (1548-1600), after Copernicus, revived the work of Democritus [the ancient Greek atomist, 5th ct. B.C.E.], arguing that the Sun was only one of an infinite number of stars. Why? Because only an infinite sphere would be consistent with the greatness of God. Both also argued that some of these other stars would have planets and would be populated.
  
  

As Pine emphasizes, these arguments are primarily philosophical and deductive - resting further, we can note, on both ancient Greek metaphysics [atomism] and specifically Medieval Christian beliefs about the nature of God:

• An infinite universe is the only one consistent with the infinite perfection of God, therefore a heliocentric system must be true. It must be real. (150)
  

This view, further, has direct political consequences:

• Bruno also argued that consistent with this scheme would be a new relationship among God's creatures. God granted each creature its own inner source of power, and these powers were more or less equal, leaving no justification for domination and servitude. (150)
  
  

As Margaret Jacob's comments make clear, such arguments work directly in favor of the rebelling Protestants - and the various emerging nation-states seeking to escape the political control of Rome.

If the Roman Catholic Church reacted negatively to these new developments in the natural sciences - sciences directly supported by the Church in the Middle Ages precisely for the religious reason that they uncover the footprints of God, and thus bring us closer to God - it should be clear that the root of this reaction is not directly religious: it is more directly political.

Pine's account:

• The Catholic Church, which had supported scientific research as a method for studying the wonders of God's creation, could have accepted this new system as evidence of God's infinite greatness. But its leaders chose the safe path of tradition, and the rest is history. Copernicus's book was edited to make sure the instrumentalist interpetation predominated, Bruno was convicted of heresy and burned at the stake (for his religious view on equality), and Galileo was persecuted for advocating the heliocentric system as real. Copernicus, Bruno, and Galileo were all religious men; they thought of themselves as attempting to read the mind of God. Their difference with the Church was not so much a battle between science and religion, as it is so often portrayed, but part of a larger battle over different conceptions of epistemology, God, and world view.
  
  Note that Psalms 93 and 104 read, respectively, "You have made the world firm, unshakable....You fixed the earth on its foundations, unshakable for ever and ever." Are these clear statements that the Earth is the center of planetary motion? Nicolas Oresme, a devout Parisian defender of the faith, argued that these passages are not meant to be taken literally, no more than those that describe God as angry or pacified, and Galileo, borrowing the thought from Cardinal Baronias, said, "The Bible teaches how to go to heaven, not how the heavens go." (151)
  
  

Galileo's appeal to the Medieval doctrine of the two-fold truth should be noted - along with its problems.

This leaves us to consider Kepler (1571-1630) who, according to Pine, played the greatest role in the so-called Copernican Revolution:

• As a Protestant and religious individualist, he was not swayed by the attitude of the Church or the views of other Protestants, such as the Lutherans, who were actually the first to attack the Copernican system as heresy. For the most part, he lived in his own world of an ardent mystical Neoplatonic [and, Kuhn suggests, Neopythagorean] faith. (151)
  
  

This Neoplatonic/Neopythagorean faith is apparent in Kepler's first publication, Mystery of the Cosmos (1596), in which he seeks to demonstrate the the five regular solids (cube, tetrahedron, dodecahedron, icosahedron, and octahedron) determine both the number of planets and the relationships between their spheres. Kepler writes:

• I undertake to prove that God, in creating the universe and regulating the order of the cosmos, had in view the five regular bodies of geometry as known since the days of Pythagoras and Plato, and that he has fixed according to those dimensions, the number of heavens, their proportions, and the relations of their movements.
  

(quoted in Morris Kline, Mathematics in Western Culture, 113f.)

[Cf. the selection from Kuhn, including the diagram on p. 218]

Kline comments that Kepler

• mingled science and mathematics with theology and mysticism in his approach to astronomy, just as he combined wonderful imaginative power with meticulous care and extraordinary patience.
  Moved by the beauty and harmonious relations of the Copernican system, he decided to devote himself to the search for whatever additional geometrical harmonies the data supplied by Tycho Brahe's observations might suggest and, beyond that, to find the mathematical relations binding all the phenomena of nature to each other. His predilection for fitting the universe into a preconceived mathematical pattern, however, led him to spend years in following up false trails. (113)
  
  

-- the first of which was precisely the hypothesis based on the five regular solids.

But it was precisely this originally Pythagorean belief in the fundamentally mathematical order of the universe that sustained Kepler's quest for such patterns "underneath" Brahe's data. (This Pythagorean belief, recall, is at once Christian in this context.) As Pine puts it:

• Because of the problems and inaccuracies of the Ptolemaic, Copernican, and Tychonic systems, Kepler was convinced that no one had yet succeeded in reading the harmonies of the world. He desired passionately to be the first to read the mind of God. (151f.)

Kepler's Neoplatonic faith, moreover, gave him a certain luxury regarding at least one kind of data. In 1610, Galileo announced his discoveries of the pits and craters on the Moon, sunspots, the four moons of Jupiter, and moon-like phases in his observations of Venus - data, in short, which bolstered the Copernican system. But:

• Kepler did not know these facts until after he made his major discovery, and even if he had known them earlier they would have had little effect on him. Kepler too knew of the telescope - he even designed one, but did not bother to construct it. As a Neoplatonist, what mattered most to him were not such qualitative physical features as new moons, new stars, or phases of a planet. What mattered were the precise quantitative data of Tycho and finding the elegant mathematical relationships that would instantly related every possible observation. (Pine, 152)
  

[As well, Pine refers to Kuhn on this point, who observes that with the exception of the phases of Venus, none of Galileo's observations constitute direct evidence for the Copernican system - i.e., the heliocentric claim and the motion of the planets about the Sun. Indeed,

• Either the Ptolemaic or the Tychonic universe contains enough space for the newly discovered stars; either can be modifed to allow for imperfections in the heavens and for satellites attached to celstial bodies; the Tychonic system, at least, provides as good an explanation as the Copernican for the observed phases of and distance to Venus. Therefore, the telescope did not prove the validity of Copernicus's conceptual scheme. But it did provide an immensely effective weapon for the battle. It was not proof, but it was propaganda. (Kuhn, The Copernican Revolution, p. 224, in Pine, 152f.)
  
  

Briefly, through a laborious process of trial and error, eventually Kepler considered the possibility of the planets moving not in perfect circles - the motion assumed by everyone since the Pythagoreans (including Galileo, according to Pine, 153). Rather, he discovered a fit between Brahe's data on the orbit of Mars and the ellipse.

This "heretical" step (so Pine), is accompanied by a second: "To account for the data, the planet's path around the Sun must be an ellipse and it must move around the Sun at a variable speed." (153)

On the basis of these new ideas about planetary motion, Kepler then discerned his Three Laws of planetary motion:

• 1: each planet revolves around the Sun in the path of an ellipse with the Sun at one focus of the ellipse.
  
  
  2: (the law of inverse-distance) each planet proceeds at a variable speed, such that if an imaginary line is connected between the planet and the Sun, the planet sweeps through equal areas during equal times. [This means that the further the distance from the Sun, the slower the orbital motion.]
  
  

These first two laws were published in 1609 - i.e., before Galileo's 1610 Sidereus Nuncius (The Starry Messenger). As Pine notes, Kepler thus deftly reduces the 80 circles of Ptolemy, the 48 of Copernicus, along with their epicycles, deferents, eccentrics, and equants to

• ...7 slightly squashed circles with the planets moving around the Sun and the Moon around the Earth at variable speeds. Kepler had solved the problem of the planets. He had, so he thought, finally read the mind of God. He had discoverd the true harmony of the motions of the worlds. (154)
  
  

This true harmony, however, is only complete with the addition of the third law:

• 3: there is a proportional relationship between the planets' distances from the Sun and their orbital period - more precisely:
  
  the square of the time of revolution is proportional to the cube of its average distance.
  
  

This law is announced in 1619, in his Harmonies of the World - a work which further

• ...elaborated a new set of Neoplatonic regularities which related the maximum and minimum orbital speeds of the planets to the concordant intervals of the musical scale. Today this intense faith in number harmonies seems strange, but that is at least partly because today scientist are prepared to find their harmonies more abstruse. Kepler's application of the faith in harmonies may seem naive, but the faith itself is not essentially different from that motivating bits of the best contemporary research. Certainly the scientific attitude demonstrated in those of Kepler's "laws" which we have now discarded [e.g., the five regular solids] is not distinguishable from the attitude which drove him to the three Laws that we now retain. Both sets, the "laws" and the Laws, arise from the same renwed faith in the existence of mathematical harmony that had so large a role in driving Copernicus to break with the astronomical tradition and in persuading him that the earth was, indeed, in motion. But in Kepler's work, and particularly in the parts of it that we have now discarded, the Neoplatonic drive to discover the hidden mathematical harmonies embedded in nature by the Divine Spirit are illustrated in a purer and more distinct form.
  
  -- Kuhn, 219
  
  

Morris Kline agrees with Kuhn on the role of an essentially religious, specifically Neoplatonic/Pythagorean conviction in the work of both Copernicus and Kepler. He comments: "These sun-struck lovers of mathematics were designing a beautful theory. If the theory did not fit all the facts, it was too bad for the facts." (118)

More generally:

• ...Copernicus and Kepler were not at all concerned with these pressing, practical problems [ i.e., of navigation, better maps, etc.]. What these men did owe to their times was the opportunity to come into contact with Greek though, an opportunity furnished by the revival of learning in Italy. Copernicus...studied there and kepler benefited by Copernicus' work. Also both men owed to their times an atmosphere certainly more favorable to the acceptance of new ideas than the one that prevailed two centuries ealier. The geographical explorations, the Protestant Revolution, and so many other exciting movements were challenging conservatism and complacency, that one new theory did not have to bear the brunt of the natural opposition to change.
  
  Actually, Copernicus and Kepler developed their most revolutionary theory to satisfy certain philosophical and religious interests. Having become convinced of the Pythagorean doctrine that the universe is a systematic, harmonious structure whose essence is mathematical law, they set about discovering this essence. Copernicus' published works give unmistakable, if indirect, indications of his reasons for devoting himself to astronomy. He valued his theory of planetary motion not because it improves navigational procedures but because it reveals the true harmony, symmetry, and design in the divine workshop. It is wonderful and overpowering evidence of God's presence. Writing of his achievement, which was thirty years in the making, Copernicus expressed his gratification:
  
  
  
  • We find, therefore, under this orderly arrangement, a wonderful symmetry in the universe, and a definite relation of harmony in the motion and magnitude of the orbs, of a kind that is not possible to obtain in any other way.
    
    
• (Kline, 119)
  
  

Kline goes on to quote Kepler's preface to the Mystery of the Cosmos:

• Happy the man who devotes himself to the study of the heavens; he learns to set less value on what the world admires the most; the works of God are for him above all else, and their study will furnish him with the purest of enjoyments.
  
  

This religious impulse, again, is perhaps clearest in his 1619 The Harmony of the World, which

• ...actually expounded a system of heavenly harmonies, a new 'musci of the spheres,' which made use of the varying velocities of the six planets. These harmonies were enjoyed by the sun which Kepler endowed with a soul specifically for this purpose. Lest it be supposed that this treatise was just a lapse into poetic mysticism, we should realize that it also announced his celebrated third law of motion. (120)
  
  

[This "music of the spheres," by the way, was finally realized, following Kepler's score, in the 1970's by Willie Ruff and John Rodgers. We will hear the opening sequences in class.]

In sum:

• The work of Copernicus and Kepler was the work of men searching the universe for the harmony which their commingled religious and scientific beliefs assured them must exist, and exist in aesthetically satisfying mathematical form....
  
  To men who were convinced that an omnipotent being designing a mathematical universe would certainly prefer these superior features [of mathematical simplicity and harmony], the new theory was necessarily right. Indeed, only a matehmatician who was assured that the universe was rationally and simply ordered would have had the mental fortitude to buck the prevailing philosophical, religious, and scientific beliefs, and the severance to work out the mathematics of such a revolutionary astronomy. Only men possessed of unshakable convictions with regard to the importance of mathematics in the design of the universe would have dared to uphold the new theory against the powerful opposition it was sure to encounter. (Kline, 120)
  
  

On Pine's showing, Kepler's third law was not only the immensely satisfying reward for decades of patient persistence: coupled precisely with Kepler's Neoplatonic convictions - the third law demonstrated for him that the heliocentric system was real:

• That such a simple equation worked was much more of an explanation than any observation Gelileo could make with the telescope. There was no logical relationship, however, between this law and the acceptance of Copernicanism. It was not needed to prove the heliocentric system. The observational data, the ellipse, and the notion of variable speed were sufficient. (157f.)
  
  

While not necessary from an empiricist perspective to support the Copernican system, however, it will turn out that the Third Law - at this point in time, thus just a satisfying bit of mathematics for Neoplatonists and Neopythagoreans - will be of central significance in the development of Newton's theory of gravity. (158)

As Pine summarizes this chapter, he points to a number of interesting ironies - ironies, at least, if we presume the positivist and popular assumption of the natural sciences as following a purely logical, data-driven, and inevitable march towards ever more realistic accounts of Nature/Reality:

• Copernicus - like Kepler after him - was attracted to the ancient Greek heliocentric hypothesis by an essentially religious conviction (Neoplatonic) regarding the central importance of the Sun as a material emblem of God;
  
  Both the Ptolemaic and Copernican system equally account for the observational "facts" - there are, initially at least, no empirical reasons, no "data' to force us to choose one over the other.
  
  On the contrary, there is much good empirical evidence against the Copernican view: Tycho Brahe, the best observationalist of his time, rejected it - and, we can add here, even Francis Bacon, another "founder" of modern science, rejected the Copernican view precisely because of his [Anglo-American] emphasis on empirical experience.
  
  By contrast, Kepler's embrace of Copernicanism likewise flows from his shared philosophical/religious convictions (Neoplatonic/Neopythagorean) - convictions which, coupled with Brahe's data, genuinely revolutionize our conception of the universe and the simple, mathematical order underlying that universe.
  
  These ironies, moreover, shed light on the larger epistemological question: does scientific theory, especially in its reliance on mathematics - indeed, its occasional preference for mathematical accounts even when the data do not precisely "fit" - provide us with a realist or instrumentalist "picture" of Nature/Reality?
  

In addition to the general theoretical problems with realism (developed above), remember here that:

• a) Copernicus - partly because of pressure from the Church - at least publically stressed the instrumentalist character of his hypothesis;
  
  b) the Copernican hypothesis was problematic in light of empirical data - i.e., it ran counter to at least some significant observations (e.g., the lack of parallactic motion);
  
  c) both Copernicus and Kepler seem to have accepted their theories as realistic accounts of Nature/Reality not only on the basis of data, but primarily on the basis of mathematical simplicity and elegance - i.e., as their theories fulfilled their original Neoplatonic/Neopythagorean assumptions about Nature/Reality.
  
  d) others, including some Medievals who adopted ancient Greek atomism to Christian frameworks, found an infinite heliocentric universe quite plausible not because of the "data" - but because such a universe fit their beliefs concerning God's greatness and infinite perfection.
  
  

Taken together, these elements of this history suggest Pine's opening point: contrary to the prevailing (19th ct., positivist) presumption of an essential hostility between science and religion,

• the religious and philosophical ideas of the ancient Greeks encouraged precise observation of the eccentric motion of the planets and sustained the belief that the use of reason would eventually result in an explanation. (134: emphasis added, CE)

Writing Assignment:

Using Pine, library reserve, and Web materials as your primary resources...

A: Choose between "1" and "2":

1. Describe Thales' account of the world ("all things are water"), paying careful attention to the structure of explanation apparent in this account. Make as clear as possible: how is this structure of explanation both similar to and different from earlier mythopoetic "explanations"?

2. Describe with some care the position, including supporting arguments and evidence, of the Pythagoreans for the fundamental importance of mathematics in the effort to know the "underlying unity" which accounts for or explains the world.

Be sure to comment here on the religious character of the Pythagorean view - both as this character seems to contribute to (a) the willingness to generalize from a few examples of correlation between mathematical relationships and sensory experiences (e.g., the simple ratios // musical relations), and (b) the fundamental motive or impulse underlying the Pythagorean quest to understand the mathematical order of things.

B. What are some of the arguments provided by Plato for the greater reality of the mathematical entities (e.g., our friend the triangle) and the Ideas?

Given these arguments, summarize Plato's resulting epistemology (account of knowledge) and metaphysics (account of what is real, not real; more real/less real), as suggested by the allegory of the Cave and the analogy of the line in the Republic.

C. Briefly summarize the two epistemological positions we have begun to examine - realism and instrumentalism.

D. Focusing on either Copernicus or Kepler, explain in some detail the relative roles of

(a) empirical data, observation, etc. and

(b) philosophical and religious beliefs

in their contributions to "the Copernican Revolution."

Be sure to explain as fully as you can just what philosophical and religious beliefs are at work in the development of their theories.

E. We have examined three periods in the emergence of natural science - the ancient Greeks (including the Alexandrian School), the Medievals, and the early moderns (Copernicus, Galileo, Kepler).

In no more than a paragraph for each period, summarize briefly the relationship between religious and philosophical beliefs, on the one hand, and the emerging "scientific" tradition of mathematical and physical inquiry - i.e., is this relationship primarily complementary (mutually supportive and beneficial) or primarily hostile (only one view can be true, not both)?

Remember: your work will be graded not only according to content, but also with regard to writing and documentation mechanics. For more information, please see the "Documentation" handout and the Departmental Policy on Grading.

DUE: Friday, Sept. 26, 1997