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Mathematics in Science

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Functions of Mathematical Description in Astronomy and Optics,

Illustrations from Antiquity —Stephen Boydstun (1999)

Use of mathematics in our descriptions of nature makes our conceptions of natures more definite, sensitive, and probative. By what pathways?

I shall take the process of scientific conception of natures to consist of three interconnected moments: observation, characterization, and explanation. Into these three moments, I shall cast work in early astronomy and optics, attending to mathematical descriptions and the epistemological functions they serve.

I. Mathematical Description in Observations

Writing and drawing can extend memory of past perceptions. They can improve the fidelity of memory to past perceptions. They can make aspects of past perceptions accessible to other people.

Records of Timocharis' observation of the location of the star Spica relative to the autumnal equinox was used by Hipparchus about 170 years later in establishing the precession of the equinox (Neugebauer 1975, 292). Aristarchus' observation of the summer solstice was used 145 years later by Hipparchus, who compared it with an observation of his own to establish an accurate length of the year (Thurston 1994, 126). And essential parameters (period relations) in Hipparchus' lunar theory had originated in Babylonian astronomy (Neugebauer 1975, 321).

In ancient astronomy, ratios between synodic and sidereal periods of planets could be established with fair accuracy from observations over a few decades because the error of individual observations would be distributed over the long intervals of integral period equations (Neugebauer 1975, 14, 386–91; Thurston 1994, 79–81). Similarly, one might establish, to fair precision in the fractional day, the number of days in a sidereal year; by counting the number of days between successive first visible heliacal risings of a certain fixed star over a large number N of such first risings; the imprecision in the fractional day in a sidereal year would be inversely proportional to N.

On a special day each summer, clouds permitting, I watch the sun rise out of Lake Michigan. That is how I perceive it—the sun rising up from the lake horizon—even though my standing belief is that the phenomenon is due to the axial rotation of the earth. I can see the phenomenon as the horizon descending below the sun, but that requires addition of an intellectual construction to the experience. It seems, then, very natural that people would first conceive the sun, moon, and stars (and planets) as moving over the earth; that is just as we experience the circumstance.

Risings, settings, eclipses, and occultations are celestial events we can observe with unaided normal vision. With normal memory over a day, we can observe as well the daily journey of the sun and moon and the journey of the shadows they induce around us. Systematic measurement and recording of locations (angular) and times of recurring events in the celestial dome, and of attendant shadow lengths, directions, and motions on our ground, can reveal patterns of motion not discernible by direct perception and memory alone. We can become cognizant of the variability of the speed of the moon through records of direct observations of the day-by-day progress of the moon with respect to the fixed stars (Neugebauer 1975, 71, 371; 1969, 210; cf. 1975, 85). By gnomon and polos and by marking records, we can determine the dates of the equinoxes, learn that the seasons are unequal (Meton and Euctemon in 432 B.C.), and determine the obliquity of the ecliptic to the equator (assuming one has apprehended those two natural distinct planes; Pedersen 1993, 37–41; Thurston 1994, 41–44; Neugebauer 1975, 371–73, 627–29; Heath 1981, 130–31). This is mathematics, and instruments and records, extending our indirect observations.

As indirect observations, I should also count interpolations, such as the Seleucid Babylonians evidently performed to approximate daily longitudes of planets (between conjunction, opposition, or stationary points) using higher-order difference sequences (Neugebauer 1969, 127; 1975, 397, 412–18). Likewise, I should count as indirect observations inferred distances and angles, such as were attained by Hipparchus' resort to procedures such as tables of chords, analemmata, and stereographic projection (Neugebauer 1975, 299–304, 868–69; 1948, 1017, 1028–37; Goldstein 1983a). I should count as indirect observation Ptolemy's use of stereographic projection to determine zodiac-sign rising times by plane trigonometry (Neugebauer 1969, 185, 220) and his use of Menelaos' theorems to infer solar declinations, thence right ascensions, for a given solar longitude (Neugebauer 1975, 30–32).

It is commonly noticed that mathematics is abstract, precise, and concise (Metaph. 982a25–29; 1061a29–61b2). I expect the latter two ride upon the former, though upon two different aspects of the former. Precision is borne by the prescinding aspect of abstraction. Conciseness is borne by the hypostatizing aspect of abstraction (Ph. 193b31–194a11).

Conciseness would seem to be a matter of cognitive economy, such as we enjoy in counting and elementary arithmetic, all the more with place-value notation (Neugebauer 1969, 18–22), and such as we enjoy by widely applicable relationships made manifest in formal mathematics. I take the following mathematical techniques as effecting cognitive economies, whether applied to observation, to characterization, or to explanation: Babylonian use of the concept of the geometrical relation similarity (Neugebauer 1969, 46) and their use of formulae for obtaining solutions of quadratic equations (ibid., 41, 149–51); use by Autolycus, Euclid, and Theodosius of representative figures of spherical astronomy, with lettered elements, for reference in text (Neugebauer 1975, 752–54); use by Menaechmus of an algebraic relationship as proxy for a geometric object, so that connections among geometric objects "may be deduced by manipulation of their algebraic equivalents" (North 1987, 176).

Though cognitive economy is an important function served by mathematical descriptions, I want to put most of my attention on precision. Precision narrows possibilities of what is in nature. That narrowing is a function of mathematical description in science in general, whether in observations, in characterizations, or in explanations. The special gift of mathematical precision to observation is as we have seen above: expansion of the range of phenomena accessible to indirect perception.

II. Mathematical Description in Characterizations

What I mean by a characterization, in the context of physics and astronomy, is the finding of an essential form of a phenomenon. The form need not characterize the phenomenon exactly, only approximately. Such forms as I have in mind are mathematical or at least antemathematical. The contrast I want to make between what we should, by our present lights, call mere characterization and what we should call explanation is that the latter is the finding of working or constituting causes. For Pythagoreans, of course, and perhaps for many thinkers today, characterization by essential forms suffices for explanation. I think rather not, although a middle course (in which what I am calling mathematical characterization of phenomena be taken as a formal explanation, though not invoking formal causes) does not seem unreasonable (e.g. Gaukroger, chp. 6).

In saying mere, I intend no necessary inferiority of mere characterization in comparison with causal explanation. The latter also is a characterization, and I intend by the qualification mere only to designate a characterization that is not a causal explanation.

One reservation I have with placement of mere characterization by essential form into one hand and causal explanation into the other hand is that the causal buck always stops somewhere. If asked why the earth continues its axial rotation, I should say this is due to conservation of angular momentum. I should also be fairly satisfied, justifying my satisfaction of this causal stopping-place by reminding us how deep are angular momentum and its conservation: their ubiquity in both classical and quantum regimes; the importance of the unit of angular momentum in quantum mechanics; the connections of angular momentum conservation to other deep principles (of physics, including physical geometry) in Hamiltonian mechanics and in general relativity. Still, I am left in the end with angular momentum conservation and its cohorts as brute, unexplained physical facts, just as the Pythagorean is left in the end with brute mathematical forms. Perhaps, ultimately, the two hands and what is in them must be brought together, but for today it seems most sensible to me to keep the two hands apart, that is, to maintain an ontological distinction between mere essential forms and working or constituting causes.

Babylonian astronomers, at least by the Seleucid era, had devised elaborate, purely arithmetical procedures (using periodic step or zigzag functions; Neugebauer 1969, 110–15; 1975, 373–79) from which they could predict dates of characteristic planetary phenomena such as oppositions or onsets of retrogression (Neugebauer 1969, 125–34; 1975, 420–31; Thurston 1994, 79–81) and times of lunar events such as full moons and eclipses of the moon, even the extent of lunar eclipses (Neugebauer 1969, 109, 117; 1975, 474, 549–50, 1094–95, 1124; Thurston 1994, 74–78). The key to Babylonian prediction of lunar eclipses was "the construction of a common period (later known as the Saros) of syzygies and latitudes that made it possible to select those syzygies which would be accompanied by eclipses" (Neugebauer 1975, 664; also 1969, 118–19). Nevertheless, Otto Neugebauer judges that this does not imply that the Babylonians had a geometric model for the sun, moon, earth, and earth shadow; only arithmetic methods were required for predicting times of full moons and for describing the lunar motions in latitude, thence predicting lunar eclipses (Neugebauer 1975, 664; 1948, 1020). Furthermore, we do not know whether Babylonian astronomers explained lunar eclipses—"to introduce the concept of the earth's circular shadow is [tantamount] to postulating the sphericity of the earth, a concept which otherwise is completely lacking in Babylonian astronomy"—nor even lunar phases (Neugebauer 1975, 550; also, 1093–94). What we do know is that they had arrived at more and less effective arithmetical characterizations of lunar, planetary, and solar phenomena. Some of these were characterizations whose precision could be improved by corrective adjustments in response to deviations from observations accumulating over time (Neugebauer 1969, 116; 1975, 484–86, 497–99).

A second example of pure characterization would be the Aristotelian schematic geometric characterization of the shape of rainbows, a pure characterization anyway when we consider it in isolation from Aristotle's full, causality-dressed picture. For a rainbow formed when the sun is at some place on the horizon, Aristotle considers the straight line from the horizon point (say, at the center of the sun's width) to some point on the rainbow, which rainbow is to the side of the observer opposite the sun, and the straight line from that rainbow point to the observer. Whatever be the reasons for color forming that point of the rainbow, the apparently crucial angle between the aforesaid two straight paths (of rays of vision) will be preserved if those two paths are together rotated about the line from the observer to the sun's horizon point. Under such a rotation, the bow point will trace a full semicircle of rainbow above the ground; as we observe. The sun's horizon point, the point of the observer, and the center of the rainbow's (semi)circle will be collinear. For rainbows formed when the sun is above the horizon, the sun's location is treated as if it were the former horizon point elevated. The center of the rainbow's circle is equally lowered beneath the ground. Those two points keep their collinearity with the point of the observer. The portion of the rainbow's circle remaining above the ground will then be less than a semicircle; as we observe (Boyer 1959, 42–44; Meteor. 371b26–29, 375b17–76b21, 376b28–77a11). The Aristotelian geometric characterization of the rainbow is primitive, but promising (Descartes!) for future geometric modeling of rainbow phenomena.

III. Mathematical Descriptions in Explanations

Bernard Goldstein and Alan Bowen have argued that Eudoxus was the first to use the fundamental two-sphere model of Greek astronomy "to account for the risings and settings of stars, to provide a framework for geographical studies, and to justify a more mathematically sophisticated sundial. At the same time, he laid the foundations for the application of geometrical argument to the study of celestial phenomena" (Goldstein 1983b, 234). The first sphere represents the stationary earth. The second, surrounding and concentric sphere rotates daily and uniformly about a fixed axis passing through the common center of the two spheres. The second sphere is just the celestial sphere, the orb of the fixed stars. If all the two-sphere model did was describe the nightly risings and settings of stars as seen from some point on the earth, I should be inclined to class this model as a mere geometric characterization of diurnal stellar motion, really a straight extraction of the essential character of that phenomenon. But the combination of rotating celestial star-studded sphere with sphericity of the earth does explain, by working causes, the changes in the nightly stars passing overhead as one travels north or south on the earth.

For quite a different reason, I should also take Eudoxus' characterization of the motions of moon, sun, and planets by interconnected homocentric uniformly rotating spheres (with axes of rotation variously oriented) to also count as explanatory. For Eudoxus, as for the Pythagoreans and Plato, uniform rotation of an isolated sphere was an elementary motion. I construe Eudoxus' portrayal of the observed motions of moon, sun, or planets each in terms of a trio or quartet of homocentric rotating spheres (nested about the spherical earth) to be a mechanically-minded way of composing as resultant the motion of the moon, sun, or planet from elementary uniform circular motions. Eudoxus' composition of the resultant motion of each the moon, sun, or planet does not have the freedom we have in composing an observed motion, say specifically an angular velocity, from its components along freely chosen basis vectors. Eudoxus' composition of motions does not have that sort of freedom. Inflexibility of the will-he-nil-he sort in our constructions suggests we are encountering autonomous realities. Eudoxus' assemblies of homocentric rotating spheres are as if the kinematics of rotary machines. It seems that each rotating sphere and its characteristics—its order in the nest of three or four and its angular speed (required to be uniform) and orientation of rotation—is evidently uniquely fixed by the required resultant motion of moon, sun, or planet. I am inclined to class Eudoxus' homocentric-sphere models as not only characterizations of the motions of moon, sun, and planets, but as constitutive causal explanations: the resultant motion of the celestial body is constituted by the elementary motions in its assembly of homocentric spheres. The elementary motions cause the motion of the celestial body in that they compose it, or constitute it.

Eudoxus' model specifies the number, order, inclinations, and periods of the homocentric spheres; the celestial body being affixed to the equator of the innermost sphere. The model is specific enough to get itself into trouble, and that is a great virtue. That, I should say, is a broad function served by mathematical description in explanation, as in pure characterization: mathematical precision begets sensitivity of characterization, causal or pure.

The Eudoxan model for the motion of the sun implies equality of the seasons; Callippus tried to remedy this defect by adding two more spheres for composing the solar motion. The Eudoxan model for the motion of the moon captures the Saros period, variations in the latitudes of the moon, and eclipses of the moon. This model of lunar motion implies (other, unknown conditions being constant) that the apparent diameter of the moon will be constant. But, as a student of Eudoxus came to realize, the apparent diameter of the moon does change. In addition, the model for the lunar motion implies that the moon's motion around the ecliptic will be uniform, whereas in actuality the speed of the moon varies in that migration (Pedersen 1993, 69; Thurston 1994, 113–14). Callipus may have known about this variation, from Babyonian astronomy. It was perhaps for the sake of bringing this variation into the homocentric account of the moon's motion, that he added two more spheres for composing the motion of the moon, as he had done for the sun (Neugebauer 1975, 625).

The Eudoxan model of the planetary motions yields motions resembling retrogradations and variations in latitude for some planets, but the model's shortcomings, qualitative and quantitative, are several (Thurston 1994, 116–17; Neugebauer 1975, 679–84). Among them are the noticeable facts of variable brightness of Mars and of Venus over their circuits. This variability, like the variability of the apparent diameter of the moon, could not be accounted for by homocentric models; each celestial body, through all its gyrations, is necessarily conjectured to remain a fixed distance from the earth under such models.

The eccenter geometric model of Apollonius was able to account for evident variations of distances and angular speeds of the sun and moon as they are observed from the earth. The eccenter model retained uniform circular motion as elementary. The moon, sun, or planet orbits at constant speed and distance about some fixed center, the eccenter, which is in the neighborhood of the fixed earth (or, as the phenomena may require, the eccenter moves uniformly in a small circle about the fixed earth).

The epicycle method of modeling motions, discovered some time before Apollonius, also retains uniform circular motion as elementary. Apollonius came to realize that all the results obtainable with an eccenter model could also be obtained by an epicycle model. These two ways of modeling the motion of moon, sun, or planet about the earth are equivalent in the sense that they both yield the same angular motion of the moon, sun, or planet about the earth. Apollonius then realized that an epicycle model can be made to display retrograde motions of planets, and he proceeded to develop both epicycle and eccenter models for the stations of the planets (Pedersen 1993, 70–73; Neugebauer 1975, 263–70).

Hipparchus suited these models with quantitative values taken from observations and from the Babylonian arithmetic characterizations of phenomena. He applied an eccentric model (with fixed eccenter) to the motion of the sun, deriving basic parameters of the model (eccentric-quotient and longitude of solar apogee) from the positions (true longitudes) of the sun at cardinal points and from the lengths of seasons between them. From this model, solar longitudes can be calculated as a function of time to an accuracy of about half a degree (Thurston 1994, 128–31; Pedersen 1993, 73–75; Neugebauer 1975, 306–8.). He applied an epicycle model to the motion of the moon, deriving basic parameters from (two sets of) three positions-at-times of the moon, defined by three lunar eclipses. With his epicycle model for lunar motion, conjoined with his model of solar motion, he could have predicted lunar eclipses with some assurance (Thurston 1994, 131–34; Pedersen 1993, 75–76; Neugebauer 1975, 319–22, 129–31).

Hipparchus' mathematical characterizations had that good quality of being specific enough to get themselves into trouble. Ptolemy's mathematical characterizations of the apparent motions of sun, moon, and planets would be more refined, more capable of accounting for further aspects of those motions. The refinements and extensions of Hipparchus upon the work of Apollonius and the refinements and extensions of Ptolemy upon the work of Hipparchus are further cases of mathematical precision begetting sensitivity of characterization. Along these courses of refinements, lasting discoveries of nature were made, notably the precession of the equinoxes, or distinctness of sidereal and tropical year, and the second lunar inequality.

Firm discoveries and fineness of fit (including predictive power) of mathematical models are indications that we are approaching the independent natures of things. How does that square with the fact that throughout the modeling by Apollonius, Hipparchus, and Ptolemy, there is equivalence between the geometric eccenter model of a particular apparent motion and a geometric epicycle model of that same particular apparent motion?

My perspective on the dual geometric characterization, by eccenter and by epicycle, is as follows: An eccenter model and its equivalent epicycle model are each, after the fashion of characterization by homocentric spheres, prima facie constitutive causal explanations of the apparent motions of sun, moon, or planet. The "resultant" motions of a celestial body are prima facie constituted by the elementary motions in both its eccenter and epicycle renditions. The elementary motions in each of the two renditions are prima facie constituting causes of the celestial body's apparent motion in that each rendition composes that motion.

Unlike the constitutive causation of a resultant physical motion by an assembly of homocentric rotating spheres, there is freedom to choose between an eccenter- or an epicycle- constitutive cause of the resultant physical motion. That is, Apollonius, Hipparchus, and Ptolemy have a freedom in their constitutive causal characterizations that Eudoxus did not have. That freedom seems moderately similar to our freedom to choose between two coordinate systems (e.g., elliptical cylindrical coordinates v. parabolic cylindrical coordinates) for a problem. With the freedom to choose—apparently indifferently to nature—between an eccenter or an epicycle characterization of the apparent motion of sun, moon, or planet, it seems very natural to reclassify these characterizations as not constitutive causes of the motions of sun, moon, or planet, but as mere compositional characterizations of those motions. According to this adjusted stance, what remains in nature (as in nature itself) from these characterizations would be whatever elements of the eccenter and epicycle characterizations are in common between them (Neugebauer 1975, 57 and 1220, fig. 51), including of course, the apparent motion of the celestial body. Invariance under transformation (eccenter-epicycle) demarcates degree of autonomous reality.

What remains most definitely physically given are those resultant (of eccenter and epicycle characterization) apparent motions of the sun, moon, and planets and the working effects (optical) of those bodies among themselves, the earth, and vice versa, such as the cast of the earth's shadow across the moon. Physically, the earth can remain central, if not dispositively at the center; with sun, moon, and planets dancing out orbits around the realm of the earth, still besinging the praises of uniform circular motion in the hymns of eccenter and epicycle characterizations (but with a discord by Ptolemy's use of equants). This would seem close to the overall picture settled upon by Ptolemy (Pedersen 1993, 87–89; Thurston 1994, 171–76).

We have at hand then a further way in which mathematical characterization can make our conceptions of natures sensitive to those natures. By what and what not we are free to choose in our so-far true characterizations, we can sort the likely facts of natures themselves from their accessories.

References

Aristotle 1984 [c. 348–22 B.C.]. The Complete Works of Aristotle. J. Barnes, editor. Princeton: University Press.

Boyer, C.B. 1959. The Rainbow: From Myth to Mathematics. New York: Thomas Yoseloff.

Gaukroger, S. 1978. Explanatory Structures: Concepts of Explanation in Early Physics and Philosophy. Atlantic Highlands, NJ: Humanities Press.

Goldstein, B.R. 1983a. The Obliquity of the Ecliptic in Ancient Greek Astronomy. In Goldstein 1985.

——. 1983b. A New View of Early Greek Astronomy. In Goldstein 1985.

——. 1985. Theory and Observation in Ancient and Medieval Astronomy. London: Variorum Reprints.

Heath, T. 1981 [1913]. Aristarchus of Samos. New York: Dover.

Neugebauer, O. 1948. Mathematical Methods in Ancient Astronomy. In Astronomy and History: Selected Essays. 1983. New York: Springer-Verlag.

——. 1969. The Exact Sciences in Antiquity. 2nd ed. New York: Dover.

——. 1975. A History of Ancient Mathematical Astronomy. Providence: Brown University Press.

North, J.D. 1987. Coordinates and Categories: The Graphical Representation of Functions in Medieval Astronomy. In Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages. E. Grant and J.E. Murdoch, editors. Cambridge: University Press.

Pedersen, O. 1993. Early Physics and Astronomy. 2nd ed. Cambridge: University Press.

Thurston, H. 1994. Early Astronomy. New York: Springer-Verlag.

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For some uses of mathematics in modern science, see:

http://www.solopassion.com/node/2361#comment-61725

Edited by Stephen Boydstun

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Stephen writes;

Still, I am left in the end with angular momentum conservation and its cohorts as brute, unexplained physical facts, just as the Pythagorean is left in the end with brute mathematical forms. Perhaps, ultimately, the two hands and what is in them must be brought together, but for today it seems most sensible to me to keep the two hands apart, that is, to maintain an ontological distinction between mere essential forms and working or constituting causes.

I respond:

angular momentum and its cohorts are not quite as brutally unexplained. By way of Noether's theorem the conservation of these quantities follow from the underlying symmetries in nature. It is the symmetries that are basic and primordial.

As for causes, causes are itch scratchers. I am with Hume on this. When we see an event of type A always preceding and event of type B we are just driven to postulate a cause connecting the two. Causes live up in our heads. Events are Out There.

Our so-called explanations are mental dispositions that follow from our acceptance of and love for our models. Remember when the cause of heat was the flow of caloric? Ah yes. Those were the days.

Ba'al Chatzaf

Edited by BaalChatzaf

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angular momentum and its cohorts are not quite as brutally unexplained. By way of Noether's theorem the conservation of these quantities follow from the underlying symmetries in nature. It is the symmetries that are basic and primordial.

Right, conservation of angular momentum is the result of space rotation symmetry, i.e. there is no preferred direction in space, all directions are equivalent. This is the simplest assumption we can make about the rotational structure of space, and therefore we should really only need an explanation if angular momentum were not conserved. Assume nothing special and you get conservation of angular momentum (in a similar way conservation of momentum follows from space translation symmetry and conservation of energy from time translation symmetry).

As for causes, causes are itch scratchers. I am with Hume on this. When we see an event of type A always preceding and event of type B we are just driven to postulate a cause connecting the two. Causes live up in our heads. Events are Out There.

Exactly.

Edited by Dragonfly

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Folks:

One of the most common fallacious "reasoning" events...after this, therefore because of this.

Adam

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Folks:

One of the most common fallacious "reasoning" events...after this, therefore because of this.

Adam

More correct is: not after this therefore not because of this.

Ba'al Chatzaf

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Reading this composition these nine years later, it took me a while to see why I was referring to the principle of conservation of angular momentum as a causal explanation of phenomena at all. Conservation principles are reasons, they fulfill because, but are they causal explanations? I had written “if asked why the earth continues its axial rotation, I should say this is due to conservation of angular momentum.” That’s pretty bad. All it really says is that unchanging continuance of the earth’s axial rotation is a case of the principle of inertia, Newton’s first law. That is a principle that sorts motions into those not requiring an explanation in terms of an external cause (force or torque) and the sorts of motion that do require an external cause. The steady spinning earth (supposing its angular velocity were exactly unvarying) is the sort of motion not requiring an external causal explanation at all.

That each cubic inch of matter composing the earth is rotating about the earth's axis, rather than traveling along some straight line at constant speed, does require causal explanation external to each such part. To begin mathematical characterization to serve causal explanations, we turn from Newton’s first law to his second.

But let us not simply toss conservation principles into the bin of characterization by an essential form. Instead of the whole spinning globe, think of the way a spinning figure skater draws in her arms to spin faster. Appealing to the conservation of angular momentum is a squarely causal explanation for what is going on in this maneuver. The lesson I draw then is that a given conservation principle can be appealed to merely to point out an essential form present in a case at hand (unvarying spin of earth) or appealed to by way of causal explanation (changing spin of skater).

In my 1991 essay “Induction on Identity,” I had written “Out of all the conditions that obtain in a situation, we typically take only one or a few as cause of some distinctive result, only a select portion of the ways in which the law of identity applies to an action or a becoming. We try to discover among antecedent conditions ones that will make a certain result under a wide range of variations in the remaining variable conditions. . . . In a primary sense, causes make things happen” (Objectivity V1N3, pp.25–26). That still goes. I’m pleased to say that this conception of causality has been developed in a really big way by James Woodward in Making Things Happen: A Theory of Causal Explanation (OUP 2003).

Looking to our further, modern understanding of why angular momentum is conserved, Bob and Peter remind us that isotropy of space joined with Hamilton’s principle implies that conservation. I say that causality is quartered in Hamilton’s principle, and the way causality can be brought out of it in classical domains, such as for the extended causal explanation of the changing spin of the skater, is through the relation between the Hamiltonian (a composition of energies in this situation) and force (or torque). With rendition of an application in terms of forces, we could turn to Newton’s second law (and third) to produce a causal analysis, in mathematical form.

As you would expect, I do not agree that causation is in the head and not given by the world. There are true causes, and mathematical characterizations help us find them and their situation in the mind-independent world.

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Following on the lesson I drew above, it appears that in Newton’s derivation of the form of the law of gravity in Book 1 of Principia, he uses the conservation of angular momentum only as an essential form. However, the principle is here doing significant explanatory work (noncausal), unlike in an appeal to conservation of angular momentum to “explain” the continued rotation of the earth on its axis.

In Newton’s dynamical analysis of planetary motion, the conservation of angular momentum was quartered in the law of equal areas. Kepler had discovered that law, you will recall, for the elliptical orbits of the planets. Newton demonstrated that the law holds more generally for any orbit due to any sort of central force.

This generalization is necessary for his grand strategy of demonstrating the various separation-dependencies of the various force laws that would be required for various supposed forms of orbits (mathematically specified of course)—to the end of showing that his demonstration of the form of the actual force for the actual planetary orbits is an instance of a general form of demonstration upon very general dynamical principles ( http://www.objectivistliving.com/forums/in...amp;#entry45789 ). (Instructive: The Key to Newton’s Dynamics by J. B. Brackenridge [u of CA 1995] and, rocking the boat, essay 3 in Reading Natural Philosophy edited by D. B. Malament [Open Court 2002])

Newton’s generalized law of equal areas in equal times is Proposition I, Theorem I of Principia Book 1. The demonstration relies on Newton’s first law alone. (Newton’s use of the second law in his proof of the parallelogram rule is dispensable.) In Proposition II, Theorem II, Newton demonstrates that any body subject to central forces (as in Prop. I) is subject, more specifically, to a centripetal force. This demonstration relies on Newton’s second law, as well as the first. Proposition IV, Theorem IV derives the general mathematical form of centripetal forces, relying on Propositions I and II. (The Scholium here mentions his alternative demonstration, from his student days, of this mathematical form [ http://www.solopassion.com/node/2361#comment-61725 ].) Proposition VI, Theorem V codes centripetal force into a characteristic of the instantaneous arc along the path of an orbiting body. This demonstration relies on Proposition I and its corollaries.

The specific form of the centripetal force (gravity) that sets an orbiting body in an elliptical orbit, where the source of the force is located at one focus of the ellipse, is demonstrated in Proposition XI, Problem VI. The demonstration relies on Proposition VI. All of these demonstrations rely on geometry. That is not a causal element here, unlike in machinery.

Newton’s demonstration of the mathematical form of the law of gravity is an explanation. His demonstration displays explanatory structure, following the causal structure of nature. Forces have sources, and the form of those forces dictate the form of orbits about the source. The form of the orbits indicates the form of their external cause.

Proposition XI relies on Newton’s first law and second law. Conservation of angular momentum enters in the way that the first law enters. Causality enters under the second law. Conservation of angular momentum is part of the Newtonian explanation of the law of gravity, but it is not the causal strand in the explanation.

When we come to general relativity, the gravitational force is seen as a particular type of curvature relation between nearby geodesics in a curved spacetime. It is sensible to say that spacetime curvature dictates that planets shall orbit the sun in the way they do. Certain aspects of spacetime (not mere space) become causes. But, of course, spacetime curvature is caused by source matter/fields. Causality runs from source mass-energy density to spacetime curvature to characteristics of the motion of bodies in that spacetime vicinity. Newton’s law of gravitation is recovered from general relativity in the joint limit of (i) velocities small in comparison to c and (ii) weak gravitational fields.

On the loss of traction from Noether’s theorem when applied in GR so as to include not only mass-energy density but energy of the (nonlocal) gravitational field, see Roger Penrose’s The Road to Reality, pp. 489–90 (Knopf 2004).

Related:

http://www.objectivistliving.com/forums/in...amp;#entry28908

http://www.objectivistliving.com/forums/in...amp;#entry27765

http://rebirthofreason.com/Forum/Objectivi...0242_3.shtml#69

~~~~~~~~~~~~~~~~~~

Using ancient astronomy for examples, I moved from the uses of mathematics in observation to its uses in characterization by essential form to its uses more particularly in characterizations that are causal explanations. In my next note, I want to look at a case, from optics, having those three, but scientific experimentation as well.

Edited by Stephen Boydstun

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Rainbows

Aristotle (384–22 B.C.)

The idea that rainbows are produced by sunlight and its reflection by clouds goes back at least as far as Anaximenes (flourished c. 575 B.C.). Anaxagoras (born c. 500 B.C.) and Democritus (fl. c. 460 B.C.) may have held more advanced views on the rainbow specifically and on the character of reflection in general. Plato (427–347 B.C.) had some sense of the fixed character of reflection angles off mirrors, though not the law of reflection itself (Timaeus 46a–c; Boyer 1987, 34–39).

Aristotle offered an explanation of the rainbow. Some sciences are related to others such that “one is under the other—e.g. optics to geometry and harmonics to arithmetic” (An.Post 75b16–17). “Geometrical demonstrations apply to mechanical or optical demonstrations, and arithmetic to harmonical” (76a 24–25). Considering optics by means of geometry, “it is for the empirical scientist to know the fact and for the mathematical to know the reason why; for the latter have the demonstrations of the explanations, and often they do not know the fact, just as those who consider the universal often do not know some of the particulars through lack of observation” (79a3–7).

“While geometry investigates natural lines but not quo natural, optics investigates mathematical lines, but quo natural, not quo mathematical” (Phys 194a10–11). Sciences considered by means of mathematics “are those which, being something different in substance, make use of forms, for its objects are not said of any underlying subject—for even if geometrical objects are said of some underlying subject, still it is not as being said of an underlying subject that they are studied. / Related to optics as this is related to geometry, there is another science related to it—viz. the study of the rainbow; for it is for the natural scientist to know the fact, and for the student of optics—either simpliciter or mathematical—to know the reason why” (An.Post 79a8–13).

What is the phenomenon of the rainbow and its circumstances for which Aristotle offers an explanation? “The rainbow never forms a full circle, nor any segment [portion of circle] greater than a semicircle. At sunset and sunrise the circle is smallest [radius is smallest] and the segment largest: As the sun rises higher the circle is larger and the segment smaller. . . . There are never more than two rainbows at one time. Each of them is three-colored; the colors are the same in both and their number is the same, but in the outer rainbow they are fainter and their position [order] is reversed. In the inner rainbow the first and largest band is red; in the outer rainbow the band that is nearest to this one and smallest is of the same color: the other bands correspond on the same principle” (Meteor 371b27–372a6). The three colors Aristotle sees in the rainbow are red, green, and purple, “though between the red and the green an orange color is often seen” (372a9–10). http://www.arrowphotos.com/Rainbows.htm

We have seen Aristotle’s geometrical explanation of the shape of the rainbow in the last paragraph of §II of the starter for this thread. I will repeat that explanation now.

For a rainbow formed when the sun is at some place on the horizon, Aristotle considers the straight line from the horizon point (say, at the center of the sun’s width) to some point on the rainbow, which rainbow is to the side of the observer opposite the sun, and the straight line from that rainbow point to the observer. Whatever the reasons for color forming that point of the rainbow, the apparently crucial angle between the aforesaid two straight paths will be preserved if those two paths are together rotated about the line from the observer to the sun’s horizon point. Under such a rotation, the bow point will trace a full semicircle of rainbow above the ground, which is as we observe. The sun’s horizon point, the point of the observer, and the center of the rainbow’s (semi)circle will be collinear. For rainbows formed when the sun is above the horizon, the sun’s location is treated as if it were the former horizon point elevated. The center of the rainbow’s circle is equally lowered beneath the ground. Those two points keep their colinearity with the point of the observer. The portion of the rainbow’s circle remaining above the ground will then be less than a semicircle, which is as we observe (Boyer 1987, 42–44; Meteor 371b26–29, 375b17–76b21, 376b28–77a11).

In the starter for this thread, I had offered Aristotle’s schematic geometrical interpretation of the shapes of rainbows as an example of pure characterization. I am going to have to take back now what I said there: that this geometrical characterization is itself not a causal explanation. This characterization is too similar to the case of the spinning figure skater to not be counted a causal explanation of the shape of the rainbow. In particular the geometry explains how a greater elevation of the sun above the horizon brings it about that the rainbow forms a smaller portion of a semicircle. We now have examples of conservation-of-angular-momentum explanations that are causal (spinning skater changing her spin rate) and not causal (planetary orbits*) as well as geometrical explanations that are causal (elevation of sun reducing rainbow circular portion) and not causal (say, Newton’s use [in Prop. II, Th. II] of Euclid’s Prop. XL).

Aristotle supposed that for a given rainbow, a particular band of color lies at some single angle between the direction of the sunlight falling on the water in the air and the line of sight of the viewer (Meteor 375b17–377a11). He thought that the way the droplets of water produce color is by reflection. Aristotle did not know of the dispersion of light into colors by refraction. He knew of reflection by manmade mirrors and by natural ones, such as the appearance of the Milky Way on a body of still water.

He figured that just as shapes are revealed in mirror reflections under tight angular rules, there is another kind of reflection—by very tiny mirrors—which yield no shapes, only colors. These colors by minute reflectors, it is assumed, also operate under tight angular rules (Meteor 372a17–b8, 373a32–b32; An 435a5–10).

Aristotle may have known the law of reflection (of shapes) from mirrors, which is the rule that the angle of light incidence (angle with the plane it falls upon) equals the angle (with that plane) of line from eye to mirror (Boyer 1987, 39–41). But if he did know that rule and carried it over to “color reflection,” he failed to see an important implication that then follows from his geometrical account of the rainbow: arcs formed under all elevations of the sun should have equal radii, contrary to his statement of rainbow phenomena. The visual observation that rainbows less than semicircular have larger radii than full semicircular rainbows is a visual illusion, which could have been deposed by reasoning that enlisted the law of reflection (43–44).

Ptolemy (c. A.D. 100–180)

Look into a plane mirror. Reflected objects appear in it. In optics we call an object in there an image.

Ptolemy sets out three general principles for reflections in plane mirrors and, by tangent planes, for reflections in spherical mirrors. (1) The image in the mirror lies along a straight line from the location e of the eye to the image. That line intersects the plane M of the mirror at some point r. Let the location of the image in the mirror be called the point i.

Consider the straight line from a point o on the real object to its image i in the mirror. (2) That line is perpendicular to the mirror. Imprecise confirmation of this perpendicularity can be obtained by holding the blunt end of a pencil against the mirror, holding the pencil perpendicular to the mirror. Observe that the line of the pencil continues into the mirror without any bend. Tilt the pencil. Its line in the mirror will bend such that the line between the actual lead point o and its image point i looks to be perpendicular to the mirror plane M.

We now have two lines intersecting the point i, the lines ei and oi. From our Euclidean geometry, we recall that two intersecting lines determine a plane. Because one of our two lines lying in that plane P is perpendicular to the mirror plane M, we know that P is perpendicular to M. Now the point o lies in the plane P, and the point r lies in the plane P (for the lines oi and ei determine and lie in P, and r lies on ei). Therefore the line or lies in the plane P. But the line er also lies in P. The lines er and or determine a plane, and that is the very plane in which oi lies. The real object, the eye, the point of reflection on the mirror, and the image in the mirror all lie on a single plane.

The intersection of two planes determines a line. Call by L the line formed by the intersection of M and P. Only L lies on both M and P, and because r lies on both M and P, r lies on L. With all this understanding, it is proposed: (3) The angle between er and L is the same as between or and L. This we know as the law of reflection. Ptolemy gives a way of imprecise confirmation of this principle, then turns to a more certain demonstration by experiment (Optics Bk. 3, [3]–[6]).

Ptolemy has inscribed on a round bronze plate a circle, two perpendicular diameters through the circle’s center, and fine marks graduated by one angular degree within each quadrant. He has made three mirrors of polished iron. One mirror is plane, one convex cylindrical, and one concave cylindrical. He mounts a diopter along a radius of the circle, at some angle from one of the diameter lines, to be able to view the center of the circle from a definite known angle along the edge of the plate. At the center he puts a mirror aligned with the other diameter line. He slides a little colored object along the inscribed circle until it comes into view in the mirror through the diopter. Lo! The angle between the object-to-center radius and the plane of the mirror at the center point equals the angle between the diopter-viewing radius and the plane of the mirror at the center point (Bk. 3, [8]–[11]). http://brunelleschi.imss.fi.it/museum/esim.asp?c=201201

Ptolemy turns to refraction, the deflection of images seen through transparent media. To begin, make a watertight cylindrical basin. The basin shall be open along its length such that the remaining lengthwise wall has cross section significantly greater than a semicircle. Set the basin horizontal. Rest a coin inside the basin such that it is just out of view when one peers over the open edge of the basin. Gently fill the basin with water. The coin will become visible from one’s same vantage point, peering just over the edge of the basin (Bk. 5, [6]).

Now to quantification. Set that round inscribed disk from the reflection experiment on its edge square in a semicylindrical horizontal basin. Align one of the disk’s perpendicular lines horizontally, and fill the vessel with water to that line. Set a marker at some angle along the circle of the disk above the water. The disk has a nodule at its center, to use for sighting (as with a rifle). Sight from the edge marker such that it and the center marker appear to coincide. At the same time, move another marker along the basin wall under water until the marker appears aligned with the center-sight and the marker on the disk arc above water. The angle between the underwater marker (its radial line) and the disk’s inscribed vertical line will be less than the above-water marker (its radial line) and the disk’s vertical line (Bk. 5, [7]–[8]).

In this manner, Ptolemy obtains the refractive bending of sight-line from air to water for angles of sight at 10, 20, . . . 80 degrees from the vertical line. (Today we would say he was observing the refractive bending of light-line from water to air in this experiment.) The amounts of bending he reports are fairly accurate, except for the one at 80 degrees. It has been argued that Ptolemy was doctoring his results to fit a sequence of numbers that could be generated by an algorithm that had been used for generating sequences in Babylonian astronomy of the Seleucid (Smith 1996, 44–45, 233n9; Neugebauer 1969, 110–14, 135).

Ptolemy learned from his experiments in refraction that the greater the angle between sight-line and the line perpendicular to the water surface, the greater will be the difference between that angle and the angle between the straight extension of the air sight-line on into the water and the actual, bent sight-line continued to objects under the water. But he does not have the sine-formula we call the law of refraction which is able to capture the amount of water-surface bending for all values of air sight-line.

If Ptolemy worked on theory of the rainbow, his work has not come down to us. Ptolemy lived in Alexandria. A point was added to the phenomenology of the rainbow by an Athenian philosopher, Alexander of Aphrodisias. He became head of Aristotle’s school, the Lyceum, between A.D. 198 and 211. He reasoned from Aristotle’s theory of the rainbow (parts of the theory not presented here) that the sky between the primary and secondary bows should be brighter than the rest of the blue sky. He noted, however, that the sky between the bows is in fact darker than the rest of the sky. Today we call this dark arc Alexander’s band, and it is one characteristic to be explained by theories of the rainbow.

(To be continued.)

Note

* I said in #6 of this thread that Kepler’s second law (and, therein, conservation of angular momentum) does not enter as causal in Newton’s dynamical analysis of planetary motion. It does not enter as causal in Kepler’s account either. Recall that a planet travels through the path of its orbit faster while nearer the sun than while farther the sun. That is explained by Kepler’s second law: equal areas are swept out in equal times throughout the (elliptical) orbit of a given planet. Kepler did not see this as a causal explanation of the speed-distance inverse variation. In his conception, the cause of the planet moving faster while nearer the sun was that the strength of the sun’s ability to move a planet along its orbit is stronger the nearer the planet is to the sun (Kozhamthadam 1994, Chapter 8; Voelkel 2001a, 2001b)

References

Aristotle 1984. The Complete Works of Aristotle. J. Barnes, editor. Princeton.

Boyer, C. B. 1987 (1959). The Rainbow: From Myth to Mathematics. Princeton.

Kozhamthadam, J. 1994. The Discovery of Kepler’s Laws. Notre Dame.

Neugebauer, O. 1969. The Exact Sciences in Antiquity. 2nd ed. Dover.

Smith, A. M. 1996. Ptolemy’s Theory of Visual Perception: An English Translation of the Optics. The American Philosophical Society.

Voelkel, J. R. 2001a. Commentary on Ernan McMullin, “The Impact of Newton’s Principia on the Philosophy of Science.” Philosophy of Science 68:319–26.

——. 2001b. The Composition of Kepler’s Astronomia Nova. Princeton.

Edited by Stephen Boydstun

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Rainbows (cont.)

Ibn Sahl (fl. after 950)

Ptolemy’s experiments measured not only the bending of sight-lines at the interface of air and water, but the bending at interfaces of air and glass and the bending at interfaces of water and glass. Ptolemy’s Optics was studied by Abū Sad al-Alā ibn Sahl, a mathematician connected to the court in Baghdad.

Ibn Sahl composed the treatise On the Burning Instruments around 984. (Rashed 1990 is my source throughout for Ibn Sahl.) Analyses of parabolic burning mirrors were performed by Diocles about 110 B.C. and by a succession of others leading to Ibn Sahl. Circularly symmetric mirrors having the cross sectional shape of a parabola concentrate light from a distant source, such as the sun, onto a single spot. Our word focus for this spot, which is Latin for fireplace, was coined by Kepler. Recall that the parabola, the ellipse, and the hyperbola are conic sections[1]. Ibn Sahl continued the tradition of analyzing the parabolic conic section and demonstrating that and why the mirror surface (the surface generated by rotating a parabola about its axis of symmetry) brings sunlight to focus at a particular distance from the mirror. Likewise, for the concentration of light from nearby sources, he advanced analysis of the ellipsoidal mirror.

Ibn Sahl was evidently the first to produce some of the geometric theory of lenses. Consider the plane surface of a crystal such as quartz. We desire a lens to receive sunlight on its plane face and to concentrate the light to a focus from out the other face of the lens. What should be the convex shape of that second side if we are to accomplish this? Ibn Sahl proved that it should be a hyperboloid, the surface of revolution of a hyperbola.

He first considers the refraction of a ray of light out of the crystal into the air at some point P on a plane surface. This analysis will also apply to light emerging from a smoothly curved surface of the crystal, when thought about with reference to the plane tangent to the surface at the point P.

Let N be the line perpendicular to the plane at P. Consider an actual, bent ray A emerging from the plane crystal surface at a certain angle α (<90°) with N. Let E be the line that is an unbroken geometrical extension, into the air at P, of the ray as it had been directed in the crystal. E will form an angle ε (<90°) with the line N. The angle ε is more narrow than the angle α, which is to say that a ray is bent away from N when passing from the crystal to air. Let both line A and line E be intercepted in the in the air by a line L perpendicular to the plane, a perpendicular line set freely at some distance l from P. The intersection of the bent ray A with L will be, along A, at some distance a from P. The intersection of the unbroken extension-line E with L will be, along E, at some distance e from P. The distance e along E will be greater than the distance a along A.

The cosine of the angle that is 90 minus α is defined by the ratio of l to a and is identical with the sine of α. The cosine of 90 minus ε is defined by the ratio of l to e and is identical with the sine of ε. The ratio of e to a is expressed e/a, which is identically (e/l) multiplied by (l/a), which is (1/sine ε) multiplied by (sine α). We have then: e/a = (sine α)/(sine ε). This ratio is a constant characteristic of a specific kind of crystal, such as the quartz available to Ibn Sahl, for any angle of light ray traveling from the crystal to its surface and on into the air. Ibn Sahl had in hand this law of refraction known to us by the name Snell’s law, named after Willebrord van Roigen Snell, who rediscovered the law in 1621.

Why did Ptolemy not discover the sine-law of refraction, 800 years before Ibn Sahl?

Ptolemy’s optics is organized to the end of explaining the formation of visual images, especially the anomalous ones. That the image in a mirror cannot be really located in the real space behind the mirror is resolved by relating the image location to real locations of object, mirror surface, and eye in real space.[2] Ptolemy’s program for refraction also aims to resolve a class of visual illusions. That the straight stick appears bent when partly submerged in water is resolved by relating the image location to real locations of object, air-water interface, and eye (Smith 1996, 32–33, 37–42, 47–49).

Ptolemy conceived of visual perception in terms of visual rays (continuous as the set of geometric rays in a solid cone) reaching out to objects somewhat like reaching one’s hand to touch an object. Under that assumption, the burning power of concave mirrors is a phenomenon remote from the phenomena of visual perception. Perhaps that is why we do not find Ptolemy working on theory of burning mirrors. Ibn Sahl, like his contemporaries (Boyer 1987, 77–78), conceived vision occurring by rays of light coming to the eye. For Ibn Sahl it is paths of light rays that capture attention for geometric analysis.

Motivated by astronomy, Ptolemy became the crowning developer of spherical trigonometry (developed in terms of chords of arcs). The sine-law of refraction requires plane trigonometry. Ptolemy had the theoretical basis of plane geometry in hand, but apparently had no incentive to develop trigonometry for the plane (Kline 1972, 125–26). The possibility of its application to the phenomenon of refraction evidently did not take hold with him.

Mark Smith (1982) has argued that Ptolemy approached his refraction experiments with a preconceived general form for the relation between angle of visual ray in one medium and the angle to which it is bent when it enters another medium. According to Smith’s thesis, Ptolemy expected a constant direct proportionality between those two angles (so, not between their sines), which was only a small generalization from the equality of angles observed for reflection. That the angles should be equal in reflection had been shown by Heron[3] to follow from an assumption that the length of line touching the mirror and connecting eye to object shall be shortest among such connecting lines. Alas, Ptolemy’s experimental data for refraction could not be made to fit a constant proportionality between the two angles for refraction, and the true law, the sine-law, is contrary a principle of least lengths.

(To be continued.)

Notes

[1] Conic Sections – http://eom.springer.de/C/c024960.htm

[2] I notice that in this way, the visual geometry into the mirror becomes like the constructions one might add to a given figure to solve a geometry problem. In the case of optics, however, the auxiliary construction that goes beyond the physically given (object, mirror surface, and eye) is provided by the visual process rather than by imagination.

[3] Heron of Alexandria – http://objectivity-archive.com/

http://en.wikipedia.org/wiki/Hero_of_Alexandria#Mathematics

References

Boyer, C. B. 1987 (1959). The Rainbow: From Myth to Mathematics. Princeton.

Kline, M. 1972. Mathematical Thought from Ancient to Modern Times, vol. 1. Oxford.

Rashed, R. 1990. A Pioneer in Anaclastics: Ibn Sahl on Burning Mirrors and Lenses. Isis 81:464–91.

Smith, A. M. 1982. Ptolemy’s Search for a Law of Refraction. Archive for History of Exact Sciences 26:221–40.

——. 1996. Ptolemy’s Theory of Visual Perception: An English Translation of the Optics. The American Philosophical Society.

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Rainbows (cont.)

Correction of a typo in the next-to-last paragraph of the preceding note: “Ptolemy had the theoretical basis of plane trigonometry in hand, but apparently had no incentive . . . .”

Ibn al-Haytham (c. 965−1041)

By the time of Ibn Sahl (Baghdad) and Ibn al-Haytham (Basra, Cairo), Arab astronomers were developing plane trigonometry. Mark Smith thinks it likely that Ibn al-Haytham was familiar with the work of Ibn Sahl. I have gathered, however, that Ibn al-Haytham did not possess the sine-law of refraction. His optical works include treatises on the foci of mirrors and lenses, a treatise on the rainbow and halos, and his influential Book of Optics (also known as Treasury of Optics).

He gave tables for the refraction of light for angles of incidence graduated by 5°. He showed that the angle of refraction is not proportional to the angle of incidence. He surmised that the velocity of light is finite, not perfectly instantaneous. It is my understanding that Ibn al-Haytham investigated successive internal reflections of light rays in spheres of glass and in spherical flasks of water. Ibn al-Haytham implicated only reflection, not refraction, in his account of the rainbow (Boyer 1987, 80−81, 112; Smith 2001a cxxxiiiN153; 2001b 446).

See also: http://www.solopassion.com/node/2390#comment-63819

Qutb al-Din al-Shirazi (1236−1311) / Kamal al-Din al-Farisi (d. 1320)

The first key of geometry for unlocking the rainbow had been shown by Aristotle. The second key of geometry was first seen in Persia. Nasir al-Din al-Tusi (1201−74) had made plane and spherical trigonometry systematic, and he had made them independent of astronomy. Nasir was the teacher of Qutb, who was evidently the first to conclude that the primary rainbow is caused by a refraction of sunlight into droplets of water, followed by a single internal reflection within the droplets, then refraction upon exit into air. He concluded the secondary bow to be due to air-to-droplet refraction, internal reflection, a second internal reflection, then droplet-to-air refraction.

Qutb conceived the back scattering of sunlight yielding rainbows to be by way of individual droplets scattering light. He undertook to investigate that scattering by using Ibn al-Haytham’s laboratory globe of water “as a glorified raindrop and studying the passage through it of rays of light” (Boyer 1987, 127).

Systematic laboratory study was made by Kamal, who was a student of Qutb. He published his findings within a commentary on Ibn al-Haytham’s Book of Optics. “Kamal made a careful study of the paths of rays of light through simulated raindrops. In a dark room, he placed a glass sphere full of water in such a position that it would be struck by rays of the sun which entered the room through a hole, and he studied the colored bow which was formed” (ibid. 128). He correctly drew paths, from a point source of light, through the sphere of water for various numbers of internal reflections between the entry and exit refractions. Sadly, the attainments of Qubt and Kamal did not reach minds in a position to develop them further.

To more fully uncover the rainbow, Kamal would have needed to trace not rays reaching the water droplet from the range of directions of emanation from a point source. Rather, he would need to trace a bundle of parallel rays reaching the water droplet, rays all aimed in the same direction, all emanating from a source effectively an infinite distance away. Additionally, the Persian account of the rainbow, like all its predecessors, gives no explanation for why the rainbow is the size it is. (What is its size, anyway?

What is the angle between Aristotle’s surface cone and its axis?) Like all its predecessors, too, the account is false concerning the causes of the colors in the rainbow. Kamal had the second key of geometry for unlocking the rainbow, but he was trying the key on the wrong lock. He thought that the variety of light-entry points on a droplet yields⎯after determinate refraction, reflection(s), refraction⎯the variety of colors. Wrong lock.

Roger Bacon (c. 1214−92)

Roger Bacon was a Franciscan friar educated at Oxford. He was versed in the optical works Aristotle, Euclid, Ptolemy, and Alhacen (Ibn al-Haytham). He thought that the rainbow must be produced by reflections, and reflections alone, in small drops of water. The colors we see in the rainbow are not colors fixed to objects. They are not such as the colors in crystals, colors that are there before they are seen and are seen to be in the same place by different viewers.

There are as many rainbows as observers. For if two people stand observing the rainbow in the north and one moves westward, the rainbow will move parallel to him; and if the other observer moves eastward, the rainbow will move parallel to him; and if he stands still, the rainbow will remain stationary. . . . No two observers can see the same rainbow. (Opus maius, part VI, quoted in Lindberg 1997, 269)

The colors of the rainbow are not in the place of the bow, but are the result of vision. The colors of the rainbow are not true colors, but only an appearance of color, as we say the image of an object in a mirror is only an appearance of an object. The colors of the rainbow only appear to be located at the points of light reflection in drops of rain. According to Bacon, these colors are created by vision. Similarly, “in the summer morning, when a person lowers their head to the ground in order to see drops of dew on the ends of the grass, and if he looks at them in a careless or easygoing manner with half closed eyes, he sees in appearance all the colors of the rainbow” (Opus maius, part VI, quoted in Hackett 1997, 301).

So far as is known, Roger Bacon was the first to measure the elevation of the rainbow. He writes in 1266:

Further, let the experimenter take the required instrument [probably an astrolab*] and look through the openings of the instrument and find the altitude above the horizon, and keeping the instrument immovable let him turn in the opposite direction and look through the openings of the instrument until he sees the summit of the bow, and let him note the altitude of the rainbow above the horizon; and he will find that the higher the sun’s altitude is, the lower is that of the bow. . . .

The experimenter . . . will find that the final altitude at which the rainbow can appear above the horizon is 42 degrees, and this is the maximum elevation of the rainbow. (Opus Maius, part VI, quoted in Hackett 1997, 299)

(To be continued.)

* Astrolab − http://en.wikipedia.org/wiki/Astrolabe

References

Boyer, C. B. 1987. The Rainbow: From Myth to Mathematics. Princeton.

Hackett, J. 1997a. Roger Bacon and the Sciences. J. Hackett, editor. Brill.

⎯⎯. 1997b. Roger Bacon on Scientia Experimentalis. In Hackett 1997a.

Lindberg, D. C. 1997. Roger Bacon on Light, Vision, and the Universal Emanation of Force. In Hackett 1997a.

Smith, A. M. 2001a. Introduction to Alhacen’s Theory of Visual Perception. Vol. 1. Transactions of the American Philosophical Society.

⎯⎯. 2001b. Alhacen’s Theory of Visual Perception. Vol. 2. Transactions of the American Philosophical Society.

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Folks:

One of the most common fallacious "reasoning" events...after this, therefore because of this.

Adam

But that is what we observe. We construct causes or necessary causal connection in our minds.

Ba'al Chatzaf

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Brant:

There was a limited amount of baby making in Rand's literature - one would almost think she was a Shaker lol.

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I thought we identified causes, not construct them...

Michael

I don't think we identify them, per se, I think we suggest them. Look at the theory of gravity for example. In it's early stages one might say that gravity causes the earth to stay in orbit around the sun but later on it is suggested that the the mass of the sun causes changes in the structure of space-time on this constains the movement of the earth. We create a causal chain of events to reflect what we observe and this changes over time as our knowledge grows.

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Interlude

I have taken Aristotle’s geometric characterization of the arc of the rainbow to be not only a characterization according to essential form, but an explanation, by working causes, of how a greater elevation of the sun brings it about that the rainbow forms a smaller portion of a semicircle. Roger Bacon’s measurement of the variation of rainbow elevation with solar elevation, and particularly the limiting value of 42 degrees for the primary rainbow, puts mathematical characterization to work in a way similar to its service in the measurement of suites of refraction angles for various media. In the refraction measurements, the precision conferred by mathematical characterization enables greater chance of empirical disconfirmation of characterizations of essential form (e.g. Ibn al-Haytham’s showing that the angle of refraction is not proportional to the angle of incidence), greater chance of true characterization of essential form (e.g. Ibn Sahl’s / Snell’s sine-law for refraction). Bacon’s 42-degree angle becomes part of the characterization of the rainbow by essential form and is a fact, like the sequences of colors or Alexander’s band, which must be explained in an adequate account of the rainbow.

Before moving into work on the rainbow in the fourteenth century, in the Latin West, I want to go back and notice the full power of mathematics in Ibn Sahl’s analysis of light concentration by mirrors and lenses. Ibn Sahl intended to construct these instruments. He not only demonstrated by geometry that the requisite shape of the convex side of a planoconvex lense should be a hyperboloid, he set forth the continuous drawing of the hyperbolic arc by a mechanical device based on properties of the hyperbola that specify it uniquely (Rashed 1990, 467, 480–83).

At this pause, too, let me set out between (i) noncausal characterizations by essential form, (ii) explanations by constitutive causes, and (iii) explanations by working causes the relations to Aristotle’s four causes insofar as they are useful in contemporary, Objectivist philosophy. (See also Causality.)

Aristotle’s material causes are both constitutive and working. The iron of the saw cutting wood is both constitutive and working in the cutting. Likewise for the stone grinding quartz to form a lense. Aristotle’s efficient causes are, of course, working causes.

I have elsewhere (pp. 188–91) analyzed the gravitropic response of the roots of certain plants in terms of Fred Dretske’s structuring causes and triggering causes. Evolutionary and developmental histories are the structuring causes, and those histories explain why this type of root-curving exists. Uprooting of the plant is the triggering cause, and this explains why a particular root-curving growth occurred when it did occur.

Aristotle’s teleological causes, insofar as they obtain in biological systems without conscious intentions, are structuring causes, which are to be identified with material causes. Where material causes are assembled by a conscious agent, they are also working, final causes of the process of assembly. If I design a machine capable of gravitropic responses, then the teleological characteristic of the machine will be a working, structuring cause of my design process. Moreover, the aim of making such a machine is itself a teleological cause known as a final cause.

I have said that Eudoxus’ characterization of the motions of moon, sun, and planets by interconnected homocentric uniformly rotating spheres was explanatory. His assembly in imagination of homocentric rotating spheres was like the kinematics of rotating machines. His characterizations were constitutive causal explanations of the motions of the celestial bodies. Elementary uniform rotations compose, or constitute, the resultant actual motions of those bodies. The constitutive causes here have a virtual character. They are like our resolutions in imagination of real velocities or forces into components. Another example of this type of constitutive cause would be minima-principles, such as Ptolemy’s apparent quest for a shortest-lengths account of refraction.

Aristotle’s formal causes are to be identified with the constitutive causes described in the preceding paragraph. Note that explanation in terms of formal causes is more than characterization by essential form. In physics formal causes lie specifically in statics, kinetics, kinematics, and dynamics. Formal causes lie in analogous representations in other sciences. Formal causes are parasitic on working causes and the other constitutive causes.

Turning all this around, we have: Working cause comprises the material causes, the efficient causes, and the teleological and final causes. Constitutive cause comprises the material causes, the teleological causes, and the formal causes. Teleological causes are collapsed into material causes, which are constitutive and working. Final causes are teleological with an added layer of working causation. In sum: Working causes comprise the material, efficient, and final causes. Constitutive causes comprise the material, final, and formal causes.

Edited by Stephen Boydstun

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Rainbows (cont.)

Theodoric of Freiberg (d. c.1310) – Part 1

Theodoric was a Dominican professor of theology in Germany and at University of Paris. He was familiar with Alhacen’s Book of Optics and with Bacon’s treatise on the rainbow. Theodoric’s book On the Rainbow and Ray-Impressions (Ir) was issued in 1307. Like Qubt and Kamal, but independently of them, Theodoric conceived of the rainbow as an aggregate of effects of individual raindrops, and he conceived of a laboratory globe of water as a magnified raindrop.

In The Scientific Methodology of Theodoric of Freiberg (T), William Wallace remarks that Albert the Great (1206–80) had maintained that individual falling drops of water, not clouds, are the adequate material causes of the rainbow. Sunlight, in Albert’s account, is the efficient cause (T 142).

Thomas Aquinas (d. 1274) takes transparency to be the subject of which light is said. The transparent is “a common nature found in things and not a strict property of air or water, nor existing apart from things as a separate nature” (T 143). In his account of it, light is “an active quality following on the substantial form of the sun or other self-illuminating body, and it is in the third species of quality, although it lacks a contrary” (T 143).

The substantial form of the sun would be the specific nature of the sun—specific nature whose determinations are made not by mere determinable matter, but by form in the solar composite of form and matter—that gives the sun its manifest characteristics, and it is that by which those characteristics are to be explained (see further, here and Garber 1992, chp. 4). Light is not the substantial form of the sun as others such as Bonaventure had maintained. “Substantial forms are not of themselves objects of the senses, for the essence is the object of the intellect, as it is said in De Anima iii [430b28]; whereas light is visible of itself. . . . / As heat is an active quality consequent on the substantial form of fire, so light is an active quality consequent on the substantial form of the sun or of any other body that is itself luminous” (ST Q67 A3). Wallace stated that in the view of Aquinas, light is in the third species of quality. In Metaphysics Aristotle distinguishes four types of quality. Qualities of this third type are any “attributes of substances in motion (e.g. heat and cold, whiteness and blackness, heaviness and lightness, and others of this sort) in virtue of which, when they change, bodies are said to alter” (1020b9–11).

It would be objected to Aquinas: “Every sensible quality has its opposite as cold is to heat and black to white. But this is not the case with light, since darkness is merely a privation of light.” Like the Greek, the medieval understanding was that cold/heat and black/white were not only opponent pairs. Both partners in those pairs were positive qualities. They did not conceive coldness to be as we conceive it: nothing but lack of heat. They did conceive darkness in that way: nothing but lack of light.

Aquinas replied that “it is accidental to light not to have a [positive] contrary, inasmuch as it is the natural quality of the first corporeal cause of change, which is itself removed from contrariety.” What is “the first corporeal cause of change”? By this phrase, I think Aquinas refers to substantial form. There can be no transformations from pole to pole of contraries until there can be transformations. There can be no transformations without substantial form. The most basic principles of a real constitution are its most generic principles. A productive work must proceed from the most generic principles (ST Q67 A4). To create the material world, first would come substantial form in its unity with prime matter. Transformations and contrary poles come later. The existence of light requires that there be substantial form of a certain type of matter. Contraries are not required for light to be divided from darkness. The Creator knew his Aristotle.

Theodoric was able to make progress on uncovering the causes of the rainbow in part by experimentation. His experiments were profitable because he took those laboratory globes of water to show the optics of individual raindrops, not the optics of whole volumes of vapor in the sky.

Like Avicenna, Albert, and Aquinas, Theodoric was largely Aristotelian. Theodoric contradicted Aristotle concerning the number of colors in the rainbow. In the primary bow, “it is manifest to the senses that four colors can be distinguished in it, and also in the upper bow when two appear” (Ir 60, quoted in T 184). These are red, yellow, green, and blue. They are always in that order, with red on the outer periphery of the primary bow, but on the inner periphery of the secondary bow.

Theodoric did not conceive the colors of the rainbow as some sort of virtual result of vision in the manner of Roger Bacon. Theodoric argued that light is real. It is a real quality of translucent bodies. It is the compression of the parts of the translucent, or transparent, body. It is a potency of such bodies, a potency made actual only in the presence of a luminous body. In the luminous body, too, the luminous “is merely the compression of transparent parts which are bright, in act” (XVI 2 in On Light and Its Origin in Transparency [Lu], quoted in T 159).

Light is a vehicle of active powers, such as the power of heat, but light in itself lacks any active powers (contra Aquinas and the Neo-Platonists). “Luminosity is the very substance of color, and it is obvious that colors do not act on one another” (T 159).

The reason that the luminous has no active powers is that compression is a passive modification of substance, this because compression is a quantitative mode of something qualitative. It is qualities themselves that are the immediate principles of change. Quantity is another genus, one subordinate to the genus quality in the operations of nature (T 29).

When a quality intensifies or diminishes, each different value of the quality is a change in the essence of the quality. One of the four arguments Theodoric gives for this proposition is that it is the essence of accidents—such is quality—“to be a disposition of substance; but a more intense quality disposes the substance differently from a less intense one, and thus has a different essence” (T 103).

Now what is lightedness in the interior of a transparent or translucent body is coloredness at the surface of the body. Light supplies a general form, which actuates optical properties of the colored body (T 164–65). Variations in the mixture of such properties in a medium yield variations of color, and indeed different species of color. Between the extremes of white and black are the intermediate species of colors: red and yellow near the white; green and blue near the black (T 167).

That much for colors of bodies subjected to pure light (white light). What of bodies bathed in colored light? The resulting color of the body is to be regarded as truly the color of the object, just as one perceives it to be in the circumstance (T 168).

I mentioned that in Theodoric’s optics light requires the presence of a luminous body. He distinguishes four modes of the required presence. The luminous body may be present to the object by shining its pure light directly on the colored object. The luminous body may be present to the object by shining through a colored medium, such as a gem, yielding in the object a color that is affected by the intervening color. The luminous body (say, the sun) may be present to the object in reflection or in a prismatic refraction that results in multiple colors on an object. In those three cases, the resulting colors reside in the illuminated object, where the colors are seen (T 168). If one looks into the light directly from a prism, one sees those colors that would be shown on an object receiving such light. Where are the colors in this fourth case?

Obviously, they are not in the sun and not in the eye. Yet these colors, too, are real. Theodoric tries to understand this situation by posing a fourth mode by which a luminous body may be present to an object, specifically, to the eye. Theodoric thought these colors must reside in the prism, but in a peculiar way somehow akin to the way an image is in a mirror. Do not both mirrors and transparent bodies make images appear in places different from the places of the imaged bodies themselves? Do not some reflectors, such as the tail of peacock or duck, produce colors? (T 169–70, 173, 181–82)

(To be continued. Theodoric will be continued, followed by Kepler, Descartes, Newton, Young, and Airy.)

References

Aquinas, T. 1997 [1945]. Summa Theologica. In Volume 1 of Basic Writings of Saint Thomas Aquinas. A.C. Pegis, editor. Hackett.

Aristotle 1984. Metaphysics. In Volume 2 of The Complete Works of Aristotle. J. Barnes, editor. Princeton.

Garber, D. 1992. Descartes’ Metaphysical Physics. Chicago.

Wallace, W.A. 1959. The Scientific Methodology of Theodoric of Freiberg. University of Fribourg.

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References

Aquinas, T. 1997 [1945]. Summa Theologica. In Volume 1 of Basic Writings of Saint Thomas Aquinas. A.C. Pegis, editor. Hackett.

Aristotle 1984. Metaphysics. In Volume 2 of The Complete Works of Aristotle. J. Barnes, editor. Princeton.

Garber, D. 1992. Descartes' Metaphysical Physics. Chicago.

Wallace, W.A. 1959. The Scientific Methodology of Theodoric of Freiberg. University of Fribourg.

Here is another good book on the subject of light:

The Fire Within the Eye by David Park.

Ba'al Chatzaf

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MSK: Spammer content deleted.

The "gravitational force" between two masses is hypothetical. The only thing ever observed were the motions of the masses (relative to a fiduciary frame). The gravitational force explains and predicts the motions. Newton's Law of Gravitational Force is not quite right as the anomalous precession of the perihelion of Mercury shows. Einstein's hypothetical of curved space time does a better job of explaining the observed motions.

David Hume pointed out that a necessary connection between two event-types was never observed, but only inferred. So much for "necessary connection".

Is that a bad thing? Not at all. Any hypothesis that predicts a future motion and explain past motions is not to be despised.

Ba'al Chatzaf

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Stephen, do you know of any Objectivist oriented folks who have attempted to assimilate Chaos theory into an objectivist framework? I'm reading 'Chaos Theory Tamed' right now which is pretty heavy duty for a non-mathematician.

Edited by DavidMcK

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Stephen, do you know of any Objectivist oriented folks who have attempted to assimilate Chaos theory into an objectivist framework? I'm readiong Chaos Theory Tamed right now which is pretty heavy duty for a non-mathematician.

Ha! They asked the famous Objectivist scientist Harriman about that, during one of Peikoff's DIM lectures:

Question from the audience: What is chaos theory, and is that an oxymoron?

...they developed their own approach and called it chaos theory basically, and they have given up causality, so the only way they can think of to describe the messy physical world we observe is in terms of statistics, so they describe random statistical outcomes of complicated systems, and that is chaos theory.

Yeah, sure...

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Stephen, do you know of any Objectivist oriented folks who have attempted to assimilate Chaos theory into an objectivist framework? I'm readiong Chaos Theory Tamed right now which is pretty heavy duty for a non-mathematician.

Ha! They asked the famous Objectivist scientist Harriman about that, during one of Peikoff's DIM lectures:

Question from the audience: What is chaos theory, and is that an oxymoron?

...they developed their own approach and called it chaos theory basically, and they have given up causality, so the only way they can think of to describe the messy physical world we observe is in terms of statistics, so they describe random statistical outcomes of complicated systems, and that is chaos theory.

Yeah, sure...

I understand that this Harriman fellow is supposed to be a physicist. Right. Sure.

Ba'al Chatzaf

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