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.
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For some uses of mathematics in modern science, see:
Edited by Stephen Boydstun, 15 November 2008 - 03:05 AM.