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The Observational Cast of Science

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#1 Stephen Boydstun

Stephen Boydstun


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Posted 15 December 2009 - 09:57 AM

Thomas S. Kuhn

I. Searching

In Thomas Kuhn's view, observation and experiment are essential to scientific understanding of the world (1990 [S], 42), but observation and experiment, in an established science, are guided and made sense of by one or another paradigm (S 109). Under the notion paradigm, Kuhn means to include theories, theoretical definitions, natural laws, particular models, and preeminently, concrete problem-solutions that exemplify theory and law, giving them empirical content (S 182–89). Scientific concepts, laws, and theories are presented and comprehended not in the abstract alone; but with applications to some concrete range of phenomena, purely natural, such as freely falling bodies, or natural-within-contrivance, such as pendulums (S 46–47, 187–91).

Normal science undertakes ever more exact and subtle experimental and observational investigation of "facts that the paradigm has shown to be particularly revealing of the nature of things" (S 25), determination of facts that "can be compared directly with predictions from the paradigm theory" (S 26), and articulation of the paradigm theory (S 27–29). These are roles of observation in science, I should say, even if we should reject Kuhn's distinction between normal and revolutionary periods of science as too sharply drawn.

One illustration of normal science would be the ongoing investigation of neutrinos. The existence of neutrinos is a fact established in 1956 (they were then detected) within the theoretical framework of quantum mechanics and detail conservation of energy. The characteristics of neutrinos are facts particularly revealing of the nature of elementary-particle interactions. The further, more refined determination of neutrino characteristics bears on the correctness and further refinement of a number of interconnected paradigms. Elaborate observations of solar neutrinos the past few decades provide quantitative constraints on models of nuclear reactions in the sun's core and on models of the sun's magnetic fields. And they provide constraints on the fundamental theory of neutrinos and of the electronuclear forces of nature (Bahcall 1990). Elaborate observations of cosmic neutrinos, these past few years [1], to ascertain whether they change flavors, hence whether they possess nonzero mass, inform efforts toward a Grand Unified Theory that may eventually supercede or subsume the Standard Model for elementary particles and their forces (Kearns, Kajita, and Totsuka 1999; Feldman and Steinberger 1991; Weinberg 1999). And they bear on current cosmology, under purview of general relativity.

Other normal-science investigations framed under the paradigm of general relativity are these: The finding of pulsar binary neutron stars has yielded, as hoped, empirical data for comparison with predictions from general relativity in the context of strong gravitational fields, predictions such as the rate of the orbital precession of the major axis of the stars' elliptical orbit and red-shifting of the pulse-clock (Piran 1995). Observation of quasi-periodic X-ray emissions from neutron stars pulling in matter from gaseous companion stars are yielding data indicating that, as predicted by general relativity (contrary the prediction from Newtonian gravitation), there is, just outside the neutron star, an innermost circular orbit for captured gas (Cowen 1998). Black holes are entities conceived and cultured solely by general relativity. Astronomical search for black holes and their distinctive features may yield an overwhelming vindication of general relativity (Lasota 1999).

There are three points made by Kuhn concerning the role of observation in science with which I should take some issue. One is his claim that "no part of the aim of normal science is to call forth new sorts of phenomena" (S 24). "Even the [normal-science] program whose goal is paradigm articulation does not aim at the unexpected novelty" (S 35). "Normal science does not aim at novelties of fact or theory and, when successful, finds none" (S 52). But scientists will be human, chronically so, hoping to catch something unexpected and momentous in their instruments, not only expected and readily comprehended phenomena. X-ray astronomer Bruno Rossi writes: "The initial motivation of the experiment which led to this discovery . . . was a subconscious trust of mine in the inexhaustible wealth of nature, a wealth that goes far beyond the imagination of man. This meant that, whenever technical progress opened up a new window into the surrounding world, I felt the urge to look through this window hoping to see something unexpected" (1977, 39).

Kuhn does say that "without the special apparatus that is constructed mainly for anticipated functions, the results that lead ultimately to novelty could not occur" (S 65, emphasis added). So I should productively construe the statements of his that I have quoted in the preceding paragraph as delineation of the strain that he calls normal science which in truth is found within a broader, richer actual practice of science.

Kuhn errs secondly, though only slightly, in his contention that "the act of judgment that leads scientists to reject a previously accepted theory is always based upon more than a comparison of that theory with the world. The decision to reject one paradigm is always simultaneously the decision to accept another, and the judgment leading to that decision involves the comparison of both paradigms with nature and with each other" (S 77, further, 147). Not always so. We continue to test empirically whether any mass-energy can be transported faster than vacuum c (Alväger, Farley, Kjellman and Wallin 1964; Brecher 1977; Chiao, Kwait, and Steinberg 1993). Some of these experiments, in the last two decades, have helped to articulate more finely the light-speed postulate of relativistic kinematics. But it is perfectly possible that such tests in the future could dispositively contradict the postulate. That would be the demise of special relativity regardless of the existence of competitor theories. Without viable alternative kinematics already on the stage or in the wings, what should we do if the light-speed postulate were empirically refuted? We should take our cues from the particulars of the failure and from our old, very successful special-relativity kinematics, and then develop a new and better kinematics.

Again, we continue to test a principle of general relativity, the principle of the equivalence of inertial and gravitational mass. These tests are not simply tests that help us articulate the paradigm, as when we search the heavens for evidence bearing on whether Einstein's field equations should include a nonzero cosmological constant (Krauss 1999; Cowen 2000). No, tests of the equivalence of inertial and gravitational mass cut to the quick of general relativity (Wald 1984, 8, 66–67; Ciufolini and Wheeler 1995, 13–18, 90–116). As I understand it, if gravitational and inertial mass are not precisely equivalent, then gravity cannot rightly be made geometric. And we have no viable alternative (nongeometric) to general relativity waiting in the wings. Were gravitational and inertial mass shown inequivalent by experiment or observation, then theoretical physicists would scramble to construct a replacement theory. We need not already have a competing theory to prefer over general relativity in order to reject the latter on experimental or observational grounds [2].

II. Seeing

Kuhn errs thirdly, and most seriously, in his (inconstant) denial that in our scientific observations we can always separate and adequately express what we literally perceive and what we take those percepts to indicate. Having learned prevailing scientific concepts, theories, and natural laws under exemplifying concrete observational applications, one is not able to see the phenomena in those applications entirely freely of the prevailing conceptual apparatus (S 46–47, 111–12, 186–89). Scientific observational phenomena are to some extent inextricably structured by the scientific, theoretical paradigm under which one is operating (S 111–35, 147–50).

"Looking at a bubble-chamber photograph, the student sees confused and broken lines, the physicist a record of familiar subnuclear events" (S 111). To enter the physicist's scientific observational world, the student undergoes "transformations of vision," like coming "to see a new gestalt." Hardly. Throughout the student's entry into the observational world of physics, all participants easily, routinely, and expressly distinguish between what of the physicist's observational world is commonsense perception and what is scientific interpretation, however automatic the latter may become (cf. S 196–98).

A bubble-chamber photograph provides detailed records of particle events "in a form that experienced physicists can interpret at a glance" (Breuker et al. 1991, 61). The photographs from bubble, cloud, or spark-streamer chambers never do yield a strictly perceptual particle-interaction gestalt in the way, say, that an X-ray photograph of a hand yields a strictly perceptual hand-skeleton gestalt. In the hand X-ray, given our ontogeny and our ordinary visual experience with hands, we are required to see the hand-in-the-image. We can tell ourselves, truly, that what is before us when we see the hand-in-the-image is only the trace of a hand, shadows of hand preserved on film, but we cannot avoid seeing the hand-in-the-image all the same. That is our perceptual constitution. Gestalt shifts too, such as in the Necker cube, are mandated by our primate perceptual constitution. We can tell ourselves that before us are only lines on paper, but we are required to see one cube or the other, with alternations every few seconds (Logothetis 1999). Contemporary elementary-particle tracking is mediated by vast electronic and computer processing systems, embodying painstaking deliberate interpretations. What is perceptually obligatory in the resulting computer-image displays are things like colors, lines, and 3D perspectives; all of these, self-conscious visual aids to scientific, interpretive observation.

Kuhn writes: "Since remote antiquity most people have seen one or another heavy body swinging back and forth on a string or chain until it finally comes to rest. To the Aristotelians, who believed that a heavy body is moved by its own nature from a higher position to a state of natural rest at a lower one, the swinging body was simply falling with difficulty. Constrained by the chain, it could achieve rest at its low point only after a tortuous motion and a considerable time. Galileo, on the other hand, looking at the swinging body, saw a pendulum, a body that almost succeeded in repeating the same motion over and over again ad infinitum." (S 118–19)

To be sure, Kuhn was "acutely aware of the difficulties created by saying that when Aristotle and Galileo looked at swinging stones, the first saw constrained fall, the second a pendulum" (S 121). Yet Kuhn will not let go his continual equivocation on see and its cognates (S 196–97). He maintains that an embracer of the new paradigm of mechanics—such was Galileo—is not an interpreter of swinging stones as pendulums, but "is like a man wearing inverted lenses," like a man who's vision has adapted to those lenses (S 122). "Galileo interpreted observations on the pendulum, Aristotle observations on [constrained] falling stones" (ibid.). That is inaccurate, I should say. Rather, Galileo and we interpret swinging stones as pendulums, on which we then make further interpretative observations. Similarly, one may interpret the swinging stone as in Aristotelian mechanics, as a constrained body working its way to the lowest feasible point. We can deliberately, with training, switch our interpretative perspectives: Aristotelian, Galilean, Newtonian, Lagrangian. Kuhn suggests that the contemporary scientist "who looks at a swinging stone can have no experience that is in principle more elementary than seeing a pendulum. The alternative is not some hypothetical 'fixed' vision, but vision through another paradigm, one which makes the swinging stone something else" (S 128). I suggest, to the contrary, that developmentally, epistemologically, and evidentially, it is a swinging stone that is most elementary for everyone. It is with respect to analysis that we "see" (take) the pendulum as most elementary.

I do not mean to contradict Kuhn's thesis that scientists do not come to reject scientific theories on account of uninterpreted observations (e.g. S 77). We can recognize that and assimilate that without conflating what we literally perceive and what we make of those percepts in thought.

III. Saying

According to Kuhn, "there can be no scientifically or empirically neutral system of language or concepts" (S 146). Moreover, since we have no rudimentary paradigm-neutral observation language, the pendulum and constrained fall must be simply different perceptions, rather than "different interpretations of the unequivocal data provided by observation of a swinging stone" (S 126). Kuhn has in mind "a generally applicable language of pure percepts," where, by the term percepts, he apparently thinks not of swinging stones, but of more primitive constituents that compose our perceptions of swinging stones. Attempts to construct such a language of pure percepts have not fully succeeded, and anyway, all such projects "presuppose a paradigm, taken either from current scientific theory or from some fraction of everyday discourse, . . . . [thereby yielding] a language that—like those employed in the sciences—embodies a host of expectations about nature and fails to function the moment these expectations are violated" (S 127).

I should say, with Willard Quine, that we do indeed have a trustworthy scientifically neutral system of observation language appropriate and necessary for the physical sciences. This is not a rarified, fully reductive language of "pure percepts," but a natural language of posited objects and events (1969 [EN], 74–79; 1995a [N], 252, 254; 1995b [SS], 10–21, 27–29, 35–42; cf. 1951, 293–98). Swinging body and pendulum are both legitimate expressions of things observed[3], the former providing a fallback in cases of dispute over the latter. "What counts as an observation sentence varies with the width of community considered. But we can always get an absolute standard by taking in all speakers of the language, or most" (EN 88; also N 255; SS 22, 42–45). Pendulum, damped harmonic oscillator, and electron-positron track may be rightly spoken of as observed in the narrower, scientific community. But when necessary, scientists can shift gears and recognize those items as interpretations of more widely accepted and developmentally prior observed items.

Our broadest and most rudimentary observation language is our language of everyday experience, in which we report "it is raining" or "the iron is on" and in which we generalize "swinging suspended bodies return to rest" or "if it is snowing, then it is cold" (N 252, 254–55; SS 22–26). That last ordinary observation sentence is an example of what Quine calls an observation categorical, which is an empirically testable hypothesis, standing (as Popper would have it) as not yet shown false. Quine supposes, reasonably I think, that an empirically testable scientific hypothesis can be cashed out as an elementary observation categorical (N 255; SS 43–47). The detection of cosmic background microwave radiation, for example, cashes to visible records of activities in an antenna (which antenna cashes to . . . ). Quine realizes, of course, that scientists do not trace all the links from their hypothesis to observational categorical. "Still, the deduction and checking of observation categoricals is the essence, surely, of the experimental method, . . . . [and it remains] that prediction of observable events is the ultimate test of scientific theory" (N 256).

Quine recognizes that some hypotheses thus far not testable are accepted, rationally, even in the hard sciences. They may be accepted because "they fit in smoothly by analogy, or they symmetrize and simplify the overall design. . . . Moreover, such acceptations are not idle fancy; their proliferation generates, every here and there, a hypothesis that can indeed be tested. Surely this is the major source of testable hypotheses and the growth of science" (N 256; also SS 49). Can we test whether spacetime is curved? Well, yes, indirectly, more and more, we can.

Kuhn overrated the difficulties of vocabulary translations between alternative paradigms (S 149, 201). He did seem to allow that eventually translation can be effected (S 201–3). Such has been effected between Newtonian gravitational theory and general relativity, and gradually physics has attained more and more tests between those deep and grand theories, tests such as that for an innermost circular orbit about a neutron star.

1. This study was composed in 2000.
2. Hilary Putnam points out that Kuhn exaggerates in asserting that a paradigm can never be overthrown in the absence of a competitor paradigm. But Putnam then deflates the demerit of the exaggeration by posing as a hypothetical counterexample to Kuhn's universal claim only a Goodmanesque scenario: the world simply starts to behave radically differently. Barring such an implausible scenario, Putnam then expressly affirms the Kuhnian generalization at issue (Putnam 1974, 69–70). My counterexample scenarios (failure of light-speed postulate or failure of principle of equivalence) are intended to be entirely, mundanely realistic.
3. Rudolf Carnap (1966) likewise recognized that what in one context of inquiry should be taken as inferred from what was observed could in another context be rightly taken as simply observed (Suppe 1977, 47).

Alväger, T., Farley, F.J.M., Kjellman, J., and I. Wallin 1964. Test of the Second Postulate of Special Relativity in the Gev Region. Physics Letters 12:260.
Bahcall, J.N. 1990. The Solar-Neutrino Problem. Sci. Amer. (May):54–61.
Brecher, K. 1977. Is the Speed of Light Independent of the Velocity of the Source? Phy. Rev. Ltrs. 39(17):1051–54.
Breuker, H., Drevermann, H., Grab, C., Rademakers, A.A., and H. Stone 1991. Tracking and Imaging Elementary Particles. Sci. Amer. (Aug):58–63.
Chiao, R.Y., Kwait, P.G., and A.M. Steinberg 1993. Faster than Light? Sci. Amer. (Aug):52–60.
Ciufolini, I., and J.A. Wheeler 1995. Gravitation and Inertia. Princeton: University Press.
Cowan, R. 1998. All in the Timing. Sci. News 154:318–19.
——. 2000. Revved-Up Universe. Sci. News 157:106–8.
Feldman, G.J., and J. Steinberger 1991. The Number of Families of Matter [= Three]. Sci. Amer. (Feb):70–75.
Kearns, E., Kajita, T., and V. Totsuka 1999. Detecting Massive Neutrinos. Sci. Amer. (Aug):64–71.
Krauss, L.M. 1999. Cosmological Antigravity. Sci. Amer. (Jan):52–59.
Kuhn, T.S. 1990 [1970, 1962]. The Structure of Scientific Revolutions. 2nd ed. Chicago: University Press.
Lasota, J-P. 1999. Unmasking Black Holes. Sci. Amer. (May):40–47.
Logothetis, N.K. 1999. Vision: A Window on Consciousness. Sci. Amer. (Nov):69–75.
Piran, T. 1995. Binary Neutron Stars. Sci. Amer. (May):53–61.
Putnam, H. 1974. The "Corroboration" of Theories. In Scientific Revolutions. I. Hacking, editor. 1981. New York: Oxford University Press.
Quine, W.V.O. 1951. Two Dogmas of Empiricism. In Philosophy of Science: The Central Issues. M. Curd and J.A. Cover, editors. 1998. New York: W.W. Norton.
——. 1969. Epistemology Naturalized. In Ontological Relativity and Other Essays. New York: Columbia University Press.
——. 1995a. Naturalism, or, Living within One's Means. Dialectica 49(2–4):251–61.
——. 1995b. From Stimulus to Science. Cambridge, MA: Harvard University Press.
Rossi, B. 1977. X-Ray Astronomy. Daedalus 106(4):37–58.
Suppe, F. 1977 [1973]. The Search for Philosophic Understanding of Scientific Theories. In The Structure of Scientific Theories. 2nd. ed. Urbana: University of Illinois Press.
Wald, R.M. 1984. General Relativity. Chicago: University Press.
Weinberg, S. 1999. A Unified Physics by 2050? Sci. Amer. (Dec):68–75.

Other Works by Thomas S. Kuhn
The Essential Tension (1977)
The Road since Structure Edited by James Conant and John Haugeland (2000)
The Copernican Revolution (1957)
Black-Body Theory and the Quantum Discontinuity, 1894–1912 (1978)

Works about Kuhn’s Work
Reconstructing Scientific Revolutions: Thomas S. Kuhn's Philosophy of Science – Paul Hoyningen-Huene (1993)
Thomas Kuhn: A Philosophical History of Our Times – Steve Fuller (2000)
Thomas Kuhn's Revolution – James A. Marcum (2005)
Kuhn’s Evolutionary Epistemology – Barbara Gabreilla Renzi (2009)
Philosophy of Science 76(2):143–59,

Bibliographic Essays on Scientific Revolution
The Scientific Revolution – Richard S. Westfall
The Scientific Revolution – Paradigm Lost? – Robert A. Hatch

Alternative Views of Scientific Revolutions
Scientific Revolutions Edited by Ian Hacking
Hacking provides a good brief summary of Kuhn’s Structure in the Introduction. The important chapters are these:
II. “Meaning and Scientific Change” by Dudley Shapere
III. “The ‘Corroboration’ of Theories” by Hilary Putnam
IV. “The Rationality of Scientific Revolutions” by Karl Popper
V. “History of Science and Its Rational Reconstructions” by Imre Lakatos
VI. “Lakatos’ Philosophy of Science” by Ian Hacking
VII. “A Problem-Solving Approach to Scientific Progress” by Larry Laudan
The Cognitive Structure of Scientific Revolutions
Hanne Andersen, Peter Barker, and Xiang Chen (2006)


This thread is to be continued with (i) assimilation of work of Harold Brown on observation and objectivity and (ii) integration with Ayn Rand’s definition of knowledge in terms of observation; her conception of empirical science; and her measurement analysis of concepts and (iii) the transition from those to the roles of mathematics in scientific observation.

#2 Stephen Boydstun

Stephen Boydstun


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Posted 14 December 2010 - 04:50 AM

Notes on the Roles of Mathematics in Scientific Observation

Functions of Mathematical Description in Astronomy and Optics,
Illustrations from Antiquity
—Stephen Boydstun (1999)
. . .
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.
. . .
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.
. . .

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

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

Neugebauer, O. 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.


Ptolemy (c. A.D. 100–180)
. . .
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.
. . .

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.

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 trigonometry 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.


[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/


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.

In his 2010 book on induction in physics, David Harriman writes:

Ptolemy conducted a systematic study in which he measured the angular deflection of light at air/water, air/glass, and water/glass interfaces. This experiment, when eventually repeated in the seventeenth century, led Willebrord Snell to the sine law of refraction. But Ptolemy did not discover the law, even though he did the right experiment and possessed both the requisite mathematical knowledge and the means to collect sufficiently accurate data.
. . .
Ptolemy’s failure was caused primarily by his view of the relationship between experiment and theory. He did not regard experiment as the means of arriving at the correct theory; rather, the ideal theory is given in advance by intuition, and then experiment shows the deviations of the observed physical world from the ideal. This is precisely the Platonic approach he had taken in astronomy. . . . [Ptolemy] began with an a priori argument that the ratio of incident and refracted angles should be constant for a particular type of interface. When measurements indicated otherwise, he used an arithmetic progression to model the deviations from the ideal constant ratio.* (37)

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

Mr. Harriman’s citation is to one of the works of A. Mark Smith that I had earlier cited in the posts quoted above. Before remarking on Harriman’s assessment of Ptolemy’s failure to discern the law of refraction, I need to show a little more physics and its history. Jerry Marion writes in Classical Dynamics (1965):

Minimal principles in physics have a long and interesting history. The search for such principles is predicated on the notion that Nature always acts in such a way that certain important quantities are minimized when a physical process takes place. The first such minimum principles were developed in the field of optics. Hero of Alexandria [Heron], in the second century B.C., found that the law governing the reflection of light could be obtained by asserting that a light ray, traveling from one point to another by a reflection from a plane mirror, always takes the shortest possible path. A simple geometrical construction will verify that this minimum principle does indeed lead to the equality of the angles of incidence and reflection for a light ray reflected from a plane mirror. Hero’s principle of the shortest path cannot, however, yield a correct law for refraction. In 1657 Fermat reformulated the principle by postulating that a light ray always travels from one point to another in a medium by a path that requires the least time. Fermat’s principle of least time leads immediately, not only to the correct law of reflection, but also to Snell’s law of refraction. (216)

Harriman states that Ptolemy’s most important obstacle to discovering the law of refraction was his incorrect view of the relation of theory and experiment. That proposition is not from Mark Smith. Harriman goes on to contend that Ptolemy’s science was “a logical application of Platonism” and that Ptolemy regarded experiment as “the handmaiden of intuition” (38). These contentions, too, are not squarely from Mark Smith.

Professor Smith has argued that Greek science worked within a methodological framework of “saving the appearances.” This framework is rather at odds, I notice, with the strain in Plato of “pooh-pooh the appearances” (Rep. 509d–513c, 517b–518d); it is less at odds with Plato’s praise of measurement as opponent of false appearances (Prt. 356c–357a; Sph. 236b; Phlb. 66a–c). Whatever the relative weights of its debts to Plato and other Greek philosophers, Smith lays out the assumptions of the appearance-saving endeavor of Greek science (in optics that would be Euclid, Heron, and Ptolemy) as follows: All irregular change is merely appearance, illusion. “Beneath the appearances, there lies a real, intelligible world that is utterly simple, changeless, and eternal” (1982, 224). That world is a Euclidean locus in which things are not as in appearance, but stand in their true spatial relationships. “Moreover, the only real relationships are those most basic ones obtaining between and among points, and they are mathematically expressible in terms solely of distances and angles” (ibid.).

What does it mean to save the appearances? What does such salvation amount to? It is the reduction of appearances “to the utter simplicity of uniformity. Such a reduction requires some absolutely simple and perfect gauge of uniformity, a salvans . . . . And what finally determines the perfection of the salvans is its conformance to what I have called the Principle of Natural Economy” (ibid.)

Like his predecessors, Ptolemy thought of the visual ray as a physically real line

through which the geometrical reality behind the visible appearances could be immediately construed. And the rectilinearity of that line was presumed to be a function of its absolute spatial brevity.

We have already seen how successfully the ray-as-least-distance was employed in the salvation not only of direct vision, but, far more important, of reflection. Hero’s demonstration of the contingency of the equal-angles law upon the Principle of Least Lines is a clear testament to that success. In the case of refraction, though, the same sort of analysis will not work. The sine-law simply cannot be established on the basis of least distances but, as Fermat eventually showed, must be grounded upon least times. In other words, if it is to save refraction, the ray must be understood to represent a temporal, not a spatial, path.* The inadequacy of Ptolemy’s analysis of refraction was therefore due to the inadequacy of his ray-concept.

* The basic flaw in Ptolemy’s refraction-analysis consists in the fact that the terms of his analysis are too concrete and specific. The spatial brevity that he supposed to be the fundamental governing principle of visual radiation is actually a function of a more profound temporal brevity. Likewise, the angular relationships that he thought governed refraction are actually functions of a more profound sine relationship. Thus, Ptolemy’s failure overall was due to his inability to conceive the phenomena more abstractly, to transcend the limitations of the simple spatial intuitionism that dictated his scientific approach. (Smith 1982, 239)

Within a widened saving-the-appearances methodological framework, Ptolemy could have arrived at the correct sine law by expanding beyond the intuitive assumption that the deep true story of visual radiation is writ by spatial extent. By the times of Descartes and Fermat, a ray of light is just a trajectory of light from object to object, not the visual ray of the ancients. Then too, their accounts of optical paths were more abstract than Ptolemy’s failed attempt, which was all too bound to “spatial intuitionism” (Smith 1982, 240). That is not to say that Ibn Sahl or (much later) Snell needed to wait on any such accounts to learn the correct law from experiment.

Ptolemy has been variously characterized by scholars as Platonist-Pythagorean, Aristotelian, Stoic, and Empiricist. Mark Smith concludes that “although the epistemological foundations of Ptolemy’s analysis may be legitimately characterized as ‘Aristotelian’, the structure of that analysis may no less legitimately be characterized as ‘Platonic’” (Smith 1996, 19; on the general Aristotelian empirical foundation, see p. 28).

David Harriman’s portrayal of Ptolemy’s conception of experiment as the “handmaiden of intuition” does not refer to Ptolemy’s confinement to abstractions all-too-low in his analysis of experiment, the confinement that was detailed by Smith and was called by Smith “spatial intuitionism.” The extent to which Harriman’s characterization of experimentation in pre-Galilean physics as "handmaiden of intuition" is true and cogent will have to wait for another occasion. More generally, I expect to assess eventually, in a review of his book in another venue, the strengths and weaknesses of Harriman’s attempt to apply Rand’s epistemology to philosophy of physics.

#3 Merlin Jetton

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Posted 14 December 2010 - 07:07 AM

I don't known your aim here, but it seems to me a section on triangulation would be worthwhile.

#4 Starbuckle



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Posted 14 December 2010 - 09:59 PM

Interesting discussion of Kuhn.

#5 Stephen Boydstun

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Posted 04 June 2012 - 11:02 AM

Experiment and the Making of Meaning
Human Agency in Scientific Observation and Experiment
David C. Gooding (Springer 1990)

Histories of Scientific Observation
Lorraine Daston and Elizabeth Lunbeck, editors (Chicago 2011)

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