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.

. . .

References

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.

[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.

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.

For knowledge of the mathematical characteristics of refraction and reflection, Theodoric relied on Alhacen. For his conception of scientific method, Theodoric relied on Aristotle.

In Theodoric’s time, logic was the grand instrument of the natural sciences, such as optics and theory of the rainbow. Theodoric was among the schoolmen for whom Aristotelian logic had been made available by Albert the Great in his paraphrases of peripatetic doctrine (T 25). Theodoric subscribed to all the Aristotelian doctrines on definition and demonstration as set forth in the Albertine paraphrase (T 42–50).

Albert (1206–80) was a contemporary of Roger Bacon (c.1214–92), Bonaventure (1221–74), and Aquinas (1225–74). The balance of Aristotle’s works had come into Latin translation, from Greek and from Arabic, in the century preceding these scholars. Albert composed paraphrases of works of Aristotle to make them more intelligible to the Latins. In his own views, Albert largely adopted Aristotle in theory of abstraction and in opposing Pythagorean and Platonic views on the relation of mathematics to nature.

Some Pythagoreans, Aristotle writes, “make all nature out of numbers” (Cael. 300a16). That cannot be. “For natural bodies are manifestly endowed with weight and lightness, but an assemblage of units can neither be composed to form a body nor possess weight” (Cael. 300a16–18).

Natural science, in Aristotle’s broad sense, “concerns itself for the most part with bodies and magnitudes and their properties and movements, but also with the principles of this sort of substance . . .” (Cael. 268a1–3). An object of nature is a substance, for a nature is a subject of predicates (Ph. 192b34). Surfaces, volumes, lines, and points are rightly predicated of natural bodies, as limits of natural bodies. (Add also Heron’s later physical limit that is a line.*) In that relation, those characters are essential elements in natural science.

“The mathematician, though he too treats of these things, nevertheless does not treat them as the limits of a natural body; nor does he consider the attributes indicated as the attributes of such bodies. That is why he separates them; for in thought they are separable from motion, and it makes no difference, nor does any falsity result if they are separated” (Ph. 193b31–35). It is by construction and division of lines and figures in thought that geometrical relations are discovered. Potentially existing relations are thereby brought to actuality in thought. An example of such relation would be that the sum of angles of any triangle equals the angle about a point on a line (180°) (Metaph. 1051a21–33; Euclid I.32).

The disciplines of optics, harmonics, and astronomy, are seen by Aristotle as branches of mathematics, but they are “the more natural of the branches of mathematics . . . . These are in a way the converse of geometry. While geometry investigates natural lines but not qua natural, optics investigates mathematical lines, but qua natural, not qua mathematical” (Ph. 194a6–11).

Light is a natural line, a line studied in optics as natural, rather than as line per se. A concord is a natural sound (of a human instrument), a sound studied in harmonics as natural, rather than as ratio per se.

Now to have demonstrative understanding, one must know the reason for the fact, as opposed to simply the fact. In a syllogism, the conclusion will contain two of the three terms appearing in the syllogism. For example, man and mortal appear in the conclusion of a syllogism in which a third term, a “middle” term, appears in the two premises. That middle term could be animal as when we reason: “All animals are mortal. Man is an animal. Therefore, man is mortal.” If mortality necessarily belongs to animality and humanity necessarily belongs to animality, then mortality necessarily belongs to humanity. Only when there is necessity for both premises of the syllogism is there necessity in the conclusion. Demonstrative understanding of the fact that all men are mortal requires understanding necessities of the middle term animal (APo. 75a11–15).

Demonstrations make clear the underlying essential attributes of a thing. Demonstrative understanding is to be had only from that which is necessary to a thing, that which is essential to itself. “Scientific demonstrations are about what belongs to things in themselves, and depends on such things . . . . The middle term must belong to the third, and the first to the middle, because of itself. / One cannot, therefore, prove anything by crossing from another genus—e.g. something geometrical by arithmetic” (APo. 75a30–39). Therefore, one cannot “prove by any other science the theorems of a different one, except such as are so related to one another that the one is under the other—e.g. optics to geometry and harmonics to arithmetic” (APo. 75b14–17; see also 75b36–76a15).

“Mathematics is about 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 [in geometry]” (APo. 79a9–11). Though one cannot prove theorems of one science by another science, thereby confounding one genus with another, one can sometimes know something of the reason for a fact in one science by considering it in light of another science. “Such are those which are related to each other in such a way that the one is under the other, e.g. optics to geometry . . . and harmonics to arithmetic . . . . Here it is for the empirical scientists 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” (APo. 78b35–79a7).

Aristotle takes theory of the rainbow to stand to optics as optics stands to geometry (APo. 79a12–13). As the reader knows, Aristotle contributed, in his Meteorology, to theory of the rainbow, appealing to observations of the bows, to optics, and to geometry. I have recently learned of an analysis of Aristotle’s treatment by James Lennox, in a paper twenty-six years ago. I shall return to Aristotle’s treatment in connection with the advances of Theodoric in Part 3.

In his commentaries on Aristotle’s Physics and Metaphysics, Albert writes that mathematics and physics are founded on being, mathematics on being with quantity, physics on being with quantity, motion, and sensible matter. However, “we need to beware the error of Plato who said that naturals were founded in mathematicals and mathematicals in divines, . . . and therefore he said that mathematicals were principles of naturals, which is completely false” (quoted in Molland 1980, 467; also Lindberg 1982, 15).

Albert reduces the power of mathematics in optics below the Aristotelian level. (Theodoric stayed closer to Aristotle on this issue—Part 3.) Commenting on the Posterior Analytics, Albert observes that it is not the geometer who can give the reason for the facts of reflection. “These properties are not caused by a line in that it is quantity, therefore lines as such are not appropriate, nor can [the properties] be produced by the geometer from the proper principles of quantity” (quoted in Molland 1980, 468). It is for the physicists of light, the perspectivists, to provide the reasons for properties of light such as capability for and character in reflection (Molland 1980, 468; see also Lindberg 1982, 15, for Albert’s doctrine applied to the spherical shape of the earth).

In contrast to Albert, Robert Grosseteste (1175–1253) and Roger Bacon kept with Aristotle’s view that geometry gives the reason for phenomena in optics. Without retreating to the platonic view and without reducing physics to mathematics, they stressed that geometry provides causal, necessary explanation for optical phenomena. Bacon: “The cause of natural things cannot be given except by means of geometry” (quoted in Lindberg 1982, 17). Bacon goes beyond Aristotle in arguing that “in one way or another all of the categories depend on quantity” (Lindberg 1982, 19; cf.)

“Bacon argues, on the basis of Aristotelian epistemology, that sense perception precedes intellection and that quantity is particularly important in sense perception because it is one of the common sensibles (that is, it is perceived by more than one external sense) and nothing can be perceived without quantity; ‘therefore, the intellect is especially occupied with quantity’” (Lindberg 1982, 19–20; cf. Cm).

The first postulate of Euclid is: to draw a straight line from any point to any point. Bacon attends to the fact that man cannot, in matter, draw a perfect line.

But it is possible for operative nature, . . . as in the diffusion of light and {visual} rays, which is made multiplicatively {extends itself} by straight lines in a single body {medium}, and also the perpendiculars to the first bodies {light sources} are made in a straight fashion. The geometer therefore considers the possible paths of nature, because geometry was first and essentially constituted for the sake of certifying the works of nature, and therefore for human works. For the authors of perspective {e.g. Alhacen} show us that lines and figures declare to us the whole operation of nature, its principles and effects. . . . The geometer therefore does not attend to tortuous sensible matter, but he understands regular nature as it is in celestials and as nature knows how to find in its operations in these inferiors {sublunaries}, and he imitates the ways of nature. (quoted in Molland 1997, 171)

The straight lines of nature—lines of light, of visual rays, of all forces—are source and end of geometry. The lines of geometry are not causal, but not causal only in abstraction from their residence in the physical lines that indeed have causal powers (further, Lindberg 1997, 250–55, 265–67). Under the conception of Bacon, perfect lines are in the world concretely, by nature. Different kinds of them have different causal powers. In the “mixed” science of mathematical optics, there is no ontological cleft between geometrical relations as revealed in geometry and as resident in observed optical phenomena. The pertinent geometrical relations truly are physically instanced in the physical phenomena and can provide causal explanation for some parts of the phenomena without mixture of genus.

To be continued.

References

Aristotle c. 348–322 B.C.The Complete Works of Aristotle. J. Barnes, editor. 1984. Princeton.

Euclid c. 300 B.C.The Thirteen Books of The Elements. T. L. Heath, translator. 1925 [1908]. Dover.

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

Lennox, J. G. 1985. Aristotle, Galileo, and “Mixed Sciences.” In Reinterpreting Galileo. W. A. Wallace, editor. Catholic University of America.

Lindberg, D. C. 1982. On the Applicability of Mathematics to Nature: Roger Bacon and His Predecessors. The British Journal for the History of Science 15:3–15.

——. 1997. Roger Bacon on Light, Vision, and the Universal Emanation of Force. In Hackett 1997.

Molland, G. 1980. Mathematics in the Thought of Albertus. In Mathematics and the Medieval Ancestry of Physics. 1995. Variorum.

——. 1997. Roger Bacon’s Knowledge of Mathematics. In Hackett 1997.

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

Mathematics plays a central role in much of contemporary science, but philosophers have struggled to understand what this role is or how significant it might be for mathematics and science. In this book Christopher Pincock tackles this perennial question in a new way by asking how mathematics contributes to the success of our best scientific representations. In the first part of the book this question is posed and sharpened using a proposal for how we can determine the content of a scientific representation. Several different sorts of contributions from mathematics are then articulated. Pincock argues that each contribution can be understood as broadly epistemic, so that what mathematics ultimately contributes to science is best connected with our scientific knowledge.

In the second part of the book, Pincock critically evaluates alternative approaches to the role of mathematics in science. These include the potential benefits for scientific discovery and scientific explanation. A major focus of this part of the book is the indispensability argument for mathematical platonism. Using the results of part one, Pincock argues that this argument can at best support a weak form of realism about the truth-value of the statements of mathematics. The book concludes with a chapter on pure mathematics and the remaining options for making sense of its interpretation and epistemology.

Thoroughly grounded in case studies drawn from scientific practice, this book aims to bring together current debates in both the philosophy of mathematics and the philosophy of science and to demonstrate the philosophical importance of applications of mathematics.

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## Guyau

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

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):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

savethe 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, asalvans. . . . And what finally determines the perfection of thesalvansis 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

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.

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## Guyau

AuthorRainbows(cont.)Theodoric of Freiberg (d. c.1310) – Part 2 (*)

For knowledge of the mathematical characteristics of refraction and reflection, Theodoric relied on Alhacen. For his conception of scientific method, Theodoric relied on Aristotle.

In Theodoric’s time, logic was the grand instrument of the natural sciences, such as optics and theory of the rainbow. Theodoric was among the schoolmen for whom Aristotelian logic had been made available by Albert the Great in his paraphrases of peripatetic doctrine (

T25). Theodoric subscribed to all the Aristotelian doctrines on definition and demonstration as set forth in the Albertine paraphrase (T42–50).Albert (1206–80) was a contemporary of Roger Bacon (c.1214–92), Bonaventure (1221–74), and Aquinas (1225–74). The balance of Aristotle’s works had come into Latin translation, from Greek and from Arabic, in the century preceding these scholars. Albert composed paraphrases of works of Aristotle to make them more intelligible to the Latins. In his own views, Albert largely adopted Aristotle in theory of abstraction and in opposing Pythagorean and Platonic views on the relation of mathematics to nature.

Some Pythagoreans, Aristotle writes, “make all nature out of numbers” (

Cael.300a16). That cannot be. “For natural bodies are manifestly endowed with weight and lightness, but an assemblage of units can neither be composed to form a body nor possess weight” (Cael.300a16–18).Natural science, in Aristotle’s broad sense, “concerns itself for the most part with bodies and magnitudes and their properties and movements, but also with the principles of this sort of substance . . .” (

Cael. 268a1–3). An object of nature is a substance, for a nature is a subject of predicates (Ph. 192b34). Surfaces, volumes, lines, and points are rightly predicated of natural bodies, as limits of natural bodies. (Add also Heron’s later physical limit that is a line.*) In that relation, those characters are essential elements in natural science.“The mathematician, though he too treats of these things, nevertheless does not treat them as the limits of a natural body; nor does he consider the attributes indicated as the attributes of such bodies. That is why he separates them; for in thought they are separable from motion, and it makes no difference, nor does any falsity result if they are separated” (

Ph. 193b31–35). It is by construction and division of lines and figures in thought that geometrical relations are discovered. Potentially existing relations are thereby brought to actuality in thought. An example of such relation would be that the sum of angles of any triangle equals the angle about a point on a line (180°) (Metaph. 1051a21–33; Euclid I.32).The disciplines of optics, harmonics, and astronomy, are seen by Aristotle as branches of mathematics, but they are “the more natural of the branches of mathematics . . . . These are in a way the converse of geometry. While geometry investigates natural lines but not

quanatural, optics investigates mathematical lines, butquanatural, notquamathematical” (Ph. 194a6–11).Light is a natural line, a line studied in optics as natural, rather than as line per se. A concord is a natural sound (of a human instrument), a sound studied in harmonics as natural, rather than as ratio per se.

Now to have demonstrative understanding, one must know the reason for the fact, as opposed to simply the fact. In a syllogism, the conclusion will contain two of the three terms appearing in the syllogism. For example,

manandmortalappear in the conclusion of a syllogism in which a third term, a “middle” term, appears in the two premises. That middle term could beanimalas when we reason: “All animals are mortal. Man is an animal. Therefore, man is mortal.” If mortality necessarily belongs to animality and humanity necessarily belongs to animality, then mortality necessarily belongs to humanity. Only when there is necessity for both premises of the syllogism is there necessity in the conclusion. Demonstrative understanding of the fact that all men are mortal requires understanding necessities of the middle termanimal(APo. 75a11–15).Demonstrations make clear the underlying essential attributes of a thing. Demonstrative understanding is to be had only from that which is necessary to a thing, that which is essential to itself. “Scientific demonstrations are about what belongs to things in themselves, and depends on such things . . . . The middle term must belong to the third, and the first to the middle, because of itself. / One cannot, therefore, prove anything by crossing from another genus—e.g. something geometrical by arithmetic” (

APo. 75a30–39). Therefore, one cannot “prove by any other science the theorems of a different one, except such as are so related to one another that the one is under the other—e.g. optics to geometry and harmonics to arithmetic” (APo. 75b14–17; see also 75b36–76a15).“Mathematics is about 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 [in geometry]” (

APo. 79a9–11). Though one cannot prove theorems of one science by another science, thereby confounding one genus with another, one can sometimes know something of the reason for a fact in one science by considering it in light of another science. “Such are those which are related to each other in such a way that the one is under the other, e.g. optics to geometry . . . and harmonics to arithmetic . . . . Here it is for the empirical scientists 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” (APo. 78b35–79a7).Aristotle takes theory of the rainbow to stand to optics as optics stands to geometry (

APo. 79a12–13). As the reader knows, Aristotle contributed, in hisMeteorology,to theory of the rainbow, appealing to observations of the bows, to optics, and to geometry. I have recently learned of an analysis of Aristotle’s treatment by James Lennox, in a paper twenty-six years ago. I shall return to Aristotle’s treatment in connection with the advances of Theodoric in Part 3.In his commentaries on Aristotle’s

PhysicsandMetaphysics,Albert writes that mathematics and physics are founded on being, mathematics on being with quantity, physics on being with quantity, motion, and sensible matter. However, “we need to beware the error of Plato who said that naturals were founded in mathematicals and mathematicals in divines, . . . and therefore he said that mathematicals were principles of naturals, which is completely false” (quoted in Molland 1980, 467; also Lindberg 1982, 15).Albert reduces the power of mathematics in optics below the Aristotelian level. (Theodoric stayed closer to Aristotle on this issue—Part 3.) Commenting on the

Posterior Analytics,Albert observes that it is not the geometer who can give the reason for the facts of reflection. “These properties are not caused by a line in that it is quantity, therefore lines as such are not appropriate, nor can [the properties] be produced by the geometer from the proper principles of quantity” (quoted in Molland 1980, 468). It is for the physicists of light, the perspectivists, to provide the reasons for properties of light such as capability for and character in reflection (Molland 1980, 468; see also Lindberg 1982, 15, for Albert’s doctrine applied to the spherical shape of the earth).In contrast to Albert, Robert Grosseteste (1175–1253) and Roger Bacon kept with Aristotle’s view that geometry gives the reason for phenomena in optics. Without retreating to the platonic view and without reducing physics to mathematics, they stressed that geometry provides causal, necessary explanation for optical phenomena. Bacon: “The cause of natural things cannot be given except by means of geometry” (quoted in Lindberg 1982, 17). Bacon goes beyond Aristotle in arguing that “in one way or another all of the categories depend on quantity” (Lindberg 1982, 19; cf.)

“Bacon argues, on the basis of Aristotelian epistemology, that sense perception precedes intellection and that quantity is particularly important in sense perception because it is one of the common sensibles (that is, it is perceived by more than one external sense) and nothing can be perceived without quantity; ‘therefore, the intellect is especially occupied with quantity’” (Lindberg 1982, 19–20; cf. Cm).

The first postulate of Euclid is: to draw a straight line from any point to any point. Bacon attends to the fact that man cannot, in matter, draw a perfect line.

The straight lines of nature—lines of light, of visual rays, of all forces—are source and end of geometry. The lines of geometry are not causal, but not causal only in abstraction from their residence in the physical lines that indeed have causal powers (further, Lindberg 1997, 250–55, 265–67). Under the conception of Bacon, perfect lines are in the world concretely, by nature. Different kinds of them have different causal powers. In the “mixed” science of mathematical optics, there is no ontological cleft between geometrical relations as revealed in geometry and as resident in observed optical phenomena. The pertinent geometrical relations truly are physically instanced in the physical phenomena and can provide causal explanation for some parts of the phenomena without mixture of genus.

To be continued.References

Aristotle c. 348–322 B.C.

The Complete Works of Aristotle. J. Barnes, editor. 1984. Princeton.Euclid c. 300 B.C.

The Thirteen Books ofThe Elements. T. L. Heath, translator. 1925 [1908]. Dover.Hackett, J., editor, 1997.

Roger Bacon and the Sciences. Brill.Lennox, J. G. 1985. Aristotle, Galileo, and “Mixed Sciences.” In

Reinterpreting Galileo. W. A. Wallace, editor. Catholic University of America.Lindberg, D. C. 1982. On the Applicability of Mathematics to Nature: Roger Bacon and His Predecessors.

The British Journal for the History of Science15:3–15.——. 1997. Roger Bacon on Light, Vision, and the Universal Emanation of Force. In Hackett 1997.

Molland, G. 1980. Mathematics in the Thought of Albertus. In

Mathematics and the Medieval Ancestry of Physics. 1995. Variorum.——. 1997. Roger Bacon’s Knowledge of Mathematics. In Hackett 1997.

Wallace, W. A. 1959.

The Scientific Methodology of Theodoric of Freiberg. University of Fribourg.## Link to comment

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Mathematics and Scientific RepresentationChristopher Pincock (Oxford 2011)

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AuthorInvestigations into the Applicability of GeometryDouglas Bertrand Marshall (Diss. Harvard 2011)

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Surprising Occurrence of a Wallis Series for Approximating PI in a QM Approximation Technique when Applied to Electron Bound-State Energies in the Hydrogen Atom

Thanks to Abhijeet Melkani for drawing attention to this announcement.

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