Wednesday, May 19, 2010

Science's Light Ages

One often hears today that everything before Galileo (1564-1642), including science, lived in the "Dark Ages" where people were unenlightened and apparently wasted their time studying theology. Yet Galileo was not alone; he indeed did "stand on the shoulders of giants." Who were some of them? What did they say? Certainly they were not theologians, or were they?

Besides Aristotle (382-322 B.C.), the first physicist, as mentioned before, there was St. Augustine (354-430 A.D.), whose theory of time even today is mentioned in the quantum cosmology literature (e.g., in Rev. Mod. Phys. 61, 1 (1989) pg. 15). He said:
For what is time? Who can easily and briefly explain it? Who even in thought can comprehend it, even to the pronouncing of a word concerning it? But what in speaking do we refer to more familiarly and knowingly than time? And certainly we understand when we speak of it; we understand also when we hear it spoken of by another. What, then, is time? If no one ask of me, I know; if I wish to explain to him who asks, I know not. Yet I say with confidence, that I know that if nothing passed away, there would not be past time; and if nothing were coming, there would not be future time; and if nothing were, there would not be present time. Those two times, therefore, past and future, how are they, when even the past now is not; and the future is not as yet? But should the present be always present, and should it not pass into time past, time truly it could not be, but eternity. If, then, time present—if it be time—only comes into existence because it passes into time past, how do we say that even this is, whose cause of being is that it shall not be—namely, so that we cannot truly say that time is, unless because it tends not to be?

—St. Augustine's Confessions XI, ch. 14

Much later (1225-1274) St. Thomas Aquinas unified Greek, principally Aristotelean, thought with that of Christendom. Some of his scientific contributions were to describe the
  1. relation of mathematics to the natural sciences,
  2. relativity of locomotion,
  3. nature of light in optics,
  4. motion of falling bodies, and
  5. foundations of the modern, Galileo-like scientific method,
among many other things.

1. Relation of Mathematics to Natural Sciences

St. Thomas commented on question V of Boethius's De Trinitate, saying:

By its very nature motion is not in the category of quantity, but it partakes somewhat of the nature of quantity from another source, namely, according as the division of motion derives from either the division of space or the division of the thing subject to motion. So it does not belong to the mathematician to treat of motion, although mathematical principles can be applied to motion. Therefore, inasmuch as the principles of quantity are applied to motion, the natural scientist treats of the division and continuity of motion, as is clear in the Physics. And the measurements of motions are studied in the intermediate sciences between mathematics and natural science: for instance, in the science of the moved sphere and in astronomy.

Simple bodies and their properties remain in composite bodies although in a different way, as the proper qualities of the elements and their proper movements are found in a mixed body. What is proper to composite bodies, however, is not found in simple bodies. And so it is that the more abstract and simple the objects of a science are, the more applicable its principles are to the other sciences. Thus the principles of mathematics are applicable to natural things, but not visa versa, because physics presupposes mathematics; but the converse is not true, as is clear in the De Caelo et Mundo. So there are three levels of sciences concerning natural and mathematical entities. Some are purely natural and treat of the properties of natural things as such, like physics, agriculture, and the like. Others are purely mathematical and treat of quantities absolutely, as geometry considers magnitude and arithmetic numbers. Still others are intermediate, and these apply mathematical principles to natural things; for instance, music, astronomy, and the like. These sciences, however, have a closer affinity to mathematics, because in their thinking that which is physical is, as it were, material, whereas that which is mathematical is, as it were, formal. For example, music considers sounds, not inasmuch as they are sounds, but inasmuch as they are proportionable according to numbers; and the same holds in other sciences. Thus they demonstrate their conclusions concerning natural things, but by means of mathematics. Therefore nothing prevents their being concerned with sensible matter insofar as they have something in common with natural science, but insofar as they have something in common with mathematics they are abstract.

In Boethium De Trinitate, q. 5, a. 3 ad 5 et ad 6

Commenting on Aristotle's Physics 193b22, St. Thomas also wrote:

161. Next where he says, ‘That is why he separates ...’(193 b 33), he concludes to a sort of corollary from what he has just said. Because the mathematician does not consider lines, and points, and surfaces, and things of this sort, and their accidents, insofar as they are the boundaries of a natural body, he is said to abstract from sensible and natural matter. And the reason why he is able to abstract is this: according to the intellect these things are abstracted from motion.

As evidence for this reason we must note that many things are joined in the thing, but the understanding of one of them is not derived from the understanding of another. Thus white and musical are joined in the same subject, nevertheless the understanding of one of these is not derived from an understanding of the other. And so one can be separately understood without the other. And this one is understood as abstracted from the other. It is clear, however, that the posterior is not derived from the understanding of the prior, but conversely. Hence the prior can be understood without the posterior, but not conversely. Thus it is clear that animal is prior to man, and man is prior to this man (for man is had by addition to animal, and this man by addition to man). And because of this our understanding of man is not derived from our understanding of animal, nor our understanding of Socrates from our understanding of man. Hence animal can be understood without man, and man without Socrates and other individuals. And this is to abstract the universal from the particular.

In like manner, among all the accidents which come to substance, quantity comes first, and then the sensible qualities, and actions and passions, and the motions consequent upon sensible qualities. Therefore quantity does not embrace in its intelligibility the sensible qualities or the passions or the motions. Yet it does include substance in its intelligibility. Therefore quantity can be understood without matter, which is subject to motion, and without sensible qualities, but not without substance. And thus quantities and those things which belong to them are understood as abstracted from motion and sensible matter, but not from intelligible matter, as is said in Metaphysics, VII:10.

Since, therefore, the objects of mathematics are abstracted from motion according to the intellect, and since they do not include in their intelligibility sensible matter, which is a subject of motion, the mathematician can abstract them from sensible matter. And it makes no difference as far as the truth is concerned whether they are considered one way or the other. For although the objects of mathematics are not separated according to existence, the mathematicians, in abstracting them according to their understanding, do not lie, because they do not assert that these things exist apart from sensible matter (for this would be a lie). But they consider them without any consideration of sensible matter, which can be done without lying. Thus one can truly consider the white without the musical, even though they exist together in the same subject. But it would not be a true consideration if one were to assert that the white is not musical.

162. Next where he says, “The holders of the theory...’ (193 b 35), he excludes from what he has said an error of Plato.

Since Plato was puzzled as to how the intellect could truly separate those things which were not separated in their existence, he held that all things which are separated in the understanding are separated in the thing. Hence he not only held that mathematical entities are separated, because of the fact that the mathematician abstracts from sensible matter, but he even held that natural things themselves are separated, because of the fact that natural science is of universals and not of singulars. Hence he held that man is separated, and horse, and stone, and other such things. And he said these separated things are ideas, although natural things are less abstract than mathematical entities. For mathematical entities are altogether separated from sensible matter in the understanding, because sensible matter is not included in the understanding of the mathematicals, neither in the universal nor in the particular. But sensible matter is included in the understanding of natural things, whereas individual matter is not. For in the understanding of man flesh and bone is included, but not this flesh and this bone.

163. Next where he says, ‘This becomes plain ...’ (194 a 1), he clarifies the solution he has given in two ways, first by means of the difference in the definitions which the mathematician and the natural philosopher assign, and secondly by means of the intermediate sciences, where he says, ‘Similar evidence ...’ (194 a 7 #164).

He says, therefore, first that what has been said of the different modes of consideration of the mathematician and the natural philosopher will become evident if one attempts to give definitions of the mathematicals, and of natural things and of their accidents. For the mathematicals, such as equal and unequal, straight and curved, and number, and line, and figure, are defined without motion and matter, but this is not so with flesh and bone and man. Rather the definition of these latter is like the definition of the snub in which definition a sensible subject is placed, i.e., nose. But this is not the case with the definition of the curved in which definition a sensible subject is not placed.

And thus from the very definitions of natural things and of the mathematicals, what was said above [#160ff] about the difference between the mathematician and the natural philosopher is apparent.

164. Next where he says, ‘Similar evidence...’ (194 a 7), he proves the same thing by means of those sciences which are intermediates between mathematics and natural philosophy.

Those sciences are called intermediate sciences which take principles abstracted by the purely mathematical sciences and apply them to sensible matter. For example, perspective applies to the visual line those things which are demonstrated by geometry about the abstracted line; and harmony, that is music, applies to sound those things which arithmetic considers about the proportions of numbers; and astronomy applies the consideration of geometry and arithmetic to the heavens and its parts.

However, although sciences of this sort are intermediates between natural science and mathematics, they are here said by the Philosopher to be more natural than mathematical, because each thing is named and takes its species from its terminus. Hence, since the consideration of these sciences is terminated in natural matter, then even though they proceed by mathematical principles, they are more natural than mathematical sciences.

He says, therefore, that sciences of this sort are established in a way contrary to the sciences which are purely mathematical, such as geometry or arithmetic. For geometry considers the line which has existence in sensible matter, which is the natural line. But it does not consider it insofar as it is in sensible matter, insofar as it is natural, but abstractly, as was said [#160ff]. But perspective conversely takes the abstract line which is in the consideration of mathematics, and applies it to sensible matter, and thus treats it not insofar as it is a mathematical, but insofar as it is a physical thing.

Therefore from this difference between intermediate sciences and the purely mathematical sciences, what was said above is clear. For if intermediate sciences of this sort apply the abstract to sensible matter, it is clear that mathematics conversely separates those things which are in sensible matter.

165. And from this it is clear what his answer is to the objection raised above [#158] concerning astronomy. For astronomy is a natural science more than a mathematical science. Hence it is no wonder that astronomy agrees in its conclusions with natural science.

However, since it is not a purely natural science, it demonstrates the same conclusion through another method. Thus, the fact that the earth is spherical is demonstrated by natural science by a natural method, e.g., because its parts everywhere and equally come together at the middle. But this is demonstrated by astronomy from the figure of the lunar eclipse, or from the fact that the same stars are not seen from every part of the earth.

In II Phys. lect. 3, nn. 5-9

He also mentioned in his Summa Theologica that:

As stated above (Question 1, Article 1), every cognitive habit regards formally the mean through which things are known, and materially, the things that are known through the mean. And since that which is formal, is of most account, it follows that those sciences which draw conclusions about physical matter from mathematical principles, are reckoned rather among the mathematical sciences, though, as to their matter they have more in common with physical sciences: and for this reason it is stated in Phys. ii, 2 that they are more akin to physics. Accordingly, since man knows God through His creatures, this seems to pertain to "knowledge," to which it belongs formally, rather than to "wisdom," to which it belongs materially: and, conversely, when we judge of creatures according to Divine things, this pertains to "wisdom" rather than to "knowledge."

II-II, q. 9, a. 2 ad 3

2. Relativity of Locomotion

Commenting on Aristotle's De Cælo II., St. Thomas preceded Galilean relativity by writing:

396. First he considers the first one [297], and says that it is impossible that both, i.e., the star and its orb, be at rest if we assume that the earth is also at rest. For the apparent motion of the stars cannot be saved if both the stars which appear to be in motion are at rest, and the men who see them. For, that motion should appear, this must be caused either by the motion of the thing seen or of the one seeing. For this reason, some, positing the stars and the whole heaven to be at rest, posited the earth on which we live to be moved from west to east around the equinoxial poles [i.e., its axis] once a day. According to this, it is due to our own motion that the stars seem to move in a contrary direction. This is said to have been the opinion of Heraclitus of Pontus and Aristarchus. However, Aristotle is supposing for the present that the earth is at rest —which fact he will later prove. Hence it remains, the first member, in which the heaven and the stars were assumed to be at rest, having been set aside, to verify one of the two others —namely, that stating that both, i.e., the star and the orb, are in motion, or that stating one to be in motion and the other at rest.

In II De Cælo, lect. 11, n. 2

405. Then he shows that the motion seen in the stars is due to neither of these two motions. First he shows that the motion seen in the stars is not one of circumgyration; and he proves this in two ways. First, because if the stellar bodies were being moved with the motion of circumgyration, then, even though the parts of the star exchanged places as to subject, the star as a whole would have to remain in the same place as to subject, the place being varied only according to notion, as is clear from what was proved in Physics VI. For that is the way things turn out for a spherical motion due to its relation to a center and to poles that are stationary. But we cannot admit such a situation in the stars, since the contrary is evident to sense —for we see stars sometimes in the east and sometimes in the west. Likewise, everyone says that the stars do not remain always in the same place but are transferred from one place to another. Therefore, the motion that appears to be in the stars is not one of circumgyration.

In II De Cælo, lect. 12, n. 4

3. Nature of Light in Optics

Commenting on Aristotle's De Anima II., St. Thomas wrote this about light, which is reminiscent of field of view or the inverse square law in optics:

§ 433. [...] For if anything is to be seen it must actually affect the organ of sight. Now it has been shown that this organ as such is not affected by an immediate object—such as an object placed upon the eye. So there must be a medium between organ and object. But a vacuum is not a medium; it cannot receive or transmit effects from the object. Hence through a vacuum nothing would be seen at all.

§ 434. Democritus went wrong because he thought that the reason why distance diminishes visibility was that the medium is of itself an impediment to the action of the visible object upon sight. But it is not so. The transparent medium as such is not in the least incompatible with luminosity or colour; on the contrary, it is proximately disposed to their reception; a sign of which is that it is illumined or coloured instantaneously. The real reason why distance diminishes visibility, is that everything seen is seen within the angle of a triangle, or rather pyramid, whose base is the object seen and apex in the eye that sees.

§ 435. It makes no difference whether seeing takes place by a movement from the eye outwards, so that the lines enclosing the triangle or pyramid run from the eye to the object, or e converso, so long as seeing does involve this triangular or pyramidal figure; which is necessary because, since the object is larger than the pupil of the eye, its effect upon the medium has to be scaled down gradually until it reaches the eye. And, obviously, the longer are the sides of a triangle or pyramid the smaller is the angle at the apex, provided that the base remains the same. The further away, then, is the object, the less does it appear—until at a certain distance it cannot be seen at all.

In II De Anima lect. 15, §433-§435

Long before the Italian physicist Macedonio Melloni (1798-1854) discovered that heat and light share similar properties, St. Thomas wrote this:
[L]ux [...] semper est effectiva caloris; etiam lux lunæ. ["Light always is effected of heat; even moonlight."]

Super Sent., lib. 2 d. 15 q. 1 a. 2 ad 5

4. Motion of Falling Bodies

Previously, many adopted Aristotle's theory that the medium—e.g., air—is what keeps a falling object in motion. Commenting on Aristotle's Physics III., St. Thomas distinguished for the first time these three things: weight, mass, and the resisting medium:

535. [...] This resistance can arise from three sources: First, from the situs of the mobile; for from the very fact that the mover intends to transfer the mobile to some certain place, the mobile, existing in some other place, resists the intention of the mover. Secondly, from the nature of the mobile, as is evident in compulsory motions, as when a heavy object is thrown upwards. Thirdly, from the medium. All three are taken together as one resistance, to constitute one cause of slowing up in the motion. Therefore when the mobile, considered in isolation as different from the mover, is a being in act, the resistance of the mobile to the mover can be traced either to the mobile only, as happens in the heavenly bodies, or to the mobile and medium together, as happens in the case of animate bodies on this earth. But in heavy and light objects, if you take away what the mobile receives from the mover, viz., the form which is the principle of motion given by the generator, i.e., by the mover, nothing remains but the matter which can offer no resistance to the mover. Hence in light and heavy objects the only source of resistance is the medium. Consequently, in heavenly bodies differences in velocity arise only on account of the ratio between mover and mobile; in animate bodies from the proportion of the mover to the mobile and to the resisting medium—both together. And it is in these latter cases that the given objection would have effect, viz., that if you remove the slowing up caused by the impeding medium, there still remains a definite amount of time in the motion, according to the proportion of the mover to the mobile. But in heavy and light bodies, there can be no slowing up of speed, except what the resistance of the medium causes—and in such cases Aristotle’s argument applies.

In IV Physica lect. 12, n. 535

5. Foundations of the Modern Scientific Method

Commenting on Aristotle's De Cælo II., St. Thomas notes that there can be multiple theories explaining given observations:
Yet it is not necessary that the various suppositions which [the astronomers] hit upon be true—for although these suppositions save the appearances, we are nevertheless not obliged to say that these suppositions are true, because perhaps there is some other way men have not yet grasped by which the things which appear as to the stars are saved.

In II De cælo, lect. 17, n. 451

Similarly, St. Thomas writes, when considering whether one can know the Trinity by natural means:

Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astrology the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them. [...]

Summa Theologica, I, q. 32, a. 1 ad 2

Following St. Thomas Aquinas came these people:
Robert Grosseteste (c. 1168-1253) did experiments (not yet of course with modern rigor) and was keen on using mathematics; he is known for his work on understanding the rainbow. Thomas of Bradwardine (c. 1295-1349) at Merton College Oxford introduced the distinction between mean velocity (x/t) and instantaneous velocity (dx/dt) [and he was the first to write a physics equation]. Bradwardine had an enthusiasm for empiriometric physics that started a whole school called the Merton school (his successors include: William Heytesbury, Richard Swineshead, and John Dumbleton) that was extremely influential throughout Europe. Among other things, they were known for the Merton mean speed theorem, by which they proved the correct formula for free fall distance was given by s=1/2 a t². Interestingly, both Bradwardine and Grosseteste at some point in their lives were Archbishops of Canterbury. Nicole Oresme (<1348-1382) and Giovanni di Casali (c. 1350) independently developed use of 2-D graphs [long before Descartes (1596-1650)]. Oresme described all change using these graphs in particular local motion, including calculating area (integrating) under velocity curves to get distance. Oresme's arguments for the sun-centered and moving earth were widely known: he said, for example, that "...not only is the earth so moved diurnally, but with it the water and the air, as was said, in such a way that the water and the lower air are moved differently than they are by winds and other causes. It is like this situation If air were enclosed in a moving ship, it would seem to the person situated in this air that it was not moved." (p. 133, Dales.)

—A. Rizzi's Science Before Science pgs. 199-200

Roger Bacon (1214-1294) advocated mathematics in the experimental sciences:
The neglect [of mathematics] for the past thirty or forty years has nearly destroyed the entire learning of Latin Christendom. For he who does not know mathematics cannot know any of the other sciences.

Opus maius IV.1.1. (ed. J.H. Bridges [Oxford 1897], I, 97-98)

Quantity is the first property of anything, so to neglect that would indeed be to miss a lot. St. Thomas said that mathematics is most connatural to man, hence its development—not the introduction of experimentation, which already existed—was what has driven the scientific boom in the past 400 years. Card. Thomas of Bradwardine said this about mathematics in the sciences:
[Mathematics] reveals every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to study physics while neglecting mathematics, should know from the start that he will never make his entry through the portals of wisdom.

Tractatus de continuo MS Erfurt Amplon Q.385, fol. 31v.

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