Tuesday, May 25, 2010

Handwriting of Geniuses

Many people have heard of the mirrored handwriting of Leonardo da Vinci (1452-1519),
Da Vinci's mirrored handwriting


but few know that another genius, St. Thomas Aquinas (1225-1274), wrote Latin with a littera inintelligibilis, an "unintelligible lettering:"
St. Thomas Aquinas's "littera inintelligibilis" or "unintelligible lettering" in a manuscript he wrote and autographed

Manuscript page showing “littera inintelligibilis,” written and autographed by St. Thomas Aquinas.

"St. Thomas Aquinas," New Catholic Encyclopedia

St. Thomas apparently dictated his writings to a handful secretaries simultaneously because he could not get his thoughts down fast enough. This might explain his littera inintelligibilis.

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.

Monday, May 17, 2010

Sense Knowledge

We have seen the amazing relationship of faith and sense knowledge in the context of "credo ut intelligam" versus "intelligo ut credam" and the story of Doubting St. Thomas the Apostle (John 20:24-29).

Here I analyze St. Thomas Aquinas's Summa Theologica article that, contrary to most moderns who are steeped in a Cartesian idealism—viz., who think everything is in our heads—intellectual knowledge does indeed come from the senses. Overcoming modern Manicheans' and Albigenses' denial of this—i.e., that they think a human being is not both a body and an intellectual soul but just a soul trapped in a body, hence rationalizing any misuse of the human body, e.g., pornography, slavery, contraception, abortion, because the apparently evil body has no affect on oneself nor on one's soul—is vitally important both for society and for the proper development of modern physics, as the Institute of Advanced Physics recognizes and promotes. Nothing is in the intellect that is not first in the senses: Nihil est in intellectu quod non prius in sensu. (Refer to this post for an overview of this "Epistemology of Modern Physics" series.)

Article 6. Whether intellectual knowledge is derived from sensible things?

Objection 1. It would seem that intellectual knowledge is not derived from sensible things. For Augustine says (QQ. 83, qu. 9) that "we cannot expect to learn the fulness of truth from the senses of the body." This he proves in two ways. First, because "whatever the bodily senses reach, is continually being changed; and what is never the same cannot be perceived." Secondly, because, "whatever we perceive by the body, even when not present to the senses, may be present to the imagination, as when we are asleep or angry: yet we cannot discern by the senses, whether what we perceive be the sensible object or the deceptive image thereof. Now nothing can be perceived which cannot be distinguished from its counterfeit." And so he concludes that we cannot expect to learn the truth from the senses. But intellectual knowledge apprehends the truth. Therefore intellectual knowledge cannot be conveyed by the senses.

[So would St. Augustine deny that anything can be sensed? Would he say that the senses always are a faulty and untrustworthy assessor of what is real?]

Objection 2. Further, Augustine says (Gen. ad lit. xii, 16): "We must not think that the body can make any impression on the spirit, as though the spirit were to supply the place of matter in regard to the body's action; for that which acts is in every way more excellent than that which it acts on." Whence he concludes that "the body does not cause its image in the spirit, but the spirit causes it in itself." Therefore intellectual knowledge is not derived from sensible things.

[This seems to be a plausible argument. If the soul animates the body, then what effect can the body have on the soul? Is it a direct effect, or does the body change such that the soul animating it must correspondingly adapt to the change? Is this how it would communicate sense knowledge to the soul? Affecting how the soul can animate the body?]

Objection 3. Further, an effect does not surpass the power of its cause. But intellectual knowledge extends beyond sensible things: for we understand some things which cannot be perceived by the senses. Therefore intellectual knowledge is not derived from sensible things.

[I suppose this is related to the fact that senses deal with particulars—"this atom," "that dog", etc.—than with universals—"atom-ness," "dog-ness," etc.—generalized from particulars. The human intellectual soul can only do this.]

On the contrary, The Philosopher says (Metaph. i, 1; Poster. ii, 15) that the principle of knowledge is in the senses.

[The principle of knowledge, not just a principle. This seems like a strong statement. Not only does the fact that 2+2=4 originate from the senses but so does the fact that something cannot both be and not be at the same time and in the same way: the law of non-contradition. So, also, does the fact that God exists come from the senses!]

I answer that, On this point the philosophers held three opinions. For Democritus held that "all knowledge is caused by images issuing from the bodies we think of and entering into our souls," as Augustine says in his letter to Dioscorus (cxviii, 4). And Aristotle says (De Somn. et Vigil.) that Democritus held that knowledge is cause by a "discharge of images." And the reason for this opinion was that both Democritus and the other early philosophers did not distinguish between intellect and sense, as Aristotle relates (De Anima iii, 3). Consequently, since the sense is affected by the sensible, they thought that all our knowledge is affected by this mere impression brought about by sensible things. Which impression Democritus held to be caused by a discharge of images.

[So, in a way, Democritus thought that we had a sort of angelic, direct knowledge of objects via a "discharge of images" coming from them; thus, we knowledge not mediated by the senses. But this is just pushing the question into the realm of the unknown: What exactly are these images that are discharged? How are they distinguished? Does every soul interpret them similarly? How would you know? This seems to be a subtle form of Cartesian idealism.]

Plato, on the other hand, held that the intellect is distinct from the senses: and that it is an immaterial power not making use of a corporeal organ for its action.[Yes, not even the brain] And since the incorporeal cannot be affected by the corporeal, he held that intellectual knowledge is not brought about by sensible things affecting the intellect, but by separate intelligible forms being participated by the intellect, as we have said above (4,5) [and it would seem in the previous paragraph, too]. Moreover he held that sense is a power operating of itself. Consequently neither is sense, since it is a spiritual power, affected by the sensible: but the sensible organs are affected by the sensible, the result being that the soul is in a way roused to form within itself the species of the sensible. [How exactly?] Augustine seems to touch on this opinion (Gen. ad lit. xii, 24) where he says that the "body feels not, but the soul through the body, which it makes use of as a kind of messenger, for reproducing within itself what is announced from without." [This might be akin to what we said in the first comment above.] Thus according to Plato, neither does intellectual knowledge proceed from sensible knowledge, nor sensible knowledge exclusively from sensible things; but these rouse the sensible soul to the sentient act, while the senses rouse the intellect to the act of understanding.

Aristotle chose a middle course. For with Plato he agreed that intellect and sense are different. But he held that the sense has not its proper operation without the cooperation of the body [Now this is more reasonable because how can the senses not be corporeal?]; so that to feel is not an act of the soul alone, but of the "composite." [Yes, the body and soul, when a human is alive, are inextricably connected.] And he held the same in regard to all the operations of the sensitive part. Since, therefore, it is not unreasonable that the sensible objects which are outside the soul should produce some effect in the "composite," Aristotle agreed with Democritus in this, that the operations of the sensitive part are caused by the impression of the sensible on the sense: not by a discharge, as Democritus said, but by some kind of operation. [Through the physical] For Democritus maintained that every operation is by way of a discharge of atoms [not in the modern sense of "atom," though], as we gather from De Gener. i, 8. But Aristotle held that the intellect has an operation which is independent of the body's cooperation. Now nothing corporeal can make an impression on the incorporeal. [Agreed] And therefore in order to cause the intellectual operation according to Aristotle, the impression caused by the sensible does not suffice, but something more noble is required, for "the agent is more noble than the patient," as he says (De Gener. i, 5). Not, indeed, in the sense that the intellectual operation is effected in us by the mere impression of some superior beings, as Plato held; but that the higher and more noble agent which he calls the active intellect, of which we have spoken above (79, 3,4) [See Sentencia De anima, lib. 3 l. 10.] causes the phantasms received from the senses to be actually intelligible, by a process of abstraction.

According to this opinion, then, on the part of the phantasms, intellectual knowledge is caused by the senses. But since the phantasms cannot of themselves affect the passive intellect, and require to be made actually intelligible by the active intellect, it cannot be said that sensible knowledge is the total and perfect cause of intellectual knowledge, but rather that it is in a way the material cause.

[Wow, so sense knowledge is like a material cause of intellectual knowledge. What would the formal cause be like, then? The phantasms? The light of reason? The active intellect? The fact that the passive and active intellect is one of the hardest things to understand definitely sheds light on the mystery of the relation between the senses and the soul, but it cannot disprove that sense knowledge is not like the form of intellectual knowledge.]

Reply to Objection 1. Those words of Augustine mean that we must not expect the entire truth from the senses. For the light of the active intellect is needed, through which we achieve the unchangeable truth of changeable things, and discern things themselves from their likeness.

["[T]he light of the active intellect is needed." What about the principles afforded by Revelation, the truths of faith?]

Reply to Objection 2. In this passage Augustine speaks not of intellectual but of imaginary knowledge. And since, according to the opinion of Plato, the imagination has an operation which belongs to the soul only, Augustine, in order to show that corporeal images are impressed on the imagination, not by bodies but by the soul, uses the same argument as Aristotle does in proving that the active intellect must be separate, namely, because "the agent is more noble than the patient." And without doubt, according to the above opinion, in the imagination there must needs be not only a passive but also an active power. But if we hold, according to the opinion of Aristotle, that the action of the imagination, is an action of the "composite," there is no difficulty; because the sensible body is more noble than the organ of the animal, in so far as it is compared to it as a being in act to a being in potentiality; even as the object actually colored is compared to the pupil which is potentially colored. It may, however, be said, although the first impression of the imagination is through the agency of the sensible, since "fancy is movement produced in accordance with sensation" (De Anima iii, 3), that nevertheless there is in man an operation which by synthesis and analysis forms images of various things, even of things not perceived by the senses. And Augustine's words may be taken in this sense.

Reply to Objection 3. Sensitive knowledge is not the entire cause of intellectual knowledge. And therefore it is not strange that intellectual knowledge should extend further than sensitive knowledge.

[What is the rest of the cause? "[T]he light of the active intellect?"]

—St. Thomas Aquinas's Summa Theologica Iª q. 84 a. 6 with my commentary in [red]

See future posts for the continuation of the commentary.

Monday, May 3, 2010

Believe, that you may understand.

Atheist scientists frequently contend that doubt, which is opposed to faith, and skepticism are the most necessary virtues for a scientist. They argue that because we are, as Carl Sagan put it, stardust living on a "pale blue dot," we have no right hubristically to value human life over anything else; we must remain "humble." Consequently, they oppose any form of dogmatism that impedes free inquiry and unrestricted academic freedom.

Even King David, after beholding "the moon and the stars," humbly asks their Creator: "What is man that thou art mindful of him?" (Psalm 8:4-5). And even St. Thomas the Apostle ("Doubting Thomas") was the first doubting scientist:
Now Thomas, one of the twelve, who is called Didymus, was not with them when Jesus came. The other disciples therefore said to him: We have seen the Lord. But he said to them: Except I shall see in his hands the print of the nails, and put my finger into the place of the nails, and put my hand into his side, I will not believe. And after eight days again his disciples were within, and Thomas with them. Jesus cometh, the doors being shut, and stood in the midst, and said: Peace be to you. Then he saith to Thomas: Put in thy finger hither, and see my hands; and bring hither thy hand, and put it into my side; and be not faithless, but believing. Thomas answered, and said to him: My Lord, and my God. Jesus saith to him: Because thou hast seen me, Thomas, thou hast believed: blessed are they that have not seen, and have believed.

John 20:24-29

The Incredulity of St. Thomas the Apostle by Michelangelo Caravaggio (c. 1571-1610)

How can we make sense of this? Does it apply to understanding everything? What does St. Anselm mean when he says "faith seeking understanding" (fides quærens intellectum) and "Nor do I seek to understand that I may believe, but I believe that I may understand [credo ut intelligam]. For this too I believe, that unless I first believe, I shall not understand [«nisi credidero, non intelligam», St. Augustine Sermo 43 7,9; cf. Isaias 7:9]." (Proslogion I)? St. Augustine—the 5th century saint whose theory of time is frequently quoted in the quantum cosmology literature, e.g., in Rev. Mod. Phys. 61, 1 (1989) pg. 15—says:
7. [...]

What were we arguing about? You were saying “Let me understand in order to believe”; I was saying “In order to understand, believe.” An argument has arisen, let us put it before a judge, let a prophet judge, or rather let God judge through a prophet. Let's both of us keep silent. What we have each said has been heard: “Let me understand,” you say, “in order to believe.” “Believe,” say I, “in order to understand.” Let the prophet make his reply: “Unless you believe, you shall not understand” (Is 7:9).


9. Just now when the gospel was being read, you heard If you can believe—the Lord Jesus said to the boy's father, If you can believe, all things are possible to one who believes (Mk 9:23). And the man took a look at himself, and standing in front of himself, not in a spirit of brash self-satisfaction but first examining his conscience, he saw that he did have some faith in him, and he also saw that it was tottering. He saw both things. He confessed he had one, and he begged for help for the other. I believe, Lord, he says. What was to follow, if not “Help my faith”? That's not what he said. “I believe, Lord. I can see this something in me, which I'm not lying about. I believe; I'm telling the truth. But I also see this other heaven knows what, and I don't like it. I want to stand, I'm still staggering. I'm standing and speaking, I haven't fallen, because I believe. But yet I'm still staggering: Help my unbelief” (Mk 9:24).

And so, beloved, that other man too whom I set up against myself, calling in the prophet as referee because of the argument that arose between us, he too isn't saying just nothing when he says “Let me understand, in order to believe.” Of course, what I am now saying, I am saying to help those people believe who do not yet believe. And yet, unless they understand what I am saying, they cannot believe. So what this person says is partly true—“Let me understand, in order to believe”; and I on my side, when I say, just as the prophet says, “On the contrary, believe, in order to understand,” am speaking the truth. Let's come to an agreement, then. So: understand, in order to believe; believe, in order to understand. I'll put it in a nutshell, how we can accept both without argument: Understand, in order to believe, my word; believe, in order to understand, the word of God.

St. Augustine's Sermo 43 7,9

So, many modern atheist scientists are missing faith, one of the "two wings on which the human spirit rises to the contemplation of truth," which are faith and reason (Fides et Ratio 1.). As a result, doubt hinders the full contemplation of the absolute truths.

Obviously, though, atheists would object to most of what has been said, e.g., the invoking of God and absolutes. They might say:
Why is any of this necessary for science, which just studies contingent—viz., non-necessary—things? It is untrue that "faith purifies reason" (fides purificat rationem) and "liberates [it] from presumption" ([fides] rationem a nimia confidentia exsolvit), no (Fides et Ratio 76.)? Faith, especially that of organized religions, has destroyed science by enslaving it to outdated, primitive presumptions, e.g., that the universe and everything in it was created in a week. [To which I would immediately respond, due to their misunderstanding: "St. Augustine, e.g., wrote that the Genesis 'day' is not necessarily 24 hours. In fact, before the earth and sun were created, how could one even define 'day' anyways?"] Modern science has enlightened us now. We no longer need blind religion or anything else besides our intellects and reason for our guide.
The problem with this sort of Cartesian "Cogito, ergo sum" ("I think; therefore, I am.") is that it denies an objective reality and makes us our own authorities even on matters of which we are often unqualified to judge; this is extreme Protestantism. The atheist scientists might maintain that truth is relative but they have no way of scientifically justifying this. In a sense, what they have faith in is what is most detrimental to science: agnosticism, in the sense that the human intellect can never know anything with certainty, even the basic premises of logic such as the law of non-contradiction or the syllogism, and that which follows from it: extreme doubt, skepticism, and nihilism. Even the pagan Aristotle realized—almost á la Kurt Gödel and his Incompleteness Theorem—that, although there is scientific knowledge, not all truths are demonstrable as through a syllogism or science experiment; Aristotle was therefore one of the first anti-positivists (on positivism, cf. Card. Schönborn's "The Designs of Science"). We quote first his Posterior Analytics—a book that even Galileo faithfuflly followed despite his disagreements with Aristotle's physics—on "The erroneous views of scientific knowledge" and "The futility of circular demonstration," both of which I believe many atheist scientists are culpable, then we follow it by St. Thomas Aquinas's commentary, which more illuminates Aristotle's work in the present context of "crede, ut intelligas:"


Some hold that, owing to the necessity of knowing the primary premisses, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premisses. The first school, assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand—they say—the series terminates and there are primary premisses, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premisses, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premisses are true. The other party agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.

Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premisses from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.

Now demonstration must be based on premisses prior to and better known than the conclusion; and the same things cannot simultaneously be both prior and posterior to one another: so circular demonstration is clearly not possible in the unqualified sense of ‘demonstration’, but only possible if ‘demonstration’ be extended to include that other method of argument which rests on a distinction between truths prior to us and truths without qualification prior, i.e. the method by which induction produces knowledge. But if we accept this extension of its meaning, our definition of unqualified knowledge will prove faulty; for there seem to be two kinds of it. Perhaps, however, the second form of demonstration, that which proceeds from truths better known to us, is not demonstration in the unqualified sense of the term.

The advocates of circular demonstration are not only faced with the difficulty we have just stated: in addition their theory reduces to the mere statement that if a thing exists, then it does exist—an easy way of proving anything. That this is so can be clearly shown by taking three terms, for to constitute the circle it makes no difference whether many terms or few or even only two are taken. Thus by direct proof, if A is, B must be; if B is, C must be; therefore if A is, C must be. Since then—by the circular proof—if A is, B must be, and if B is, A must be, A may be substituted for C above. Then ‘if B is, A must be’=’if B is, C must be’, which above gave the conclusion ‘if A is, C must be’: but C and A have been identified. Consequently the upholders of circular demonstration are in the position of saying that if A is, A must be—a simple way of proving anything. Moreover, even such circular demonstration is impossible except in the case of attributes that imply one another, viz. ‘peculiar’ properties.

Now, it has been shown that the positing of one thing—be it one term or one premiss—never involves a necessary consequent: two premisses constitute the first and smallest foundation for drawing a conclusion at all and therefore a fortiori for the demonstrative syllogism of science. If, then, A is implied in B and C, and B and C are reciprocally implied in one another and in A, it is possible, as has been shown in my writings on the syllogism, to prove all the assumptions on which the original conclusion rested, by circular demonstration in the first figure. But it has also been shown that in the other figures either no conclusion is possible, or at least none which proves both the original premisses. Propositions the terms of which are not convertible cannot be circularly demonstrated at all, and since convertible terms occur rarely in actual demonstrations, it is clearly frivolous and impossible to say that demonstration is reciprocal and that therefore everything can be demonstrated.


Since the object of pure scientific knowledge cannot be other than it is, the truth obtained by demonstrative knowledge will be necessary. And since demonstrative knowledge is only present when we have a demonstration, it follows that demonstration is an inference from necessary premisses.

—Aristotle's Posterior Analytics bk. 1 ch. 3-4 (72b5-24)

Lecture 7

b5. Some hold that, owing— b8. The first school— b15. The other party agree— b18. Our own doctrine is that

After determining about the knowledge of the principles of demonstration, the Philosopher now excludes the errors which have arisen from these determinations. Concerning this he does three things. First, he states the errors. Secondly, the reasons they erred (72b8). Thirdly, he removes the roots of these reasons (72b 18).

He says therefore first (72b5), that two contrary errors have arisen from one of the truths established above. For it has been established above that the principles of demonstration must be known and must be even better known. But the first of these is sufficient for our purpose. For some, basing themselves on this first statement, have come to believe that there is no science of anything, whereas others believe that there is science, even to the extent of believing that there is science of everything through demonstration. But neither of these positions is true and neither follows necessarily from their reasons.

Then (72b8) he presents the reasons why they have fallen into these errors. And first of all he presents the reason given by those who say that there is no science, and it is this: The principles of demonstration either proceed to infinity or there is a halt somewhere. But if there is a process to infinity, nothing in that process can be taken as being first, because one cannot exhaust an infinite series and reach what is first. Consequently, it is not possible to know what is first. (They are correct in thus arguing, for the later things cannot be known unless the prior ones are known).

On the other hand, if there is a halt in the principles, then even so, the first things are still not known, if the only way to know scientifically is through demonstration. For first things do not have prior principles through which they are demonstrated. But if the first things are not known, it follows again that the later things are not known in the strict and proper sense, but only on condition that there are principles. For it is not possible for something to be known in virtue of something not known, except on condition that that unknown be a principle. So in either case, whether the principles stop or go on to infinity, it follows that there is no science of anything.

Secondly (72b15), he presents the reasoning of those who say that there is science of everything through demonstration, because to their basic premise—the only way to know scientifically is by demonstration—they added another, namely, that one may demonstrate circularly. From these premises it followed that even if a limit is reached in the series of the principles of demonstration, the first principles are still known through demonstration, because, they said, those principles were demonstrated by previous ones. For a circular demonstration is one which is reciprocal, i.e., something which was first a principle is later a conclusion, and vice versa.

Then (72b18) he cuts away the false bases of these arguments. First, their supposition that the only way to know scientifically is by demonstration. Secondly, their statement that it is legitimate to demonstrate circularly (72b25).

He says therefore first (72b18), that not all scientific knowledge is demonstrative, i.e., obtained through demonstration, but the scientific knowledge of immediate principles is indemonstrable, i.e., not obtained by demonstration. However, it should be noted that Aristotle is here taking science in a wide sense to include any knowledge that is certain, and not in the sense in which science is set off against understanding, according to the dictum that science deals with conclusions and understanding [intuition] with principles.

But that it is necessary for some things to be held as certain without demonstration he proves in the following way: It is necessary that the prior things from which a demonstration proceeds be known in a scientific way. Furthermore, these must be ultimately reduced to something immediate; otherwise one would be forced to admit that there is an actual infinitude of middles between two extremes—in this case between the subject and predicate. Again, one would have to admit that no two extremes could be found between which there would not be an infinitude of middles. But as it is, the middles are such that it is possible to find two things which are immediate. But immediate principles, being prior, must be indemonstrable. Thus it is clear that it is necessary for some things to be scientifically known without demonstration.

Therefore, if someone were to ask how the science of immediate principles is possessed, the answer would be that not only are they known in a scientific manner, but knowledge of them is the source of a science. For one passes from the knowledge of principles to a demonstration of conclusion on which science, properly speaking, bears. But those immediate principles are not made known through an additional middle but through an understanding of their own terms. For as soon as it is known what a whole is and what a part is, it is known that every whole is greater than its part, because in such a proposition, as has been stated above, the predicate is included in the very notion of the subject. And therefore it is reasonable that the knowledge of these principles is the cause of the knowledge of conclusions, because always, that which exists in virtue of itself is the cause of that which exists in virtue of something else.

Lecture 8

b25. Now demonstration must— b33. The advocates of circular— b38. Thus by direct proof— a1. Since then— a6. Moreover, even such

After excluding one false basis by showing that not all science depends on demonstration, the Philosopher now excludes another by showing that it is not possible to demonstrate circularly.

To understand this it should be noted that a demonstration is circular when the conclusion and one of the premises (in converted form) of a syllogism are used to prove the other premise. For example, we might form the following syllogism:

Every rational mortal animal is risible;
Every man is a rational mortal animal:
Therefore, every man is risible.

Now if the conclusion were to be used as one principle and the minor in converted form as the other, we would get:

Every man is risible;
Every rational mortal animal is a man:
Therefore, every rational mortal animal is risible—which was the major of the first syllogism.

Accordingly, he presents three arguments to show that it is not possible to demonstrate circularly. The first of these (72b25) is this: In a circular syllogism the same thing is at once a conclusion and a principle. But a principle of a demonstration is prior to and better known than the conclusion, as has been shown above. Therefore, it follows that a same thing is both prior to and subsequent to one same thing, and also more known and less known. But this is impossible. Therefore, it is impossible to demonstrate circularly.

But someone might say that a same thing can be both prior and subsequent, although not in the same way. For example, this might be prior in reference to us, but that prior absolutely. Thus singulars are prior in reference to us and subsequent absolutely: and conversely for universals. Again, induction makes something known in one way and demonstration in another way. For demonstration proceeds from things that are prior absolutely, but induction from things that are prior in reference to us.

Now if a circular demonstration were so constructed that something is first concluded from things that are absolutely prior, and then from things that are prior in reference to us, it would follow that our doctrine on scientific knowing was not well established. For we stated that to know scientifically is to know the cause of a thing. From this it followed that a demonstration which causes scientific knowledge must proceed from the absolutely prior. But if demonstration were at one time to proceed from the absolutely prior and at another time from things which are prior in reference to us, we would be forced to admit that scientific knowing is not confined to knowing the cause of a thing, but that there is another, namely, that form of knowing which proceeds from what is later. Therefore, one must either admit both or admit that the second form, namely, the demonstration which proceeds from what is better known to us is not a demonstration in the absolute sense.

The aforesaid also reveals why a dialectical syllogism can be circular. For it proceeds from things which are probable. But things are said to be probable if they are better known to the wise or to a great number of persons. Consequently, a dialectical syllogism proceeds from things that are better known to us. However, it happens that a same thing is better known to some and less known to others. Consequently, there is nothing to hinder a dialectical syllogism from being circular. But a demonstration is formed from things that are absolutely prior. Therefore, as we have already stated, there cannot be circular demonstration.

Then he sets forth the second argument (72b33) and it is this: If there were circular demonstration, it would follow that a same thing is demonstrated by the same thing, as if I were to say: If it is this, it is this. In this way it is easy for anyone to demonstrate everything, for anyone, wise or ignorant, will be able to do this. Accordingly, science is not acquired through demonstration. But this is against the definition of demonstration. Therefore, there cannot be circular demonstration.

He proves the truth of the first consequence in the following way: It is obvious, first of all, that with a circular demonstration the same thing is proved by a same thing, as has been stated above, i.e., if only three terms are employed; although it makes no difference whether the reflexion be made with fewer terms or more. (By reflexion he means the process whereby one goes from principle to conclusion in a demonstration, and then from conclusion to principle). In such a reflexion it makes no difference, so far as the force of the argument is concerned, whether it involves several or fewer terms or even two. For an argument has the same force if one proceeds thus: “If it is A, it is B, and if it is B, it is C, and if it is C, it is D,” and then by reflecting continues, “If it is D, it is C, and if it is C, it is B, and if it is B, it is A”; or if he proceeds by reflecting at the very start, saying: “If it is A, it is B, and if it is B, it is A.” (Although he spoke above of three terms, he restricted himself to two terms in this example, because in the deduction he is about to make he will use a third term, which is the same as the first).

Then (72b38) he gives the form of the argument in three terms, namely: “If it is A, it is B, and if it is B, it is C; therefore, if it is A, it is of necessity C.”

Then (73a1) he shows by the aforesaid form of arguing that in a circular demonstration a same thing is proved by a same thing, using only two terms. For it consists in saying, “If it is A, it is B,” and then reflecting, “If it is B, it is A”—which is a circular demonstration. Now according to the above given form it follows from these two, that “if it is A, it is A.”

That it does follow is obvious: for just as in the first deduction which involved three terms’ C followed from B, so in the reflex deduction of two terms, A followed from B. Let us suppose, then, that the A of the second deduction, i.e., the reflex, signifies the same thing that C signified in the first, i.e., in the direct deduction which was composed of three terms. Therefore, to state in the second deduction that “if it is B it is A” is to state the same thing as was stated in the first deduction, namely, that “if it is B, it is C.” But when it was stated in the first deduction that “if it is B, it is C,” it followed that “if it is A, it is C.” Therefore, in the circular deduction it follows that “if it is A, it is A,” since C is assumed to be the same as A. In this way, it will be easy to demonstrate all things, as has been said.

Then he presents the third argument (73a6) which is this: Those who suppose that everything can be known through demonstration on the ground that demonstration is circular, must grant that anything can be demonstrated by a circular demonstration and, as a consequence, grant that in a circular demonstration each of the premises can be concluded from the conclusion. However, the only cases in which this can be done are those in which mutual conversion is possible, i.e., in things that are convertible, as properties. But not all things are so related. Therefore, it is ridiculous to say that everything can be demonstrated on the ground that there are such things as circular demonstrations.

Now the reason is obvious why in a circular demonstration all the propositions must be convertible. For it has been shown in the book of Prior Analytics that if one thing is laid down, another does not follow of necessity, whether the thing laid down be one term or one proposition. For every syllogism must start with three terms and two propositions as a minimum. Therefore, in a circular demonstration three terms which are convertible must be taken, namely, A, B, C, such that A is in every B and in every C, and these, namely, B and C, must inhere in each other, so that every B is C and every C is B, and also inhere in A so that every A is B and every A is C. And so, the terms being thus related, it is possible, when using the first figure, to derive any one from any two circularly, i.e., the conclusion from two premises and each premise from the conclusion and the remaining premise, as we pointed out in the Prior Analytics, where we treated the syllogism formally.

The way it is done is this: take the three convertible terms, “risible,” “rational mortal animal” and “man,” and form the syllogism:

Every rational mortal animal is risible;
Every man is a rational mortal animal:
Therefore, every man is risible.

Then from the conclusion it is possible to conclude both the major and the minor; the major thus:

Every man is risible;
But every rational mortal animal is a man:
Therefore, every rational mortal animal is risible

and the minor thus:

Every risible is a rational mortal animal;
But every man is risible:
Therefore, every man is a rational mortal animal.

However, it has also been proved in the Prior Analytics that in figures other than the first, namely, in the second and third, one cannot form a circular syllogism, i.e., one through which each of the premises can be syllogized from the conclusion; or if one is formed, it is done not by using the premises already used but by using propositions other than those which appear in the first syllogism.

That this is so is obvious. For the second figure always yields a negative conclusion. Consequently, one premise must be affirmative and the other negative. However, it is true that if both are negative, nothing can be concluded; and if both are affirmative, a negative conclusion cannot follow. Therefore, it is not possible to use the negative conclusion and the negative premise to obtain the affirmative premise as a conclusion. Hence, if this affirmative is to be proved, it must be proved through propositions other than the ones originally used. Again, in the third figure the only conclusion ever obtained is particular. However, at least one premise must be universal; furthermore, if either premise is particular, a universal cannot be concluded. Hence it cannot occur that in the third figure each of the premises can be syllogized from the conclusion and the remaining premise.

For the same reasons it is obvious that such a circular syllogism (through which each premise could be concluded) cannot be formed in the first figure except in the first mode, which is the only one that concludes to a universal affirmative. Furthermore, even in this mode the only case in which a circular syllogism could be formed such that each of the premises could be concluded, is when the three terms employed are equal, i.e., convertible. The proof is this: The premise must be concluded from the conclusion and the converse of the other premise, as has been stated. But such a conversion of each premise is impossible (for each is universal), except when the terms happen to be equal.

St. Thomas Aquinas's Expositio Posteriorum, lib. 1 l. 7 et 8

Therefore, contrary to the atheist scientists who nihilistically think doubt is the supreme intellectual virtue—even though they are correct that humility is good, just not the sort they conceive as devaluing human life, which is really denying the reality of the sacredness of life—we can see the necessity of belief in the ultimate principles upon which science bases itself. If "faith is more certain than science and the other intellectual virtues" (Summa Theologica IIª-IIae q. 4 a. 8), then science—especially the supreme science of theology that unifies all others—is possible because "demonstration must be based on premisses prior to and better known than the conclusion," as Aristotle said above.

I have claimed that science, a handmaiden of theology, is grounded not only by reason but also by faith—i.e., faith that knowledge of God from the created world (Romans 1:20) is possible through the senses because God's grace builds on our human nature. "[F]aith is the substance of things to be hoped for, the evidence of things that appear not" (Hebrews 11:1); it is an intellectual assent; it "is a kind of knowledge, inasmuch as the intellect is determined by faith to some knowable object" (Summa Theologica Iª q. 12 a. 13 ad 3). I do not mean that with sola fide (faith alone) one obtains intellectual knowledge as a fideist, one who agnostically "affirms that the fundamental act of human knowledge consists in an act of faith" and "denies intellectual knowledge," maintains (Sauvage, G.); no, intellectual knowledge comes from the senses (Nihil est in intellectu quod non prius in sensu.). St. Thomas the Apostle, through his desire to know, had faith that he could, and Christ, knowing that humans obtain intellectual knowledge via the senses, told him physically to touch Him, and he believed. For the same reason, Christ instituted the sacraments, physical signs of an invisible reality.

If, however, the lower sciences are indeed handmaidens of theology (Summa Theologica Iª q. 1 a. 5 s.c.), unified in this study of God, the supreme science, and if the "articles of faith" afford knowledge of God and His revealed truths for theology to adopt as its principles (Summa Theologica Iª q. 1 a. 7 c.), then why does it seem the truths of theology do not help the lower sciences like physics, chemistry, and biology, or vice versa? If they would—were modern science, far from being true science, not generally atheistic—then the ultimate source of all the sciences' principles would be the "deposit of faith," Revelation. But how? How do these truths from Revelation come to us as a result of having faith? They still come through the senses, except in very rare visions "accompanied by abstraction from the senses" (Summa Theologica IIª-IIae q. 173 a. 3).

On the necessity of believing things which can be proven by natural reason, St. Thomas says
It is necessary for man to accept by faith not only things which are above reason, but also those which can be known by reason: and this for three motives. First, in order that man may arrive more quickly at the knowledge of Divine truth. Because the science to whose province it belongs to prove the existence of God, is the last of all to offer itself to human research, since it presupposes many other sciences: so that it would not be until late in life that man would arrive at the knowledge of God. The second reason is, in order that the knowledge of God may be more general. For many are unable to make progress in the study of science, either through dullness of mind, or through having a number of occupations, and temporal needs, or even through laziness in learning, all of whom would be altogether deprived of the knowledge of God, unless Divine things were brought to their knowledge under the guise of faith. The third reason is for the sake of certitude. For human reason is very deficient in things concerning God. A sign of this is that philosophers in their researches, by natural investigation, into human affairs, have fallen into many errors, and have disagreed among themselves. And consequently, in order that men might have knowledge of God, free of doubt and uncertainty, it was necessary for Divine matters to be delivered to them by way of faith, being told to them, as it were, by God Himself Who cannot lie.

Summa Theologica IIª-IIae q. 2 a. 4 co.

The Italian writer and apologist Alessandro Manzoni (1785-1873) said this about faith, which really reminds me of what is happening in science where it meets with Catholic morality; true science always conforms to true faith:
Have you examined all these objections [against Revelation]? Objections of fact, of chronology, of history, of natural history, of morals etc. Have you discussed all the arguments of the adversaries, have you recognized their falsity, unfoundedness?... this is not enough to have faith in Scripture. It is possible, it is unfortunately possible that in the generations to come... there will be some men who will study new arguments against the truths of the Scriptures; they will rummage through history, ... they will pretend to have discovered truth of fact for which the things affirmed in the Scriptures have to appear false. Now you must swear that these arguments that are not yet found, will be false, that these books that are not yet written, will be full of error: do you swear it? If you deny it, you admit to not having faith.

—Alessandro Manzoni's Morale cattolica, vol. II, pp. 544-545 [my translation]

If one understands this, then one can understand why Pope Paul VI—the pope of the controversial encyclical Humanæ Vitæ which correctly condemned as immoral any form of contraception—would advocate St. Thomas Aquinas's "perennial philosophy" for anyone seeking truth, including, e.g., modern physicists:
[...] to be a faithful disciple of St. Thomas today, it is not enough to want to do in our time and with the means available today that which he did in his. Contenting oneself with imitating him, like walking on a parallel street without anything to draw from him, one would with difficulty arrive at a positive result or, at least, offer to the Church and to the world that contribution of wisdom which they need. One cannot, in fact, speak of true and fecund loyalty if one does not receive, almost from his own hands, his principles which also illuminate the most important problems of philosophy and, to be more precise, to understand better the faith in these our times and, similarly, the fundamental notions of his system and the force of his ideas. Only so, the thought of the Angelic Doctor, confronted always with new contributions of profane science, will meet—through a sort of mutual osmosis—a new, thriving, lively development.

—Pope Paul VI's 1974 letter Lumen ecclesiæ 29. [my translation]

Hence, in light of all we have said, one must indeed have faith in order to come to a fuller understanding.