Wednesday, December 15, 2010

Mathematics: Its Importance and Limitations

Mathematics is most co-natural to man and it is also the most exact science according to St. Thomas Aquinas, says Bernard I. Mullahy in his excellent thesis Thomism and Mathematical Physics. Armand Maurer on pg. xxxv of his translation of St. Thomas Aquinas's Division and methods of the sciences, a commentary on Boethius's De Trinitate questions V and VI, says that
St. Thomas also attributes a major role to reasoning in mathematics. In this respect it is like natural philosophy. The difference in their methods lies in the causes employed in reasoning. Mathematical demonstrations begin with definitions and principles, from which conclusions are deduced by way of formal causality. For example, a certain property of the triangle is shown to follow from its very definition. Unlike natural philosophy, mathematics does not demonstrate through final or efficient causes.
Fr. McCool, S.J., wrote in his book on the history of 20th century Thomism that (pg. 154-155):
A new edition of St. Thomas' In Librum Boethii de Trinitate, Questiones quinta et sexta, published in 1948, led to a major revision of the accepted understanding of St. Thomas' theory of abstraction [which that of Boethius inspired]. Apart from the abstractio totius, abstraction of a sensible whole from its particulars required for any form of conceptual thought, the only other type of abstraction proposed by St. Thomas confines itself to the level of mathematics. This is an abstractio formae, the mind's separation of the form of quantity from the rest of the sensible whole which the mind disregards in mathematics.
Thus we can see that medieval philosophers recognized the great importance and certainty of mathematics. Yet, according to Armand Maurer,
St. Thomas saw that there is a particular temptation to single out the mathematical method for this role [of being a common method for all the sciences, as Descartes wished], since it is the most exact and certain. But he warns against this, insisting on the specificity of method in each of the sciences [natural philosophy (physics in the broad sense), mathematics, and metaphysics; modern physics is a natural philosophy / mathematics hybrid, a scientia media or "middle science"].
St. Thomas warns about this in In II Meta. lect. 5, n. 335-337, which says, commenting on Aristotle's answer regarding "The Method to Be Followed in the Search for Truth:"

ARISTOTLE’S TEXT Chapter 3: 994b 32-995a 20

171. The way in which people are affected by what they hear depends upon the things to which they are accustomed; for it is in terms of such things that we judge statements to be true, and anything over and above these does not seem similar but less intelligible and more remote. For it is the things to which we are accustomed that are better known.

172. The great force which custom has is shown by the laws, in which legendary and childish elements prevail over our knowledge of them, because of custom.

173. Now some men will not accept what a speaker says unless he speaks in mathematical terms; and others, unless he gives examples; while others expect him to quote a poet as an authority. Again, some want everything stated with certitude, while others find certitude annoying, either because they are incapable of comprehending anything, or because they consider exact inquiry to be quibbling; for there is some similarity. Hence it seems to some men that, just as liberality is lacking in the matter of a fee for a banquet, so also is it lacking in arguments.

174. For this reason one must be trained how to meet every kind of argument; and it is absurd to search simultaneously for knowledge and for the method of acquiring it; for neither of these is easily attained.

175. But the exactness of mathematics is not to be expected in all cases, but only in those which have no matter. This is why its method is not that of natural philosophy; for perhaps the whole of nature contains matter. Hence we must first investigate what nature is; for in this way it will become evident what the things are with which natural philosophy deals, and whether it belongs to one science or to several to consider the causes and principles of things.


[St. Thomas's] COMMENTARY

335. For this reason one must be trained (174).

He exposes the proper method of investigating the truth. Concerning this he does two things. First (335), he shows how a man can discover the proper method of investigating the truth. Second (336), he explains that the method which is absolutely the best should not be demanded in all matters (“But the exactness of mathematics”) .

He says, first, that since different men use different methods in the search for truth, one must be trained in the method which the particular sciences must use to investigate their subject. And since it is not easy for a man to undertake two things at once (indeed, so long as he tries to do both he can succeed in neither), it is absurd for a man to try to acquire a science and at the same time to acquire the method proper to that science. This is why a man should learn logic before any of the other sciences, because logic considers the general method of procedure in all the other sciences. Moreover, the method appropriate to the particular sciences should be considered at the beginning of these sciences.

336. But the exactness of mathematics (175).

He shows that the method which is absolutely the best should not be demanded in all the sciences. He says that the “exactness,” i.e., the careful and certain demonstrations, found in mathematics should not be demanded in the case of all things of which we have science, but only in the case of those things which have no matter; for things that have matter are subject to motion and change, and therefore in their case complete certitude cannot be had. For in the case of these things we do not look for what exists always and of necessity, but only for what exists in the majority of cases.

Now immaterial things are most certain by their very nature because they are unchangeable, although they are not certain to us because our intellectual power is weak, as was stated above (279). The separate substances are things of this kind. But while the things with which mathematics deals are abstracted from matter, they do not surpass our understanding; and therefore in their case most certain reasoning is demanded.

Again, because the whole of nature involves matter, this method of most certain reasoning does not belong to natural philosophy. However, he says “perhaps” because of the celestial bodies, since they do not have matter in the same sense that lower bodies do.

337. Now since this method of most certain reasoning is not the method proper to natural science, therefore in order to know which method is proper to that science we must investigate first what nature is; for in this way we will discover the things which natural philosophy studies. Further, we must investigate “whether it belongs to one science,” i.e., to natural philosophy, or to several sciences, to consider all causes and principles; for in this way we will be able to learn which method of demonstration is proper to natural philosophy. He deals with this method in Book II of the Physics, as is obvious to anyone who examines it carefully.

Wednesday, December 1, 2010

Heisenberg: "concept of the soul [...] more natural and less forced"

Heisenberg said in his Physics and Philosophy that the probability wave concept in quantum mechanics "was a quantitative version of the concept of 'potentia' in Aristotelian philosophy" (p. 41) and that the "concept of the soul for instance in the philosophy of Thomas Aquinas was more natural and less forced than the Cartesian concept of 'res cogitans,' even if we are convinced that the laws of physics and chemistry are strictly valid in living organisms." (p. 80).

Friday, November 19, 2010

Crisis of Faith in Science

The Crisis of the Faith in Science

The resistance of creation to its manipulation by men has become a new factor in the intellectual situation in the last decade. It is impossible to evade the question of the limits of science and of the criteria it must follow. The change in the way in which the case of Galileo is evaluated seems to me characteristic of the change of climate. This event, to which little attention was paid in the seventeenth century, was elevated in the following century to nothing less than the my of the Enlightenment: Galileo appears as the victim of the medieval obscurantism in which the Church persists. Good and evil stand in a distinct confrontation: on the one side, we find the Inquisition as the power of superstition, as the opponent of freedom and knowledge; on the other side stand the natural sciences, represented by Galileo, as the power of progress and of the liberation of man from the fetters of ignorance that kept him powerless vis-à-vis nature. The star of the modern period arises over the darkness of the Middle Ages.

Strangely enough, Ernst Bloch with his romantic Marxism was one of the first to oppose this myth openly and to offer a new interpretation of the events. For him, the heliocentric world-system, just like the geocentric system, rests on unprovable presuppositions, including above all the supposition of motionless space, which has since been shattered by the theory of relativity. He states:

Consequently, since an empty motionless space no longer exists, no movement toward it occurs, but merely a relative movement of bodies toward one another, the determination of which depends on the choice made of the body that is to be taken to be at rest. Thus, if it were not for the fact that the complexity of the calculations involved makes this appear infeasible, the earth could continue to be taken as stable and the sun as moving. [E. Bloch, Das Prinzip Hoffnung (frankfurt am Main, 1959), 920]
According to this view, the advantage of the heliocentric system over the geocentric does not consist in a greater degree of objective truth but merely in an easier calculability for us. Up to this point Bloch is doing no more than expressing the insight of the modern natural sciences; but the conclusion he derives from this now is astonishing:
Since the relativity of the motion is beyond doubt, an older man-centered Christian reference system does not indeed have the right to involve itself in the astronomical calculations and their heliocentric simplification; but it does have its own methodological right to hold fast to the earth as far as the question of the importance of mass is concerned and to impart an ordered structure to the world around what happens and has happened on the earth. [Bloch, 920f.]

The two methodological spheres are clearly distinguished from one another here, and the rights, as well as the limitations, of each are acknowledged. But the summary of the skeptical agnostic philosopher P. Feyerabend sounds much more aggressive when he writes:

The Church at the time of Galileo kept much more closely to reason than did Galileo himself, and she took into consideration the ethical and social consequences of Galileo's teaching too. Her verdict against Galileo was rational and just, and the revision of his verdict can be justified only on grounds of what is politically opportune. [P. Feyerabend, Wider den Methodenzwang (Against Method) (Frankfurt am Main, 1976, 1983), 206.]

C. F. von Weizäcker (to take one example) goes even one step farther in considering the prictical effects when he sees a "perfectly straight path" leading from Galileo to the the atomic bomb. To my surprise, when I was interviewed recently about the case of Galileo, I was not asked (for instance) why the church had presumed to hinder the knowledge of the natural sciences but, quite to the contrary, why the Church had not taken up a clearer position against the disasters that were bound to result when Galileo opened Pandora's box. It would be foolish to construct an impulsive apologetic on the basis of such views; faith does not grow out of resentment and skepticism with respect to rationality, but only out of a fundamental affirmation and a spacious reasonableness; we shall come back to this point. I mention all this only as a symptomatic case that permits us to see how deep the self-doubt of the modern age, of science and of technology goes today.

—Then-Cardinal Ratzinger, now Pope Benedict XVI, A Turning Point for Europe, pg. 95-98 (Cf. "Ratzinger's 1990 remarks on Galileo")

Monday, November 15, 2010

Albertus Magnus' Feast Day

Deus, qui beatum Albertum Pontificem tuum atque Doctorem in humana sapientia divinae fidei subicienda magnum effecisti: da nobis, quaesumus; ita eius magisterii inhaerere vestigiis ut luce perfecta fruamur in caelis.

O God, who didst make blessed Albert, thy bishop and doctor, great in subjecting human wisdom to divine faith, grant we pray, that we may so adhere to the footprints of his authorative teaching that in heaven we may enjoy perfect life.

—today's collect for St. Albertus Magnus

Vide also James A. Weisheipl, O.P., "Albert the Great and Medieval Culture," The Thomist 44 (1980) 500, cited here.

Wednesday, October 27, 2010

St. Albertus Magnus's Commentary on Euclid's Elements

St. Albertus Magnus (1206-1280) wrote extensively on physics methodology, among other fields like theology and the experimental sciences, and is credited for discovering arsenic. He said: "The aim of natural science is not simply to accept the statements [narrata] of others, but to investigate the causes that are at work in nature" (De Miner., lib. II, tr. ii, i) and "Experimentum solum certificat in talibus" ("Experiment is the only safe guide in such investigations;" De Veg., VI, tr. ii, i). Against what today one would call "intelligent design," St. Albert says: "In studying nature we have not to inquire how God the Creator may, as He freely wills, use His creatures to work miracles and thereby show forth His power: we have rather to inquire what Nature with its immanent causes can naturally bring to pass" (De Coelo et Mundo, I, tr. iv, x).
     The Dominican Scholastic philosopher Albertus Magnus (1193?-1280) acquired the title Doctor Universalis from the breadth of his learning, which astonished his contemporaries. His fame earned him a place in the Divine Comedy, where his most famous student, Thomas Aquinas, introduces him to Dante:
Questi che m'è a destra più vicino
Frate e maëstro fummi, ed esso Alberto
È di Cologna, e io Thomas d'Aquino. (Paradiso, X, 97-99)

He who is nearest to me on the right
Was my colleague and teacher, namely, Albert
Of Cologne, and I am Thomas Aquinas.
The program of the scholastic philosophers was to use the deductive method of mathematics to demonstrate by reason the existence of Deity and to describe His attributes, to prove the immortality of the soul, to assert free will, and in general to establish thereby the truth of the Catholic religion. Their first axiom was, that this was possible. Even Russell, who considered theology nothing more than organized ignorance, could nevertheless respect the Medieval mastery of logic:
     The medieval outlook of educated men had a logical unity which has now been lost. We may take Thomas Aquinas as the authoritative exponent of the creed which science was compelled to attack. He maintained—and his view is still that of the Roman Catholic Church—that some of the fundamental truths of the Christian religion could be proved by the unaided reason, without the help of revelation. Among these was the existence of an omnipotent and benevolent Creator. From His omnipotence and benevolence followed that He would not leave His creatures without knowledge of His decrees, to the extent that might be necessary to obey His will. There must therefore be a Divine revelation, which, obviously, is contained in the Bible and the decisions of the Church. This point being established, the rest of what we need to know can be inferred from the Scriptures and the pronouncements of oecumenical Councils. The whole argument proceeds deductively from premisses formerly accepted by almost the whole population of Christian countries, and if the argument is, to the modern reader, at times faulty, its fallacies were not apparent to the majority of learned contemporaries.
     Now logical unity is at once a strength and a weakness. It is a strength because it insures that whoever accepts one stage of the argument must accept all later stages; it is a weakness because whoever rejects any of the later stages must also reject some, at least, of the earlier stages. The Church, in its conflict with science, exhibited both the strength and the weakness resulting from the logical coherence of its dogmas [Religion and Science, 12-13].
Within this program, it was the special enterprise of Albertus Magnus to interpret the works of Aristotle for his Medieval contemporaries, insofar as the writings of that philosopher could be known through the veil of Latin translations of Arabic translations of the Greek texts, which at that time covered the hearts of the learned. In so doing, he made a major contribution to culture, for Aristotle was interested in the natural world, and investigation into his activities tended to promote the rebirth of knowledge, which was accomplished in the period called the Renaissance. The labors of pople like Albertus Magnus were appreciated by Macaulay, who noticed them in one of the most famous chapters in literature:
     Whatever reproach may, at a later period, have been justly thrown on the indolence and lukury of religios orders, it was surely good that, in an age of ignorance and violence, there should be quiet cloisters and gardens, in which the arts of peace could be safely cultivated, in which gentle and contemplative natures could find an asylum, in which one brother could employ himself in transcribing the Aeneid of Virgil, and another in meditating the Analytics of Aristotle, in which he who had a genius for art might illuminate a martyrology or carve a crucifix, and in which he who had a turn for natural philosophy might make experiments on the properties of plants and minerals [The History of English from the Accession of James II, I, 9]
     Albertus Magnus is one of those personalities who are appreciated even by those whith a critical attitude towarsd the Catholic religion. For example, White, cofounder and first president of Cornell University, a declared enemy of dogmatic theology, had a sympathetic opinion of him:
First among these was Albert of Bollstädt, better known as Albert the Great, the most renowned scholar of his time. Fettered though he was by the methods sanctioned in the Church, dark as was all about him, he had conceived better methods and aims; his eye pierced the mists of scholasticism; he saw the light, and sought to draw the world toward it. He stands among the great pioneers of physical and natural science; he aided in giving foundations to botany and chemistry, he rose above his time, and struck a heavy blow on those who opposed the possibility of human life on the opposite sides of the earth; he noted the influence of mountains, seas, and forests upon races and products, so that Humboldt justly finds in his works the germs of physical geography as a comprehensive science [A History of the Warfare of Science with Theology in Christendom, I, 377].
The most recent biography of Albert, written on the occasion of the seven hundredth anniversary of his death, is by Weisheipl ["The Life and Works of St. Albert the Great". Albertus Magnus and the Sciences — Commemorative Essays].
     Albertus Magnus conceived and carried out the plan of commenting on all human knowledge by beginning with the natural sciences, proceeding to mathematics, and finishing in philosophy and theology. Thus, in his writings in the natural sciences, he refers to mathematics as the object of future activity:
     Primo complebimus, Deo adiuvante, scientiam naturalem, et deinde loquemur de mathematicis omnibus, et intentionem finiemus in scientia divina (Physica, Book I, Treatise 1, Chapter i).
     With God's help, we shall first complete natural science, and then we shall talk about all of mathematics, and we shall finish our program in divine science.

     Longum esset demonstrare, sed in geometria hoc docebitur et in astronomia, Domino concedente (Op. cit., I, 2, i).
     It would be long to prove, but, God willing, this will be taught in geometry and astronomy.

     Haec autem omnia supponenda sunt, probanda autem in libris de visu in Perspectivis, quae scientia compleri non potest, nisi primum consideremus ea quae pertinent ad geometriam (De Sensu et Sensato, Treatise 1, Chapter 14).
     All these things, however, must be supposed for now; they are to be proven, though, in the books on sight in Perspective, which science cannot be completed unless we shall first consider those things that pertain to geometry.
In his compositions on philosophy and theology, however, Albert refers to his work on geometry as at hing of the past, for he accepted the idea of Plato, that one could not competently study philosophy without having first mastered mathematics.
     Naturalibus et doctrinalibus iam, quantum licuit, scientiis elucidatis, iam ad veram philosophiae sapientiam accedamus (Metaphysica, Book I, Treatise I, Chapter i).
     Now that the natural and mathematical sciences have been elucidated as much as was possible, let us proceed to the true wisdom of philosophy.

     Hoc autem iam a nobis in geometricis est demonstratum (Ibid., I, 2, x).
     For this [sc. that the diameter and side of a square are incommensurable] has already been proven by us in the geometrical [works].

     Sicut in XV and XVI tertii geometriae nostrae demonstratum est (Ibid., III, 2, iii).
     Just as has been proven in the fifteenth and sixteenth [propositions] of the third [book] of our Geometry [sc. namely, that a tangent line to a circle intersects it at only one point].

     Sicut nos in I nostrae geometriae ostendimus (Ibid., V, 3, i).
     As we showed in the first [book] of our Geometry [sc., that two straight lines do not enclose a surface].

Albertus Magnus on Euclid's Elements of Geometry, pgs. xi-xv

Here is St. Albertus Magnus's commentary on Euclid's famous proof of the Pythagorean Theorem, Proposition 46 here:

Saturday, September 11, 2010

Galileo Truly Recanted.

Was Galileo really a martyr of modern science, the theories and explanations of which are in a constant state of flux, or did he ultimately seek an absolute, unchanging, objective Truth and recant of holding a changeable scientific theory to be objectively true? Galileo wrote to Francesco Rinuccini, Arcetri, 29 March 1641, the year before his death:
The falsity of the Copernican system needs not be called into doubt, and especially by us Catholics, having the irrefragable authority of Sacred Scripture, interpreted by the supreme masters in Theology, whose concordant consensus renders us certain of the stability of the Earth placed in the center, and of the mobility of the Sun around it. The conjectures then for which Copernicus and his other followers have professed the contrary, are all lifted with that most solid argument of the Omnipotence of God, Who can do in diverse—rather, in infinite ways—that to our opinion and observation seem done in one particular way; we should not want to shorten the hand of God and tenaciously sustain that in which we can be deceived.

Le opere di Galileo Galilei, vol. 7 edited by Vincenzio Viviani [my translation]

Galileo, were he alive in the 19th and 20th centuries, respectively, would agree with these statements:
[...] if writers on physics travel outside the boundaries of their own branch, and carry their erroneous teaching into the domain of philosophy, let them be handed over to philosophers for refutation.

—Pope Leo XIII's Providentissimus Deus

Human science gains greatly from revelation, for the latter opens out new horizons and makes known sooner other truths of the natural order, and because it opens the true road to investigation and keeps it safe from errors of application and of method. Thus does the lighthouse show many things they otherwise would not see, while it points out the rocks on which the vessel would suffer shipwreck.

—Pope St. Pius X's Iucunda Sane

Sunday, August 15, 2010

Consecration of the Chapel: Our Lady of Guadalupe Seminary

Assumpta est Sedes Sapientiæ in cælum; gaudet exercitus Angelorum.

Friday, August 13, 2010

Interface between Theology and the Other Sciences

Related to this is again the question of why intellectual knowledge must come through the senses (Nothing is in the intellect that is not first in the senses: Nihil est in intellectu quod non prius in sensu.). The faithful hold by faith that:
For the invisible things of him, from the creation of the world, are clearly seen, being understood by the things that are made; his eternal power also, and divinity.

Romans 1:20

As well as these:
If anyone shall have said that the one true God, our Creator and our Lord, cannot be known with certitude by those things which have been made, by the natural light of human reason: let him be anathema

Denzinger 1806

The same Holy Mother Church holds and teaches that God, the beginning and end of all things, can be known with certitude by the natural light of human reason from created things; "for the invisible things of him, from the creation of the world, are clearly seen, being understood by the things that are made" [Rom 1:20]; nevertheless, it has pleased His wisdom and goodness to reveal Himself and the eternal decrees of His will to the human race in another and supernatural way, as the Apostle says: "God, who at sundry times and in divers manners, spoke in times past to the fathers by the prophets, last of all, in these days hath spoken to us by His Son" [Heb. 1:1].

Denzinger 1785

Most relevant is this:
All knowledge of God, even natural knowledge, even in the pagan philosophers, cannot come except from God; and without grace knowledge produces nothing but presumption, vanity, and opposition to God Himself, instead of the affections of adoration, gratitude, and love.

Denzinger 1391

How does the study of God's creatures, by doing physics in the broad sense of studying the natural world, lead to knowledge of Him? St. Thomas elucidates this:

Chapter 1


“I meditated upon Your works: I meditated upon the works of Your hands” (Ps. 142-5).

[1] Of no thing whatever can a perfect knowledge be obtained unless its operation is known, because the measure and quality of a thing’s power is judged from the manner and type of its operation, and its power, in turn, manifests its nature; for a thing’s natural aptitude for operation follows upon its actual possession of a certain kind of nature.

[2] There are, however, two sorts of operation, as Aristotle teaches in Metaphysics IX [8]: one that remains in the agent and is a perfection of it, as the act of sensing, understanding, and willing; another that passes over into an external thing, and is a perfection of the thing made as a result of that operation, the acts of heating, cutting and building, for example.

[3] Now, both kinds of operation belong to God: the former, in that He understands, wills, rejoices, and loves; the latter, in that He brings things into being, preserves them, and governs them. But, since the former operation is a perfection of the operator, the latter a perfection of the thing made, and since the agent is naturally prior to the thing made and is the cause of it, it follows that the first of these types of operation is the ground of the second, and naturally precedes it, as a cause precedes its effect. Clear evidence of this fact, indeed, is found in human affairs; for in the thought and will of the craftsman lie the principle and plan of the work of building.

[4] Therefore, as a simple perfection of the operator, the first type of operation claims for itself the name of operation, or, again, of action; the second, as being a perfection of the thing made, is called making so that the things which a craftsman produces by action of this kind are said to be his handiwork.

[5] Of the first type of operation in God we have already spoken in the preceding Book of this work, where we treated of the divine knowledge and will. Hence, for a complete study of the divine truth, the second operation, whereby things are made and governed by God, remains to be dealt with.

[6], In fact, this order we can gather from the words quoted above. For the Psalmist first speaks of meditation upon the first type of operation, when he says: “I have meditated on all your operations”; thus, operation is here referred to the divine act of understanding and will. Then he refers to meditation on God’s works: “and I meditated on the works of Your hands”; so that by “the works of Your hands” we understand heaven and earth, and all that is brought into being by God, as the handiwork produced by a craftsman.

Chapter 2


[1] This sort of meditation on the divine works is indeed necessary for instruction of faith in God.

[2] First, because meditation on His works enables us in game measure to admire and reflect upon His wisdom. For things made by art are representative of the art itself, being made in likeness to the art. Now, God brought things into being by His wisdom; wherefore the Psalm (103:24) declares: “You made all things in wisdom.” Hence, from reflection upon God’s works we are able to infer His wisdom, since, by a certain communication of His likeness, it is spread abroad in the things He has made. For it is written: “He poured her out,” namely, wisdom, “upon all His works” (Eccli. 1:10). Therefore, the Psalmist, after saying: “Your knowledge is become wonderful to me: it is high, and I cannot reach it,” and after referring to the aid of the divine illumination, when he says: “Night shall be my light,” etc., confesses that he was aided in knowing the divine wisdom by reflection upon God’s works, saying: “Wonderful are Your works, and my soul knows right well” (Ps. 138:6, 11, 14).

[3] Secondly, this consideration [of God’s works] leads to admiration of God’s sublime power, and consequently inspires in men’s hearts reverence for God. For the power of the worker is necessarily understood to transcend the things made. And so it is said: “If they,” namely, the philosophers, “admired their power and effects,” namely of the heavens, stars, and elements of the world, “let them understand that He that made them is mightier than they” (Wis. 13:4). Also it is written: “The invisible things of God are clearly seen, being understood by the things that are made: His eternal power also and divinity” (Rom. 1:20). Now, the fear and reverence of God result from this admiration. Hence, it is said: “Great is Your name in might. Who shall not fear You, O King of Nations?” (Jer. l0:6-7).

[4] Thirdly, this consideration incites the souls of men to the love of God’s goodness. For whatever goodness and perfection is distributed to the various creatures, in partial or particular measure, is united together in Him universally, as in the source of all goodness, as we proved in Book I. If, therefore, the goodness, beauty, and delightfulness of creatures are so alluring to the minds of men, the fountainhead of God’s own goodness, compared with the rivulets of goodness found in creatures, will draw the enkindled minds of men wholly to Itself. Hence it is said in the Psalm (91:5): “You have given me, O Lord, a delight in Your doings, and in the works of Your hands I shall rejoice.” And elsewhere it is written concerning the children of men: “They shall be inebriated with the plenty of Your house,” that is, of all creatures, “and You shall make them drink of the torrent of Your pleasure: for with You is the fountain of life” (Ps. 35:9-10). And, against certain men, it is said: “By these good things that are seen,” namely, creatures, which are good by a kind of participation, “they could not understand Him that is” (Wis. 13:1), namely, truly good; indeed, is goodness itself, as was shown in Book I.

[5] Fourthly, this consideration endows men with a certain likeness to God’s perfection. For it was shown in Book I that, by knowing Himself, God beholds all other things in Himself. Since, then, the Christian faith teaches man principally about God, and makes him know creatures by the light of divine revelation, there arises in man a certain likeness of God’s wisdom. So it is said: “But we all beholding the glory of the Lord with open face, are transformed into the same image” (2 Cor. 3:18).

[6] It is therefore evident that the consideration of creatures has its part to play in building the Christian faith. And for this reason it is said: “I will remember the works of the Lord, and I will declare the things I have seen: by the words of the Lord are His works” (Sirach 42:15).

Chapter 3


[1] The consideration of creatures is further necessary, not only for the building up of truth, but also for the destruction of errors. For errors about creatures sometimes lead one astray from the truth of faith, so far as the errors are inconsistent with true knowledge of God. Now, this happens in many ways.

[2] First, because through ignorance of the nature of creatures men are sometimes so far perverted as to set up as. the first cause and as God that which can only receive its being from something else; for they think that nothing exists beyond the realm of visible creatures. Such were those who identified God with this, that, and the other kind of body; and of these it is said: “Who have imagined either the fire, or the wind, or the swift air, or the circle of the stars, or the great water, or the sun and moon to be the gods” (Wis. 13: 2).

[3] Secondly, because they attribute to certain creatures that which belongs only to God. This also results from error concerning creatures. For what is incompatible with a thing’s nature is not ascribed to it except through ignorance of its nature—as if man were said to have three feet. Now, what belongs solely to God is incompatible with the nature of a created thing, just as that which is exclusively man’s is incompatible with another thing’s nature. Thus, it is from ignorance of the creature’s nature that the aforesaid error arises. And against this error it is said: “They gave the incommunicable name to stones and wood” (Wis. 14:21). Into this error fell those who attribute the creation of things, or knowledge of the future, or the working of miracles to causes other than God.

[4] Thirdly, because through ignorance of the creature’s nature something is subtracted from God’s power in its working upon creatures. This is evidenced in the case of those who set up two principles of reality; in those who assert that things proceed from God, not by the divine will, but by natural necessity; and again, in those who withdraw either all or some things from the divine providence, or who deny that it can work outside the ordinary course of things. For all these notions are derogatory to God’s power. Against such persons it is said: “Who looked upon the Almighty as if He could do nothing” (Job 22:17), and: “You show Your power, when men will not believe You to be absolute in power” (Wis. .12: 17).

[5] Fourthly, through ignorance of the nature of things, and, consequently, of his own place in the order of the universe, this rational creature, man, who by faith is led to God as his last end, believes that he is subject to other creatures to which he is in fact superior. Such is evidently the case with those who subject human wills to the stars, and against these it is said: “Be not afraid of the signs of heaven, which the heathens fear” (Jer. 10:2); and this is likewise true of those who think that angels are the creators of souls, that human souls are mortal, and, generally, of persons who hold any similar views derogatory to the dignity of man.

[6] It is, therefore, evident that the opinion is false of those who asserted that it made no difference to the truth of the faith what anyone holds about creatures, so long as one thinks rightly about God, as Augustine tells us in his book On the Origin of the Soul [De anima et ejus origine, IV, 4]. For error concerning creatures, by subjecting them to causes other than God, spills over into false opinion about God, and takes men’s minds away from Him, to whom faith seeks to lead them.

[7] For this reason Scripture threatens punishment to those who eff about creatures, as to unbelievers, in the words of the Psalm (27:5): “Because they have not understood the works of the Lord and the operations of His hands, You shall destroy them, and shall not build them up”; and: “These things they thought and were deceived,” and further on: “They did not esteem the honor of holy Souls” (Wis. 7:2122).

Chapter 4


[1] Now, from what has been said it is evident that the teaching of the Christian faith deals with creatures so far as they reflect a certain likeness of God, and so far as error concerning them leads to error about God. And so they are viewed in a different light by that doctrine and by human philosophy. For human philosophy considers them as they are, so that the different parts of philosophy are found to correspond to the different genera of things. The Christian faith, however, does not consider them as such; thus, it regards fire not as fire, but as representing the sublimity of God, and as being directed to Him in any way at all. For as it is said: “Full of the glory of the Lord is His work. Did the Lord not make the saints declare all His wonderful works?” (Sirach 42: 16-17)

[2] For this reason, also, the philosopher and the believer consider different matters about creatures. The philosopher considers such things as belong to them by nature-the upward tendency of fire, for example; the believer, only such things as belong to them according as they are related to God—the fact, for instance, that they are created by God, are subject to Him, and so on.

[3] Hence, imperfection is not to be imputed to the teaching of the faith if it omits many properties of things, such as the figure of the heaven and the quality of its motion. For neither does the natural philosopher consider the same characters of a line as the geometrician, but only those that accrue to it as terminus of a natural body.

[4] But any things concerning creatures that are considered in common by the philosopher and the believer are conveyed through different principles in each case. For the philosopher takes his argument from the proper causes of things; the believer, from the first cause—for such reasons as that a thing has been handed down in this manner by God, or that this conduces to God’s glory, or that God’s power is infinite. Hence, also, [the doctrine of the faith] ought to be called the highest wisdom, since it treats of the highest Cause; as we read in Deuteronomy (4:6): “For this is your wisdom and understanding in the sight of nations.” And, therefore, human philosophy serves her as the first wisdom. Accordingly, divine wisdom sometimes argues from principles of human philosophy. For among philosophers, too, the first philosophy utilizes the teachings of all the sciences in order to realize its objectives.

[5] Hence again, the two kinds of teaching do not follow the same order. For in the teaching of philosophy, which considers creatures in themselves and leads us from them to the knowledge of God, the first consideration is about creatures; the last, of God. But in the teaching of faith, which considers creatures only in their relation to God, the consideration of God comes first, that of creatures afterwards. And thus the doctrine of faith is more perfect, as being more like the knowledge possessed by God, who, in knowing Himself, immediately knows other things.

[6] And so, following this order, after what has been said in Book I about God in Himself, it remains for us to treat of the things which derive from Him.

Contra Gentiles, lib. 2 cap. 1-4

Tuesday, July 27, 2010

Friday, July 16, 2010

Galileo's Giant: Nicole Oresme

Nicole Oresme (c. 1320 - 1382) argued, a couple hundred years before Galileo, that a rotating earth is a simpler explanation than that of Ptolemy. He invented the coordinate system long before Descartes (1596-1650). He also investigated fractional powers and determined that the distance a freely falling body travels (x) is proportional to the square of the time (t) it has been traveling, viz., x = ½ g t², where g is the acceleration due to gravity. He used a slightly different notation, however, beginning in his first page of Algorismus proportionum:

From the first page of Oresme's Algorismus proportionum (fourteenth century)

The original Latin
The original Latin

An English translation of the Latin
An English translation of the Latin

—Cajori's A History of Mathematical Notation, Vol. 1 pgs. 91-93

From the excellent introductory physics textbook Physics for Realists (available here) by the M.I.T. and Princeton physicist Dr. Anthony Rizzi, founder of the Institute for Advanced Physics:

Nicole Oresme (o'rem) was born c. 1320 AD in Normandy, France, and died in 1382. He was, among other things, a mathematician, a physicist and a priest (made bishop of liseaux, France, in 1377). His major mathematical work is "Tractatus de Difformitatum." Among his accomplishments are the use of rectangular coordinates and graphing of the intensity of a quality then called latitudo (say, temperature of a rod), against a length then called longitudo (e.g., distance along the same rod), on such a rectangular coordinate system. He also discovered that speed versus time graphs can be constructed. Further, he investigated, in his own notation, fractional powers, saying 43/2 = 8.

In dynamics, he shows, following Jean Buridan [...], that the movement of the Earth is consistent with immediate experience, though it does not seem so at first. And, he points out that the movement of the Earth, not the geocentric hyothesis, is, indeed, the simpler one. These arguments we be employed by his successors, including Copernicus and Galileo.

He also made an argument for an international dateline. His argument fundamentally rests on noticing that only the relative motion of the earth and sun defines a conventional day (i.e., based on the rising and setting of the sun for a given observer). The principle can be understood by noting that if one moved fast enough to say "under" the sun, one would never experience a single conventional day. The roundness of the earth, which is also needed in the argument, was commonly assumed in the Middle Ages.

He proved, using graphical methods, the mean speed theorem [...] which Galileo imported into his work without reference. He determined the distance = ½ acceleration time² law for uniformly accelerated motion that was later applied by Fr. Dominic De Soto (1494-1560 [...]) to free falling objects.

PFR pg. 48

A page from Tractatus de latitudinibus formarum (1505)

—"Nicole Oresme" in the MacTutor History of Mathematics archive: A page from Oresme's Tractatus de latitudinibus formarum (1505 reprint)

Could not have Oresme, being a theologian, progressed science and mathematics more had he not seemingly wasted his time becoming a Master of Theology at the University of Paris? The answer is no. Fr. Benedict Ashley, O.P.—a key contributer and reviewer of Physics for Realists—explains that philosophy, which includes mathematics and science, is a handmaiden of theology:

Benedict Ashley, O.P.

True philosophy, the perennial philosophy of St. Thomas Aquinas, supports true theology, both of which demand a pursuit of truth. Mentioning the horrible state of the philosophy that Pope Pius XII condemned in Humani Generis, "Nouvelle Théologie" (New Theology)—which is opposed to true, scientific theology and definitely merits the stereotypes of theology being an irrational, unscientific discipline, proceeding haphazardly from from shaky first-principles and jumping to non sequitur conclusions, ignoring that parvus error in principio magnus est in fine ("a small error in principle is a big error in conclusion"), and ultimately upholding the relativism of truth, that it is based on the changing state of man—the great 20th century Thomistic philosopher Fr. Reginald Garrigou-Lagrange, O.P., wrote that
no new definition of truth is offered in the new definition of theology: “Theology is no more than a spirituality or religious experience which found its intellectual expression.” And so follow assertions such as: “If theology can help us to understand spirituality, spirituality will, in the best of cases, cause our theological categories to burst, and we shall be obliged to formulate different types of theology…For each great spirituality corresponded to a great theology.” Does this mean that two theologies can be true, even if their main theses are contradictory and opposite? The answer will be no if one keeps to the traditional definition of truth. The answer will be yes if one adopts the new definition of truth, conceived not in relation to being and to immutable laws, but relative to different religious experiences. These definitions seek only to reconcile us to modernism.

It should be remembered that on December 1, 1924, the Holy Office condemned 12 propositions taken from the philosophy of action, among which was number 5, or the new definition of truth: “Truth is not found in any particular act of the intellect wherein conformity with the object would be had, as the Scholastics say, but rather truth is always in a state of becoming, and consists in a progressive alignment of the understanding with life, indeed a certain perpetual process, by which the intellect strives to develop and explain that which experience presents or action requires: by which principle, moreover, as in all progression, nothing is ever determined or fixed.” The last of these condemned propositions is: “Even after Faith has been received, man ought not to rest in the dogmas of religion, and hold fast to them fixedly and immovably, but always solicitous to remain moving ahead toward a deeper truth and even evolving into new notions, and even correcting that which he believes.

Many, who did not heed these warnings, have now reverted to these errors.

—“Where is the New Theology Leading Us?” by Fr. Reginald Garrigou-Lagrange, O.P.

Oresme did not fall into these errors. He studied true theology because he did true science.

Monday, June 28, 2010

Fr. Coyne & Dawkins

Nota bene: The Church does not necessarily endorse all Fr. Coyne's opinions, and he is not necessarily speaking infallibly.
A related speech's transcript.

Saturday, June 26, 2010

Vatican Astronomer on "Intelligent Design Theory"

Fr. Coyne's presentation and question-and-answer session at the American Association for the Advancement of Science (AAAS)'s "Dialogue on Science, Ethics, and Religion" lecture. Can modern science truly do without God, as he says? Are true philosophy and theology necessary for modern science to be true science, i.e., to search for absolute truth?
Part 1 and Part 2
Nota bene: The Church does not necessarily endorse all Fr. Coyne's opinions, and he is not necessarily speaking infallibly.

Current Methodologies of Physics & Astronomy

Regarding the correct division and method of the sciences, this lecture, given at the American Astronomical Society (AAS)'s January 2008 meeting in Austin, Texas, highlights the current state of the methodologies in two branches of physical science (specif. scientia media), astronomy and physics.
[Duhem] seems to regard [the non-falsifiability theses, which "is that 'if the predicted phenomenon is not produced, not only is the questioned proposition put into doubt, but also the whole theoretical scaffolding used by the physicist' (Duhem 185),"] as an obvious corollary of another thesis, which could be called the non-separability thesis, that the physicist can never submit an isolated hypothesis to experimental test: “To seek to separate each of the hypotheses of theoretical physics from the other assumptions upon which this science rests, in order to subject it in isolation to the control of observation, is to pursue a chimera” (Duhem 199-200).

—Roger Ariew, "Pierre Duhem"

Modern scientists do indeed consider issues pertaining to this, else their science's arguments risk being circular.

Monday, June 21, 2010

Creation Mingled with Works of Nature? No.

My responses are red and Dr. Tkacz's are green.

Hello Prof. Tkacz,

Thank you for the excellent article "Thomas Aquinas vs. The Intelligent Designers." [...] I understand the flaws of ID [Intelligent Design] as it is presently conceived by people like Behe, but I am confused when you wrote about the hippo:

Consider another example: a large quadrapedic mammal, such as a hippopotamus, gives live birth to its young. Why? Well, we could answer this by saying that “God does it.” Yet, this could only mean that God created hippopotamuses—indeed the mammalian order, the whole animal kingdom, and all of nature—such that these animals have the morphology, genetic make-up, etc. that are the causes of their giving live birth. [So God is a deist's God who only sets up the natural conditions and leaves them alone?] It cannot be that God “reaches into” the normal operations of hippopotamuses to cause them to give live birth. [Why not? Is He not involved at all past some level?] Were one to think that “God does it” must mean that God intervenes in nature in this way, one would be guilty of the Cosmogonical Fallacy. [Or I just wouldn't be a deist?]

Isn't this an issue of proximate versus remote causes? A proximate cause (what biology would study) of a hippo giving birth is the female hippo, and a remote cause (what theology would study) is God, so both the Thomists and ID folk are right; to me it just seems to be an issue of epistemology. What exactly are you trying to know: how the hippo's existence is due ultimately to God or how it's due to its hippo nature—its "hipponess"?

From St. Thomas's Summa Contra Gentiles:

Accordingly if we be asked the wherefore of a particular natural effect, we can assign the reason to some proximate cause: provided, however, that we refer all things to the divine will as their first cause. Thus if it be asked: Why was the wood heated at the presence of fire? we reply: Because to heat is fire's natural action: and this, because heat is its proper accident: and this results from its proper form: and so on until we come to the divine will. Hence if we reply to the question Why was the wood made hot? by saying: Because God so willed: we shall answer rightly, if we intend to trace the question back to its first cause, but incorrectly if we intend to exclude all other causes.

Summa Contra Gentiles, lib. 3 cap. 97 n. 17

His response:

Thank you for your message. I understand your question about my “hippo” example. I certainly do not intend to claim that St. Thomas is a deist. He most certainly is not—indeed, he would consider deism heretical because it denies God’s omnipresence and immanence to creation. The point of my example is that God’s creation is not a substitute for natural causes. Any natural thing or process has natural causes of the kinds that we discover in our scientific research. At the same time, any natural thing or process requires God’s immediate act of creation to keep it in existence. Both are true at once. This is St. Thomas’ view. So, the hippopotamus is created by God in the sense that it is immediately caused to exist and is kept in existence by God. What God causes here, however, is the animal as caused by its natural causes. So, divine creation is not a substitute for natural causes, nor is it like a natural cause. It is a unique kind of cause in a class of its own.

As you know, deism is the view that creation is some primordial cause. Thomas rejects this view. He holds that creation is not an event that took place at some primordial time. [But some creation could; it's not necessary, though, right? No. This is the point St. Thomas is making: it is impossible that God creates the way human beings and other natural things create. God’s act of creation does not take place in time and it must be immediately present to the thing being created. Thus, it cannot be that God created way back when. God creates here and now at every here and now of the universe, whether that here and now is, from our temporal point of view, a present here and now or a future here and now or a past here and now. Remember: According to St. Thomas, creation is not an event, but a relationship of absolute dependence of creature on creator. I think you misunderstood me; I was implying that creatio ex nihilo is possible at a certain moment in time for some creatures, such as the soul of a newly conceived human, or is that false? I think you would say it just appears that way form our temporally-bound viewpoint? I see. You are right and St. Thomas would agree, but he would point out that the creation of the individual human soul must be true in a way that does not commit the Cosmological Fallacy (the confusion of natural cause with God’s divine agency). Because God is absolutely immanent to all of nature, then he is omnipresent with his power to the conception of the individual human being making it be. Basically, this is no different from God’s being present to any other new natural event. But there is a sense in which the creation of the individual human soul is different from God’s general immanence: the human soul is the image of God in a more perfect way than is any other natural thing or process. God’s creation of the individual rational human soul, then, is God sharing his own divine essence in a particular manner that is different from his creation of other natural things. We often tend to think of this in an event manner, but this is, as you say, just part of our “temporally-bound viewpoint.”] Rather, creation is the radical dependence of everything on God for its existence. So, creation is not an event at all, but a relationship of absolute dependence. As in the text you cite from SCG, St. Thomas often uses the term “first cause” to refer to God’s act of creation. He does not here mean “first” in the sense of first in time, but in the sense of absolutely fundamental. God’s actions do not occur in time, they are eternal or to put it another way from our point of view, God is always making us be, he is always creating us. Were God to crease being our creator, we would pass out of existence. So, there is no deism here.

The problem with ID theorists such as Dr. Behe is that he confuses divine causation with natural causation. [As I understand it, he thinks that biology, e.g., can understand the supernatural's effect on the natural order? I get the feeling that he considers life to be a miracle of the first degree:

The highest degree in miracles comprises those works wherein something is done by God, that nature can never do: for instance, that two bodies occupy the same place, that the sun recede or stand still, that the sea be divided and make way to passers by. Among these there is a certain order: for the greater the work done by God, and the further it is removed from the capability of nature, the greater the miracle: thus it is a greater miracle that the sun recede, than that the waters be divided.

Summa Contra Gentiles, lib. 3 cap. 101

How does science know when it's dealing with the natural or supernatural if miracles do indeed happen? Miracles is something I have never seen discussed in the ID debate. Maybe they have been, and I'm just unaware. According to St. Thomas, a miracle is not just a wonderful event, but a revelation of God—it is one of the ways in which God tells us about himself. Miracles must meet three criteria: [1] they must be unusual events out of the regular order of nature (this rules out Behe’s notion that “irreducible complexity” is a miracle), [2] they must draw attention of human beings and evoke wonder in them, and [3] they must have theological significance (that is, they must reveal to us something that is part of the deposit of faith). So, the raising of Lazarus is a miracle because [1] it is out of the regular order of nature (this is not the usual way in which nature creates life), [2] it evokes wonder in human witnesses, and [3] it is a type of the resurrection of Christ.] He treats God’s agency as a sort of super-powerful natural cause. [Yes, definitely] From the Thomistic perspective this is incorrect, for it implies that God requires a material potentiality in order to create, as do natural causes. God’s agency is not just a more powerful sort of cause, but it is totally unlike natural causes. God does not actualize a potentiality when he creates, but he simple is the reason why things are. How can this be? Well, we would have to be God to understand how something can be caused to be without the actualization of a potentiality, but then we are not God. God is the transcendent creator and his very transcendence means that we cannot comprehend how he does what he does. But we can distinguish God’s action from the actions of created things and, therefore, know that God is the transcendent creator. This is important, because if we are not careful, we can be slip into error in our thinking about who God is. The natural universe is intelligible and scientific research is the means by which we know it. God made it that way. But the fact that we can explain natural things in terms of natural causes does not rule out their being created by God. In fact, the only way to explain why there are natural things and their natural causes at all is to understand that they radically depend on God for their existence.

M. Tkacz

Dr. Michael W. Tkacz
Associate Professor of Philosophy
Gonzaga University
I think every ID advocate correctly understands that God and only God creates ex nihilo ("out of nothing"), but they do not understand that God does not override nature. Thus, every ID advocate should read this article in St. Thomas Aquinas's Summa Theologica: "Whether creation [i.e., creatio ex nihilo] is mingled with works of nature and art?" While ID advocates would say "Yes," St. Thomas says "No." He says that "in the works of nature creation does not enter, but is presupposed to the work of nature." This is fully consistent with God simply letting things be. He does not say, e.g., "I create light!" but "Let there be light." (Genesis 1:3). Nor does He say "I bring forth the living creature!" but "Let the earth bring forth the living creature." (Genesis 1:24).

Tuesday, June 15, 2010

Tuesday, June 8, 2010

Assenting to Things Above Natural Reason

Intellectual assent to things that are above human reason is not contrary to human reason nor to science.


Now those who believe this truth, of which reason affords a proof, believe not lightly, as though following foolish fables (2 Pet. i. 16). For divine Wisdom Himself, Who knows all things most fully, deigned to reveal to man the secrets of God's wisdom: and by suitable arguments proves His presence, and the truth of His doctrine and inspiration, by performing works surpassing the capability of the whole of nature, namely, the wondrous healing of the sick, the raising of the dead to life, a marvellous control over the heavenly bodies, and what excites yet more wonder, the inspiration of human minds, so that unlettered and simple persons are filled with the Holy Ghost, and in one instant are endowed with the most sublime wisdom and eloquence. And after considering these arguments, convinced by the strength of the proof, and not by the force of arms, nor by the promise of delights, but—and this is the greatest marvel of all—amidst the tyranny of persecutions, a countless crowd of not only simple but also of the wisest men, embraced the Christian faith, which inculcates things surpassing all human understanding, curbs the pleasures of the flesh, and teaches contempt of all worldly things. That the minds of mortal beings should assent to such things, is both the greatest of miracles, and the evident work of divine inspiration, seeing that they despise visible things and desire only those that are invisible. And that this happened not suddenly nor by chance, but by the disposition of God, is shown by the fact that God foretold that He would do so by the manifold oracles of the prophets, whose books we hold in veneration as bearing witness to our faith. This particular kind of proof is alluded to in the words of Heb. ii. 3, 4: Which, namely the salvation of mankind, having begun to be declared by the Lord, was confirmed with us by them that heard Him, God also bearing witness by signs and wonders, and divers. . . distributions of the Holy Ghost.

Now such a wondrous conversion of the world to the Christian faith is a most indubitable proof that such signs did take place, so that there is no need to repeat them, seeing that there is evidence of them in their result. For it would be the most wondrous sign of all if without any wondrous signs the world were persuaded by simple and lowly men to believe things so arduous, to accomplish things so difficult, and to hope for things so sublime. Although God ceases not even in our time to work miracles through His saints in confirmation of the faith.

On the other hand those who introduced the errors of the sects proceeded in contrary fashion, as instanced by Mohammed, who enticed peoples with the promise of carnal pleasures, to the desire of which the concupiscence of the flesh instigates. He also delivered commandments in keeping with his promises, by giving the reins to carnal pleasure, wherein it is easy for carnal men to obey: and the lessons of truth which he inculcated were only such as can be easily known to any man of average wisdom by his natural powers: yea rather the truths which he taught were mingled by him with many fables and most false doctrines. Nor did he add any signs of supernatural agency, which alone are a fitting witness to divine inspiration, since a visible work that can be from God alone, proves the teacher of truth to be invisibly inspired: but he asserted that he was sent in the power of arms, a sign that is not lacking even to robbers and tyrants. Again, those who believed in him from the outset were not wise men practised in things divine and human, but beastlike men who dwelt in the wilds, utterly ignorant of all divine teaching; and it was by a multitude of such men and the force of arms that he compelled others to submit to his law.

Lastly, no divine oracles of prophets in a previous age bore witness to him; rather did he corrupt almost all the teaching of the Old and New Testaments by a narrative replete with fables, as one may see by a perusal of his law. Hence by a cunning device, he did not commit the reading of the Old and New Testament Books to his followers, lest he should thereby be convicted of falsehood. Thus it is evident that those who believe his words believe lightly.

St. Thomas Aquinas's Summa Contra Gentiles, lib. 1 cap. 6

∃ Two Separate Laws?

Section a1.4 of the top-selling book of Islamic Sharia law says:
a1.4: The measure of good and bad [...] is the Sacred Law, not reason.
This is totalitarian, diabolic, and irrational, "For all the gods of the Gentiles are devils: but the Lord made the heavens." (Ps. 95:5). Catholic Christians believe that the natural law,
written in [even the Gentiles'] hearts,

Romans 2:16

"is a participation in us of the eternal law," of the Divine law (Summa Theologica Iª-IIae q. 91 a. 4 arg. 1). A gloss on this verse says: "'written in their hearts,' as to the existence of a God [and that] their reason tells them that many sins are unlawful." Moral law is founded on reason and perfected by faith; grace builds on nature. Muslims apparently think that Allah and Sharia law is so far above human reason that it can even contradict it. It is no wonder Sharia law also condemns as
a7.2 Unlawful knowledge [...] (2) philosophy [...] (5) the science of the materialists [...] (6) and anything that is a means to create doubts (n: in eternal truths).
Of course truth to Muslims must means whatever Allah's messengers say, a sort of relativism of truth. It is within this context of faith and reason that Pope Benedict so skillfully gave his Regensburg lecture in 2006 which affirmed that the Trinity, Λόγος, is the God of reason.

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.