Thursday, November 5, 2009

Fathers of Modern Mathematics and Physics

Noting both the difficulty and the ease in investigating the truth, as does science (from Latin scire, "to know" or "to understand"), Aristotle says "that we should be grateful" to those thinkers before us, our scientific heritage:
The investigation of the truth is in one way hard, in another easy. An indication of this is found in the fact that no one is able to attain the truth adequately, while, on the other hand, we do not collectively fail, but every one says something true about the nature of things, and while individually we contribute little or nothing to the truth, by the union of all a considerable amount is amassed. Therefore, since the truth seems to be like the proverbial door, which no one can fail to hit, in this respect it must be easy, but the fact that we can have a whole truth and not the particular part we aim at shows the difficulty of it.

Perhaps, too, as difficulties are of two kinds, the cause of the present difficulty is not in the facts but in us. For as the eyes of bats are to the blaze of day, so is the reason in our soul to the things which are by nature most evident of all.

It is just that we should be grateful, not only to those with whose views we may agree, but also to those who have expressed more superficial views; for these also contributed something, by developing before us the powers of thought. It is true that if there had been no Timotheus we should have been without much of our lyric poetry; but if there had been no Phrynis there would have been no Timotheus. The same holds good of those who have expressed views about the truth; for from some thinkers we have inherited certain opinions, while the others have been responsible for the appearance of the former.

It is right also that philosophy should be called knowledge of the truth. For the end of theoretical knowledge is truth, while that of practical knowledge is action (for even if they consider how things are, practical men do not study the eternal, but what is relative and in the present). Now we do not know a truth without its cause; and a thing has a quality in a higher degree than other things if in virtue of it the similar quality belongs to the other things as well (e.g. fire is the hottest of things; for it is the cause of the heat of all other things); so that that causes derivative truths to be true is most true. Hence the principles of eternal things must be always most true (for they are not merely sometimes true, nor is there any cause of their being, but they themselves are the cause of the being of other things), so that as each thing is in respect of being, so is it in respect of truth.

Metaphysics 993a30-993b19

Who are the analogous Phrynises in science, the giants on whose shoulders we stand? Besides the obvious—e.g., Galileo, Copernicus, Newton, Kepler, Einstein—let us note some lesser-known characters.

First among these is Aristotle, the first physicist (Physics) and developer of the scientific method of knowing causes through their effects (Posterior Analytics). We have mentioned him already, e.g., in this post on the dehellenization of modern science. Second is St. Thomas Aquinas, student of the pro-science patron saint St. Albert the Great, both scholastics. St. Thomas's contributions to modern scientific thought, such as his knowledge of Euclid's Elements and the empiriological sciences of at his time, display a profound respect for the scientific method and methodological naturalism. In discussing how one cannot know by natural reason that God is triune (three in one), he reflects his support of the modern scientific method, often attributed to Galileo yet more deservingly to Aristotle's Posterior Analytics. St. Thomas mentions that one can outmode scientific theories for better, truer ones, i.e., for ones that conform with reality better.
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. In the first way, we can prove that God is one; and the like. In the second way, reasons avail to prove the Trinity; as, when assumed to be true, such reasons confirm it. We must not, however, think that the trinity of persons is adequately proved by such reasons. This becomes evident when we consider each point; for the infinite goodness of God is manifested also in creation, because to produce from nothing is an act of infinite power. For if God communicates Himself by His infinite goodness, it is not necessary that an infinite effect should proceed from God: but that according to its own mode and capacity it should receive the divine goodness. Likewise, when it is said that joyous possession of good requires partnership, this holds in the case of one not having perfect goodness: hence it needs to share some other's good, in order to have the goodness of complete happiness. Nor is the image in our mind an adequate proof in the case of God, forasmuch as the intellect is not in God and ourselves univocally. Hence, Augustine says (Tract. xxvii. in Joan.) that by faith we arrive at knowledge, and not conversely.

Summa Theologica Iª q. 32 a. 1 ad. 2

St. Thomas also mentions the interplay between physics and mathematics, shedding light on the mystery of the connection between mathematics and the physical world, a topic fascinating to Einstein, who wrote:
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?
To the latter question, no; we have no knowledge without prior sense experience. In the former question, we can see Kant's idealism—i.e., agnosticism of an objective reality—creeping into his thought when says that mathematics is independent of experience. Our ideas of it may be independent of external sense experience, but mathematical properties of matter such as quantity exist in the objective reality outside one's mind ("intellectual soul" or simply "soul"). As Boethius says in his De Trinitate II., "Mathematics does not deal with motion and it is not abstract, for it inquires into the forms of bodies apart from matter and therefore apart from motion [viz., change], which forms, however, since they exist in matter, cannot be separated from bodies." This is how St. Thomas explains how mathematics and physics differ:
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 vice 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 number. 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.

Super De Trinitate, pars 3 q. 5 a. 3 ad 5 et 6

Modern physics is what St. Thomas would call an intermediate science because it is intermediate between a truly physical science—physics as Aristotle conceived it—and mathematics, which abstracts from physical matter.
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. 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.

In Physic., lib. 2 l. 3 n. 8

What makes the Catholic philosophy of St. Thomas so efficacious to the advancement of science? Catholics embrace the physical world—especially through the necessarily physical Sacraments, the physical manifestations of a hidden reality—because it is with the world, through our five external senses, that one obtains knowledge of God, a human's first Beginning and ultimate End. Catholics do not despise the human body nor do they consider it intrinsically evil. From a review of the popular science book The Tao of Physics, a book about how modern physics and Eastern thought relate:
But it is least of all to history that we should look for confirmation of Capra's thesis. In the early chapters he blames Aristotle and Christianity for the ensuing "lack of interest in the material world" (p. 22). But what cultures ever displayed a more profound and studious disregard for the material world than the Eastern mystical traditions? And why would they hold in high regard something that is at best a creation of the human mind and at worst a deceptive illusion?
Hence the philosophies of the Eastern religions are fundamentally at odds with understanding the physical world. The philosophy of the Greeks is better. E.g., the word "science" in English is equivocal; however, the Greeks distinguished ἐπιστήμη, "knowledge of an event or a thing through its causes" (Weisheipl 183), from τέχνη (art, skill, craft; root of the word "technology"), νόος (verbal: νοέω; "understood" in Rom. 1:20), and σοφία (wisdom). Thus scientific knowledge as we moderns conceive it, the ἐπιστήμη, is not the only form of knowing. Fr. Georges-Henri Lemaître—with his background in the supreme science, theology—recognized this. He was the inventor of the Big Bang theory and pupil of the cosmologist Fr. Désiré Nys at the University of Louvain, a university Pope Leo XIII established to promote St. Thomas's philosophy in the context of modern scientific discoveries. From The Heavens Proclaim: Astronomy and the Vatican:
Astronomy has long featured in Christian theology. Indeed, astronomy was one of the seven subjects of the medieval university that all scholars were expected to master before they could begin their studies of philosophy and theology. At the beginning of this book we examined two specific instances in the history of the Church and astronomy: the successful reform the calendar under Pope Gregory XIII in 1582, and the tragic conflict just fifty years later between the Church and Galileo. Here, however, we would like to take a look at more recent statements of Popes concerning the modern science of astronomy. Much of the Church’s interest has had an overt apologetic slant, using science to support its philosophical ideas or using its support of science to refute those who would accuse the Church of opposing progress and fearing newly-discovered truths. Even in Roman times, the apologetic need for the Church’s teachers to have an up-to-date knowledge of the physical universe, to give credibility to the theological truths of the Church, was evident to St. Augustine. Writing in AD 400, he commented:
Even a non-Christian knows something about the Earth, the heavens, and the other elements of this world, about the motion and orbit of the stars and even their size and relative positions, about the predictable eclipses of the Sun and Moon, the cycles of the years and the seasons... and this knowledge he holds to as being certain from reason and experience. Now, it is a disgraceful and dangerous thing for an infidel to hear a Christian, presumably giving the meaning of Holy Scripture, talking nonsense on these topics; and we should take all means to prevent such an embarrassing situation, in which people show up vast ignorance in a Christian and laugh it to scorn.

St. Augustine's The Literal Meaning of Genesis, pgs. 42-43

The irony is, of course, that the cosmology the learned men of Rome knew so well, was the very Ptolemaic cosmology later overthrown by Copernicus and Galileo! But through the writings of these modern Popes one begins to see developing a second realization: that, as the Psalmist knew, the Heavens themselves do proclaim the greatness of the Creator. The simple act of seeking truth in the natural sciences is in and of itself a religious act, independent of any apologetic agenda.

from
AETERNI PATRIS, 1879 (POPE LEO XIII)
In an encyclical letter proclaimed in 1879, subtitled “On the Restoration of Christian Philosophy in Catholic Schools in the Spirit (ad mentem) of the Angelic Doctor, St. Thomas Aquinas,” Pope Leo XIII endorsed the study of scholastic philosophy and ignited a new interest in the rational understanding of the faith. In passing, he reflects on the role of the physical sciences, in a way that foreshadows his establishment, twelve years later, of the Vatican Observatory itself:
Our philosophy can only by the grossest injustice be accused of being opposed to the advance and development of natural science. For, when the Scholastics, following the opinion of the holy Fathers, always held in anthropology that the human intelligence is only led to the knowledge of things without body and matter by things sensible, they well understood that nothing was of greater use to the philosopher than diligently to search into the mysteries of nature and to be earnest and constant in the study of physical things. And this they confirmed by their own example; for St. Thomas, Blessed Albertus Magnus, and other leaders of the Scholastics were never so wholly rapt in the study of philosophy as not to give large attention to the knowledge of natural things; and, indeed, the number of their sayings and writings on these subjects, which recent professors approve of and admit to harmonize with truth, is by no means small. Moreover, in this very age many illustrious professors of the physical sciences openly testify that between certain and accepted conclusions of modern physics and the philosophic principles of the schools there is no conflict worthy of the name.
THE REFOUNDATION AND RESTRUCTURING OF THE VATICAN OBSERVATORY, 1891 (POPE LEO XIII)
Here is the text of Leo XIII’s Motu Proprio [Ut mysticam Sponsam], a personal decree that re-established the Vatican Observatory. In it he explains the apologetic need for supporting a scientific institution at that time, and also outlines the previous history of papal support for astronomy.
So that they might display their disdain and hatred for the mystical Spouse of Christ, who is the true light, those borne of darkness are accustomed to calumniate her to unlearned people and they call her the friend of obscurantism, one who nurtures ignorance, an enemy of science and of progress, all of these accusations being completely contrary to what in word and deed is essentially the case.
Right from its beginnings all that the Church has done and taught is an adequate refutation of these impudent and sinister lies. In fact, the Church, besides her knowledge of divine realities, in which she is the unique teacher, also nourishes and gives guidance in the practice of philosophy which is essential to understanding the scientific foundations of knowing – to make its principles clear, to suggest the criteria necessary for rigorous research and for a systematic presentation of the results, to investigate the soul’s faculties, to study life and human behavior – and she does this so well that it would be difficult to add anything worth mentioning and it would be dangerous to dissociate oneself from her teachings.
Furthermore, it is to the great merit of the Church that the legal code has been completed and perfected, nor can we ever forget how much she has contributed through her doctrine, her example and her institutions to addressing the complex issues arising in the so-called social sciences and in economics.
In the meantime the Church has not neglected those disciplines which investigate nature and its forces. Schools and museums have been founded so that young scholars might have a better opportunity to deepen those studies. Among the Church’s children and ministers there are some illustrious scientists whom the Church has honored and assisted as much as she could by encouraging them to apply themselves with complete dedication to such studies.
Among all of these studies astronomy holds a preeminent position. It proposes to investigate those inanimate creatures which more than all others proclaim the glory of God and which gave marvelous delight to the wisest of beings, the one who exulted in his divinely inspired knowledge, especially of the yearly cycles and of the positions of the heavenly bodies (Wisdom VII.19).
The Church’s pastors were motivated, among other considerations, to see to progress in this science and to support its followers by the possibility that it alone offered to establish with certainty those days on which the principal religious solemnities of the Christian mystery should be celebrated. So it was that the Fathers at Trent, well aware that the calendar reform done by Julius Caesar had not been perfect so that time calculations had changed, urgently requested that the Roman Pontiff would, after consulting experts in the field, prepare a new and more perfect reform of the calendar.
It is well known from historical documents how zealously and generously committed was Our Predecessor Gregory XIII in responding to this request. He saw to it that at the place judged to be best for an observatory within the confines of the existing Vatican buildings an observing tower was constructed and he equipped it with the best instruments of those days. It was here that he held the meetings of the experts he had selected for the reform of the calendar. This tower still exists today and it brings back the memory of its illustrious and generous founder. The meridian constructed by Ignazio Danti from Perugia is to be found there. Along the meridian line there is a round marble tablet whose lines are designed with such wisdom that when the suns rays fall on them it becomes obvious how necessary it was to reform the old calendar and how well the reform conformed to nature.
That tower, a splendid memory to a Pontiff who is to be much praised for his contribution to the progress of literary and scientific studies, was, toward the end of the last century after a long period of inactivity, restored to its original use as an astronomical observatory by the auspicious orders of Pius VI. Through the initiatives of a Roman Monsignor Filippo Gilii, other types of research were also undertaken on terrestrial magnetism, meteorology and botany. But, after the death in 1821 of this very capable scientist, this monument to astronomical research went into neglect and was abandoned. Right after this Pius VII died and the energies of Leo XII were completely taken up with the reform of studies in the worldwide Church, a huge undertaking aimed at promoting all branches of learning. Such a reform, which had already been planned by his immediate and immortal predecessor, came by his efforts to a happy ending with the Apostolic Letter, Quod divina sapientia. In this letter he established certain rules with respect to astronomical observatories, the observations which were to be made regularly, the daily list of data to be made, and the information that was to be distributed internally concerning discoveries made by others.
The fact that the tower in the Vatican was no longer used as an observatory, after others in Rome had been equipped for that very purpose, came about because those who were competent to judge were of the opinion that the nearby buildings, and especially the dome which crowns the Vatican basilica, would have obstructed observations. And so it was deemed preferable to have observatories in other higher places where unobstructed observations could be carried out.
It then happened that, after those observing sites along with the whole city of Rome fell into the hands of others, we were given, on the occasion of our 51st anniversary as a priest, many excellent instruments for research in astronomy, meteorology, and earth physics, as well as other gifts. It was the opinion of the experts that no place was better to house them than the Vatican tower, where, it seems, Gregory XIII had already in some way made preparations. After having evaluated this proposal and having examined the structure itself of the building, the history of its past glories, and the equipment already gathered there, as well as the opinions of persons renowned for their knowledge and judgement, we were persuaded to give orders that the observatory be restored and that it be equipped with all that would be required to carry out research not only in astronomy but also in earth physics and in meteorology. As to the lack of an unobstructed view of the heavens in all directions from this Vatican tower, we saw fit to consider providing the nearby ancient and solid Leonine fortification where there is a quite high tower which, since it rises on the summit of the Vatican hill, provides for complete and perfect observation of the heavenly bodies. We, therefore, added this tower to the one of Gregory and we had installed there the large equatorial telescope for photographing the stars.
To this purpose we chose conscientious men, prepared to do all that was necessary for such an undertaking, and we proposed to them a most competent scholar in astronomy and physics, Father Francesco Denza of the Clerks Regular of Saint Paul, also called the Barnabites. Relying on their dedicated work, we agreed wholeheartedly that the Vatican Observatory be chosen to collaborate with other renowned astronomical institutes in the project to reproduce from photographic plates an accurate map of the whole sky.
Considering the fact that we wish this work of restoring the Specola to be a lasting one and not one that terminates after a short time, we have established bylaws for it with rules to be observed both for internal administration and for the services which others require of it. Furthermore, we have appointed a Board of carefully selected persons whose responsibility it is to govern the observatory and they have the highest authority after our own for all decisions respecting the internal administration.
And so with the present letter we confirm those bylaws and that Board and we also assign the various jobs and all that, with our order or consent, has been done with respect to the Specola. And we desire that the Specola be considered at the same level as the other Pontifical Institutes founded to promote the sciences. In order to provide in a more secure way for the stability of this work, we even designate a sum of money which should suffice to cover the expenses required to keep it operating and to maintain it. Nevertheless, we trust that such a work will find its justification and support in the favor and help of Almighty God more than in what humans can do. In fact, in taking up this work we have become involved not only in helping to promote a very noble science which, more than any other human discipline, raises the spirit of mortals to the contemplation of heavenly events, but we have in the first place put before ourselves the plan which we have energetically and constantly sought to carry out right from the beginning of Our Pontificate in talks, writings, and deeds whenever we were provided the opportunity. This plan is simply that everyone might see clearly that the Church and her Pastors are not opposed to true and solid science, whether human or divine, but that they embrace it, encourage it, and promote it with the fullest possible dedication.
We wish, therefore, that everything that has been established and announced in the present letter will remain into the future confirmed and ratified as it is proposed herein and we declare null and void any attempt at changes by whatsoever person. And it remains established and confirmed, despite any previous contrary declaration.

Given in Rome at St. Peter’s, 14 March 1891

Pope Leo XIII, through his establishing the Catholic University at Louvain and the Vatican Observatory, was a very pro-science pope. Even our current pope, Benedict XVI, has been very pro-science. Thus the fathers of modern mathematics and physics have been true (albeit ordained) Catholic fathers!

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