Monday, November 23, 2009

Beauty in Science

Is a genuine aesthetics necessary for science, or does it just pertain to art, e.g., music? Paul Dirac "said that his belief in what he called the principle of mathematical beauty became 'like a religion' to him and his friend Schrödinger." Occam's razor says that the simplest explanation is often the best, which makes sense if the explanation is beautiful, i.e., if it "results from the concurrence of clarity and due proportion," which is how St. Thomas Aquinas, following Dionysius, defines beauty in context of honesty and truth; "God is said to be beautiful, as being 'the cause of the harmony and clarity of the universe.'" (Deus dicitur pulcher sicut universorum consonantiæ et claritatis causa; Summa Theologica IIª-IIae, q. 145 a. 2 co.).

Let us first look at how St. Thomas Aquinas shows that God is responsible for the beauty in the universe, specifically beauty in music; then we will look at how Paul Dirac sees God's beauty in mathematics. Although it was said of Dirac that his religious stance is "There is no God, and Dirac is His prophet" (Heisenberg's Physics and beyond: encounters and conversations pg. 87), he nevertheless studied God's creation even though he objected to organized religion on ethical grounds. From Aquinas, however, one must be virtuous to experience beauty, and virtue requires ethics.
The analysis of beauty by St. Thomas Aquinas helps us appreciate the value of the musician because for Aquinas, the beautiful stimulates not only the pleasure of the ear but the delight of the mind. The three characteristics or properties of beauty—clarity, order and proportion, splendor of form—cannot be simply reduced to any laws of music or the supposed laws of the other arts. These properties transcend any laws, which is the key to appreciating the openness of Aquinas's thought to artistic evolution within any of the arts, notwithstanding the misunderstanding of the critics. Given his hints about the possibility of a virtue regulating the pleasure of the arts, there is a virtue of music appreciation which regulates one's choice and attitudes about the music one listens to. As music lovers grow in the ability to distinguish beautiful music, they are able to turn the aesthetic experience of music into a preparation for contemplation of other things that may answer certain important questions regarding the meaning of life. Likewise, the virtue of music appreciation will lead them to know when to get refreshment from music and when someone feels he is becoming too attached to this pleasure and so must moderate its use in the overall life of virtue.

Aquinas used the notion of beauty to help understand that the creation of the world is shot through with beauty. [De Potentia, IV, 2c; S.T., I, 65, 4; 66, 4, ad 2; 70, 1; 73, 1; 93, 1.] Looking at his commentary on Pseudo-Dionysius we discover that the reason for God's creative act is reduced to his beauty. [In Div. Nom., c. 4, lect. 5, nn. 352 & 353.] God wanted to make things like to himself who is Beauty per se. Hence the beauty of creation is spoken of in the following manner: "The beauty of the creature is nothing else than the likeness of the divine beauty participated in things"; [In Div. Nom., c. 4, lect. 5, n. 337.] "... whence it is evident that from the divine beauty is derived the existence of all things." [Ibid. n. 349.] So, it follows that each thing is beautiful in its own way. [Π44, 2; In Div. Nom. IV, 5.] Aquinas also says that this divine beauty gives unity, mutual adaptations, agreements in ideas and friendship. [In Div. Nom., c. 4, lect. 5, n. 337.]

From another point of view, beauty of spirit consists in conversations and actions which are well formed and suffused with intelligence. [S.T., II-II, 145, 2c.] Therefore, from the point of view of morals and spirituality, the beauty of an entire life is founded upon the virtuous life which consists in the co-ordination of many human acts and emotions according to reason. [II-II, 145, 2 & 4.] Because the instincts and emotions are brought under the order of reason, this inner activity of the human person, like a musician's, harmonizes, and sets in proportion the human life of the person. [Cf. S.T., II-II, 180, 2 ad 3 where Thomas says that the moral life is beautiful insofar as it participates in reason; see also Contra Gentes, III, 37.] On the other hand, immoderate pleasure sought for its own sake "... dulls the light of reason, from which comes all clarity and beauty of virtue." [S.T., II-II, 145, 2c.]

Music and Spirituality: To the Tune of St. Thomas Aquinas by Fr. Basil Cole, O.P.

Now let us turn to how Paul Dirac sees beauty in mathematics in the context of his 1939 paper "The Relation between Mathematics and Physics" (Proceedings of the Royal Society (Edingburgh), vol. 59, 1938-1939, part II, pp. 122-9). He mentions not only mathematics but also how it relates to quantum, General Relativity, and even Lemaître's Big Bang cosmology. Dirac also believes that a mechanistic world-view is inadequate for understanding quantum.
The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with remarkable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature's scheme.

One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from a thorough examination of the various factors that make it up. The main aim of my talk to you will be to give you such an appreciation. I propose to deal with how the physicist's views on this subject have been gradually modified by the succession of recent developments in physics, and then I would like to make a little speculation about the future.

Let us take as our starting-point that scheme of physical science which was generally accepted in the last century—the mechanistic scheme. This considers the whole universe to be a dynamical system (of course an extremely complicated dynamical system), subject to laws of motion which are essentially of the Newtonian type. The role of mathematics in this scheme is to represent the laws of motion by equations, and to obtain solutions of the equations referring to observed conditions.

The dominating idea in this application of mathematics to physics is that the equations representing the laws of motion should be of a simple form. The whole success of the scheme is due to the fact that equations of simple form do seem to work. The physicist is thus provided with a principle of simplicity, which he can use as an instrument of research. If he obtains, from some rough experiments, data which fit in roughly with certain simple equations, he infers that if he performed the experiments more accurately he would obtain data fitting in more accurately with the equations. The method is much restricted, however, since the principle of simplicity applies only to fundamental laws of motion, not to natural phenomena in general. For example, rough experiments about the relation between the pressure and volume of a gas at a fixed temperature give results fitting in with a law of inverse proportionality, but it would be wrong to infer that more accurate experiments would confirm this law with greater accuracy, as one is here dealing with a phenomenon which is not connected in any very direct way with the fundamental laws of motion.

The discovery of the theory of relativity made it necessary to modify the principle of simplicity. Presumably one of the fundamental laws of motion is the law of gravitation which, according to Newton, is represented by a very simple equation, but, according to Einstein, needs the development of an elaborate technique before its equation can even be written down. It is true that, from the standpoint of higher mathematics, one can give reasons in favour of the view that Einstein's law of gravitation is actually simpler than Newton's, but this involves assigning a rather subtle meaning to simplicity, which largely spoils the practical value of the principle of simplicity as an instrument of research into the foundations of physics.

What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature. The restricted theory changed our ideas of space and time in a way that may be summarised by stating that the group of transformations to which the space-time continuum is subject must be changed from the Galilean group to the Lorentz group.

The latter group is a much more beautiful thing than the former—in fact, the former would be called mathematically a degenerate special case of the latter. The general theory of relativity involved another step of a rather similar character, although the increase in beauty this time is usually considered to be not quite so great as with the restricted theory, which results in the general theory being not quite so firmly believed in as the restricted theory.

We now see that we have to change the principle of simplicity into a principle of mathematical beauty. The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should still take simplicity into consideration in a subordinate way to beauty. (For example Einstein, in choosing a law of gravitation, took the simplest one compatible with his space-time continuum, and was successful.) It often happens that the requirements of simplicity and of beauty are the same, but where they clash the latter must take precedence.

Let us pass on to the second revolution in physical thought of the present century—the quantum theory. This is a theory of atomic phenomena based on a mechanics of an essentially different type from Newton's. The difference may be expressed concisely, but in a rather abstract way, by saying that dynamical variables in quantum mechanics are subject to an algebra in which the commutative axiom of multiplication does not hold. Apart from this, there is an extremely close formal analogy between quantum mechanics and the old mechanics. In fact, it is remarkable how adaptable the old mechanics is to the generalization of non-commutative algebra. All the elegant features of the old mechanics can be carried over to the new mechanics, where they reappear with an enhanced beauty.

Quantum mechanics requires the introduction into physical theory of a vast new domain of pure mathematics—the whole domain connected with non-commutative multiplication. This, coming on top of the introduction of new geometries by the theory of relativity, indicates a trend which we may expect to continue. We may expect that in the future further big domains of pure mathematics will have to be brought in to deal with the advances in fundamental physics. Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. It is difficult to predict what the result of all this will be. Possibly, the two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics. At present we are, of course, very far from this stage, even with regard to some of the most elementary questions. For example, only four dimensional space is of importance in physics, while spaces with other numbers of dimensions are of about equal interest in mathematics.

It may well be, however, that this discrepancy is due to the incompleteness of present day knowledge, and that future developments will show four-dimensional space to be of far greater mathematical interest than all the others.

The trend of mathematics and physics towards unification provides the physicist with a powerful new method of research into the foundations of his subject, a method which has not yet been applied successfully, but which I feel confident will prove its value in the future. The method is to begin by choosing that branch of mathematics which one thinks will form the basis of the new theory. One should be influenced very much in this choice by considerations of mathematical beauty. It would probably be a good thing also to give a preference to those branches of mathematics that have an interesting group of transformations underlying them, since transformations play an important role in modern physical theory, both relativity and quantum theory seeming to show that transformations are of more fundamental importance than equations. Having decided on the branch of mathematics, one should proceed to develop it along suitable lines, at the same time looking for that way in which it appears to lend itself naturally to physical interpretation.

This method was used by Jordan in an attempt to get an improved quantum theory on the basis of an algebra with non-associative multiplication. The attempt was not successful, as one would rather expect, if one considers that non-associative algebra is not a specially beautiful branch of mathematics, and is not connected with an interesting transformation theory. I would suggest, as a more hopeful-looking idea for getting an improved quantum theory, that one take as basis the theory of functions of a complex variable. This branch of mathematics is of exceptional beauty, and further, the group of transformations with which it is connected, namely, the group of transformations in the complex plane, is the same as the Lorentz group governing the space-time of restricted relativity. One is thus led to suspect the existence of some deep-lying connection between the theory of functions of a complex variable and the space-time of restricted relativity, the working out of which will be a difficult task for the future.

Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply. The latter are considered to be not amenable to theoretical treatment and to be determinable only from observation. The enormous complexity of the universe is ascribed to an enormous complexity in the initial conditions, which removes them beyond the range of mathematical discussion.

I find this position very unsatisfactory philosophically, as it goes against all ideas of the unity of Nature. Anyhow, if it is only to a part of the description of the universe that mathematical theory applies, this part ought certainly to be sharply distinguished from the remainder. But in fact there does not seem to be any natural place in which to draw the line. Are such things as the properties of the elementary particles of physics, their masses and the numerical coefficients occurring in their laws of force, subject to mathematical theory? According to the narrow mechanistic view, they should be counted as initial conditions and outside mathematical theory. However, since the elementary particles all belong to one or other of a number of definite types, the members of one type being all exactly similar, they must be governed by mathematical law to some extent, and most physicists now consider it to be quite a large extent. For example, Eddington has been building up a theory to account for the masses. But even if one supposed all the properties of the elementary particles to be determinable by theory, one would still not know where to draw the line, as one would be faced by the next question—Are the relative abundances of the various chemical elements determinable by theory? One would pass gradually from atomic to astronomic questions.

This unsatisfactory situation gets changed for the worse by the new quantum mechanics. In spite of the great analogy between quantum mechanics and the older mechanics with regard to their mathematical formalism, they differ drastically with regard to the nature of their physical consequences. According to the older mechanics, the result of any observation is determinate and can be calculated theoretically from given initial conditions; but with quantum mechanics there is usually an indeterminacy in the result of an observation, connected with the possibility of occurrence of a quantum jump, and' the most that can be calculated theoretically is the probability of any particular result being obtained. The question, which particular result will be obtained in some particular case, lies outside the theory. This must not be attributed to an incompleteness of the theory, but is essential for the application of a formalism of the kind used by quantum mechanics.

Thus according to quantum mechanics we need, for a complete description of the Universe, not only the laws of motion and the initial conditions, but also information about which quantum jump occurs in each case when a quantum jump does occur. The latter information must be included, together with the initial conditions, in that part of the description of the universe outside mathematical theory.

The increase thus arising in the non-mathematical part of the description of the universe provides a philosophical objection to quantum mechanics, and is, I believe, the underlying reason why some physicists still find it difficult to accept this mechanics. Quantum mechanics should not be abandoned, however, firstly, because of its very widespread and detailed agreement with experiment, and secondly, because the indeterminacy it introduces into the results of observations is of a kind which is philosophically satisfying, being readily ascribable to an inescapable crudeness in the means of observation available for small-scale experiments. The objection does show, all the same, that the foundations of physics are still far from their final form.

We come now to the third great development of physical science of the present century—the new cosmology. This will probably turn out to be philosophically even more revolutionary than relativity or the quantum theory, although at present one can hardly realize its full implications. The starting-point is the observed red-shift in the spectra of distant heavenly bodies, indicating that they are receding from us with velocities proportional to their distances. [The recession velocities are not strictly proved, since one may postulate some other cause for the spectral red-shift. However, the new cause would presumably be equally drastic in its effect on cosmological theory and would still need the introduction of a parameter of the order 2×10⁹ years for its mathematical discussion, so it would probably not disturb the essential ideas of the argument in the text.] The velocities of the more distant ones are so enormous that it is evident we have here a fact of the utmost importance, not a temporary or local condition, but something fundamental for our picture of the universe.

If we go backwards into the past we come to a time, about 2×10⁹ years ago, when all the matter in the universe was concentrated in a very small volume. It seems as though something like an explosion then took place, the fragments of which we now observe still scattering outwards. This picture has been elaborated by Lemaître, who considers the universe to have started as a single very heavy atom, which underwent violent radioactive disintegrations and so broke up into the present collection of astronomical bodies, at the same time giving off the cosmic rays.

With this kind of cosmological picture one is led to suppose that there was a beginning of time, and that it is meaningless to inquire into what happened before then. One can get a rough idea of the geometrical relationships this involves by imagining the present to be the surface of a sphere, going into the past to be going in towards the centre of the sphere, and going into the future to be going outwards. There is then no limit to how far one may go into the future, but there is a limit to how far one can go into the past, corresponding to when one has reached the centre of the sphere. The beginning of time provides a natural origin from which to measure the time of any event. The result is usually called the epoch of that event. Thus the present epoch is 2×10⁹ years.

Let us now return to dynamical questions. With the new cosmology the universe must have been started off in some very simple way. What, then, becomes of the initial conditions required by dynamical theory? Plainly there cannot be any, or they must be trivial. We are left in a situation which would be untenable with the old mechanics. If the universe were simply the motion which follows from a given scheme of equations of motion with trivial initial conditions, it could not contain the complexity we observe. Quantum mechanics provides an escape from the difficulty. It enables us to ascribe the complexity to the quantum jumps, lying outside the scheme of equations of motion. The quantum jumps now form the uncalculable part of natural phenomena, to replace the initial conditions of the old mechanistic view.

One further point in connection with the new cosmology is worthy of note. At the beginning of time the laws of Nature were probably very different from what they are now. Thus we should consider the laws of Nature as continually changing with the epoch, instead of as holding uniformly throughout space-time. This idea was first put forward by Milne, who worked it out on the assumptions that the universe at a given epoch is roughly everywhere uniform and spherically symmetrical. I find these assumptions not very satisfying, because the local departures from uniformity are so great and are of such essential importance for our world of life that it seems unlikely there should be a principle of uniformity overlying them. Further, as we already have the laws of Nature depending on the epoch, we should expect them also to depend on position in space, in order to preserve the beautiful idea of the theory of relativity that there is fundamental similarity between space and time. This goes more drastically against Milne's assumptions than a mere lack of uniformity in the distribution of matter. We have followed through the main course of the development of the relation between mathematics and physics up to the present time, and have reached a stage where it becomes interesting to indulge in speculations about the future. There has always been an unsatisfactory feature in the relation, namely, the limitation in the extent to which mathematical theory applies to a description of the physical universe. The part to which it does not apply has suffered an increase with the arrival of quantum mechanics and a decrease with the arrival of the new cosmology, but has always remained.

This feature is so unsatisfactory that I think it safe to predict it will disappear in the future, in spite of the startling changes in our ordinary ideas to which we should then be led. It would mean the existence of a scheme in which the whole of the description of the universe has its mathematical counterpart, and we must suppose that a person with a complete knowledge of mathematics could deduce, not only astronomical data, but also all the historical events that take place in the world, even the most trivial ones. Of course, it must be beyond human power actually to make these deductions, since life as we know it would be impossible if one could calculate future events, but the methods of making them would have to be well defined. The scheme could not be subject to the principle of simplicity since it would have to be extremely complicated, but it may well be subject to the principle of mathematical beauty.

I would like to put forward a suggestion as to how such a scheme might be realized. If we express the present epoch, 2×10⁹ years, in terms of a unit of time defined by the atomic constants, we get a number of the order 10³⁹, which characterizes the present in an absolute sense. Might it not be that all present events correspond to properties of this large number, and, more generally, that the whole history of the universe corresponds to properties of the whole sequence of natural numbers? At first sight it would seem that the universe is far too complex for such a correspondence to be possible. But I think this objection cannot be maintained, since a number of the order 10³⁹ is excessively complicated, just because it is so enormous. We have a brief way of writing it down, but this should not blind us to the fact that it must have excessively complicated properties.

There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would lead to a connection between atomic theory and cosmology.
Beauty is therefore apparent in the sciences as well as the arts. On contemplating the truth, in any of its forms, e.g. in music or mathematics, St. Thomas Aquinas says:
Beauty [...] consists in a certain clarity and due proportion. Now each of these is found radically in the reason; because both the light that makes beauty seen, and the establishing of due proportion among things belong to reason. Hence since the contemplative life consists in an act of the reason, there is beauty in it by its very nature and essence; wherefore it is written (Wisdom 8:2) of the contemplation of wisdom: "I became a lover of her beauty."

Summa Theologica IIª-IIae, q. 180 a. 2 ad 3

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.

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.
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!

Tuesday, November 3, 2009

Pope: Understanding the universe brings us closer to God

Faith and reason cooperate in the service of an integral understanding of man and his place in the universe. This is what Benedict XVI underlined in his speech to the participants in the conference sponsored by the Specola Vaticana, the Vatican Observatory, for the International Year of Astronomy. The Pope expressed his gratitude for the careful studies which have clarified the precise historical context of Galileo's condemnation, and noted that understanding the universe brings us closer to the marvels of creation and to the contemplation of the Creator.

Read the full speech here. For more information, read about the "Astronomers of the Pope."

Monday, November 2, 2009