Thursday, December 17, 2009

On Evolution & Intelligent Design

The fifth of St. Thomas Aquinas's "five ways" (quinque viæ) of proving God's existence is as follows:
The fifth way is taken from the governance of the world. We see that things which lack intelligence, such as natural bodies, act for an end, and this is evident from their acting always, or nearly always, in the same way, so as to obtain the best result. Hence it is plain that not fortuitously, but designedly [ex intentione], do they achieve their end. Now whatever lacks intelligence cannot move towards an end, unless it be directed by some being endowed with knowledge and intelligence; as the arrow is shot to its mark by the archer. Therefore some intelligent being exists by whom all natural things are directed to their end; and this being we call God.

Summa Theologica Iª q. 2 a. 3 co.

This is a teleological (Greek telos = "end") argument for God's existence because it argues that the whole world, its creatures—all created things—are directed toward an end; viz., they have a "final cause," to use the philosophical term. It, like Cardinal Schönborn's controversial essay "Finding Design in Nature," is "based neither on theology nor modern science nor ‘intelligent design theory.’"
In “Finding Design in Nature,” I said:
  • The Church “proclaims that by the light of reason the human intellect can readily and clearly discern purpose and design in the natural world, including the world of living things.”
  • “Any system of thought that denies or seeks to explain away the overwhelming evidence for design in biology is ideology, not science.”
  • Quoting our late Holy Father John Paul II: “The evolution of living beings, of which science seeks to determine the stages and to discern the mechanism, presents an internal finality which arouses admiration. This finality, which directs beings in a direction for which they are not responsible or in charge, obliges one to suppose a Mind which is its inventor, its creator.”
  • Again quoting John Paul II: “To all these indications of the existence of God the Creator, some oppose the power of chance or of the proper mechanisms of matter. To speak of chance for a universe which presents such a complex organization in its elements and such marvelous finality in its life would be equivalent to giving up the search for an explanation of the world as it appears to us. In fact, this would be equivalent to admitting effects without a cause. It would be to abdicate human intelligence, which would thus refuse to think and to seek a solution for its problems.”
  • Quoting the Catechism: “Human intelligence is surely already capable of finding a response to the question of origins. The existence of God the Creator can be known with certainty through his works, by the light of human reason. . . . We believe that God created the world according to his wisdom. It is not the product of any necessity whatever, nor of blind fate or chance.”
  • Referring to the Church's teaching on the importance and reach of metaphysics: “But in the modern era, the Catholic Church is in the odd position of standing in firm defense of reason as well. In the nineteenth century, the First Vatican Council taught a world newly enthralled by the ‘death of God' that by the use of reason alone mankind could come to know the reality of the Uncaused Cause, the First Mover, the God of the philosophers.”
My argument was based neither on theology nor modern science nor “intelligent design theory.” In theology, although the mind's ability to grasp the order and design in nature is adopted by, taken up into, and elevated to new heights by the faith of Christianity, that ability precedes faith, as Romans 1:19-20 makes clear. In science, the discipline and methods are such that design—more precisely, formal and final causes in natural beings—is purposefully excluded from its reductionist conception of nature. [...] [T]rue science is impossible unless we first grasp the reality of natures and essences, the intelligible principles of the natural world. We can with much profit study nature using the tools and techniques of modern science. But let us never forget, as some modern scientists have forgotten, that the study of reality via reductive methods leads to incomplete knowledge. To grasp reality as it is, we must return to our pre-scientific and post-scientific knowledge, the tacit knowledge that pervades science, the knowledge that, when critically examined and refined, we call philosophy. [...] First of all, we must observe that the role of randomness in Darwinian biology is quite different from its role in thermodynamics, quantum theory, and other natural sciences. In those sciences randomness captures our inability to predict or know the precise behavior of the parts of a system (or perhaps, in the case of the quantum world, some intrinsic properties of the system). But in all such cases the “random” behavior of parts is embedded in and constrained by a deeply mathematical and precise conceptual structure of the whole that makes the overall behavior of the system orderly and intelligible. [...] [T]he overwhelming trend of Catholic commentators on the question of neo-Darwinian evolution [...] gladly discuss[es] its compatibility with the truths of faith but seldom bother to discuss whether and how it is compatible with the truths of reason. [...] Let us return to the heart of the problem: positivism. Modern science first excludes a priori final and formal causes, then investigates nature under the reductive mode of mechanism (efficient and material causes), and then turns around to claim both final and formal causes are obviously unreal, and also that its mode of knowing the corporeal world takes priority over all other forms of human knowledge. Being mechanistic, modern science is also historicist: It argues that a complete description of the efficient and material causal history of an entity is a complete explanation of the entity itself—in other words, that an understanding of how something came to be is the same as understanding what it is. But Catholic thinking rejects the genetic fallacy applied to the natural world and contains instead a holistic understanding of reality based on all the faculties of reason and all the causes evident in nature—including the “vertical” causation of formality and finality.

—Cardinal Schönborn's "The Designs of Science"

There are two important ideas that the cardinal mentions: randomness and teleology. In the case of randomness, what appears to us as random is not necessarily so when understood more deeply, so randomness in nature is really an epistemological issue possibly due to Original Sin, our finite nature, and our imperfect intellects. E.g., the motions of atoms in a gas are not random in the sense that they obey no underlying rules; one can in principle derive the statistical properties of a gas by analyzing its trillions of atoms individually. Taken as a collection they have very predictable characteristics—e.g., temperature and entropy—indicative of an underlying logic. The same goes for quantum mechanics. The neo-Darwinists' biology, on the contrary, makes no predictions from random genetic mutations toward an increase in the information in DNA, which one would expect if more complex beings were to evolve from lesser complex ones, contrary to what the second law of thermodynamics says—that disorder tends to increase in closed systems. They call what they do not understand "random." Therefore, randomness in the former sense in no way precludes teleology as does randomness in the neo-Darwinists' sense. As for teleology:
Again, we should notice that, although every agent, both natural and voluntary, intends an end, still it does not follow that every agent knows the end or deliberates about the end. To know the end is necessary in those whose actions are not determined, but which may act for opposed ends as, for example, voluntary agents. Therefore it is necessary that these know the end by which they determine their actions. But in natural agents the actions are determined, hence it is not necessary to choose those things which are for the end. Avicenna gives the following example. A harpist does not have to deliberate about the strings that he will pluck, since these are already determined for him; otherwise there would be a delay between the notes which would cause uneveness. However, it seems more reasonable to attribute deliberation to a voluntary agent than to a natural agent. Thus it is plain, by reasoning a maiori, that, if a voluntary agent, for whom deliberation is more proper, sometimes does not deliberate, therefore neither does the natural agent. Therefore it is possible for the natural agent to intend the end without deliberation; and to intend this is nothing else than to have a natural inclination to something.

—St. Thomas Aquinas's Introduction to the Philosophy of Nature

The exclusion of final causes from science is a limitation because chance cannot be a cause:
It is necessary, no doubt, that the causes of what comes to pass by chance be indefinite; and that is why chance is supposed to belong to the class of the indefinite and to be inscrutable to man, and why it might be thought that, in a way, nothing occurs by chance. For all these statements are correct, as might be expected. Things do, in a way, occur by chance, for they occur accidentally and chance is an accidental cause. But it is not the cause without qualification of anything; for instance, a housebuilder is the cause of a house; accidentally, a fluteplayer may be so. [...] The question 'why', then, is answered by reference to the matter, to the form, and to the primary moving cause. For in respect of coming to be it is mostly in this last way that causes are investigated—'what comes to be after what? what was the primary agent or patient?' and so at each step of the series. [...] A difficulty presents itself: why should not nature work, not for the sake of something, nor because it is better so, but just as the sky rains, not in order to make the corn grow, but of necessity? (What is drawn up must cool, and what has been cooled must become water and descend, the result of this being that the corn grows.) Similarly if a man's crop is spoiled on the threshing-floor, the rain did not fall for the sake of this—in order that the crop might be spoiled—but that result just followed. Why then should it not be the same with the parts in nature, e.g. that our teeth should come up of necessity—the front teeth sharp, fitted for tearing, the molars broad and useful for grinding down the food—since they did not arise for this end, but it was merely a coincident result; and so with all other parts in which we suppose that there is purpose? Wherever then all the parts came about just what they would have been if they had come to be for an end, such things survived, being organized spontaneously in a fitting way; whereas those which grew otherwise perished and continue to perish, as Empedocles says his 'man-faced oxprogeny' did. Such are the arguments (and others of the kind) which may cause difficulty on this point [of teleology]. Yet it is impossible that this should be the true view. For teeth and all other natural things either invariably or for the most part come about in a given way; but of not one of the results of chance or spontaneity is this true. We do not ascribe to chance or mere coincidence the frequency of rain in winter, but frequent rain in summer we do; nor heat in summer but only if we have it in winter. If then, it is agreed that things are either the result of coincidence or for the sake of something, and these cannot be the result of coincidence or spontaneity, it follows that they must be for the sake of something; and that such things are all due to nature even the champions of the theory which is before us would agree. Therefore action for an end is present in things which come to be and are by nature.

—Aristotle's Physics 197a8-197a15; 198a33-198a35; 198b17-199a8

Addressing "I turned me to another thing, and I saw that under the sun, the race is not to the swift, nor the battle to the strong, nor bread to the wise, nor riches to the learned, nor favour to the skilful: but time and chance in all." (Eccles. 9:11), St. Thomas touches on what I called above randomness due to our imperfect intellects and lack of knowledge of all causes of things; i.e., that we are not able to perceive things perfectly holistically, atemporally, and omnisciently as can God:
These things are said to be under the sun which are generated and corrupted according to the sun's movement. In all such things we find chance: not that everything is casual which occurs in such things; but that in each one there is an element of chance. And the very fact that an element of chance is found in those things proves that they are subject to government of some kind. For unless corruptible things were governed by a higher being, they would tend to nothing definite, especially those which possess no kind of knowledge. So nothing would happen unintentionally; which constitutes the nature of chance. Wherefore to show how things happen by chance and yet according to the ordering of a higher cause, he does not say absolutely that he observes chance in all things, but "time and chance," that is to say, that defects may be found in these things according to some order of time.

—St. Thomas Aquinas's Summa Theologica Iª q. 103 a. 5 ad 1

Pope Pius XII's encyclical Humani Generis, the first papal document addressing evolution, sums up the Catholic position on evolution:
5. If anyone examines the state of affairs outside the Christian fold, he will easily discover the principle trends that not a few learned men are following. Some imprudently and indiscreetly hold that evolution, which has not been fully proved even in the domain of natural sciences, explains the origin of all things, and audaciously support the monistic and pantheistic opinion that the world is in continual evolution. Communists gladly subscribe to this opinion so that, when the souls of men have been deprived of every idea of a personal God, they may the more efficaciously defend and propagate their dialectical materialism. 6. Such fictitious tenets of evolution which repudiate all that is absolute, firm and immutable, have paved the way for the new erroneous philosophy which, rivaling idealism, immanentism and pragmatism, has assumed the name of existentialism, since it concerns itself only with existence of individual things and neglects all consideration of their immutable essences.
Hopefully science will free itself from some of its self-limiting practices, and, as Cardinal Schönborn says in his essay, "awaken Catholics from their dogmatic slumber about positivism in general and evolutionism in particular."

Monday, December 7, 2009

Science and Modernism

Because modern science has proven very effective at explaining the physical world, some have succumbed to the error of Modernism, which is not only harmful to religion but also actually harmful to the development of a true science which can more accurately explain nature.
The cognitional theoretical basis of Modernism is agnosticism, according to which human rational cognition is limited to the world of experience. Religion, according to this theory, develops from the principle of vital immanence (immanentism) that is, from the need for God which dwells in the human soul. The truths of religion are, according to the general progress of culture, caught up in a constant substantial development (evolutionism).
—Dr. Ludwig Ott's Fundamentals of Catholic Dogma pgs. 16-17
It appears that Modernism is only the subject of religion, the dogmas of which some may think the progress of modern science threatens. Yet it relates to science, too, since even though it is true that knowledge begins in the senses—i.e., from experience—it does not end there; rather, it ends in the intellect, the human's soul, in which exists the universal knowledge that science discovers. Agnosticism nihilistically states that it is futile "to know the reality corresponding to our ultimate scientific, philosophic, and religious ideas." (Shanahan 1908); one therefore cannot be agnostic and interpret quantum mechanics, for example, except possibly with the instrumentalist interpretation. Immanentism basically says that God and religion are manifestations of man, not realities apart from man and given to him by God, respectively; and evolutionism says man's nature changes. Although science has failed to prove any of these propositions, some assume them—consciously or not—in order to remove God from science and consequently also to ignore the study of God or theology, which is the noblest science of which the "Other sciences [e.g., the natural sciences] are called the handmaidens." (Summa Theologica Iª q. 1 a. 5 s. c.).

Before discussing how Modernism leads to pathological science, let us first give some historical context and see how Modernism affected the French Catholic physicist of the "Gibbs-Duhem Equation," Pierre Duhem (1861-1916), who, interestingly, thought that all fields of physics are reducible to thermodynamics.
2. Fideism or Rational Obedience
In the encyclical Pascendi Dominici Gregis of 1907, two years after Duhem's 'Physics of a Believer', the official position was made clear in the name of Pope Pius X. Of the dangerous aspects of the heresy it called "modernism" identified by the encyclical two concern me here, what it called the "agnosticism" of the modernists, and the separation of science and faith. The first for example was dangerous because of the damage it did to natural theology:
human reason is confined entirely within the field of phenomena, that is to say, to things that are perceptible to the senses, and in the manner in which they are perceptible; it has no right and no power to transgress these limits. [...] Given these premises, all will readily perceive what becomes of Natural Theology, of the motives of credibility, of external revelation.
The second was under suspicion of fideism:
Having reached this point [...] we have sufficient material in hand to enable us to see the relations which Modernists establish between faith and science. [I]n the first place it is to be held that the object of the one is quite extraneous to and separate from the object of the other. For faith occupies itself solely with something which science declares to be unknowable for it. Hence each has a separate field assigned to it: science is entirely concerned with the reality of phenomena, into which faith does not enter at all; faith on the contrary concerns itself with the divine reality which is entirely unknown to science.
Pascendi goes on to suggest that the modernists really meant to subject faith to science but were afraid to say so, and was even to find pantheistic implications in the position. It can be assumed that one target of this passage was Alfred Loisy's attempt to separate the results of the critical analysis of Scripture from the dogmatic claims of the Catholic Church, and that another was the memory of late mediaeval and Renaissance theories of double truth, truth in philosophy separate from truth in faith; but the concern of the first passage to preserve the integrity of natural theology shows the importance of wider considerations. Duhem is not one of those identified as targets of the encyclical. Whether his work was even known to those who drafted it must be a matter of speculation. But quite apart from his explicit disapproval of the enterprise, it is hard to envisage the kind of natural theology that could be accommodated to Duhem's account of the aim and structure of physical theory.
The issue was basic, the subject even of dogmatic definition, by the First Vatican Council of 1870-71, the Council that, in non-Catholic circles at least, is more famous for the definition of Papal infallibility. That Council, using the double negatives usual in such definitions, had declared anathema anyone who should deny that the knowledge of God was accessible to human reason. Though what that meant is not easily determined, most of the bishops present must have meant to say that the knowledge of God was accessible to human demonstration, not of course that knowledge of Him contained in the creeds and dogmatic formulations of the Church, but the knowledge that there is a good God who created all things: the rest belonged to Revelation, not reason.
But whatever the bishops thought they were doing thus making the demonstrability of God's existence a matter of faith, the general strategy is clear enough: to offer the faithful what are technically known as motives of credibility, reasons that would make it rational to accept the Catholic faith and ecclesiastical authority. The faithful could be assured that, God's existence being demonstrable by reason, it was rational to accept the dogmatic formulations of the faith concerning Him offered by the church, and rational also to accept the authority of the Church that propagated this faith, rational to support the Church as it defended its temporal power against the new kings of a reunited Italy, and rational also to defend the Church in its resistance to the Prussian rulers of Germany and republican rulers of France. At the same time, of course, it turned Catholics into a disaffected element in all three states, a disaffected element the authorities had to disarm at the price of their own survival.
So it emerges that by propagating a system of physics that undermines natural theology Duhem has rendered himself suspect of the heresy (for that is the effect of the Council's decree) of fideism, the belief that the faith rests on faith and nothing else, and that conclusion was explicitly drawn by F. Mentré, one of those who wrote on Duhem's work after his death. In his eyes, Duhem's views were of no use on religion because of this fideist taint, connected with what he identified as its Pascalian sources, the subject of the next chapter. Furthermore, the counterpart of fideism is philosophical scepticism, the doubt about the reliability of knowledge of any kind, about its ultimate guarantees. Thus in 1893 Eugène Vicaire detected it Duhem's views "the poison of scepticism" and was appalled that such views should appear in a Catholic journal, the Revue des Questions Scientifiques, in which Duhem's early articles appeared.
3. The Revival of Scholasticism
But the consequences go even further, and it is here that neo-Scholasticism falls to be considered. There is no point in asserting the demonstrability of God's existence, or of anything else for that matter, unless there is available a philosophical system to do it in, just as to demonstrate God's non-existence a philosophical system, such as that provided by the various brands of positivism, was equally necessary. In 1878 the Encyclical Aeterni Patris of Leo XIII hit the nail on the head by citing St. Paul in favour of its view that 'false philosophy' was the source of the modern apostasy. The 'true philosophy' offered in its place was a revived Scholastic philosophy, the philosophy associated by the encyclical indiscriminately with Thomas Aquinas and the latter's thirteenth-century Franciscan contemporary Bonaventure, but in practice looking more to Aquinas as interpreted by such sixteenth-century commentators as Cajetan and Suarez.
The groundwork for the encyclical had been laid over the previous half-century. At least since the sixteenth century, there had always been a tendency towards Scholasticism in Catholic philosophy and theology, and corresponding difficulties with 'modern' philosophies. For example, an episode better known that some because of the attention Leibniz paid it, is the persecution of the followers of Descartes in the 1670s and 1680s because of the difficulties their philosophy created for the Eucharistic doctrine of transubstantiation defined by the sixteenth-century Council of Trent, the doctrine that at the words of consecration the substance of the bread and wine are transformed into the substance of the body and blood of Christ, leaving only the so-called accidents of colour, taste, and texture unchanged. Despite the intentions of the Fathers of the Council of Trent, this doctrine can hardly be understood outside the philosophy of substance and accident in which it is stated, still less if matter is defined, as it was by Descartes, as essentially, in substance that is, the space it occupies—and nobody suggested that that changed at the words of consecration!
Just as at the end of the nineteenth century Duhem was to turn the philosophy of positivism against the anti-religious conclusions of the positivists, so, in the early nineteenth century a serious of attempts was made, all of them resulting in condemnations for their authors, to adapt philosophies of non-Catholic origin, such as those of Kant and Schelling, for the purposes of Catholic apologetic. As interpreted by many, the problem seems to have lain in perceived violations of the balance between faith and reason demanded of Catholic orthodoxy: reason was to offer motives of credibility, to prepare the ground for faith by making credible the acceptance of a faith such as that revealed in the Scriptures and the decisions of the Councils of the Church, but not to go further. A group of Rome-based Jesuits, of whom the most prominent were Matteo Liberatore and Joseph Kleutgen, argued that only a scholastic philosophy, looking to that of Thomas Aquinas, would do, and they convinced Gioacchino Pecci, the future Pope Leo XIII, of the merits of their case.
In the Pope's mind the main object of the scholastic revival was theological, but there were at least two others: a revaluation of the thought of the Middle Ages, and of the Church's rôle in it, and indeed a re-affirmation of its value in the face of widespread denigration; and the thought that a revived Scholasticism, judiciously interpreted, might be found of use in questions of modern philosophy and science, despite prevalent expectations to the contrary. In short, by this action the Pope intended to re-affirm, against all the apparent odds, what he saw as the Church's heritage, and from that would no doubt flow improved morale among the troops confronted by a hostile world and, even, greater respect for the Church among those who did not belong to it.
The encyclical thus led to a theological programme to re-examine and restate Aquinas's 'five ways' for proving God's existence, to a historical programme to recover the mediaeval materials on which scholastic thought to contemporary questions, whether social, ethical, or scientific. These different programmes were not of course independent: the scholasticism available to the Pope had been mediated by the work of commentators at least two centuries distant from Aquinas's own time, so that a genuine revival of his thought had to go back past these to the sources, to find out what Aquinas had actually said; Aquinas's work was also apparently embedded in an obsolete natural philosophy, of generally Aristotelian character, so that if his theology was recoverable, it had in some way to be reconciled with later scientific ideas; and any application to contemporary questions had to depend on answers to the prior question of what the philosophy was that was to be applied.
The later progress of scholarship was in due course to call into question most of the answers initially given, but it is these initial answers that concern Duhem: the work of Aquinas was supposed to consist principally in the reconciliation of Christianity with Aristotle, and in the mid-twentieth century Dom David Knowles, a historian with a low opinion of the scholars of the century following Aquinas, was still presenting Aquinas's supposedly successful achievement of this goal as the crowning intellectual achievement of the Middle Ages. It was this Aristotelianism of Aquinas, and of scholastic philosophy as it was generally understood, that was perceived to be the main stumbling block in the way of a Scholastic revival, and it will be a main problem for the remainder of this essay. To the extent that Duhem was involved in neo-Scholasticism, if he was so involved, that involvement can be expected to show itself in Aristotelian themes in his work, and by a sympathetic treatment of mediaeval Aristotelianism, as well as by associations with journals with generally neo-Scholastic policies.
4. Duhem the Scholastic?
Given the obstacles to scholasticism mentioned in the previous paragraph, it is understandable that different would-be scholastics adopted different strategies to meet them. The Roman seminaries, for example, apparently maintained an integral Thomism with few compromises towards modern science, but this was hardly a serious option for a practising mathematical physicist interested in having his work taken seriously by his contemporaries. Another obvious possibility would have been to reject the whole thing root and branch, either outright, or in the manner of the Italian Agostino Gemmelli, who in 1904-05 proposed a Scholasticism that included modern thought. I believe that something like this was Duhem's final position, but it was, as will be seen, decisively rejected by Pope Pius X in his Encyclical Pascendi of 1907, and something will be said below about how Duhem got there: it is by no means obvious that outright rejection was his position when he wrote them, and his earlier reactions to Blondel seem to point the other way.
An obvious intermediate position was to try to adapt scholastic natural philosophy to make it conform to the discoveries of modern science, the position that seems to have suited the eirenic temperament of Désiré Mercier, the future Cardinal Archbishop of Malines, and his group at Louvain. This was the programme behind the Société Scientifique de Bruxelles, which Duhem seems to have joined as a lecturer at Lille, and he seems to have appreciated its attitudes and approaches enough to attempt to recruit Paul Tannery into its membership. Apart from its Annales, which published his Les Théories Électriques de J. Clerk Maxwell, its principal organ was the Revue des Questions Scientifiques, a heavyweight quarterly carrying in depth discussions of the scientific questions of the day, but intended for an educated lay audience. it was not afraid of carrying long multipart articles, and of these Duhem became a major contributor: it seems to have been his preferred place of publication for general philosophical and historical pieces throughout the 1890s. This was the journal that carried his 'Réflexions' of 1892, Vicair's critique, and Duhem's replies. One of the latter, not considered so far in this essay, seems as good a place as any to begin a consideration of Duhem's possible relation to neo-Scholasticism.
'Physique et Métaphysique' of 1893 was the first of Duhem's replies to Vicaire. It addressed the suggestion, made by more than one of Duhem's readers, that his rigourous separation of physics from metaphysics was no more than a cover for denigration of the latter: the metaphysician was free to get on with it in his corner while the Duhemian physicist got on with it in his without interference, the implication being the positivist one that physics was the only real knowledge to be had. Duhem insisted that this was not his intention. On the contrary, metaphysics was for him a genuine form of knowledge, indeed "more excellent" than physics, but separated from physics by having different objects and being governed by different methods. The Scholastic expertise with which he set out his views seems to have impressed not a few of his readers enough to make them wonder whether he had a scholastic mentor, for in the normal course of events this was not the sort of expertise a physicist could be expected to have. Be that as it may, Duhem was claiming to be classifying independent and legitimate science, not distinguishing sense from nonsense in the manner of earlier and later positivists.
Repeated in Duhem's later writings, this move has been the main source for the view that Duhem's prime philosophical inspiration was neo-Scholastic. But initially plausible as this interpretation may seem, it becomes less so when Duhem is compared with a genuine neo-Scholastic like Jacques Maritain, who did indeed distinguish his sciences, but only so that thereafter he could unite them, assign each of them its place in the overall system of the sciences, and say which sciences could and could not establish what on the foundations of which others. The basis for Maritain's scheme, as of numberless others of like provenance, is the view that some sciences can be subordinated, or sub-alternated, to others in the Aristotelian scheme of things. A science is conceived of as a deductive system of syllogisms, deduced from one or more definitions of the essences that are the subject matter of that science, and remaining within its genus or natural kind, and it is supposed that the conclusions of one science can serve as principles for another, as when the sciences of equilibria and music take their principles, as subaltern science, from the superior sciences of arithmetic and geometry. Famously, this scheme ran into difficulties with the applied mathematical sciences, such as astronomy in ancient times and terrestrial physics in modern: if Aristotle was right, natural philosophy should have been subordinate to 'physics', or, in Duhem's terminology, 'metaphysics', but the mathematical science of nature soon left Aristotelian metaphysics far behind, a point that will be considered further below.
I mention now two aspects of such schemes: they were only achieved at the price of distorting Aristotle, for whom mixed sciences would have meant mixing genera, something his methodological principles forbade; and their rationalistic atmosphere, not to say hubris, is remote indeed from a scientific world in which, as in the physics of Duhem, mathematical formulae are devised to meet the problems thrown up by experiment, not those suggested or deduced from a priori theory. The reconstruction of Aristotle that would reconcile his views to modern science would have to be pretty radical, radical both at the level of method and of content.
Nevertheless, there seems to have been one aspect of Aristotle's system that Duhem found somewhat promising: its freedom from a priori selection of the primary qualities by which, in the manner of the mechanical philosophy of the seventeenth century, all secondary qualities had to be explained. For him, what qualities were primary and what secondary ought to be a purely pragmatic matter, decided by the progress of theory and experiment as successive theories succeeded in classifying wider and wider collections of data. Where, though, he differed from Aristotle is that his physics was to be a mathematical science; it was to classify qualities, not explain them, and do so by replacing their measured intensities by symbols subject to mathematical manipulation; it was to be mathematical science whose form no metaphysical system could decide a priori: that form too was to emerge from the progress of physics, as successive theoretical classifications of the mathematical intensities of qualities, and the implied classifications of the qualities these represented, hopefully converged on the natural classification that was the goal of physics.
Such was the 'Aristotelianism' that Duhem advertised in a variety of articles in the middle to late 1890s, particularly in his historical works culminating in Le Mixte et la Cominaison Chimique and L'Évolution de la Mécanique, before repeating it yet again in 'Physicque de Croyant' after which it disappears from view. It was a pretty minimal Aristotelianism, but after 1905 even that disappears from view. To my knowledge, Duhem never withdrew such views, but their disappearance from his later writings is indicative of his final decisive rejection of neo-Scholasticism and all it stood for, of those of his earlier attitudes that had made it reasonable for Blondel in 1893 to tease him as a peripatetic. Duhem's ultimate reasons for this shift are not completely clear—some possible answers will be explored later in this chapter—yet there can be no question but that it came at a critical time, when the so-called modernist crisis was at its height, and neo-Scholasticism lay at the hearth of that crisis.
I have already referred more than once to the Encyclical Pascendi Dominici Gregris of 1907: the prominent rôle neo-Scholasticism played in it can hardly escape the notice of any reader; implicit in the doctrinal part, it becomes explicit in the disciplinary part that follows. We are told that a distaste for the scholastic method is the surest sign of modernism in any writer (What else would be expected of an adherent of a modern philosophy but opposition to scholasticism?), and the text goes on to insist that scholastic philosophy is henceforth to be the basis for Catholic thought:
We will and ordain that scholastic philosophy be made the basis of the sacred sciences. [...] And let it be clearly understood above all things that the scholastic philosophy We prescribe is that which the Angelic Doctor has bequeathed to us, and We, therefore, declare that all the ordinances of Our Predecessor on this subject continue fully in force [...] Further let Professors remember that they cannot set St. Thomas aside, especially in metaphysical questions, without grave detriment.
The Angelic Doctor is a Scholastic appellation for Thomas Aquinas.
From all this Duhem had largely stood apart, and events will show him moving yet further away from it. Even in his historical work to date, unusual for its time in that Aristotle is taken seriously, there is no trace of the work on mediaeval science that was to be expected of a historically-minded Catholic scientist in that environment, and was in the end to do more than anything else to perpetuate Duhem's fame. But, as will be seen, when he does get involved in mediaeval science, what he finds is not perhaps what the Pope had in mind. While Pascendi was insisting on the centrality of scholastic philosophy Duhem was increasingly associated with a journal committed to opposing that philosophy. The best place to illustrate Duhem's developing distance from neo-Scholasticism is his association with the Annales de Philosophie Chrétienne.
—R.N.D. Martin's Pierre Duhem pgs. 38-49 with quotes from Pope Pius X's encyclical Pascendi Domenici Gregis
Duhem, like Heisenberg, thought there was a "sharp division between faith and science." Even though Duhem made noble discoveries in the history of science related to mediaeval science's contributions, due in large part to the Catholic Church, to modern science; he refused to see the importance of adopting a Scholastic-Thomistic philosophy not especially for theology but for science. Why is this philosophy needed? Primarily because it opposes Modernism and sees that faith and reason "are like two wings on which the human spirit rises to the contemplation of truth (bina quasi pennæ videntur quibus veritatis ad contemplationem hominis attollitur animus):"
III. Relation Between Faith and Science
Faith and Science
Q.—Can we now have some idea of the relations which the Modernists establish between faith and science, including, under this latter term, history?
A.—'16. Having reached this point [...] we have sufficient material in hand to enable us to see the relations which Modernists establish between faith and science, including history also under the name of science.'
Q.—What difference do they make between the object of the one and of the other?
A.—'And in the first place it is to be held that the object of the one is quite extraneous to and separate from the object of the other. For faith occupies itself solely with something which science declares to be unknowable for it. Hence each has a separate field assigned to it: science is entirely concerned with the reality of phenomena, into which faith does not enter at all; faith on the contrary concerns itself with the divine reality which is entirely unknown to science.
Q.—Then, according to them, no conflict is possible between faith and science?
A.—'Thus the conclusion is reached that there can never be any dissension between faith and science, for if each keeps on its own ground they can never meet and therefore never be in contradiction.
Q.—'And if it be objected that in the visible world there are some things which appertain to faith, such as the human life of Christ'?
A.—'The Modernists reply by denying this.'
Q.—How can they deny it?
A.—They say: 'For though such things come within the category of phenomena, still in as far as they are lived by faith and in the way already described have been by faith transfigured and disfigured, they have been removed from the world of sense and translated to become material for the divine.
Q.—'Hence should it be further asked whether Christ has wrought real miracles, and made real prophecies, whether He rose truly from the dead and ascended into heaven,' what do they answer?
A.—'The answer of agnostic science will be in the negative.
'The answer of faith in the affirmative'
Q.—But is not that a flagrant contradiction between science and faith?
'There will not be, on that account, any conflict between them. For it will be denied by the philosopher as philosopher, speaking to philosophers and considering Christ only in His historical reality; and it will be affirmed by the speaker, speaking to believers and considering the life of Christ as lived again by the faith and in the faith.'
Q.—Faith and science acting thus in entirely separate fields, will there be, according to the Modernists, no subordination of the one to the other.
A.—'17. [...] it would be a great mistake to suppose that, given these theories, one is authorised to believe that faith and science are independent of one another. On the side of science the independence is indeed complete, but it is quite different with regard to faith, which is subject to science.'
Q.—Faith subject to science! But on what ground?
A.—'Not on one but on three grounds.'
Q.—According to the Modernists, what is the first ground?
A.—'For in the first place it must be observed that in every religious fact, when you take away the divine reality and the experience of it which the believer possesses, everything else, and especially the religious formulas of it, belongs to the sphere of phenomena and therefore falls under the control of science. Let the believer leave the world if he will, but so long as he remains in it he must continue, whether he like it or not, to be subject to the laws, the observation, the judgments of science and of history.'
Q.—What is the second ground of the subordination of faith to science?
A.—'Further, when it is said that God is the object of faith alone, the statement refers only to the divine reality not to the idea of God. The latter also is subject to science which while it philosophises in what is called the logical order soars also to the absolute and the ideal. It is therefore the right of philosophy and of science to form conclusions concerning the idea of God, to direct it in its evolution and to purify it of any extraneous elements which may become confused with it.'
Q.—What is the third ground?
A.—'Finally, man does not suffer a dualism to exist in him, and the believer therefore feels within him an impelling need so to harmonise faith with science, that it may never oppose the general conception which science sets forth concerning the universe.'
Q.—Then, according to the Modernist doctrine, faith is in bondage to science?
A.—Yes. 'It is evident that science is to be entirely independent of faith, while on the other hand, and notwithstanding that they are supposed to be strangers to each other, faith is made subject to science.'
Q.—How did Pius XI and Gregory IX stigmatize such doctrine?
A.—'All this [...] is in formal opposition with the teachings of Our Predecessor, Pius IX, where he lays it down that: "In matters of religion it is the duty of philosophy not to command but to serve, but not to prescribe what is to be believed but to embrace what is to be believed with reasonable obedience, not to scrutinise the depths of the mysteries of God but to venerate them devoutly and humbly."
'The Modernists completely invert the parts, and to them may be applied the words of another Predecessor of Ours, Gregory IX., addressed to some theologians of his time: "Some among you, inflated like bladders with the spirit of vanity strive by profane novelties to cross the boundaries fixed by the Fathers, twisting the sense of the heavenly pages . . .to the philosophical teaching of the rationals, not for the profit of their hearer but to make a show of science ... these, seduced by strange and eccentric doctrines, make the head of the tail and force the queen to serve the servant."'
IV. Practical Consequences
The Methods of Modernists
18. [...]
Q.—Is this ["the mutual separation of science and faith"] done also in other scientific work?
A.—'So, too, acting on the principle that science in no way depends upon faith, when they treat of philosophy, history, criticism, feeling no horror at treading in the footsteps of Luther [Prop. 29, condemned by Leo X, Bull, Exsurge Domine, May 16, 1520: 'It is permissible to us to invalidate the authority of Councils, freely to gainsay their acts, to judge of their decrees, and confidently to assert whatever seems to us to be true, whether it has been approved or reprobated by any Council whatsoever.'], they are wont to display a certain contempt for Catholic doctrines, or the Holy Fathers, for the Ecumenical Councils, for the ecclesiastical magisterium; and should they be rebuked for this, they complain that they are being deprived of their liberty.'
[...]
I. Rules Relative to Studies
The Study of Scholastic Philosophy
[...]
Q.—Would it be a great disadvantage to set aside St. Thomas?
A.—'Further let Professors remember that they cannot set St. Thomas aside, especially in metaphysical questions, without grave detriment.'
[...]
Q.—According to what law ought the study of natural sciences to be regulated?
A.—'47. With regard to profane studies suffice it to recall here what Our Predecessor has admirably said: "Apply yourselves energetically to the study of natural sciences: the brilliant discoveries and the bold and useful applications of them made in our times which have won such applause by our contemporaries will be an object of perpetual praise for those that come after us" (Leo XIII. Alloc., March 7, 1880). But this do without interfering with sacred studies, as Our Predecessor in these most grave words prescribed: "If you carefully search for the cause of those errors you will find that it lies in the fact that in these days when the natural sciences absorb so much study, the more severe and lofty studies have been proportionately neglected—some of them have almost passed into oblivion, some of them are pursued in a half-hearted or superficial way, and, sad to say, now that they are fallen from their old estate, they have been disfigured by perverse doctrines and monstrous errors" (loco cit.). We ordain, therefore, that the study of natural science in the seminaries be carried on under this law.'
[...]
Conclusion
The Church and Scientific Progress
Triennial Returns
'57. This, Venerable Brethren, is what we have thought it our duty to write to you for the salvation of all who believe. The adversaries of the Church will doubtless abuse what we have said to refurbish the old calumny by which we are traduced as the enemy of science and of the progress of humanity. In order to oppose a new answer to such accusations, which the history of the Christian religion refutes by never failing arguments, it is Our intention to establish and develop by every means in our power a special Institute in which, through the co-operation of those Catholics who are most eminent for their learning, the progress of science and other realms of knowledge may be promoted under the guidance and teaching of Catholic truth. God grant that we may happily realise our design with the ready assistance of all those who bear a sincere love for the Church of Christ. But of this we will speak on another occasion.
'58. Meanwhile, Venerable Brethren, fully confident in your zeal and work, we beseech for you with our whole heart and soul the abundance of heavenly light, so that in the midst of this great perturbation of men's minds from the insidious invasions of error from every side, you may see clearly what you ought to do and may perform the task with all your strength and courage. May Jesus Christ, the author and finisher of our faith, be with you by His power; and may the Immaculate Virgin, the destroyer of all heresies, be with you by her prayers and aid. And We, as a pledge of Our affection and of divine assistance in adversity, grant most affectionately and with all Our heart to you, your clergy and people the Apostolic Benediction.
'Given at St. Peter's, Rome, on the 8th day of September, 1907, the fifth year of our Pontificate.'
—Fr. John Fitzpatrick's Catechism of Modernism with quotes from Pope Pius X's encyclical Pascendi Domenici Gregis
Even though Pope St. Pius X says that one "cannot set St. Thomas aside [...] without grave detriment," he was primarily referring to the teaching and study of theology, and in section 47. appears to think the natural sciences should be regulated by a different set of rules and "without interfering with sacred studies." Yet in both cases he is stressing the importance of St. Thomas's "perennial philosophy" in order to avoid "perverse doctrines and monstrous errors" not only in sacred sciences but also in the "profane" or secular natural sciences as well. A notable example of a "perverse doctrine" or teaching in physics is the "many worlds" interpretation of quantum mechanics. In the solid "perennial philosophy," only one world can exist. (Summa Theologica Iª q. 47 a. 3).

The mathematician and philosopher Wolfgang Smith (author of "From Schrödinger's Cat to Thomistic Ontology" in The Thomist) calls Modernism's influence on science the "plague of scientistic belief:"
Yet, despite the fact of quantum indeterminism, not a few eminent scientists continue to champion the mechanistic tenet. Albert Einstein himself, as one knows, so far from admitting that the discoveries of quantum physics have overthrown the classical postulate, argued precisely in the opposite direction: it is the principle of determinism, he said in effect, that invalidates quantum mechanics as a fundamental theory. This illustrates quite clearly the philosophical and indeed a priori character of the tenet in question, and the fact that propositions of this kind can neither be verified nor falsified by empirical findings. This fact, however, remains generally unrecognized, with the result that the postulate of universal mechanism has retained to this day its status as a major article of scientistic belief.
My second example pertains to a more fundamental stratum of philosophical thought, and is consequently still more far-reaching in its implications: “physical reductionism,” let us call it (for reasons which will presently become clear). The thesis hinges upon an epistemological assumption, an idealist postulate, one could say, which affirms that the act of sense perception terminates, not in an external object as we commonly believe, but in a subjective representation of some kind. According to this view, the red apple which we perceive exists somehow in our mind or consciousness; it is a subjective image, a fantasy which mankind has all along mistaken for an external object. Thus thought René Descartes, to whom we owe the philosophical foundations of modern science. Descartes sought to correct what he took to be the mistaken notions of mankind concerning perceptible entities by distinguishing between the external object, which he termed res extensa, and its subjective representation existing in the mind or so-called res cogitans. What was previously conceived as a single object (and what in daily life is invariably regarded as such) has therefore become split in two; as Whitehead has put it: “Thus there would be two natures, one is the conjecture and the other is the dream.” It is to be noted that this Cartesian differentiation between the “conjecture” and the “dream” goes not only against the common intuitions of mankind, but is equally at odds with the great philosophical traditions, including especially the Thomistic, where the opposition becomes as it were diametrical. Now, it is this questionable Cartesian doctrine—which Whitehead refers to as “bifurcation”—that has served from the start as the fundamental plank of physics, or better said, of the scientistic world-view in terms of which we habitually interpret the results of physics. And once again we find that the two disparate factors—the operational facts of physics and their customary interpretation—have become in effect identified, which is to say that the tenet of bifurcation does indeed function as a scientistic belief.
I would like to emphasize that in addition to the fact that bifurcation contradicts the most basic human intuitions as well as the most venerable philosophical traditions, there is also not a shred of empirical evidence in support of this heterodox position. Nor can there be, as follows from the fact that physics can be perfectly well interpreted on a non-bifurcationist basis, as I have shown in a recent monograph. It turns out, moreover, that the moment one does interpret physics in non-bifurcationist terms, the so-called quantum paradoxes—which have prompted physicists to invent the most bizarre ontologies—vanish of their own accord. It seems that quantum physics has thus implicitly sided with the pre-Cartesian world-view.
—Wolfgang Smith's "The Plague of Scientistic Belief"
This mechanism has far-reaching philosophical consequences which in turn influence how subsequent science is performed and interpreted. E.g., in Galileo's description of tickling, he adopts a subjectivist, agnostic philosophy because he denies that the nature of "feather," something beyond the senses, can be known; for him, sense knowledge begins and ends in the senses.
I move my hand first over a marble statue and then over a living man. To the effect flowing from my hand, this is the same with regard to both objects and my hand [Here he assumes spatial motion is all that matters.]; it consists of the primary phenomena of motion and touch, for which we have no further names. But the live body which receives these operations feels different sensations according to the various places touched. When touched upon the soles of the feet, for example, or under the knee or armpit, it feels in addition to the common sensation of touch a sensation on which we have imposed a special name, "tickling." This sensation belongs to us and not to the hand. Anyone would make a serious error if he said that the hand, in addition to the properties of moving and touching, possessed another faculty of "tickling," as if tickling were a phenomenon that resided in the hand that tickled. A piece of paper or a feather drawn lightly over any part of our bodies performs intrinsically the same operations of moving and touching, but by touching the eye, the nose, or the upper lip it excites in us an almost intolerable titillation, even though elsewhere it is scarcely felt. This titillation belongs entirely to us and not to the feather [therefore subjectivism]; if the live and sensitive body were removed it would remain no more than a mere word. I believe that no more solid an existence belongs to many qualities which we have come to attribute to physical bodies-tastes, odors, colors, and many more.
—Galileo's Il Saggiatore pg. 275
As interesting as Gailileo's description of tickling may be, he is wrong. Here is why:
Those who hold for the subjectivity of sensible qualities maintain that such qualities have no existence independently of the sensing subject, and on this ground effectively deny the very existence of objective intentions for such qualities. They find convincing Galileo's example of the movement of a feather across the skin to explain the tickle. So they introduce a distinction between primary qualities such as movement and secondary qualities such as the sensed tickle, and hold that the primary qualities have objective existence whereas secondary qualities do not. As a result they populate the universe with particles in motion and attempt to explain all sensations by the various kinds of movement these particles undergo, meanwhile denuding the objective world of sensible qualities in their traditional understanding.
The source of the difficulty here is an improper grasp of the role of the mental representation in the knowledge act. To think of the concept as what is known, rather than seeing that the nature is what is known, though by means of the concept, is to cut oneself off from intellectual knowledge of the real, for one is always left wondering about any extra-mental reality to which the concept might correspond. Similarly, to think of the sensation or the percept as itself what is known, rather than seeing the sensible quality as what is known, though by means of the sensation or percept, is to be imprisoned within one's sense organs and brain. The result is a radical solipsism that prohibits individuals from ever making statements about the objects of experience, leaving them to dwell in a world of their own imaginings.
The tickle may be something sensed on the surface of the skin, but that admission surely does not permit the inference that there is no movement there, or extending the argument further to hold that there is no heat in boiling water, no color in a ruby or a rose, no sound in the cry of a bird, or no odor or taste in an onion. All of these are accidents or accidental modifications of the subjects in which they are sensed. Just as those subjects have natures (inorganic, plan or animal in kind), so accidents may be said to have natures in an analogous sense. And even if we cannot know precisely the nature of heat, of color, and so on, we can at least model those natures in terms of the modalities they introduce in the components of the substantial natures in which they exist, namely the electrons, atoms, and molecules [...]
With the adoption of a Thomistic philosophy, we can avoid all these sorts of philosophical and scientific methodological errors; we can interpret quantum mechanics correctly, for example; and, in short, we can prevent scientistic belief from plaguing science.

Wednesday, December 2, 2009

Is Knowledge Divided? No.

Regarding Pasteur, Heisenberg, Pauli, Gödel and the limits of human knowledge, we mentioned Werner Heisenberg's "sharp division between knowledge and faith," with which, according to him, a society cannot cope very long. Why must not there be a "sharp division between knowledge and faith," something that St. Thomas Aquinas thinks when he says that "science and faith are not about the same things?" Is this true always? Does this also entail a sharp division between philosophy and science?
[I]t is impossible that one and the same thing should be believed and seen by the same person. Hence it is equally impossible for one and the same thing to be an object of science and of belief for the same person. It may happen, however, that a thing which is an object of vision or science for one, is believed by another: since we hope to see some day what we now believe about the Trinity, according to 1 Corinthians 13:12: "We see now through a glass in a dark manner; but then face to face": which vision the angels possess already; so that what we believe, they see. On like manner it may happen that what is an object of vision or scientific knowledge for one man, even in the state of a wayfarer, is, for another man, an object of faith, because he does not know it by demonstration. Nevertheless that which is proposed to be believed equally by all, is equally unknown by all as an object of science: such are the things which are of faith simply. Consequently faith and science are not about the same things.

Summa Theologica IIª-IIae q. 1 a. 5 co.

This does not, however, deny that "God's existence is not merely an object of natural rational knowledge, but also an object of supernatural faith (De fide)" because
As ground for [it being "impossible for the same truth to be known and believed by the same person": impossible [...], quod ab eodem idem sit scitum et creditum (S. th. 2 II, 1, 5),] [St. Thomas] submits that the clear insight into the truth associated with knowledge cannot co-exist with the obscurity of faith. It is, however, possible, that the same truth could be known by one person and believed by another. According to the teaching of St. Thomas, it is also possible for the same person at the one time to have a natural knowledge of the existence of God as the originator of the natural order, and a supernatural faith in the existence of God as the originator of the supernatural order, because the supernatural faith comprehends truths which are not contained in natural knowledge (difference of material object).

—Dr. Ludwig Ott's Fundamentals of Catholic Dogma pg. 17

Yet faith is not obscure; the Christian faith "so quickens the human mind that without difficulty it pierces the heavens, and, illumined with divine light, contemplates first, the eternal source of light, and in its radiance all created things" (Trent Catechism) and it "is a kind of knowledge, inasmuch as the intellect is determined by faith to some knowable object" (Summa Theologica Iª q. 12 a. 13 ad 3). Possibly because of this, St. Albertus Magnus, St. Thomas's teacher, believed that it is indeed possible "for the same truth to be known and believed by the same person." But why does knowledge of God, regardless of how one obtains it, even matter with regard to the physical sciences? The reason is that God is the cause of the physical world, and one can know effects by their causes and vice versa (cf. Rom. 1:20 and 1 Cor. 2:10). Even though humans are finite beings, we have immaterial, intellectual souls, viz., intellects or minds. Human souls animate the human body, and, unlike animals' souls, still exist after the death of the body because they do not depend on matter for their existence. That human souls are eternal we prove as follows: Because a knowing power, e.g. the eye, contains the form of the the thing known, color; similarly, the human intellectual soul—the self-subsisting, eternal, God-created form of a human being's material body—is also a knowing power. The things it knows are universal concepts, unique to humans. We, unlike animals, can abstract from particular sense-knowledge, e.g. "this bird," to the universal concept of "bird." Because universal concepts are immaterial, so too must our souls be because universal concepts are proper to them. St. Thomas explains it thus:
[W]e may proceed from the specific notion of the human soul inasmuch as it is intellectual. For it is clear that whatever is received into something is received according to the condition of the recipient. Now a thing is known in as far as its form is in the knower. But the intellectual soul knows a thing in its nature absolutely: for instance, it knows a stone absolutely as a stone; and therefore the form of a stone absolutely, as to its proper formal idea, is in the intellectual soul. Therefore the intellectual soul itself is an absolute form, and not something composed of matter and form. For if the intellectual soul were composed of matter and form, the forms of things would be received into it as individuals, and so it would only know the individual: just as it happens with the sensitive powers which receive forms in a corporeal organ; since matter is the principle by which forms are individualized. It follows, therefore, that the intellectual soul, and every intellectual substance which has knowledge of forms absolutely, is exempt from composition of matter and form.

Summa Theologica Iª q. 75 a. 5 co.

Therefore, the mind does not depend on an organ like the brain; the brain, along with the other sense organs, informs the soul because all knowledge originates in the senses. Since the human intellect is capable of knowing so many great universal concepts and because the soul is not divided—e.g., into a soul or power of the soul for subjective knowledge, scientific knowledge, faith knowledge, etc.—it behooves empiriological scientists to combine the various modes of knowledge—philosophy, science, faith, and reason—to arrive at a more accurate, truer knowledge of reality. With God's grace, which builds on man's finitude and human nature, humans can indeed reach truly amazing intellectual heights, perhaps even devise a "theory of everything" in a new, paradigm-shifting physics.

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. [Œ 44, 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