Galileo in the first half of Vatican II

Below are three examples from the first half of Vatican II where Galileo is explicitly mentioned by name.

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Bp. André Charue of Namur, Belgium, on 17 Nov. 1962, regarding a revision (p. 114-115) of the schema De sacra liturgia, said (p. 145):

Attendite, venerabiles Patres, ad conditionem eorum omnium, qui cum fide catholica componere debent scientificum laborem in universitatibus, in omnibus scientiarum circulis. Exemplum Galilaei et alia exempla recentiora sufficiant! Immaturae declarationes alicuius Concilii, propter earum solemnitatem, onerare possent, dicamus in semisaeculum, conditionem scientificorum.

[Beware, venerable Fathers, of the condition of all those who with catholic faith must compose scientific work, in all scientific circles. Let Galileo and the other more recent examples suffice! The immature declarations of some in this Council, because of their solemnity, could aggravate—we speak in the mid-century—the condition of the sciences.]

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Bp. emeritus of Innsbruck, Austria, Paulus Rusch (1903-1986) explicitly mentioned Galileo during the 22^nd meeting, 19 Nov. 1962, in his intervention (p. 356-357) against ch. 2, #12 ("Inerrancy") of the first schema the fathers voted on: De fontibus revelationis. Cdl. Siri's intervention, which mentioned Pope St. Pius X and Modernism, is on p. 38-39. The very next day, 61% of the council fathers rejected the schema (cf. Ratzinger Reader pp. 258 ff.). John XXIII thereafter called upon a mixed commission (incl. Cdl. Frings, whom Fr. Ratzinger advised, and Rahner) to redraft it. Cdl. Frings said (p. 34-35) the original schema was too scholastic and professorial in tone, "nec aedificans nec vivificans" ("neither edifying nor vivifying")!

Here is De fontibus revelationis ch.2, #12 on inerrancy:

Because divine Inspiration extends to everything, the absolute immunity of all Holy Scripture from error [PTC had said "the infallibility and inerrancy"] follows directly and necessarily. For we are taught by the ancient and constant faith of the Church that it is utterly forbidden to grant that the sacred author himself has erred, since divine Inspiration of itself as necessarily excludes and repels any error in any matter, religious or profane, as it is necessary to say that God, the supreme Truth, is never the author of any error whatever. [Pius XII, Divino afflante (EB 539), using the words of Leo XIII, Providentissimus Deus (D 1950); see also EB 44, 46, 125, 420, 463, etc.]

After giving an example of how Matt. 27:9 allegedly errs by quoting Jeremiah when it apparently was really quoting the prophet Zachary, Bp. Rusch said (p. 357):

Accedit nostram Ecclesiam hac in re iam duram passam esse experientiam. Anno 1633 Galilei sub Urbano VIII damnatus est, quia defendit doctrinam contra Scripturam. Doctrinam autem quam defendit erat, sicut notissimum est, terram circa solem rotare et non viceversa. [Additionally, our church has already suffered a hard experience in this matter. In 1633 Galileo was condemned under Urban VIII because he defended a doctrine contrary to Scripture. But the doctrine that he defended was, as is well-known, that the earth revolves around the sun and not vice versa.]

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Bp. Michel Darmancier (1918-1984), titular of Augurus, commenting on the "De ecclesiæ magistero" section (p. 47-54) of the 23 Nov. 1962 schema De ecclesia (p. 12 ff.), wrote in his "written animadversion" (p. 452):

De illis enim contingentibus elementis sicut in fide et theologia proprie dicta consentire possunt theologi per saecula et per totum orbem catholicum, illa intimius coniungentes cum dogmatibus, quin exinde oriatur quaevis certitudo de illorum veritate, etsi concludi potest fidem ex illis detrimentum non timere. Sic, usque ad saeculum XVI, unanimiter docuerunt theologi terram centrum universorum esse, unde Galileus quidam satis notas difficultates cum sancta Inquisitione expertus est.

[The theologians can agree, throughout the ages and the whole catholic world, on those contingent elements in faith and theology properly speaking which are intimately connected with dogmas, which might not arise from any certainty of their truth, although to fear a loss of faith from them cannot be concluded. Thus, until the 16^th century, theologians unanimously taught that the earth was the center of the universe, whence Galileo experienced some well-known difficulties with the holy Inquisition.]

Prof. Karsten Danzmann, beantworten Sie bitte 3 Fragen über das LIGO Experiment!

The Principle Podcast Episode 9: Extended Interview Bernard Carr

The Principle Podcast Episode 9: Extended Interview Bernard Carr: Episode 9 from “The Principle” podcast is extended interview excerpts with Bernard Carr. About Bernard Carr: Bernard J. Carr is a professor of mathematics and astronomy at Queen Mary University of London (QMUL). He completed his BA in mathematics in 1972 at Trinity College, Cambridge. For his doctorate, obtained in 1976, he studied relativity and cosmology under Stephen Hawking at the Institute of Astronomy in Cambridge and the California Institute of …

Ampère's force law masterpiece in English

Ampère's Electrodynamics: Analysis of the Meaning and Evolution of Ampère's Force between Current Elements, together with a Complete Translation of ... Phenomena, Uniquely Deduced from Experience by André Koch Torres Assis
My rating: 5 of 5 stars

Free complete PDF version of this book available here.

This is the first published English translation of Ampère's masterpiece, Mémoire sur la théorie mathématique des phénomènes électrodynamiques uniquement déduite de l’expérience (cf. Godfrey's unpublished one).

Assis & Chaib give an excellent introduction to Ampère's work, as well as many computer-generated illustrations of Ampère's ingenious experiments.

It is enough to quote Maxwell on the importance of Ampère's work:

The experimental investigation by which Ampère established the laws of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the ‘Newton of electricity.’ It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electrodynamics.

Also, Ampère invented the word "electrodynamics"! ☺

This work shows how Ampère experimentally determined the so-called Ampère force law (not to be confused with one of Maxwell's equations, called Ampère's circuital law, which has nothing to do with Ampère besides dealing with currents, as Ampère did not deal with the field concept).

View all my reviews

Was Galileo persecuted?

Before Copernicus, Bishop Nicole Oresme (d. 1382) advanced the hypothesis that the earth, not the heavens, rotates diurnally. He was not condemned because he did not reinterpret Holy Scripture to support his scientific view.

Galileo was condemned because he ventured into Scriptural exegesis—in, e.g., his 1615 Letter to the Grand Duchess Madame Christina Lorraine—contrary to the unanimous consent of the Fathers of the Church and the Council of Trent. Copernicus did not do Scriptural exegesis regarding heliocentrism.

Galileo was condemned as "vehemently suspected of heresy" for holding "The proposition that the Sun is the center of the world and does not move from its place[, which] is absurd and false philosophically and formally heretical, because it is expressly contrary to Holy Scripture." (1633 Condemnation).

To say Galileo was persecuted seems to imply he adhered to a different religion than Catholicism. He was Catholic, hence the Church had jurisdiction over him in moral or religious matters. His house arrest was quite unusual; it was really a paid retirement, during which he wrote his most important physics work, The Two New Sciences (1638).

As the Tuscan ambassador Francesco Niccolini wrote on 27 February 1633 (p. 225 of Maurice A. Finocchiaro's The Galileo Affair: A Documentary History):

His Holiness [Pope Urban VIII] answered that he had done Mr. Galilei a singular favor, not done to others, by allowing him to stay in this house [the Tuscan embassy] rather than at the Holy Office, and that this kind procedure had been used only because he is a dear employee of the Most Serene Patron [the Pope] and because of the regard due to His Highness [the Pope]; for a Knight of the House of Gonzaga, son of Ferdinando, had been not only placed in a litter and escorted to Rome under guard but was taken to the Castle and kept there for a long time til the end of the trial. I showed myself to be aware of the nature of the favor, and I humbly thanked His Holiness [the Pope];

and on 16 April 1633 (p. 250-51 of ibid.):

Indeed, there is no precedent of anyone ever having been interrogated during a trial without being detained in a prison cell, and in this regard he has profited from being employed by His Highness [Pope Urban VIII] and from being lodged at this house; nor is there knowledge of anyone else (whether bishop, prelate, or nobleman) who, immediately upon his arrival in Rome, has not been kept at the Castle or at the same palace of the Inquisition, subject to all rigor and strictness. Furthermore, they even allow his servant to wait on him, to sleep there, and, what is more, to come and go as he pleases, and they allow my own servants to bring him food to his room from here and to return to my house morning and evening.

This singular treatment can hardly be considered a persecution.

His house arrest began at the same Tuscan embassy on 24 June 1633. On 1 December 1633, the Pope allowed Galileo to return to his villa in Arcetri, near Florence, where he stayed for the rest of his life.

Physics: A Holistic Perspective

My friend wrote the following; it's an excellent summary of modern physics from a Catholic perspective. Homeschoolers would also benefit from it because of the summary questions at the end:

I. The Nature of Physical Science

A. Truth, Opinion, Science, and Faith

Before commencing the study of any subject matter, one ought to know its definition, scope, and methods. We must therefore examine the nature of physics, both in itself, and in relation to other branches of study. Let us attempt to understand it by first defining what we mean by “science.”

Physics is one among many different sciences. Today, when we say “science,” we usually mean one of the physical sciences. But that is simply a prejudice of the modern mind. “Science” is derived from the Latin “scientia,” which means “knowledge.” This clearly distinguishes it from technology, which is an application of this knowledge to the practical needs of life. Truth is the correspondence between our judgments and reality. Truth does not lie in our ideas, but in the affirmations and denials we make with those ideas. If our judgment about the world accurately reflects reality, then we call it a true judgment; otherwise, we call it a false judgment. Opinion can therefore in some sense be called “knowledge,” inasmuch as some opinions are indeed true. So what distinguishes science from true opinion?

We are often correct in our assertions about the world, but cannot say why. We cannot give a reasoned account of our knowledge. We merely have true opinions. Science, then, to distinguish it from true opinion, is defined as “knowledge through causes.” That is to say, if we know the causes (or reasons) for something, then we understand what it is and how it behaves in varying circumstance. If we remove the causes, then we necessarily remove these behaviors, and perhaps even the thing itself. In this way we have added to our true opinion a reasoned account of our knowledge. Naturally, some events cannot even in principle be the subject of human science. Any contingent event, that is, any event that may or may not occur in a fixed set of circumstances, cannot be the subject of a science. For instance, since human will is free to choose, one cannot prove or predict its acts. For if the will were determined by the circumstance in which it must choose, it would cease to be free. One consequence then of “rationalism” (the false opinion that all reality is subject to human science) is that free will is denied. God, by way of comparison, does have knowledge of these “future contingents,” for it is God Who moves our wills directly. In like manner, history cannot be a human science, because its course is not governed solely by material factors, but by human beings acting freely. In general, if the causes of anything are inaccessible to our human intellects, that thing cannot be the subject of a human science.

A question that will naturally occur to students concerns the place of the Catholic Faith: Is their act of faith merely true opinion, science, or something else entirely? For the theological modernist, faith is an opinion that answers to nothing objective, but to the subjective needs of men. In other words, it is even less than true opinion. It is opinion that corresponds not to a single reality, but to the desires of many different believers, which can be and usually are contradictory. This modernist conception of faith is utterly false, but before we examine the true nature of faith, we ought to consider why the modern world has taken refuge in this empty notion. It is clear that faith cannot be the same as science, because we do not know the causes of much that is comprehended under faith. There are two reasons for this. First, many articles of the Faith concern contingent events: the fall of man, the Incarnation, the Redemption, and many others. These belong properly to the study of salvation history, and depend upon not only the free choices of creatures, but the absolute freedom of the Creator. Second, some of the articles of faith transcend our human capacities of rational demonstration. Examples here must include the doctrine of the Trinity and the miracles performed by Christ. At this point the modernist goes wrong, because he is really a rationalist. Logically speaking, if the whole universe is knowable by human science, and human science has no way of evaluating the articles of faith, these articles must not correspond to anything in reality. These false friends of religion therefore seek some substitute for objective reality. All that remains to them is their subjective consciousness, so they set up a correspondence between faith and subjective feeling instead. Faith for the modernist is not about the world “out there” (reality), but about the world “in here” (feeling). Consequently authentic religion is for the modernist not a life in conformity with the Divine Will, but one in conformity with personal longings.

If, however, we are honest about the limits of human reason, our humility will lead us to a better account of the act of faith and, through this act of faith, to a more profound understanding of God and the universe of beings created by God. The Catholic Faith, which makes dogmatic claims about reality that can and often do conflict with faulty claims of human science, is most certainly true in every respect. Such claims as conflict with the Catholic Faith are not really science at all, or even true opinion. They are falsehoods. But what allows us to profess the Faith with this absolute certainty?  What character does it have beyond true opinion? An act of faith is an act of trust in the authority of a revealer. It is an entirely reasonable and rational act if we can ascertain that the revealer both has the knowledge in question and will not deceive us.

Modern scientists ask for just such an act of faith from their pupils. The student has no way of verifying the immense number of observations and inferences that have been incorporated into modern science. He must depend upon the knowledge and veracity of his instructor, who himself depended upon another. Thus, there develops a scientific tradition (handing down) of previous observation and inference. The student, for his part, has good reason to have faith in his instructor. The instructor is likely to have credentials indicating his proficiency in the subject matter. He may have accrued many years of service at a prestigious university or scientific institution. Moreover, he would be severely penalized for deviating in essentials from that tradition. Peer reviews and academic evaluations continually guarantee that the scientist conforms to the generally accepted views of the scientific community. The instructor has few credible incentives to misrepresent physical theories to the student and will suffer serious consequences if he systematically does so. It is therefore entirely reasonable for the student to submit his mind to that of his teacher, so long as the scientist remains within the boundaries of his competence.

The act of faith that the Catholic makes is similar, but more certain and absolute, depending also upon the gift of God. First he determines, either by his own reason, by his common sense, or by following the reasoning of another, that there is a God, Who has every perfection of Being, including Truth and Goodness. These truths are sometimes called the “Preambles of the Faith,” because they are the reasonable foundation on which our act of faith depends. Now God has intervened in human history, initially by sending the prophets, but finally by sending His Eternally-Begotten Son, Jesus Christ, born into time of the Blessed Virgin Mary. By His public miracles and especially by His Resurrection from the dead, He has proven His Divinity. We may therefore conclude two things: First, as perfect Truth, Christ has knowledge of all things created and uncreated. Second, as perfect Goodness, He can have no desire to deceive us and wills only the good. His Doctrine is therefore true in all details and best calculated for the good of those who will receive it. We cannot pick and choose which doctrines to believe, but are obliged to assent to Revelation in its entirety because of the Authority of the Revealer. To assure that the Tradition (handing down) of the Faith is maintained until the end of time, He established a visible Church that is One, Holy, Catholic, and Apostolic.

Faith and science can never be in contradiction, for all truth has its ultimate source in God. But we must be careful to distinguish their relative certainties. It is science that cannot contradict Faith, and not vice-versa, for the Divine Revealer is more perfect in truth and goodness than any creature, and Christ has demonstrated His Divinity with unshakable proofs. We must reject, therefore, whatsoever we find in human doctrines that either contradicts or yields a conclusion contradicting Divine Revelation, bearing in mind always the Magisterium (Teaching Authority) of the Catholic Church.

B. Three Orders of Scientific Abstraction

Since science is “knowledge through causes,” we naturally seek to understand what is meant by causes, for we cannot use these as instruments of understanding unless we can accurately recognize and classify them. But it is important to first distinguish clearly between the three principal sciences applied to the material world: metaphysics, mathematics, and physics. In the classical and medieval traditions, these sciences study real beings, but each studies beings under a different aspect. Metaphysics is the highest of these, for it is “the science of being as being.” In saying that it is a “science of being,” we simply mean that we are studying real beings. In saying that metaphysics is “the science of being as being,” we indicate that the particular aspect under which we are studying beings is precisely their being. The first is materially what we are studying, the second is the formal aspect under which it is studied. In other words, we are studying beings just insofar as they are beings and no further. We are not interested in their particular nature, whether they are living or inert, but only what can be inferred about them from their existence. This definition will become clearer if we compare metaphysics to mathematics. Mathematics likewise studies real beings, but under the aspect of quantity. So mathematics can be defined as “the science of being as quantified.” It is concerned with beings only insofar as they can be measured. Finally, the classical and medieval traditions understand physics as “the science of being as movable.” In other words, physics considers ens mobile, which is Latin for “mobile being.”

These three sciences are traditionally understood to form a hierarchy, with metaphysics at the top, followed by mathematics, and then physics. This hierarchy corresponds to differing orders of abstraction, with physics being the least abstract, mathematics more abstract, and metaphysics the most abstract. When we say that one is more abstract than another, we do not necessarily mean that one is more complicated than another, or even more difficult, but rather that as we progress upwards, we leave out certain aspects of real beings. Let us consider a concrete example. Real material beings possess individual matter that makes each thing to be this thing and no other. For example, Socrates is individuated (made an individual) by this particular matter (these hands, this snub nose). But the human intellect can only apprehend things abstractly; there can be no science of individuals. The first abstraction is to leave out the individuality (or thisness) of the matter. What is left is matter insofar as it is sensible (able to be perceived by the senses). The study of beings under this aspect is called “physics.” Physics studies Socrates not insofar as he is Socrates, but insofar as he is a man. It pertains to a man to have matter, indeed, but not any particular matter.

When objects are studied mathematically, they are studied only insofar as their matter is intelligible, for the intellect leaves out of consideration their sensibility. When we imagine a mathematical triangle, we consider it extended in two dimensions, but this extension is only intelligible to the understanding, not perceptible to the five senses.  Likewise, when we study Socrates mathematically, we consider him only insofar as he has a measurable body. (e.g. He is 68 inches tall. He weighs 160 pounds.) Finally, metaphysics abstracts from matter entirely, for it considers only the existence of the object and any other attributes that accrue to it from that existence alone. So we consider Socrates insofar as he has being and the unity, truth, and goodness, that necessarily accompany being.

Science:	Studies:	Example:	Leaves Out:	Supplies Principles To:
metaphysics	“being as being”	Socrates as being	all matter	mathematics, physics
mathematics	“being as measurable”	Socrates as body	sensible matter	physics
physics	“being as movable”	Socrates as man	individual matter

Table I.A:  The Hierarchy of Abstraction in the Sciences

This hierarchy of abstraction among metaphysics, mathematics, and the physical sciences, is also a hierarchy of principles. A superior (more abstract) science is able to supply principles of study to its subordinate sciences. The definition of motion in physics must be taken from metaphysics, for physics cannot define its own subject matter. Likewise, physical science depends upon mathematics for its understanding of quantity, which is absolutely crucial to the study of bodies in motion. The lower science cannot supply the higher examples with principles for study, but only concrete material for the higher science to study. So physical science provides bodies to be studied mathematically, and both physical science and mathematics provide beings to be investigated by metaphysics. The principles of study descend downward from higher science to lower, while the matter to be studied ascends from lower science to higher. Before we proceed to a study of physical science, then, it is necessary to discover in metaphysics some basic principles to guide us.

C. Motion and the Four Causes

We know from experience that there is change or motion. It is inescapably part of this world. There are also many different kinds of motions. So we would do well to define motion first.

A little thought will convince the student that the task is no easy one. It is one thing to recognize motion when we see it, but to define it requires a good deal of genius, such as that possessed by Aristotle. He recognized that to define motion, we need to say something about the terms of motion, that is, the before and after of every change. Aristotle saw clearly that change is only possible when something actual is also potentially something else. For example, a piece of bronze may actually be a bust of Julius Caesar, but it is potentially a bust of Marcus Brutus, that is, if it were melted down and recast in a different shape. Before any such change takes place, there must be these two principles in a thing: First, the matter (bronze), which can potentially be given a different form. Second, the form itself (Caesar), which determines it to be this thing and no other. The bronze can potentially become many different things; we say that this matter has a potency to become all of them, but can be actually only one of them at a time. Perhaps now we can understand what is meant when it is said that “being is divided into act and potency.” These are two principles that all created beings possess, and it is in terms of them that we must define motion.

We might be tempted to say that “motion is a change from potency to act,” and we would be correct, but we would have failed to define “motion.” Why? Because we would then be required to define “change,” which, as a little thought will make clear, is really nothing more than the “motion” we set out to define. This is an example of a logical fallacy called “petitio principii,” or “begging the principle.” Motion cannot be defined as a particular kind of motion! Aristotle, on the other hand, saw that in every continuous motion the object is in varying degrees of reaching its final actuality. To return to our example, as the molten bronze is poured into the cast, it gradually assumes the form of the bust of Brutus. At each moment it is actually some particular shape on its way to becoming Brutus. Aristotle saw that something like this was happening in every continuous motion. He therefore defined “motion” as “the entelechy of a being in potency insofar as it is in potency.” This term, “entelechy,” means something close to “act,” but it also includes the idea of reaching its end. As it has come down to us through the medieval Latin tradition, motion is defined simply as “the act of a being in potency insofar as it is in potency.” In our example, the motion from Caesar to Brutus would be “the shape of the bust of Caesar which is potentially Brutus, insofar as the bust is potentially Brutus.” The potency of being Brutus lies in the bronze, not in the current shape of Caesar. In other words, it is not “the shape of the bust of Caesar which is potentially Brutus insofar as it is actually Caesar.” This shape would be its present form, the shape of Caesar’s bust. Motion is something’s act precisely in its ability to be something else, that is, just insofar as it is in potency. Philosophers may legitimately debate the merits and difficulties of such a definition, but a better one has yet to be proposed.

We are half way to understanding Aristotle’s four causes in material substances. We have already discussed two kinds of cause: the material cause and the formal cause. In our example, the material cause is the bronze which can potentially become many different shapes, and the formal cause is the shape of Caesar’s bust. These two causes, matter and form, are called “intrinsic causes” because they lie within the material substance. Both the bronze and its present shape are included in the bust itself, but two other causes do lie outside of the bust. The latter causes are therefore called “extrinsic causes.” The first of these, called the “efficient cause,” is the working of a sculptor. A bust of Caesar cannot become a bust of Brutus without an external agent (one acting) to impress upon it a new form. This agent is the efficient cause. The second, or “final cause,” is the end to which the sculpture must conform. In forming the bust of Brutus from the bronze in the bust of Caesar, there must be some standard toward which the bronze is directed. This is the ideal shape which the sculptor holds in his mind. The sculptor strives to bring the shape of the bronze into conformity with his idea. Without any one of these four causes we are unable to explain why things change.

There is another way to view these four causes. We have seen that the bust is in act insofar as it has a determined shape. The shape (form) in some sense acts upon the bronze (matter) to determine it to this one thing and no other. The form is active in relation to the matter; the matter is passive in relation to the form. Or, in other words, the form acts upon matter; the matter is acted upon by form. Now the efficient cause operates through form because it imparts form to matter. That is, it causes form to act upon matter. The final cause operates through matter because it directs the matter to its new form. That is, it causes the matter to be acted upon by form. The difference is subtle, but the efficient and final causes can be likened respectively to pushing and pulling. Whereas the efficient cause moves by action from a beginning, the final cause moves by attraction to an end.

Cause:	Location:	Example:	Causal Character:
Formal	intrinsic	shape of bust	acts upon matter
material	intrinsic	Bronze	is acted upon by form
efficient	extrinsic	action of the sculptor	causes form to act upon matter
Final	extrinsic	sculptor’s idea of the bust	causes matter to be acted upon by form

Table I.C:  The Four Causes

D. Science and Method

Clearly, the methods of each science will differ. Physical science, because it abstracts from particular matter, but not from sensible matter, can make extensive use of the senses in its demonstrations (experiments). Mathematics, having abstracted from sensible matter, cannot use sensible objects in its demonstrations (proofs). It can only use the ideal properties of bodies in intelligible matter. This does not mean that the mathematician cannot use drawings or diagrams to help him remember. It only means that the diagram, which in fact always departs from the ideal, cannot supply the principles of his demonstration. Metaphysics, having abstracted from matter entirely, cannot use either sensible objects or ideal bodies. It must use principles of pure intellect and judgment alone.

We have said already that the superior science supplies principles to the lower sciences. Metaphysics studies beings only insofar as they have being. One principle of metaphysics is the principle of non-contradiction: a thing cannot be and not be at the same time under the same aspect. Clearly, both mathematics and physical science make ample use of this metaphysical principle. But mathematics adds new principles, for instance, the postulates of Euclidean geometry, without which it would not be possible to reason about figure. Physical science, the least abstract of the three, borrows from mathematics as well as metaphysics, for without the conclusions of mathematics, the physical sciences would be unable to measure change. But it also must seek additional principles from the senses.

The modern mind calls a science only that which can be studied experimentally, that is to say, through the experience of the senses. Mathematics, abstracting from sensible matter as it does, has consequently become today an exercise in logic having no necessary connection to reality. Indeed, the attempt has been repeatedly made to resolve all of mathematics into merely logical relationships, without any recourse to imagination and its intelligible matter. Naturally, metaphysics is rejected outright by those who do not understand the orders of abstraction proper to each science, and who therefore exclude whatever does not invoke a material cause. This scientific demand is nonsense, especially where the science of revealed theology is concerned. For the various elements of Divine Revelation can serve as principles (causes) of demonstration no less than those of geometry or physics. Moreover, such theological demonstrations are more dignified inasmuch as the principles of demonstration often transcend what can be obtained by any human science.

Leaving these higher sciences aside, the experience of the senses should be an important part of method in the physical sciences. The “scientific method” is usually presented as a clear-cut, even mechanical, order of investigation. But this order does not always hold in practice. The investigations of physical science bring the whole man, with all his prejudices and experiences, inclinations and aversions, into a new relation with the physical world. Many physical discoveries have been the result of accident, blind trial and error, or sudden brilliant insight. Examining this method will, nevertheless, help the student understand an important part of the physical scientist’s work.

1.	Define a problem that requires a solution.
2.	Produce an hypothesis to explain the problem.
3.	Predict new consequences of the hypothesis
4.	Test the hypothesis experimentally.
5.	Formulate a valid theory that yields new problems.

Figure I.D: The Steps of the “Scientific Method”

First, the investigator must begin with something he wishes to explain. He produces an hypothesis (educated guess) about the causes responsible. He then predicts new consequences from the hypothesis that can be tested experimentally. The testing of these leads, in turn, to a more comprehensive theory that produces new questions or problems for the scientist. Theories that have been thoroughly tested and have attained a high degree of certainty are sometimes called “scientific laws” or “laws of nature.” The third step is truly the great advance which the moderns have made upon classical Greek science. For the ancient Greeks were keen observers, profound thinkers, and quite imaginative in their hypotheses. Yet, each one having explained some physical phenomenon to his own satisfaction, they ceased to progress. It is really no wonder if an hypothesis accounts for all present observations. That is precisely why it was formed! If it did not, it could be easily discarded. But how can one determine which of the remaining hypotheses is the best? The modern method demands not only that the hypothesis account for prior observations, as did the ancient method, but also that it yield new predictions, which can then be tested. The linear method of the Greeks has been replaced by a circular one that, if followed diligently and with intelligence, will constantly check the accuracy of our physical theories. This modern scientific method has, for both better and worse, radically transformed the intellectual and material culture of the modern world.

There remains to say a few things about experiments. An experiment is not simply a passive listening to nature; it is an active interrogation of nature. Nature cannot be allowed to speak at random, but must be made to answer the questions we put to it, which means that an experiment must be carefully conceived and executed. Prior to the experiment, we should already have an hypothesis about the causes that are operating behind the appearances. The experiment must be designed to verify or reject that hypothesis with certainty. This is what Francis Bacon called an experimentum crucis (critical experiment). The physical circumstance of the experiment will typically be an uncommon or contrived one in which the presence or absence of the these causes will be easily seen. It will very often require a physical apparatus to put nature into such a condition and scientific instruments to amplify and record the results of the experiment.

The classic example of the experimentum crucis is Sir Isaac Newton’s experiment validating his theory of color mixture. It was well known from ancient times that some transparent materials, such as raindrops and some glasses, have the power to produce all the colors of the rainbow from an initial beam of white light. Some investigators were of the opinion that white light is modified to produce these colors. Newton, on the other hand, thought white light a composition of all of the colors, and that the prism is just separating white light into its colored components. He performed a simple experiment to decide which of the hypotheses was true. He first used a glass prism to produce the whole rainbow (the visible spectrum of light) from a single beam of white light. But then he cleverly positioned another prism to take in all of the colored beams and reunite them into a single beam. If the material of the prism were really modifying the light, the second prism would either further modify the light or have no more effect on the already modified light. On the other hand, if his own hypothesis were true, the various colors would be gathered back together to form white light again. The experiment proved him correct. He could separate white light into colored components and then mix them back together again to form white light. The competing theory of color was overturned.

E. Physics and the Physical Sciences

Were we to leave off our discussion of the physical sciences here, we would be greatly deceived about the scope of modern physics. For in early modern times a great change took place in the whole conception of the physical world.

The classical physics of Aristotle regards the world as filled with various natures. For instance, there is the nature of a stone, the nature of a tree, and the nature of a horse. The first is a mineral, the second a plant, and the third an animal. The Greek “φύσις,” from which we derive the word “physics,” and the Latin “natura,” from which we derive the word “nature,” have the same basic meaning. They refer to the essence of a thing insofar as it is a principle of operation and motion. In other words, having a particular nature implies having a particular kind of operation and motion. So the inert stone has its principles of motion, the living plant another set of principles, and the sensitive animal still another set of principles. These principles of operation are understood to be embodied in the highest form of the individual substance, which is called the substantial or essential form. When the substantial form is also a principle of the operations of life, it is called a soul. So a stone has the ability to act and be acted upon through contact. In addition to these operations of lifeless matter, a plant has operations proper to its own degree of being: It can transform inert matter into living matter through growth and reproduce to form new plants. An animal adds to these operations those of the senses: touch, taste, smell, vision, and hearing, or some combination of these. So it has a more perfect operation and a more perfect being than the plant. The study of these substances reveals a hierarchy of natures in the world, with man, whose soul is both the substantial form of the body and a spiritual (non-material) substance in its own right, at the pinnacle of the material creation.

A change in thinking is evident when we consider that we no longer understand physics to be the study of natures, but the study of nature. The mechanical philosophies of the early modern period had a leveling effect. The entire universe was conceived as one vast machine with interacting material parts, all essentially inert. The difference in perfection between a living being and an inert one was reduced to a mere difference in complexity. René Descartes, the first significant proponent of this philosophy, dispensed with the souls of animals and plants, but perceiving that man possesses spiritual operations (intellect and will) that do not involve matter, he retained the human soul. The human body was an “extended substance,” the human soul a “spiritual substance.” Man had become a body accidentally united to an angel. There is no way in his philosophy to rejoin the soul and body into a single human being. Later thinkers, taking Descartes to his logical conclusion, dispensed entirely with a soul that had ceased to have any relation to the physical world: Man is a machine, a marvelously intricate one to be sure, but just a machine.

So plants, animals, and men might exhibit extraordinary complexity of structure and behavior, but they can ultimately be reduced to the mechanical interaction of their material parts. This mechanical philosophy of nature is therefore called “reductionist.” The idea of individual substance has disappeared altogether. The operations of inert matter, now understood to be universal and, even more importantly, complete descriptions of all matter, have been stripped from substantial forms and turned into “laws of nature.”  Where then do these “laws” exist? In God? Scientists of the early modern period were commonly of this frame of mind, but later agnostic and atheistic thinkers could not avail themselves of this option. Form had become nothing but a particular arrangement of matter. The only remaining option was to identify these operations with matter itself. So the leveling of reality became an inversion: Matter is the basic reality; form and spirit, so-called, are but fleeting arrangements of matter. In this way were born the pernicious doctrines of Darwinism and Marxist materialism.

Since matter is understood to be the ultimate reality, and since mathematics is able to study the arrangements of matter in bodies, the application of mathematics is central to modern physics. Whatever cannot be reduced to quantity is dismissed as incapable of scientific study. There are indeed legitimate investigations corresponding to today’s mathematical physics. In the middle ages, such sciences were called “scientiae mediae,” or “intermediary sciences.” St. Thomas Aquinas taught that these sciences have a physical subject (matter), and a mathematical aspect of study (form). Medieval examples are optics and astronomy. What is objectionable in modern physics is not that bodies should be studied mathematically, but that the philosophies and even the mathematical doctrines underlying modern physics make it exclusively mathematical. These doctrines contain a complete restructuring of the physical sciences. Modern physics considers the ultimate “laws” and material constituents of inert bodies. Other physical sciences, for their part, are considered legitimate only insofar as they are thought to be reducible to physics. In accordance with reductionist thinking, they cannot invoke principles superior to those that govern inert matter. Most striking of all, since physics now has regard only for lifeless matter, there is today neither in theory nor in practice a true science of biology! All that remains is the name.

The Catholic student must always be aware of this inversion in modern physics and cautious of its influence in his thinking. Contrary to what many Catholic apologists claim today, modern physics implies a philosophy that cannot be reconciled to the Catholic Faith, for it is false in its very principles. This is not to say that modern physics’ mathematical predictions are inaccurate, for these conform well to reality. It is not experience that is faulty, but the formulation and interpretation of experience. There remains for orthodox Catholic scientists and philosophers a task of immense scope, severe intellectual discipline, and unremitting opportunity. The physical sciences, and indeed the various branches of mathematics, must be reconceived and reordered, all the while preserving the great multitude of legitimate modern discoveries and observations. No individual or small group of individuals will suffice; this project will demand the labor of legions of talented and dedicated philosophers, theorists, experimentalists, teachers, and popular expositors. Catholic Tradition affirms that “grace builds upon nature.” With a sound philosophy of nature again in hand, the Church will find more fertile ground in which to plant the seeds of faith. Let us be sure that the credit for this is referred to God alone:

Ad Majorem Dei Gloriam.

Chapter I.A Review Questions:

1.        What is meant by “truth”?

2.        What is a “science”? How does it differ from “true opinion”?

3.        What is meant by a “contingent” event? Why cannot these be the subject of a human science?

4.        What does the theological modernist mean by “faith”?

5.        What does the orthodox Catholic believer mean by an “act of faith”?

6.        Does the scientist demand an act of faith from his student? Why?

7.        Why is the Catholic’s act of faith in Jesus Christ entirely reasonable?

8.        How must all conflicts between scientific theory and the Catholic Faith be resolved? Why?

Chapter I.B Review Questions:

1.        How are the subject matters of physics, mathematics, and metaphysics defined?

2.        What do we mean when we say that one science is more “abstract” than another?

3.        In what way do physics, mathematics, and metaphysics abstract from individual beings?

4.        Under what aspects would physics, mathematics, and metaphysics study a diamond?

5.        In what way does the hierarchy of sciences yield also a hierarchy of principles?

Chapter I.C Review Questions:

1.        What are meant by “act” and “potency”?

2.        How is “motion” defined?

3.        How does the act of motion differ from the act of form?

4.        What is the difference between an intrinsic and extrinsic cause?

5.        What are the four causes? Which are intrinsic and which extrinsic?

6.        How should we understand the difference between an efficient cause and final cause?

7.        Give an example of a motion not found in the text and identify its four causes.

Chapter I.D Review Questions:

1.        In what way does the order of abstraction of each science determine its method?

2.        Give examples of principles that physics borrows from metaphysics and mathematics.

3.        What are the five steps of the modern “scientific method”?

4.        How do the physical investigations of the modern scientists differ from those of the ancient Greeks?

5.        Why cannot the “scientific method” of the physical sciences be applied to mathematics, metaphysics, and theology?

6.        What is an experimentum crucis? What distinguishes it from mere observation of nature?

Chapter I.E Review Questions:

1.        How does the classical science of Aristotle view the world?

2.        What did the Greeks and Latins understand by the terms “φύσις” and “natura”?

3.        What is a “substantial form”? What is a “soul”?

4.        Describe the “mechanical philosophy” of nature.

5.        Describe Descartes’ philosophy of the body and soul. What is the great problem with it?

6.        How did the notion of a “law of nature” come about?

7.        In what way has modern science limited the idea of form?

8.        How do Darwinism and Marxist materialism find their justification in modern physics?

9.        Why has mathematics become so central to the modern study of the world?

10.        What are “scientiae mediae”? Give some examples.

11.        Why is there today no true science of biology?

12.        Can modern physics in its present formulation be reconciled to the Catholic Faith? Why or why not?

Review of «To Explain the World: The Discovery of Modern Science» by Steven Weinberg

To Explain the World: The Discovery of Modern Science by Steven Weinberg
My rating: 2 of 5 stars

It is good to see a mainstream physicist somewhat dispelling the myth that the Middle Ages were a scientifically dark era; however, he dismisses, with not much proof, the "continuity thesis" that modern physics is a natural, continuous result of 2000+ years of scientific thought (cf. Hannam's God's Philosophers: How the Medieval World Laid the Foundations of Modern Science ). Weinberg seems to think the Middle Age physicists were groping in the dark, stumbling upon discoveries they didn't know the meaning of. This couldn't be farther from the truth. The Middle Age physicists were able to formulate precise, very modern questions and offer penetratingly clear answers to questions on infinity (laying the foundations of calculus) and on the fundamentals undergirding even modern physics: place, time (and its relativity), void, and the "plurality of the worlds" (i.e., what's called "parallel universes" today), as shown in Medieval Cosmology: Theories of Infinity, Place, Time, Void, and the Plurality of Worlds .

Weinberg begins with Aristotle and also mentions prominent High Middle Ages physicists like Bishop Oresme and Buridan, in addition to those at Merton College known for the Mean-Speed Theorem, but overall his treatment of the Middle Age physicists and the question of modern vs. pre-modern physics was treated sloppily.

View all my reviews

Neo-Pythagoreanism

Here is the exchange between my friend who is interested in Duhem:

this comes from a recent article my "Duhem" Google alert found for me:

Babette Babich, “Heidegger’s Jews: Inclusion/Exclusion and Heidegger’s Anti-Semitism,” Journal of the British Society for Phenomenology 47, no. 2 (April 2, 2016): 133–56, doi:10.1080/00071773.2016.1139927.

Babich, as fn. #62 says, is a student of Lonergan.

Here's where he discusses Duhem, ending with an interesting observation. I haven't read Duhem's La science allemande in its entirety; just the part I sent you regarding Einstein and the "geometric/'mathematical' mind."

By contrast, especially for those of us up on our Laruelle or our Stiegler or Meillassoux or our Simondon, and even in conjunction with the present theme, thinking of identities and differences, thinking of Heidegger and his Jews in a French context, who reads Pierre Duhem, author of the posthumous German Science?⁵⁰ Have we today – even those of us interested, as I am interested, in the history and philosophy and sociology of science – any political geography of theory in science in connection to or with hermeneutic reflection? In connection to Heidegger, ah, yes, but not with respect to questioning science much less technology? We leave to those in power their game plan, and we do so without remainder. Critical theory has thus managed not to be critical for years.

Where Duhem in 1916 criticizes the German turn of mind as it finds expression in Gustav Kirchoff, a theorist of mathematical physics, we could find Heidegger's words along with Duhem's critique: “We can and will posit [poser] …  Wir können und wollen setzen … ”⁵¹ Note that this stipulative posing is by no means limited to a dogmatic and axiomatic controversy. The mathematician David Hilbert made this the watchword of the so-called Göttingen programme, which project included Husserl.⁵² As Duhem continues to refute Heinrich Hertz's explicitly deductive construction of mechanics,⁵³ the problem is not that the postulate is arbitrary but rather that it is, out of context and history, thereby articulated, “imperiously”: “Sic volo, sic jubeo, sit pro ratione voluntas. [I will it thus, I order it thus; let my will stand in the place of reason.]”⁵⁴ What Duhem ultimately sought, fierce as his gainsaying was, was only the inclusion of specifically French science—this would be the torch later taken up by Bachelard and Canguilhelm and today, if less and less, Serres—finally admitted to the table along with German science: “Scientia germanica ancilla scientiae gallicae.”⁵⁵

In the published version we can read Duhem citing Nietzsche's contemporary and fellow philologist, Hermann Diels: “the German is, here and now, on this earth, the sanctuary in which the principle of order takes refuge.”⁵⁶ Yet it is Duhem's extended citation of Wilhelm Ostwald that is arguably the most disturbing, even using the language of a “great secret” with respect to the German:

Germany wants to organize Europe which, until now, has not been organized. I shall now explain to you the great secret of Germany. We, or perhaps rather the German race, have discovered the factor of organization. Other people still live under the regimes of individualism, when we are under that of organization.⁵⁷

When Duhem asks “[w]as Scholasticism not essentially, as German science is, a work of the mathematical mind”⁵⁸ he can seem to approximate Heidegger's standpoint in his Beiträge with respect to what Heidegger names Machenshaft, where Heidegger writes in GA 95 (Überlegungen VIII, 5) of the Black Notebooks currently under discussion. “One of the most secret forms of the gigantic, and perhaps the oldest, is the persistent skillfulness in calculating, pushing, and intermingling through which the worldlessness of Jewry is grounded.”⁵⁹ Or else and still more troublingly, when Heidegger writes:

the temporary increase in the power of Jewry has its ground in the fact that the metaphysics of the West, especially in its modern development, served as the point of attachment for the diffusion of an otherwise empty rationality and calculative skill, which in this way lodged itself in the “spirit” without ever being able to grasp the concealed domains of decision on its own. The more original and inceptive the coming decisions and questions become, the more inaccessible will they remain to this “race.”⁶⁰

This last is only a prelude to the most infamous of these quotes:

That in the age of machination, race is elevated to the explicit and specially erected “principle” of history (or just of historiology) is not the arbitrary stipulation of “doctrinaires” but a consequence of the power of machination, which must cast down beings, in all their regions, into planned calculation.⁶¹

It can be argued that what Duhem calls “German science” corresponds to what would come to be called “Jewish science”.⁶²

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52 I discuss this with attention to the time that was the first few decades of the twentieth century in the philosophy of science (and mathematics) in Babich, “Early Continental Philosophy of Science.

53 “Let us agree that this point – which is itself nothing but an algebraic expression, only a world of geometric consonance take to designate an ensemble of n numbers – changes, from one instant to another, by an algebraic formula. From this convention, so perfectly algebraic in nature, so completely arbitrary in appearance, we deduce, with perfect rigor, the consequences that calculation can draw from it, and we say that we are setting forth mechanics.” Duhem, “Some Reflections on German Science”, p. 93

54 Ibid.

55 Ibid., p. 112.

56 Duhem, “German Science and German Virtues”, here p. 122.

57 Ibid.

58 Ibid., p. 123

59 Heidegger, Überlegungen VIII, 5, GA 95, p. 97.

60 Heidegger, GA 96, pp. 46 (from Überlegungen XII, 24).

61 Heidegger, Ibid., p. 38.

62 That argument can be made, but for his part, Duhem is talking about “scholasticism”, that is what my old Jesuit teacher, the Canadian Thomist, Bernard Lonergan, author of the conspicuously named Method in Theology (1972) and Insight: A Study of Human Understanding (1957), with its famous listings of points to the seemingly nth degree, no mere sic et non, took the mid-twentieth century to an extraordinary pitch well beyond the tradition of generalized empirical method of “transcendental Thomism” inaugurated by the Belgian Jesuit philosopher, Joseph Maréchal. Indeed, Maréchal was probably one of the reasons Lonergan was able to answer my questions regarding the intersection of mysticism and empiricism as well as he did. Maréchal's initial main works included: Le point de départ de la métaphysique: leçons sur le développement historique et théorique du problème de la connaissance, 5 vols (Bruges-Louvain, 1922–47) and Études sur le psychologie des mystiques, 2 vols (1926, 1937).

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German/Jewish science (my attempt at a compact definition): a fabricated (magical) new “physical” science derived from hypothetical equations—e.g. relativistic dynamics and its geometric implications dictating the replacement of a planar space with a gravitational field by a curved spacetime without gravitational field. The transition from the former to the latter is essentially magical, thanks to the concoction of highly elegant algebraic tools. It would therefore be worth investigating how the actual magic, in the truly occult sense of the term, has impacted “Jewish science” and its use of mathematics (as the intellectual Talmudic culture is riddled with magic practices related to gematric ideas).

Also, as I read a little bit of Duhem’s writings and get a sense of his Ampèrian affinities, I’m wondering whether he was aware of (and, if so, what he thought of) the tensor-based reformulation of electrodynamics (relativistically rewriting Maxwell’s equations to account for the electromagnetic potential in terms of R⁴ using an electromagnetic field tensor and a current tensor). Electrodynamics assuming GR has always bothered me a great deal because it essentially unifies Einsteinian gravitation (its spacetime continuum) and electromagnetism by way of geometry, not physics (as though the physics of E&M was ultimately dependent upon and constrained by the non-physical relativistic dogma of gravitation). 

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Yes, that was fascinating. And the author quoted Duhem's German. Duhem published in French, English, and German, and knew Latin and Greek as well as reading comprehension in Italian! He was a true polymath and polyglot.

"Ampèrian affinities":
Duhem certainly admired Ampère's experimental and theoretical genius, but he disagreed with Ampère's Newtonian inductivism, which held that "phénomènes électrodynamiques" are "uniquement déduite de l’expérience," as Ampère subtitled his famous work. Duhem discusses this in his Théorie physique ch. 6,

• §4 (PDF p. 154): a critique of Newton himself ("Critique de la méthode newtonienne. - Premier exemple : La Mécanique céleste")
• §5 (PDF p. 158): a critique of Ampère ("Critique de la méthode newtonienne (suite ). – Second exemple : L’Électrodynamique").

(Duhem was very good at making everyone on all sides of a debate very uncomfortable. ☺)

Duhem was familiar with Riemann's mathematics. We know this based on a citation he made to a paper by the Italian mathematician Enrico Betti. Duhem's 2nd doctoral dissertation (the accepted one) was, after all, in the mathematics department; his 1st (the rejected one) was in the physics department.

By "tensor-based reformulation of electrodynamics," are you referring to Maxwell's quaternion way of writing things? For example, in his Treatise on Electricity & Magnetism (vol. 2), p. 232-233, where he introduces the displacement current, he wrote the Maxwell-Ampère Law
$$\nabla \times \vec{B} = \mu_{0} \vec{ J} + \mu_{0} \varepsilon_{0} \frac{{\partial \vec{E}}}{\partial t$$as
$$4\pi \mathfrak{C} = V \nabla \mathfrak{H$$
where
$${\mathfrak{C}} = \Re + \dot{\mathfrak{D}}$$i.e., (true current) = (conduction current) + (displacement current);
$\mathfrak{H}$ is the magnetic force;
$V\nabla$ is the vector part of $\nabla$ (i.e., curl).

Regarding "occult" and "gematric [geometric?] ideas" in physics:
    I like Steiner (2009)'s term "Pythagorean analogies," i.e., "analogies inexpressible in any other language but that of pure mathematics" (Karam 2014); he cites Maxwell's displacement current as the first example of a "Pythagorean analogy" in physics.

Here are Duhem's views of quaternions and vector analysis (Théorie physique ch. 4, §6 "L’École anglaise et la Physique mathématique," PDF p. 62-63):

Mais chez les Anglais seuls l’amplitude d’esprit se trouve d’une manière fréquente, habituelle, endémique ; aussi est-ce seulement parmi les hommes de science anglais que les Algèbres symboliques, le calcul des quaternions, la vector-analysis, sont usuels ; la plupart des traités anglais se servent de ces langages complexes et abrégés. Ces langages, les mathématiciens français ou allemands ne les apprennent pas volontiers ; ils n’arrivent jamais à les parler couramment ni surtout à penser directement sous les formes qui les composent ; pour suivre un calcul mené selon la méthode des quaternions ou de la vector-analysis, il leur en faut faire la version en Algèbre classique. Un des mathématiciens français qui avaient le plus profondément étudié les diverses espèces de calculs symboliques, Paul Morin, me disait un jour : « Je ne suis jamais sûr d’un résultat obtenu par la méthode des quaternions avant de l’avoir retrouvé par notre vieille Algèbre cartésienne. »

Also, you would be very interested in the

Notice sur les Titres et Travaux scientifiques de Pierre Duhem rédigée par lui-même lors de sa candidature à l'Académie des sciences (mai 1913)

While Duhem wrote most of it—summarizing all his scientific, philosophical, and historical researches—, Jordan wrote the biography section, Hadamard wrote the section on the mathematical aspects of Duhem's works, and Darbon (whom you may not have heard of) wrote the section on Duhem's history of physics.

By the way, some considered Duhem "anti-Semitic" because of his stance on the Dreyfus affair, yet he was close friends with the Jew Hadamard, who held a very high opinion of Duhem.

Also, Duhem's influence has been vast, across many fields. For example, the economist Schumpeter, in his preface to Fr. Dempsey, S.J.'s erudite defense of the moderns' understanding of interest and the medievals' arguments against usury, Interest & Usury, mentions how Fr. Dempsey did for economics what Duhem did for physics; they both showed the medievals' contributions to their respective modern disciplines.

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Thanks for this rich piece of Duhemian insights and many references! Quite dense, owing to the vast extent (“across many fields”) of Duhem’s multi-layer thought and writings.

Yes Maxwell’s equations can be reformulated using matrix operators to represent quaternions. But, in field theories, quaternions are broader algebraic tools than tensors and need not include any “relativistic” alteration. Thus “Maxwell’s quaternion way of writing things” is not exactly the same as a tensor-based reformulation of electrodynamics. The tensor version of Maxwell’s equations (deriving E&M from the deformation of R⁴ geometry) describes the relationship between the electromagnetic potential A^µ (which, from the perspective of quaternion operators, would be defined as a quadri-vector potential), the electromagnetic field strength tensor F^µv (when F^µv = ω^µv), and the current tensor j^µ, yielding the “homogenous” form of Maxwell’s equations:

∂_kF_µv + ∂_µF_vk + ∂_vF_kµ = 0.

∂²A^µ = j^µ ,

where ∂² = ∂₀² - ∂₁²- ∂₂² - ∂₃² = c^-2∂_t² - ∇_x²

“Gematric”, referring (adjectively) to the Jewish gematria and its many occult misuses of mathematics and numbers.

Fascinating quote pertaining to “Duhem’s views of quaternions and vector analysis”! Besides Cauchy and his stress tensor, I cannot think of many French mathematicians and natural philosophers with a taste for the kind of algebraic operations used in quaternion, vector, and tensor analyses.

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Oh, I see. You were referring to relativity theory's tensorial "simplification" of Maxwell's equations. To my knowledge, Duhem never wrote about that (at least not directly, by writing relativity's "'homogenous' form of Maxwell’s equations").

Re: "Duhem’s multi-layer thought and writings":
    Duhem even spoke about Loti, Corneille, Shakespeare, and Dickens in his Théorie physique. Duhem contrasts Corneille with Shakespeare (∵ he considers both as not strictly having an esprit de géométrie) and Loti with Dickens (∵ both are prime examples of an esprit de géométrie). I read Tale of Two Cities because I was curious if Dickens indeed has an "English mind" (esprit de géométrie), and he certainly does; I wasn't that impressed with Tale of Two Cities because it didn't have much of an "Ariadne's thread" (coherent idea/theme) running through it. It was a disconnected smorgasbord of events and myriads of characters. As Hertz said, Maxwell's theory is nothing more than Maxwell's equations. Maxwell did not even derive these equations from a single principle, like an energy law, which was customary to do in E&M in the era between Ampére and Maxwell; thus, Maxwell's theory is an example par excellence of the English/German/geometrical mind, juggling many disconnected ideas around—which Poincaré said, in that quote I sent you awhile back [translated on p. 8 of this], makes French minds ill-at-ease when reading Maxwell for the first time.
    Have you read any Corneille? I know he wrote Le Cid; have you seen/read that? Does it exemplify the French esprit de finesse?

Duhem certainly is not opposed to analogies in physics. Classification is impossible without the ability to form analogies, and Duhem defines physical theory as a classification of experimental laws (not a classification of equations!). Duhem explicitly mentions "analogie" in Physique du croyant p. 146 ff. (on the analogy between cosmology [natural philosophy] and physical theory; Duhem essentially proves Aristotle Physica 191a7-8: "The underlying nature is known by analogy."), which you may have already read. He describes very well what Fr. Wallace, O.P., says is the "teaching that is distinctive of Thomism," i.e., that "analogical middle terms are sufficient for a valid demonstration" (cf. this). This is vital for there to be "mixed sciences" or scientia media, where minor and major premises are taken from distinct fields, like mathematics and physics, with distinct principles of their own. Fr. William A. Wallace, O.P., who pioneered research into Galileo's logical treatises, describes this very well in the best logic-of-science work I've ever read: The Modeling of Nature (if Duhem wrote a logical work, which I wish he did!, it would probably be similar to Fr. Wallace's).

Thus, what I think best describes "German" or "Jewish science" is not that it uses analogy, which all physical theory does, but that it's Neo-Pythagorean, inverting the 1st (physical) and 2nd (mathematical) degrees of abstraction. Duhem, where he mentions Einstein in the La science allemande, makes it clear that one cannot define time from a mathematical equation as Einstein does. The inversion of the first two degrees of abstraction has become so extreme that Max Tegmark, who is a hardcore Pythagorean, even wrote a paper on the "Mathematical Universe Hypothesis," i.e., that the universe is mathematics! Pythagoreans appear to be the first gematrists.

I was pleased to see Steiner (2009) quote Peirce regarding analogies between physical theories (p. 52fn9):

These universal super-laws were, to Peirce's thinking, the key to the formal mathematical analogies we see between laws—such as the inverse square laws in gravity and electricity—analogies that demand explanation (7.509-7.511). But Peirce looked to these super-laws also to explain, not only the mathematical form of laws, but even the specific values of the constants (like the gravitational constant) appearing in them.

This reminds me of what St. Thomas said is impossible in that Super Iob quote I sent you, where he says some things cannot have a natural explanation but are up to the will of God. A "physical" theory being the analogy between two mathematical laws is not a physical theory, but a mathematical one, at best, and a confusing of mathematics with the will of God, at worst.

This reminds me: I need to read the article ["De Analogia secundum Doctrinam Aristotelico-Thomisticam"] by Fr. Ramírez, O.P., that formed the foundation of his multi-volume De Analogia. Fr. Rimírez is the master of analogy, and I'm curious what he has to say about Duhem claim that (p. 147):

…si l'on prononce à cet endroit les mots de preuve par analogie, il convient d'en fixer exactement le sens et de ne point confondre une telle preuve avec une véritable démonstration logique. Une analogie se sent ; elle ne se conclut pas ; elle ne s'impose pas à l'esprit de tout le poids du principe de contradiction. Là où un penseur voit une analogie, un autre, plus vivement frappé par les contrastes des termes à comparer que par leurs ressemblances, peut fort bien voir une opposition ; pour amener celui-ci à changer sa négation en affirmation, celui-là ne saurait user de la force irrésistible du syllogisme…

How is "preuve par analogie" not "une véritable démonstration logique" thet "ne se conclut pas" and "ne s'impose pas à l'esprit de tout le poids du principe de contradiction"?

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The formalism of quaternions was very much Maxwellian in spirit and has actually been extended in Germano-Anglo-American electromagnetic field theory on the basis of the tensor field equations of Einstein. But it is the introduction of the electromagnetic tensor field F_µv, combined with the spin connection vector ω^a_b (the electrodynamics simplification of this amounts to equating F and ω by retranslating the latter into a rotation tensor, ω_µv) that makes up for “relativity theory’s tensorial "simplification" of Maxwell’s equations”, which consists in the merging of electrodynamics with GR I was referring to (and wondering whether Duhem had had any thought on this “tensorization” of E&M, which is all the rage today among pan-relativists).

About tragedians and the difference between “esprit de finesses” and “esprit de géométrie”, Duhem was probably keen on his contrasting assessment of Corneille and Shakespeare on the one hand and Loti Dickens on the other. Corneille, as Molière, also was a comedy writer and therefore, to my view, does exemplify something of “un esprit de finesse”, since finesse was rather characteristic of the kind of spirit 17th century French political/social satires would famously convey through penetrating comedies (hardly the under-the-belt level of our contemporary late wisecracking shows on T.V.).     

Regarding the centrality of analogies to the life of the created intellect, you correctly write: “Classification is impossible without the ability to form analogies”, and logically justify the validity of Duhem’s definition of “physical theory as a classification of experimental laws”, namely by way of analogous abstraction (which is what modeling physical data in the logical formal of a physical theory really amounts to). In a broader (cross-field) sense, I would define analogicity (ἀνα-λογία, meaning quite literally the logic of comparative relation between that which is lower to that which is higher) as a critical way of both intuitively and intellectually seeing the similitude of the invisible (the higher) mirrored in the visible (the lower). Both the Hebrew word for intellectus (בִינָה, “understanding” as the attitude of the intellect seeing in between two things, meaning re-cognizing the like features by which two essentially distinct things can coherently be related) and the Aramaic word for “comparison” (ܒ݁ܡܰܬ݂ܠܶܐ ,מתלא) clearly suggest that an analogy is actually more than simply a ratio-nal proportion (in the arithmetic, Aristotelian sense). In revealed anthropology, it appears (my theory) that the created intellect (whose life is intellectus) is constitutionally analogic (together in inner structure and cognitive motion).

I did notice Duhem’s mention of analogy in Physique du croyant.  The distinction he makes (referring to your final question) between “preuve par analogie” and “une véritable démonstration logique” does not imply that the first is less intellectually powerful and epistemologically meaningful than the second. In fact, a mere logical demonstration, however valid, may not have the same epistemological value as a proof by analogy, even though the latter’s proof value is technically more limited than a syllogistic demonstration. What this means is that analogical knowledge is not reducible to logical validity. Thus a little like he did in La science allemande—first lesson (Les Sciences de Raisonnement)—when distinguishing between axioms and theorems (which the following sentence on p. 6 summarizes like an aphorism: “Les principes se sentent, les propositions se concluent…”), Duhem is essentially right to say on p. 147 of Physique du croyant: “Une analogie se sent ; elle ne se conclut pas…”      

The confusion of degrees of abstraction you talk about is indeed critical! My sense is that it is typically indulged in because there is actually more to numbers and their multifaceted relations and properties than their simply being abstract entities bereft of positive existence outside the mind (entia rationis).¹ However, no one really knows how much more, and what the true nature of this “more” is. That is the reason why mathematical physics can really lead to ontological problems (as the nature of the 2nd degree of abstraction is so evasive), but without possibly providing a solution. If you remember, it was the sense of my comparative (analogical) “syllogism” used here to extend Gödel’s incompleteness results from mathematics to all-encompassing Neo-Pythagorean physical theories whose ultimate Galilean assumption is that “the universe is mathematics.”

“Duhem, where he mentions Einstein in the La science allemande, makes it clear that one cannot define time from a mathematical equation as Einstein does.”

Yes, and it is very significant that Newton did not include “time” in his descriptive “universal super-law” of gravitation, while Einstein did. The Neo-Pythagorean thinking undergirding GR and its inclusion of the time dimension times the square root of - 1 was never meant to account for empirical results (contrary to the regular claims appealing to the “countless corroborations” of the curved geometry of Einsteinian gravitational field). GR was intended to provide a mathematical way out of the contradiction between the instantaneous Newtonian gravitational field and the new principles couched in the Einsteinian formulas pertaining to spacetime in SR (implying action at a distance).

I read Fr. Ramírez’s The authority of St. Thomas, not his De Analogia.