Saving the appearances

Saving the appearances: De Koninck, Maritain, and Eddington’s Two Tables

The theme of this volume concerns how history can illuminate contemporary philosophical matters.  Consequently, this essay looks to history to see how it might help shed light on a topic that continues to entangle physics and philosophy: the relation between science and reality, as specifically informed by the relation between the abstract mathematical objects used by physicists and the physical objects in the world to which they purport to pertain.   Specifically, this essay investigates the history of astronomy from the ancient Greeks through the early 20th Century.  It ends by considering the philosophical critique that young Thomist Charles de Koninck posed to Jacques Maritain in his 1934 Ph. D. thesis[1] on the great British physicist Sir Arthur Stanley Eddington, who was a popularizer in the 1920s and 30s of the new physics of Einstein’s relativity and Bohr’s quantum mechanics.  De Koninck argued that Eddington’s articulation of the New Physics was consistent with Scholastic philosophy but that Maritain had significantly misunderstood the New Physics in what he said in his 1932 book, The Degrees of Knowledge.[2] The essay will come back to this at the end, after looking at some matters from history.

Keep the following questions in mind.  Are the beings of reason used by physicists real, telling us about the nature of things, or are they merely convenient fictions? Or is phrasing the question as requiring such a stark binary choice already a problem, as the theoretical physicist and science historian Pierre Duhem argued in his classic 1908 book on the history of astronomy, entitled To Save the Phenomena?[3]  Duhem used the Greek phrase “sozein ta phainomena” in the original title of the French edition of his book.  The infinitive sozein means to save, preserve, or keep, and the noun phainomena comes from the verb phaino, which has a set of related meanings centered around to make visible or to be visible, to manifest, to show, or to appear.[4] Thus, the Greek phrase is sometimes rendered “to save the appearances.”

Imagine you are an ancient Greek looking at the night sky.  One sees little points of light appearing in different places in the sky, varying with the time of day and the seasons of the year.  Five of these, called planets, from the Greek word meaning “wanderer,” have rather strange variation of their place in the sky.  What is one to make of all this?  What is moving and how might one account for its motion?  The appearance of the planets posed three problems: (1) a planet’s motion was odd, sometimes reversing direction in the sky, a phenomenon known as retrograde motion. (2) a planet’s brightness varied, presumably due to a variable distance from the earth.  (3) the earth’s seasons were not uniform, indicating the sun’s motion with respect to the earth did not remain the same throughout the year.  

Duhem’s book explained how the Greeks, and then later the Arab philosophers, the Medieval Scholastics, and the thinkers of the Renaissance and early modern science came to understand the various motions of sun, moon, the five planets, and earth. He found a great continuity in both the ideas and the controversies in the astronomical sciences in the nearly 2000-year span from the time of Plato to that of Galileo.  The 6th Century Neoplatonist philosopher Simplicius (480-540CE): stated the key idea: “Plato lays down the principle that the heavenly bodies’ motion is circular, uniform, and constantly regular. Thereupon he sets the mathematicians the following problem: What circular motions, uniform and perfectly regular, are to be admitted as hypotheses so that it might be possible to save the appearances presented by the planets?”[5] Furthermore, Aristotle insisted that any circular motion in the heavens had to be centered on the earth as the center of the universe.  Both Plato and Aristotle held as a basic principle of physics (physis) that the heavens, where the stars and planets moved, were a region of perfection made out of a fifth element that was different from the four elements earth, air, fire, and water that make up the “sublunary” region on earth beneath the orbit of the moon.  Thus, Simplicius distinguished between the theories of the physicists, basing their causes on principles of natural philosophy, and those of the astronomers, who ignore nature and only use calculational fictions to calculate apparent movements in the skies. 

Around 200 BCE Apollonius of Perga succeeded in “saving the appearances” by assuming the planets moved on a circle (the deferent) centered on the earth, with additional motion on a second smaller circle, the epicycle.  A half century later, Hipparchus of Rhodes showed that the sun’s motion could be accounted for equally well if the deferent is centered on a point known as the eccentric that is displaced from the center of earth.  Now we are faced with the problem of two quite different mathematical hypotheses that could both save the appearances. The most accurate system of Greek astronomy was set up in the 2^nd Century after Christ by Claudius Ptolemy (100-170 CE) of Alexandria, Egypt. His complex geometric system used epicycles, eccentrics, and equants to mimic the observed motions of the planets, moon and sun, including the phenomena of retrograde motion, variable brightness, and variable speed of the planets. The mathematical model contained in his book The Almagest (Arabic for “The Greatest”) enabled predictions to be made for astronomical phenomena that were still quite accurate over a thousand years later at the time of Copernicus.

Duhem’s work documents at great length the tension and contradictions between the mathematical “saving of appearances” method that came to dominate Greek astronomy and the physically realistic attempt based on the supposed true nature of things that the Peripatetics following Aristotle insisted upon. The Peripatetic physicists never developed mathematics that saved the appearances as well as Ptolemaic mathematicians did. The tension between understanding physical reality through a mathematical description or through a metaphysically defined scheme of what nature is persists to this day and is evident in the 20^th Century physics that concerned De Koninck and Maritain, or in 21^st Century discussions of so-called “artificial intelligence.”

Ptolemy had no guide except the rule of maximum simplicity.  Ptolemy himself said: “We must, as best we can, adapt the simplest hypotheses to the heavenly movements. But if these prove insufficient, we must select others that fit better.”[6]  Duhem summarizes the view of the 5^th Century Neoplatonist Proclus (410-485) regarding Ptolemaic astronomy: 

Astronomy cannot grasp the essence of heavenly things. It merely gives us an image of them. And even this image is far from exact: it merely comes close. … The geometric contrivances we use to save the phenomena are neither true nor likely. They are purely conceptual. … Very different hypotheses may yield identical conclusions, one saving the appearances as well as the other. Nor should we be surprised that astronomy has this character: It shows us that man’s knowledge is limited and relative, that human science cannot vie with divine science. Such is Proclus’ teaching, surely, a far cry from the ambitious physics of Aristotle’s On the Heavens and Metaphysics, the physics that claims to have carried speculation on the essence of heavenly things so far as to have arrived at the fundamental principles of astronomy.[7]

The later Arab Peripatetic philosophers were consummate realists, and thus, great enemies of Ptolemy.  The great Andalusian polymath Averroes (1125-1198CE) said in the 12^th Century: “Actually, in our time astronomy is nonexistent; what we have is something that fits calculation but does not agree with what is.”[8]  Interestingly, another Andalusian, Moses Maimonides, followed Ptolemy and Proclus in saying that “The knowledge of heavenly things, in their essence and true nature, is beyond man’s capacities; sublunary things alone are accessible to our feeble understanding.”[9]

When we come to the Scholastics, Duhem approvingly quotes Thomas Aquinas’s understanding, given in several of his works, often using very similar language to Simplicius.  In Summa Theologiae I.32.1, Thomas says: 

in [astronomy] the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them.[10]

The key phrase here is that “some other theory might explain them.”  Thomas, like Ptolemy, Proclus, and Simplicius knew the logical point that it is false to insist that a theory is true because it saves the appearances.  Failing to appreciate this point of logic is one of the reasons that Galileo got into trouble: he insisted on a realism that was not yet adequately demonstrated, whereas his opponents insisted on a realism that was false to the way nature is.

Duhem tells us that: “The principles that Thomas Aquinas, following Simplicius, laid down enabled astronomers to use Ptolemy’s hypotheses without scruple in their study of the apparent movements of the planets, despite the fact that their metaphysical opinions might force them to reject these hypotheses.”[11]  Furthermore, “In the fourteenth century, at Paris, the system of Ptolemy was accepted without argument.”[12]  But more was afoot, since a number of the Parisians, familiar with the thought of Nicholas of Cusa, came to think that the heavens and the earth were not composed of different elements but the same and that scientific hypotheses about earthly phenomena should be treated in a similar manner to astronomical ones, namely to save the appearances.  Thus, John Buridan developed his new counter-Aristotelian notion of impetus to explain the motion of projectiles, very similar to the later Newtonian notion of linear momentum.  In rejecting Aristotle’s physics, Buridan made this point: “It seems to me that this is the assumption that should be adopted because the other assumptions do not seem correct and because all the phenomena agree with this one.”[13] To Buridan, it is more important to account for the phenomena than to conform to preconceived notions of the nature of things.

Renaissance Italian Averroists took a different tact and insisted on realism and refused astronomy the right to use purely convenient but fictitious hypotheses that do not conform to the nature of things.  They had a dilemma—they could not save the appearances with their natural philosophy but would not grant to the Ptolemaists the right to do with their fictive mathematics.  Duhem points out that the Peripatetic realists “were victims of the illusion that one can deduce an astronomical theory from a metaphysical doctrine.”[14]

Now enters Copernicus (1473-1543), whose book On the Revolutions of the Celestial Spheres was published literally as he lay on his deathbed in 1543. He had seen the impasse between the Ptolemaists and the Peripatetics in his student days in Italy.  Copernicus was willing to try novel mathematical hypotheses.  By assuming the sun was at the center of the universe, and the earth like other planets revolved around it in a perfect circle, he could save the appearances better than the Ptolemaic system.  Copernicus was a physical realist—he truly believed that his system was truly the way the world was.  But Copernicus, following Plato like everyone before him, assumed perfect circles for orbits but had to introduce corrective epicycles, since as Johannes Kepler was soon to discover, the real orbits are not circles but ellipses.  The specific mathematics of Copernicus would eventually have to be abandoned.

There were problems on the horizon, for it was in the early decades of the Protestant Reformation and all Europe was in intellectual ferment.  Luther declared war on the hypotheses of Copernicus—in the name of Scripture. Melanchthon, his faithful disciple, followed him.  Melancthon’s colleague, Andreas Osiander of Wittenberg, who helped with the publication of Copernicus’s book, in order to protect Copernicus’s idea, inserted a Preface–but unknown to Copernicus–that stated the well-established notion of Ptolemy, Proclus and contemporary

Ptolemaists that Copernicus’s hypotheses was just another mathematical fiction for the sake of saving the appearances.  For the next few decades the use of fictive astronomical hypotheses was permissible, but hearts and minds were hardening towards a requirement of realism in astronomy.  This meant to the critics of Copernicus that astronomy henceforth must be based on both sound Peripatetic philosophy and sound interpretation of Scripture.  This eventually led to Galileo’s warning in 1516 and later condemnation of his book in 1533.

Duhem looked with great favor on the wisdom of Cardinal Bellarmine, who looked into the Copernican writings of Galileo and Foscarini.  On April 12, 1615, Bellarmine had written Foscarini a letter which, according to Duhem, was “full of wisdom and prudence:”

It seems to me that Your Reverence and Signor Galileo would act prudently by contenting yourselves with speaking ex suppositione and not absolutely, as I have always believed Copernicus to have spoken. To say that by assuming the earth in motion and the sun immobile, one saves all the appearances better than the eccentrics and epicycles ever could, is to speak well indeed. This holds no danger and it suffices for the mathematician. But to want to affirm that the sun really remains at rest at the world’s center … that it turns only on itself without running from East to West, and that the earth is situated in the third heaven and turns very swiftly around the sun, that is a very dangerous thing. Not only may it irritate all philosophers and scholastic theologians, it may also injure the faith and render Holy Scripture false.[15]

Bellarmine went on to say:

If it had been demonstrated with certainty that the sun keeps to the center of the world, …, then one would have to proceed with much circumspection in explicating Scripture. … But not until someone has demonstrated this to me will I believe that it exists. … in a case of mere doubt you should not diverge from Holy Scripture as the holy fathers have expounded it.[16]

Bellarmine’s caution at the time was warranted, based on well-established principles.  First, the fact that the math saves the appearances does not prove the hypothesis, and second, if the hypothesis can in fact be demonstrated, he would have to accept it, and then judiciously, by accepted principles, reinterpret the Scriptures to accord with established facts. This is precisely what the Church did in the end.   In Galileo’s day, there was scientific counterevidence of the motion of the earth—for example, the absence of stellar parallax, which was only observed 200 years later in the 1830s when scientific instrumentation was good enough to measure the very small effect.

Galileo’s contemporary, the Protestant German astronomer Johannes Kepler (1571-1630), finally put the math and the physics together. Keppler had access to the new data from the Danish astronomer Tycho Brahe on the orbit of Mars.  Wrestling with this data forced Kepler to give up the Aristotelian notion of circular orbits.  Using heliocentric astronomy Kepler was able to determine that planetary orbits were ellipses, not circles, and to deduce his three quantitative laws of planetary motion that still hold today.  He published his two books on this in 1609 (New Astronomy) and 1618 (Summary of Copernican Astronomy).  The scientific revolution was now well underway, and its new mathematical physics came to be seen, especially after Sir Isaac Newton, as the theory of the physically real. Only 6 decades after Kepler’s New Astronomy came out, Newton published his Mathematical Principles of Natural Philosophy (1687) in which he gave his famous laws of motion and specifically deduced all three of Kepler’s laws based on the new law of universal gravitation.  Gravity appeared in the system as an apparent force between two masses that was proportional to the product of their masses divided by the square of the distance between them.  It was a remarkably simple assumption.

Newton was quite circumspect about gravity—he did not claim to know its causes—it was a just a mathematical principle that saved many appearances.  In his General Scholium at the end of his Mathematical Principles of Natural Philosophy of 1687, he said:

…hitherto I have not been able to discover the cause of those properties of gravity from phænomena, and I frame no hypotheses. …  to us it is enough, that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.[17]  

Newton’s new natural philosophy had its appeal because it saved a quite large range of appearances, from earthly and heavenly motion to the familiar tides that came ashore twice every day.  Newton’s triumph was a theoretical one, based on mathematics, explaining motion in the heavens and on earth.  Newton’s success was stunning and gave rise to a whole new conception of reality: the universe came to be seen by many as a giant machine ruled by ironclad laws of physics that determined every movement.  Was such mechanical hyperrealism warranted?  Was it any more warranted than the Peripatetic hyperrealism of Galileo’s opponents?  Pierre Duhem thought not in both cases.   He states his own view in the last two sentences of his book:

Despite Kepler and Galileo, we believe today, with Osiander and Bellarmine, that the hypotheses of physics are mere mathematical contrivances devised for the purpose of saving the phenomena. But thanks to Kepler and Galileo, we now require that they save all the phenomena of the inanimate universe together.[18]

The 19^th Century, brought a new idea to physics, that of electric and magnetic fields that permeated space by which light was explained as an electromagnetic wave phenomenon.  The 20^th Century opened with Planck and Einstein showing that light also manifests clear “particle-like” characteristics.  Furthermore, the rapidly developing quantum physics following Niels Bohr’s 1913 quantum theory of the atom introduced an uncertainty principle that dramatically overturned the “classical” mechanistic physics of Newton, while Einstein’s general relativity in 1916 accounted for Newton’s universal gravitational as local motion in a curved space-time continuum warped by the presence of matter.  Mathematical physics was rewriting our most basic views of reality.

The Cambridge physicist Arthur Stanley Eddington lead a 1922 total eclipse expedition that demonstrated that the bending of starlight by the sun’s gravitational field that conformed to and thus helped verify Einstein’s mathematical predictions.  Eddington in the 1920s and 1930s popularized the “New Physics” of relativity and quantum theory in books and public lectures.  Eddington opened his 1927 Gifford Lectures by describing the two views of the table on which he is writing his lectures.[19]  One is the substantial and solid table of the familiar ordinary world.  The other is the mathematically described table of the new physicists, mostly empty space full of tiny positive or negatively charged particles that moved according to rules that were utterly foreign to Newton’s “classical” physics and familiar common-sense notions.  What are we to make of Eddington’s “scientific table” in relation to his ordinary everyday table?  Is the physicist’s table as real as the familiar one, and if so, in what sense?  Can we know what underlies the familiar?

In his 1934 Ph. D. thesis, young Thomist Charles De Koninck vigorously defended Eddington’s ideas about physics and philosophy, finding them quite consistent with Scholastic philosophy, although Eddington was not nor ever claimed to be a philosopher.[20]  Yet, De Koninck said of him: “I do not hesitate to call him one of the greatest philosophers of our time.”[21]  In his thesis, De Koninck sharply criticized Jacques Maritain’s 1932 book The Degrees of Knowledge for misunderstanding Eddington and the philosophical significance of his thought concerning the New Physics.[22] At issue is what are we to make of the formal “beings of reason” (electrons, atoms, curved space. etc.) that Maritain, De Koninck, and Eddington each saw in his own way as symbols or images associated with the mathematics that explains the empirical measurements of the “real world” that scientists make.

Maritain in many ways expressed a great appreciation for the New Physics, which had made great advances in “saving the appearances” of a wide variety of phenomena in the macroworld and the microworld.  In particular it had completely upended the clockwork universe of Newtonian mechanism.  Maritain saw this as a good thing.[23]  But Maritain claimed that the empirico-mathematical physics only generated abstract “beings of reason with a foundation in reality” (entia rationis cum fundamento in re) that were in effect, like Ptolemy’s epicycles and eccentrics, only convenient mathematical fictions that could not get at the true natures or essences of things that were only accessible to the philosopher.[24] They do tell us something about the real world outside of the physicist’s mind; after all, they were based on observations and did “save the appearances.” De Koninck critiqued Maritain’s views on the metric of space in Einstein’s relativity and on the probabilistic nature of quantum theory.   Maritain argued that the philosopher could know for certain that the metric of space had to be Euclidian, even if Einstein required non-Euclidian curved space in his mathematical theory that “saved the appearance” of gravitational phenomena.   De Koninck sharply argued on Thomistic principles that Maritain was wrong about this, that Maritain did not fully appreciate the nature of physical measurement, and that Eddington had articulated correctly how to understand the nature of measurement. 

In a talk given at a previous American Maritain Association conference and published in the conference volume, Facts are Stubborn Things,[25] John Brungardt argued that De Koninck’s Thomistic critique was right and Maritain had made a mistake, as De Koninck had pointed out, by confusing extension with quantity.  This first mistake caused a second mistake with respect to the degrees of abstraction the human mind makes from entities in the world.  To a Thomist, in the first degree of abstraction, abstract physical beings of reason retain reference to matter as a condition of change, whereas in the second degree, abstract mathematical being of reason make no reference to matter.  If matter—the root of variability and change–is removed in this second degree of abstraction, the metric of space must be Euclidean before anything else, as Maritain thought.  But if matter can cause metric variability, the metric structure of any physical quantity can only be known by empirical discovery.  Brungardt makes the key point:

finding the metric of physical space requires one to join the second degree of abstraction to the first insofar as this is possible. This is done through the act of measurement, which is both physical and mental. De Koninck is correct about the former: this “joining” requires an account of measurement with a referentially defined measurement standard if we wish to know the metric of the physical cosmos. Maritain is correct about the latter: insofar as this “joining” by mind unites in its consideration two types of abstraction which are defined in opposition to each other, the physicist must utilize a being of reason to achieve his ends.[26]   

De Koninck, Eddington, and Maritain each in his own way all knew that in order to do their work physicists must use abstracted “beings of reason founded in reality.”  Maritain called such beings of reason symbols, while Eddington called them shadows.  Brungardt concisely summarized the heart of the matter this way: “Maritain’s view grants physics a knowledge of the real only in its symbols, while De Koninck’s view claims for physics a knowledge of the real through its symbols. [the italics are in the original]”[27]  De Koninck and Eddington both make the point that scientists really do gain real, even if incomplete, knowledge about the nature of physical reality from mathematically articulated empirical discoveries.  That is, science is doing far more than merely saving the appearances. 

The story is not over yet: for example, Andrew Younan’s 2023 book, Matter and Mathematics, carries the project forward using Aristotelian/Thomistic principles to argue—one may say successfully–that mathematical physics is grounded in the real natures of things known through the abstractive capacity of the human mind.[28] How we abstract from matter remains an important theme.  Following Aristotle’s and Aquinas’s understanding of nature as an internal principle of motion and rest, Younan sees the laws of physics as relations of mathematical abstractions describing material things or their motions.[29] While such laws are human productions, they “are not independently existing or causally governing entities … nor are they arbitrary descriptions singled out for their congeniality to human preferences.  One element of them is based on the activity of the mind—abstraction—and another element is based truly in reality—the quantitative aspects of natural things.”[30] Younan sees this capacity of the human mind to abstract universal principles from its encounter with particular things to be consistent with Thomistic noetics, that is, how we come to have a knowledge (science) of things in general.  The mathematical laws of physics “are the products of a science that is a combination of natural philosophy and mathematics, and as such they will never have the certainty and finality of mathematical proofs: physics is mathematical … but it is not mathematics.”[31]  One might say that the equations of physics account for mathematical relations pertaining to material phenomena–that is, they “save the appearances” of observed activity in the world—and thus represent a human abstraction of formal principles inherent in material things themselves, offering a true although limited understanding of the nature of things under the aspects of mathematical quantity and relation.

Duhem’s reading of history alerts us to two dangers: one is the need for caution in imposing one’s metaphysics on the world—we try to tell the world how we know it must act and the world consistently refuses to oblige.  The other danger is the refusal of metaphysics at all, to try to think pragmatically that we can get along quite nicely in the world without metaphysics, as unrepentant positivists and reductionists insist.  The second danger is surely the greatest danger in our time.  Maritain, De Koninck, Eddington, and Duhem, and St. Thomas all knew in their bones that there was more to the world than mathematics and physics, that mathematical physics is only possible by virtue of a greater background in which it takes place and which gives it meaning.  De Koninck and Eddington were explicit about this.  Seeing this greater background in which all appearances are implicate brings us to metaphysics, and even beyond that to theology.  All 5 of these figures were Christian, four Catholics and one, Eddington, an English Quaker of a mystical sort. In spite of some of the things they said, neither Eddington nor Duhem were positivists. Duhem ended an essay on the Value of Physical Theory with these words:

 the physicist is compelled to recognize that it would be unreasonable to work for the progress of physical theory if this theory were not the increasingly better defined and more precise reflection of a metaphysics; the belief in an order transcending physics is the sole justification of physical theory. [the italics are in the original][32]

The purpose of this essay has been to point out that the lessons of history can give light for a fruitful path ahead for those who follow the way of St. Thomas.  The dispute of De Koninck with Maritain in the former’s 1934 thesis still has philosophical gems to be mined from it.  For example, De Koninck, following Eddington, gives an excellent defense of an objective indeterminism in physics, which however, is not a lack of determinate order in the world.  In spite of Duhem’s warning of danger, physics and metaphysics cannot be essentially disentangled, as Maritain himself knew, for physics always transpires under a metaphysical sky which colors all that one sees. Cloudy skies obscure vision of the true sky.  De Koninck’s study of Eddington and Maritain shows that while Duhem’s warning about entangling physics and metaphysics is warranted–for the historical record shows that this can be done badly—still, a mutual engagement of these distinct ways of knowing must proceed with great care with intellectual humility and a respect for logic.  Younan’s recent work is a constructive step in this direction.  Reality speaks to us in many languages, and mathematical physics is one of them.  We must learn to speak it well, so we can articulate accurately what reality is actually showing us about the true nature of things.

 

 Citations

[1] Charles De Koninck, “The Philosophy of Sir Arthur Eddington,” thesis presented to the Cardinal Mercier Institute of Louvain for the obtention of the doctorate in philosophy, in The Writings of Charles De Koninck: Volume 1, edited and translated by Ralph McInerny (University of Notre Dame Press, 2008) 142-356.

[2] Jacques Maritain, Distinguish to Unite, or the Degrees of Knowledge, translated by Gerald Phelan (University of Notre Dame Press, 1995); the French original is Distinguer pour unir; ou, Les degrés du savoir (Desclée de Brouwer, Paris, 1932).

[3] Pierre Duham, To Save the Phenomena: An Essay on the Idea of Physical Theory from Plato to Galileo, translated by Edmund Dolan and Chaninah Maschler (University of Chicago Press, 1969); the French original is ΣΩZEIN TA ΦAINOMENA:essai sur la notion de théorie physique de Platon a Galilée (A. Hermann et Fils, Paris, 1908).

[4] Greek meanings are taken from Brill Dictionary of Ancient Greek, ed. by F. Montanari, et al. (Koninklijke Brill, Leiden, The Netherlands, 2015).

[5] Duhem, To Save the Phenomena, 5

[6] Ibid, 17

[7] Ibid, 17

[8] Ibid, 31

[9] Ibid, 33

[10] Thomas Aquinas, Summa Theologiae, I, q. 32, a. 1, ad 2

[11] Duhem, To Save the Phenomena, 43

[12] Ibid, 44

[13] Ibid, 60

[14] Ibid, 52

[15] Ibid, 107

[16] Ibid, 107

[17] Isaac Newton, The Mathematical Principles of Natural Philosophy, trans. by Andrew Motte (London, 1729), 393.

[18] Duhem, To Save the Phenomena, 116.

[19] A. S. Eddington, The Nature of the Physical World (Macmillan, New York, 1929), ix-xvii.

[20] De Koninck, Writings of Charles De Koninck, Vol. 1, 300.

[21] Ibid., 240.

[22] Ibid., 284-294.

[23] Maritain, Degrees of Knowledge, 164-166.

[24] Ibid., 47; see also Chapter V, “Knowledge of Sensible Nature,” in which Maritain deals extensively with the “New Physics” and Eddington in particular.

[25] John G. Brungardt, “A Thomistic Reply to Grunbaum’s Critique of Maritain on the Reality of Space,” in Facts are Stubborn Things: Thomistic Perspectives in the Philosophies of Nature and Science,” edited by Matthew K. Minard, American Maritain Association Book Series, Volume 30 (Catholic University of American Press, 2021), 109-123.

[26] Ibid, 123.

[27] Ibid, 121.

[28] Andrew Younan, Matter and Mathematics (The Catholic University of America Press, Washington, 2023).

[29] Ibid., 168; see 165-172.

[30] Ibid., 169.

[31] Ibid., 170.

[32] Pierre Duham, “The Value of Physical Theory,” an Appendix to The Aim and Structure of Physical Theory, translated by Philip P. Wiener (Princeton University Press, 1982), 335.