Above:  Photograph “frosty patterns on a window pane”, taken by Aleksandr Frolov. [Copyright information can be found at https://commons.wikimedia.org/wiki/File:Frost_on_the_window_04.jpg]

“And God said …” The world is a product of the Word, of the Logos, as Saint John expresses it, using a key term from the Greek language. “Logos” means “reason,” “sense,” “word.” It is not reason pure and simple, but creative Reason, that speaks and communicates itself. It is Reason that both is and creates sense. The creation account tells us, then, that the world is a product of creative Reason. Hence it tells us that, far from there being an absence of reason and freedom at the origin of all things, the source of everything is creative Reason, love, and freedom. Here we are faced with the ultimate alternative that is at stake in the dispute between faith and unbelief: are irrationality, lack of freedom and pure chance the origin of everything, or are reason, freedom and love at the origin of being? Does the primacy belong to unreason or to reason? This is what everything hinges upon in the final analysis. — Pope Benedict XVI  ¹

In 1960, the Nobel Prize-winning physicist Eugene Wigner published a classic essay entitled  “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” ²  In it, he reflected on a fact that, while easy to take for granted, is quite surprising and even awe-inspiring, namely that mathematics plays a deep and powerful role in our scientific descriptions of the natural world.  In his words, “the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena,” indicating that mathematics “is, in a very real sense, the correct language” in which to formulate the laws of physics.  There is no obvious reason within physics why this should be so, and Wigner marveled at it, arguing that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.”  His essay set off decades of discussion among physicists, mathematicians, and philosophers.

But is it really “unreasonable” that mathematics should be so well-suited to describing the physical world?  It seems especially questionable whether the effectiveness of mathematics should be unexpected if God created the universe.  Interestingly, both theists and atheists have tended to think that the existence of God would provide an explanation of the effectiveness of mathematics.  The Christian philosopher William Lane Craig, for example, has argued that it is the best explanation.³  And even some prominent atheists, such as Leonard Mlodinow,⁴ are willing to concede that it is at least some kind of explanation.  Indeed, Wigner’s original article (and Wigner was not religious) used the word “miracle(s)” about a dozen times and called the effectiveness of mathematics a “wonderful gift” — which suggests a giver of that gift.

Contemplating the intricacy, precision, and beauty of the cosmos (a word which, in ancient Greek, could be used to mean order as well as world) can indeed draw one’s mind toward God, providing a compelling reason for the skeptic to begin questioning his skepticism and for the believer to deepen her wonder at Creation and her love of God the Creator.  But even if we posit that God is the Creator of the universe, one might still wonder: why math?  Why is it mathematics that works so well to describe the natural world?  Is there any clear reason why God would make certain aspects of reality in such a way that human beings could discern mathematical structure therein?

It would perhaps be a disservice to the depth of these questions, and to the admirably open-ended spirit of seeking truth in which Wigner’s essay and many subsequent contributions to this discussion have been composed, to claim to have reached any final answers.  But I will share some reflections on why a mathematical order to physical reality seems suitable, or fitting — or, even better, reasonable — if the universe really does proceed from the “transcendent, intelligent and good” ⁵ Creator God.  Perhaps the effectiveness of mathematics as a tool for understanding the physical world only seems unreasonable if you think the universe has no transcendent First Cause.  On the other hand, if you think the universe has a transcendent Creator with an unfathomable, infinite “mind” of which the most ingenious human mind is but a pale shadow (even while human beings are made in the “image and likeness” of God), it seems rather reasonable that the universe should be endowed with deep structure and regularity that the greatest scientific and mathematical minds working over hundreds of years would have succeeded only partially in unravelling.  With God in the picture, the effectiveness of mathematics in the natural sciences might still be beautiful and compelling and amazing.  What it won’t be, is unreasonable or unexpected.

Examples of the Deep Effectiveness of Mathematics in the Natural Sciences

Let’s illustrate in some detail the phenomenon that we might call, as an alternative to Wigner’s phrase, the deep effectiveness of mathematics in the natural sciences.

Here is a relatively famous physics equation: F = GmM/r², or in the simplified and less general form in which you may have encountered it in a high-school physics course, F = mg.  In words, F = mg indicates that the gravitational force on an object falling near the surface of the Earth is equal to its mass m multiplied by a constant, g (which has a numerical value of approximately 9.8 meters per second-squared).  The fact that g is a constant implies that every object near Earth’s surface falls with the same acceleration due to gravity.  Any differences in acceleration between one falling object and another are due to something other than gravitation (such as air resistance).

This is relatively simple math, expressing a relatively straightforward regularity in the world, albeit one that is counterintuitive and that remained largely hidden until the last few centuries.  But there are many examples where the math that describes the physical world is extremely abstract.  For example, consider the following equation:

G_μν + Λg_μν = κT_μν.

This equation, which takes only a few seconds to write down, represents a fundamental set of relationships between the curvature of spacetime and the energy and momentum of matter in Einstein’s theory of gravity (called general relativity), which is an explanation of gravity that makes even more accurate predictions than Newton’s.  The mathematical quantities with Greek-letter subscripts in this equation do not stand for simple numbers but rather something much more abstract — tensors, which can be represented as matrices with several variables within them.  A great deal of complexity and information are conveyed by this equation, which has been experimentally verified as accurately describing phenomena across an enormously wide range of distance scales.

The remarkable thing in all this is not just that laws of physics can be formulated mathematically, it is that the equations are compact, elegant, and tremendously powerful. For example, Maxwell’s equations of classical electromagnetism⁶ together with the Lorentz force law are a marvel of elegant compactness that contain all of classical electromagnetic theory — which successfully accounts for the results of hundreds of thousands of experiments performed across many decades — in a few mathematical expressions that can be written down on a postcard.

Part of the power of mathematics in physics stems from the fact that the same mathematical forms often appear in many seemingly disparate places.  For example, purely by manipulating and combining the equations of electromagnetism that now bear his name, the 19th-century physicist James Clerk Maxwell was able to show that one could derive from them a “wave equation,” a kind of equation already familiar to physicists as describing such phenomena as sound waves and waves in water.  On this basis, Maxwell predicted that there ought to be such things as electromagnetic waves, i.e. waves where it is electric and magnetic fields that are doing the “waving” or oscillating.  Moreover, the equations allowed him to calculate the speed at which these waves would travel, and he found that it was numerically the same as the speed of light, which had previously been measured.  He therefore concluded — just on the basis of the mathematics — that light must actually consist of electromagnetic waves.  All of this was in the 1860s, a quarter-century before the existence of electromagnetic waves was experimentally verified.⁷

Physicists have so much faith in the predictive power of their equations that it is now routine for many millions and even billions of dollars to be spent building huge experimental facilities to detect things like the Higgs boson or gravitational waves, the existence of which (like electromagnetic waves) was predicted solely on the basis of mathematical physics theories decades before experiments succeeded in confirming them.  Particle accelerators like the LHC and gravitational-wave observatories like LIGO would never have been brought into existence if physicists had lacked the confidence that they would detect something, and in these and many other cases, the success of these experiments has vindicated the physicists’ confidence.  As Richard Hamming summarizes in a 1980 article entitled “The Unreasonable Effectiveness of Mathematics” (20 years after Wigner’s): “constantly what we predict from the manipulation of mathematical symbols is realized in the real world … For glamour, I can cite transistor research, space flight, and computer design, but almost all of science and engineering has used extensive mathematical manipulations with remarkable successes.” ⁸

While physics equations make precise and accurate predictions across a huge range of phenomena and a huge range of scales, from the microscopic to the galactic, their precision and accuracy are not unlimited.  For example, Newtonian gravity was superseded by Einstein’s general relativity because it failed to make fully accurate predictions about phenomena like Mercury’s orbit or the bending of starlight by the Sun’s gravity.  Even Maxwell’s equations fail when we are dealing with photons, the smallest “particles” of light.  As Wigner pointed out, even theories known to have serious shortcomings can sometimes give “alarmingly accurate descriptions of groups of phenomena.”  So physicists constantly try to push further, to find mathematical theories that have all the predictive success of existing theories but that have greater accuracy and succeed across an even wider range of phenomena and scales.  But within physics, mathematical laws have generally been found to have domain-specific validity.  Moreover, even a fundamental law that applied exactly, always and everywhere, would not necessarily specify what concrete substances and situations actually exist for the law to apply to.  As Wigner put it,

“[T]he laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world. Thus, classical mechanics, which is the best known prototype of a physical theory, gives the second derivatives of the positional coordinates of all bodies, on the basis of the knowledge of the positions, etc., of these bodies. It gives no information on the existence, the present positions, or velocities of these bodies.”

Why Mathematical Laws Might Be Reasonable: Regularity, Consistency, and Intelligibility

How might positing the reality of God the Creator help us make sense of why it is “reasonable” that there would be mathematical laws of physics with great but not necessarily unlimited generality?

First, the fact that one can use mathematics in physics is a reflection of the tremendous regularity in how the universe works.  The laws have great predictive power, because certain abstract patterns are repeated throughout the natural world, at least approximately.  It is not obvious that the universe had to have any order, or lawfulness, or patterns, or predictability, or stability, or regularity at all, let alone an order “far deeper than anyone ever suspected” ⁹ that lends itself surprisingly often to mathematical analysis and connections.

Another way to say this is that the natural world operates in a profoundly consistent way, and one of the defining characteristics of mathematics is its self-consistency.  Even moderate and partial applicability of mathematics to the real world reflects a profound consistency within the natural order: when the same mathematical structures appear in multiple places in physics, it reveals similar underlying structures and patterns, as when a wave equation appears in fluid dynamics and in electromagnetism. As Hamming writes, “We have tried to make mathematics a consistent, beautiful thing, and by so doing we have had an amazing number of successful applications to the real world.”  This consistency fits naturally with divine creation by the transcendently wise and profoundly consistent God who is “the same yesterday and today and for ever” ¹⁰ and who “accomplishes all things according to the counsel of His will,” ¹¹ whereas without God in the picture such principles of order would seem like they appear rather out of nowhere — a “miracle,” ¹²  to use Wigner’s word.

The universe seems not only to have more regularity and consistency than it “had” to, but also more intelligibility: we can use mathematics to make sense of aspects of the world that are far beyond ordinary human experience and otherwise unintuitive, from the structure of a hydrogen atom to how galaxies formed in the early universe.  Using existing mathematics or developing new math for the purpose, physicists probe the underlying structure of things and find that even when the scales and contexts are radically unfamiliar to us, there is order, beauty, and at least partial comprehensibility, allowing sciences like cosmology and nuclear physics to be possible.  It seems highly nontrivial that the universe, especially on its largest and smallest scales, should be so comprehensible to us.  Wigner refers to “the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them” — but perhaps these appear as disconnected “miracles” only if one does not believe in actual miracles and the God who performs them,¹³ who gives to us humans, who are made in His image, our intelligence as a small and partial reflection of His own comprehension of all things, and who “wanted to make his creatures share in his being, wisdom and goodness” ¹⁴ and so makes an intelligible and consistent world that we can make sense of and thereby live and learn and grow in.¹⁵  As the second-century Church Father St. Irenaeus wrote, God “has fitted and arranged all things by his wisdom” and “comprehends all,” ¹⁶ and as Pope Benedict XVI put it, God’s “creative reason” “precedes everything and we are truly the reflection of creative reason.” ¹⁷

Why Mathematical Laws Might Be Reasonable: Relationships

Mathematics allows us to penetrate more deeply into the relationships that pervade the structure of the cosmos.  Every equation expresses a relationship among the quantities appearing in it.  And the laws of physics themselves are all about relationships: between subatomic particles or between galaxies, between matter and spacetime, between electric and magnetic fields, between different measurable quantities.  The applicability of equations (or, for that matter, inequalities like the second law of thermodynamics) written in the precise language of mathematics reveals an abstract order expressing a consistency, depth, and richness of relationship well beyond what initially meets the eye.  The laws of physics are a little like the Sun’s corona (which can be seen only during a total solar eclipse): present all the time, and deeply beautiful, but usually hidden from view and manifest to us only under very specific circumstances.  Our capacity for mathematics gives us more insight into a certain subset of these relationships, their deep consistencies and hidden beauties, than we would otherwise be capable of.

Christianity proposes that God is One, and that God is also Three, and so is inherently relational yet “supremely undivided” ¹⁸ within Himself.  Classical Trinitarian theology holds that (in the words of the Athanasian Creed) “we worship One God in Trinity and Trinity in Unity, neither confounding the Persons [Father, Son, and Holy Spirit] nor dividing the substance.” ¹⁹  The relationality of God is reflected also in how He relates to Creation: “lov[ing] all things that exist,” ²⁰ offering covenants to human beings in the Bible, and even entering into Creation in the person of Jesus “in [Whom] all things hold together” ²¹ to restore humanity to a relationship of love with our Creator.  According to the work of Christian thinkers like the physicist John Polkinghorne and the chemist and theologian Ross Hastings, God’s inherent relationality and unity are also mirrored in Creation itself, in “the dynamic interrelatedness of the cosmos.” ²²  Creation features many distinct things that work in relationship with one another.  Order in general is about relationships, about connections; and the cosmos is a place of coordinated activity, not mere chaos.  Understood in this light, mathematically ordered relationships in the material world give us glimpses of a “flawless harmony which is in conformity with sublime reason,” in the words of the mathematician and physicist Hermann Weyl, a harmony that is readily understood as the work of “an all-merciful and all-mighty God.” ²³  The insight that relationships are fundamental to being fits rather naturally with Trinitarian theology, ²⁴ and mathematical laws characterize relationships.

Why Mathematical Laws Might Be Reasonable: Beauty and Goodness

As suggested by Weyl’s words, the search for deep, elegant underlying principles for the inner workings of the universe is a search for hidden beauty.  In practice, the abstract mathematical beauty of the laws of physics is an important source of motivation for theoretical physicists.  Accepting a Nobel Prize for his work in particle physics, the Pakistani Muslim physicist Abdus Salam quoted a passage from the Qur’an about how observing the seamlessness of God’s creation leaves one’s gaze dazzled and spoke of “the faith of all physicists”: “the deeper we seek, the more is our wonder excited, the more is the dazzlement for our gaze.” ²⁵

In addition to its mathematical structure being beautiful, the ordered lawfulness of the universe seems deeply good.  Speaking about mathematicians, astronomers, and physicists of the past, the physicist who first proposed the Big Bang — who was also a Catholic priest, Georges Lemaître — once said, “The deeper they penetrated into the mystery of the universe the deeper was their conviction that the power behind the stars and behind the electrons of atoms is one of law and goodness.” ²⁶ It is not uncommon to see the kind of lawful order that we see in the laws of physics associated with good, and with the good God who creates and sustains a reliable natural order and who, as the Book of Genesis narrates, “saw that [Creation] was good,” ²⁷ in contrast with the many ancient creation myths that present the world as a product of “a confused jumble of forces in conflict with one another” or “hidden deities at war with one another.” ²⁸ We all go about our lives assuming that the laws of physics will not suddenly and arbitrarily change; and perhaps this is a form of implicit trust — even for those who do not think of themselves as believers — in the faithfulness of God who not only gives His creatures “being and existence, but also, and at every moment, upholds and sustains them in being.” ^29,30

Why Mathematical Laws Might Be Reasonable: Non-Determinism

Scope-dependent, domain-specific mathematical laws are good also in the sense that they are not all-encompassing or constraining to the point of complete determinism.³¹  While positing a lack of determinism in the structure of the universe is somewhat controversial among physicists, and the precise way in which apparent determinism at some levels fits together with openness and contingency at other levels is still a matter of debate among practically everyone who thinks about these things, modern physics is not inconsistent with the commonsense understanding that there is both real constraint and real contingency in how things work³² — and this is partly because, as Wigner puts it, the laws of nature are “of almost fantastic accuracy but of strictly limited scope.”  And even for the types of things that are within their scope, they do not specify which particular things and situations actually exist in the real world.³³  Also, as Hamming points out, not only are there limits to how widely mathematical laws can be applied within physics, but physics itself only addresses certain very specific aspects of how the world works:

“Science in fact answers comparatively few problems … Indeed, to generalize, almost all of our experiences in this world do not fall under the domain of science or mathematics. … It has been an act of faith on the part of scientists that the world can be explained in the simple terms that mathematics handles. When you consider how much science has not answered then you see that our successes are not so impressive as they might otherwise appear.” ³⁴

Physicists like to think of physics as fundamental, and there is some truth in that, but what if physics captures only the underpinnings of something much bigger, and life and freedom are among the bigger realities which the laws of physics are intended to enable?

Perhaps the laws of physics are like the laws of a country in this regard, in that they indicate some basic constraints on behavior, but many different actions and behaviors are “allowed” within those basic constraints.  Just as an overly constraining set of national laws would be tyrannical, so an overly constraining physics would be deterministic, the kind of creation that leaves no room for freedom.³⁵  In any case, mathematical laws give structure, regularity, and predictability to many things, but they are also consistent with freedom, openness, and the working out of purposes within the created world.³⁶  There are limits to the descriptive and predictive power of the laws of physics, yet the world is not chaotic and purposeless.  Whatever determinism there is provides structure in which freedom can act.

If God is really “the free and sovereign Creator, the first cause of all that exists,” ³⁷ then “freedom and love” are the primordial realities baked in from the beginning, not the strict constraints of necessity or determinism — as Pope Benedict XVI points out in the quotation at the beginning of this article.  This is the God who is capable of creating all things from nothing, endowing them with structure and causal order.  The impressive yet bounded effectiveness of mathematics in the natural sciences then flows naturally from the infinite intelligence and creativity of God the Creator, who has “fashioned all [His] works in wisdom and in love” ³⁸ and given His creatures freedom in an ordered, intelligible, yet open cosmos.³⁹

Conclusion

This essay has attempted to explore why the effectiveness of mathematics in the natural sciences does not appear so unreasonable if one is willing to contemplate the possibility that the cosmos is the work of God.  If the “mind of God” has the depth and power that the Creator of such an infinitely fascinating universe must have, perhaps the profound consistency, relationality and interconnectedness, and ordered lawfulness expressed in the laws of physics should not surprise us so much.  And perhaps it should likewise not surprise us that we can comprehend some of the great regularities of the universe, in part but not in whole.  Christian faith proposes the reality of a Creator who is far greater than us, transcendent, unlimited by space and time, not composed of anything,⁴⁰ the granter of being to all that has being, the uncaused cause, the originator not only of the universe’s first moments but the One who sustains everything in existence in every moment, not subject to the laws of nature but, rather, their originator.  Yet we are made in His image and so can, with great effort and partial understanding, grapple with the deep truths that He comprehends effortlessly and completely even as He calls them into existence out of nothing.  If the world was created for Love and Freedom and Beauty, these are exactly the kinds of regularities we should expect: not overwhelming the possibilities of contingency and free will, but providing structure for a lawful order.  Our ability to formulate mathematical laws of physics then derives from a profoundly consistent Creator who designed the universe to reflect His great and unfathomable consistency, relationality, and goodness, and to be intelligible to us.  As the Catechism of the Catholic Church puts it, quoting passages from the biblical books of Wisdom, Colossians, and Genesis:

“Because God creates through wisdom, his creation is ordered: “You have arranged all things by measure and number and weight.” The universe, created in and by the eternal Word, the “image of the invisible God,” is destined for and addressed to man, himself created in the “image of God” and called to a personal relationship with God. Our human understanding, which shares in the light of the divine intellect, can understand what God tells us by means of his creation, though not without great effort and only in a spirit of humility and respect before the Creator and his work. Because creation comes forth from God’s goodness, it shares in that goodness – “and God saw that it was good. . . very good” – for God willed creation as a gift addressed to man, an inheritance destined for and entrusted to him.” ⁴¹

It may make some physicists, even those who believe in God, rather uneasy to invoke God as a solution to a question about physics, even a rather clearly philosophical question like that of the effectiveness of mathematics.  But in at least one regard, this is exactly how physics works: physicists often expand their theoretical frameworks by positing something that was not in their previous theoretical frameworks — like dark matter, or new subatomic particles, or new interactions — and seeing whether doing so yields results consistent with what they know already, and with what they can find out.  If the new model gives results that are more consistent with reality than the old model, that is a significant point in favor of the new model.  This is exactly how this essay suggests we might proceed with respect to the effectiveness of mathematics as a philosophical question: posit that the Creator God is responsible for it and see whether this “new model” makes better sense of the phenomenon we are considering — whether the effectiveness of mathematics goes from looking unreasonable to looking reasonable.  When it comes right down to it, this model is not even new: Newton himself and many others, after all, posited the Creator God as the ultimate origin of a lawful world that could be investigated through rational and mathematical investigation, and so confidently carried physics into new territory.

I think we should take Wigner’s words quite literally when he states that the effectiveness of mathematics in physics is a “wonderful gift.”  Perhaps mathematical order that is comprehensible to us is literally a gift, not necessary but all the more wonderful because it did not have to be, and comes to us from the hand of the freely creating, ordering, loving God.  If this is the case, the solution was there all along, implicit in the famous essay of Wigner that formulated the problem: the deep effectiveness of mathematics seems so “unreasonable” precisely because it is gratuitous, rightly regarded as a miracle and a gift.  And perhaps that is exactly what we should call it: the gratuitous effectiveness of mathematics in the natural sciences.

The author thanks Nicole Gibelyou for extremely helpful comments on multiple drafts of this essay, and Ronald Larson, Lamese Saab, Ross Hastings, Abhi Shukul, Leo Ellenson, and Chris Fink for thoughtful reviews as well.

References

1. Link goes to https://www.vatican.va/content/benedict-xvi/en/homilies/2011/documents/hf_ben-xvi_hom_20110423_veglia-pasquale.html.
2. Link goes to https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html. Wigner’s essay is also available at https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf.
3. See his various articles and videos and blog posts on this topic.
4. Link goes to Closer To Truth, “Leonard Mlodinow – Why the ‘Unreasonable Effectiveness’ of Mathematics?,” https://www.youtube.com/watch?v=uqGbn4b3LPM.
5. Catechism of the Catholic Church 284
6. Link goes to  https://en.wikipedia.org/wiki/Maxwell%27s_equations
7. A fun connection: https://physics.stackexchange.com/questions/154984/and-god-said-and-there-was-light-what-does-these-equations-mean
8. See R.W. Hamming, “The Unreasonable Effectiveness of Mathematics,” at https://ned.ipac.caltech.edu/level5/March02/Hamming/Hamming.html or https://math.dartmouth.edu/~matc/MathDrama/reading/Hamming.html
9. Stephen M. Barr, “God and Cosmic Order,” https://catholicscientists.org/articles/evidence-of-god/
10. Hebrews 13:8
11. Ephesians 1:11
12. Wigner refers to it as a miracle “that in spite of the baffling complexity of the world, certain regularities in the events could be discovered.”
13. See, e.g., Joe Heschmeyer, “How Science Proves God’s Existence,” https://www.catholic.com/audio/sp/how-science-proves-gods-existence
14. Catechism of the Catholic Church 295
15. If there were no consistency and intelligibility in the way the world works, or if the predictability of the natural order were far less comprehensive than the mathematical character of the laws of physics enables it to be, it is difficult to see how there could be a sufficient cause-and-effect framework for human beings (or anything else) to be “able to be agents who can act with intention. … The hunter must know that his spear will proceed to its target and not suddenly sprout wings and fly away or make a right-angle turn. … If one of God’s aims is to have a world where there are rational agents, then this is important.” (Stephen Barr, personal correspondence)
16. Against Heresies 2:30:9, see e.g. https://www.catholic.com/tract/the-one-true-god
17. https://www.vatican.va/content/benedict-xvi/en/speeches/2007/july/documents/hf_ben-xvi_spe_20070724_clero-cadore.html
18. Thomas Aquinas, Summa Theologiae, First Part, Question 11, Article 4: https://www.newadvent.org/summa/1011.htm#article4
19. https://www.ewtn.com/catholicism/teachings/athanasian-creed-209
20. Wisdom 11:24
21. Colossians 1:16
22. W. Ross Hastings, Echoes of Coinherence, p. 166
23. http://www.weylmann.com/2011archive.shtml
24. One of the “surprising number of features of the modern scientific account of reality [that] fit well with the being of the Creator God of Christian theology,” in Hastings’ words. See Echoes of Coinherence, p. 11.
25. See https://www.nobelprize.org/prizes/physics/1979/salam/speech/. Interestingly, Wigner also uses the word “faith” multiple times.
26. From a New York Times interview with Duncan Aikman: https://www.nytimes.com/1933/02/19/archives/lemaitre-follows-two-paths-to-truth-the-famous-physicist-who-is.html or see reprint at https://catholicscientists.org/articles/lemaitre-follows-two-paths-to-truth-new-york-times-1933-interview-with-lemaitre/.
27. Repeated several times in Genesis 1.
28. Joseph Cardinal Ratzinger (Pope Benedict XVI), The Divine Project: Reflections on Cre29ion and Church, p. 18
29. Catechism of the Catholic Church 301
30. The objection may be raised that the lawful order of the universe enables evils as well as goods: natural disasters and diseases alongside pleasant weather and bodily health, for example. Yet lawful order is still a good in and of itself, especially as compared to the alternatives of chaos and disorder. While disease and disaster may be permitted by the laws of physics just as much as health and sunshine, the lawful order of the universe is a good on a higher “plane” than these goods and evils. The idea that “natural evil” is under the sovereignty of a higher good, and limited by that higher good, fits well with the principle articulated by St. Augustine, St. Thomas Aquinas, and many other contributors to the Catholic intellectual tradition, that evil is a privation of the good, not an independently subsisting “thing”; that “every being, as being, is good” (Aquinas, Summa Theologiae, First Part, Question 5, Article 3: https://www.newadvent.org/summa/1005.htm), and that God permits evil precisely because in His “infinite goodness” He can “bring good even out of evil” (Summa, First Part, Question 2, Article 3: https://www.newadvent.org/summa/1002.htm) in ways that would not be possible without permitting the evil, as He did especially in the Crucifixion and Resurrection of Jesus – bringing about the redemption of the world from the incalculable evil of human beings crucifying God incarnate. As C.S. Lewis put it, “evil is a parasite, not an original thing. The powers which enable evil to carry on are powers given it by goodness” (Mere Christianity, p. 51). Evil may twist and corrupt things within boundaries, but it doesn’t get to establish the boundaries, and no one would like it very much if it did. For a fascinating reflection on how these observations connect with a biblical understanding of Creation and the sovereignty of the good God, see Joseph Cardinal Ratzinger (Pope Benedict XVI), The Divine Project: Reflections on Creation and Church, pp. 15-37.
31. This is especially true if by “determinism” we also include probabilistic determinism of the sort found in quantum mechanics.
32. E.g. if I tell you “jump up as high as you can,” you have the ability to choose whether to do what I am asking and attempt to jump, but if you do jump, you will fall back down to the surface of the Earth: gravity (along with many other physical realities) constrains what is possible for your body without fully determining what you do. For a more detailed consideration of free will, contingency, and physical constraint, see Cameron Gibelyou and Douglas Northrop, Big Ideas: A Guide to the History of Everything, especially Chapter 6, “Causation.”
33. Often the way this manifests itself in practice is that initial conditions must be specified in order to apply a mathematical law to a real situation.
34. See Hamming, “The Unreasonable Effectiveness of Mathematics”: https://ned.ipac.caltech.edu/level5/March02/Hamming/Hamming.html. For example, it is by no means clear that science and mathematics could ever, by their nature, answer the questions of “what Truth, Beauty, and Justice are” – to take just one example, offered by Hamming.
35. Under-constraining national laws are also not good; they provide little structure and leave too much room for the raw exercise of undisciplined power. Notice, e.g., that one of the first points mentioned in George Orwell’s Nineteen Eighty-Four is that the dystopian, totalitarian society of Oceania has no laws. Perhaps here too there is an analogy with laws of physics, which need to be “strong” enough to provide order and pattern, without being so constraining as to fully and completely determine everything that happens.
36. A profound line from St. Thomas Aquinas: “whatsoever divine providenceordains to happen infallibly and of necessityhappens infallibly and of necessity; and that happens from contingency, which the plan of divine providence conceives to happen from contingency.” (Summa Theologiae, First Part, Question 22, Article 4: https://www.newadvent.org/summa/1022.htm#article4) Again, for a more detailed consideration of free will, contingency, and physical constraint, see Cameron Gibelyou and Douglas Northrop, Big Ideas: A Guide to the History of Everything, especially Chapter 6, “Causation.”
37. Catechism of the Catholic Church 300
38. Eucharistic Prayer IV, see e.g. https://catholic-resources.org/ChurchDocs/RM3-EP1-4.htm
39. Link goes to Word on Fire Institute, “Wonder: God and Nature”: https://www.youtube.com/watch?v=fen-a6SLiSE.
40. See Thomas Aquinas, Summa Theologiae First Part, Question 3: https://www.newadvent.org/summa/1003.htm
41. Catechism of the Catholic Church 299: see https://www.vatican.va/archive/ENG0015/__P19.HTM