God & Math

You might think that if there were ever a field of intellectual endeavor free from the convulsions of the culture wars, it would be mathematics.  Two and two make four and that's all there is, you might suppose.  Not so, not so. 

You see, there is a debate raging--it has been raging off and on for centuries--about numbers. Not about whether high school algebra is useful enough for the majority of students forced to take it to warrant its manditory status in the curriculum--most people would, wrongly, vote "no".  The debate, even if it is not put in so many words, is about whether numbers are artifacts or givens of reality. Is math a human invention or a discovery of how things are and must be?

If math is a human creation then the most fundamental bit of what we think we know is not real at all and not therefore knowledge as such. It is a human construct. On the other hand, if numbers are real--that is, they exist independent of what you or I think about them--how are they real, in what sense can they be said to "exist" when, say, there is no one around to think about them?

And what has any of this to do with God?

Timothy Larsen addresses these issues in his review of two book at Books & Culture:

It's not hard to predict how eagerly the new atheists would pounce if an orthodox Christian theologian were to concede that the notion that God is three-in-one could be labeled "irrational." Or that the doctrine of the Trinity is so far beyond our normal ways of thinking that one might refer to the three persons of the Godhead as "imaginary." Yet mathematicians quite unapologetically speak of "imaginary" and "irrational" numbers. Moreover, they are content to assume that if others think they have thereby lapsed into nonsense, so much the worse for them.

The rest of the article is here: http://www.christianitytoday.com/bc/2009/sepoct/godandmath.html?start=1.

Not content to leave the aforementioned debate to be one about a single transcendental, truth, I would like to mention an Ignatius Press title worth reading in connection with the aforemention points is Father Thomas Dubay's The Evidential Power of Beauty.  It is available in print here and e book here.  The book develops the thesis of the great physicist Richard Feynman, which is summarized in his statement, "You can recognize truth by its beauty and simplicity."

Of course if mathematics does not give us truth about the real, then the recognition of beauty in mathematics or in the quantative sciences adds nothing. 

For those interested in how modern thought got into the situation where it questions whether mathematics is invention or discovery, Etienne Gilson provides a overview in one of his famous books, The Unity of Philosophical Experience.  Gilson uses his insights as a philosopher and a historian of philosophy to consider as "experiments" the key approaches to philosophy that have emerged in the last seven hundred years or so.  He examines the results of these "experiments", including the Cartesian and "modern" experiments, which ground the thinking behind the discussions Timothy Larsen explores in his reviews.  Cartesian mathematicism, Kantian physicism and Comtean sociologism are foundational to understanding so much of contemporary thinking, whether modernism, scientism, or postmodernism.  As is often the case, little errors in the beginning can lead to major mistakes later on.

For popular, highly accessible treatments of Gilson's subject matter (though not substitutes for his book), readers should see Peter Kreeft's Socrates Meets Descartes and Socrates Meets Kant, two books in the popular Socrates Meets series.  Kreeft masterfully uses the conversation format to communicate great ideas in a highly readable way.  In the afterlife, Socrates confronts various philosophers and thinkers regarding their ideas.  He shows readers the strengths and weaknesses of these ideas. 

Comments

[Mack]

Okay,yeah, sure, but Jesus never said "Solve for 'X.'"

[Mark Brumley]

No, but he did talk about receiving back thirty-, sixty- and a hundredfold.

[Brian J. Schuettler]

Thanks, Mark, for this excellent post.

Mack:

You are right that Jesus never discussed Math, at least not that we know of. However, you comment implies a fundamental disconnect between the natural sciences and faith. There is no disparity between faith and reason, nor between the Holy Scriptures and the faith of the Church. Tertullian considered faith in Christ's revelation of himself to be the only thing worth knowing whereas Augustine emphasized both the priority of faith and its incompleteness without the help of reason. One of his great insights is that faith is the foundation for all knowledge. Christians are often ridiculed for their faith, as if "faith" and "gullibility" were synonyms. But Augustine reminds us that each of us must trust some authority when making any truth claim, and that "faith" and "trust" are synonyms. But Augustine distinguished himself from Tertullian by acknowledging that philosophy does have a role in how the Christian understands God's revelation. Because humanity is made in the image of God, we are all capable of knowing truth. Augustine found in pagan philosophy helpful ideas that enabled him to elaborate God's Word. But it must be emphasized that his interest in pagan philosophy was not an end it itself, but rather a tool by which to grasp more deeply the meaning of Scripture.To take it one step further, Thomas Aquinas believed that all people could have a basic knowledge of God purely through natural reason. He did not agree with Augustine that the human mind had been totally corrupted by sin at the Fall. This belief led to his elevation of the power of the mind and his appreciation of philosophy. Theology is the higher form of wisdom, but it needs the tools of science and philosophy in order to practice its own trade. Theology learns from philosophy, because ultimately theology is a human task. How we view the relationship between faith and reason can have powerful implications for how the Christian engages society with the gospel. One of the problems with the apologetics of Tertullian is that he seemed to view all that opposed him to be enemies of the gospel, rather than as potential converts. This is in stark contrast to the behavior of the Apostle Paul in Acts 17, when he proclaimed the gospel among the Greeks at Mars Hill. He did not condemn them for their initial failure to accept the Resurrection. Instead, he attempted to reach common ground with them by quoting some of their own philosophers, picking out isolated statements from pagan thinkers which were consistent with Scripture, while still maintaining the absolute truth of Scripture as his foundation. In this way, he was able to gain a hearing with some of his listeners. But this presupposes some familiarity with pagan thought. This familiarity made Paul a more effective witness to his audience.

[Howard]

I read THE EVIDENTIAL POWER OF BEAUTY, and it is deeply flawed. It gives no serious attention to the many cases in which beauty is deceptive: the pretty woman (or handsome man) who leads others into sin, the evil policy packaged in flawless rhetoric, the false philosophy packaged in a movie with great acting and special effects. Then there are any number of plausible, very beautiful scientific theories that, contrary to the "evidence" of their beauty, turn out to be hopelessly wrong.

The book also does not do a good job of making the distinction between "beautiful" meaning "well suited to me" as opposed to "well suited to a Platonic ideal". Is the banana pudding "beautiful" because it embodies some Platonic ideal, or because it suits my physiology, experience, current state of hunger, etc.? Almost certainly the latter. What about a "beautiful proof" in math? Almost certainly it is the former -- closer to a Platonic ideal. But with many, probably most, beautiful things it's very difficult to be sure if the beauty is really objective or subjective.

[Mark Brumley]

Hmmm. There are lots of things in your post, Howard. I'm not sure where is the best place to begin to discuss the issues you raise.

Would you agree that the issue of the pretty woman who leads others into sin is not really the same issue as the beauty of a seemingly plausible theory that turns out to be false? The beauty of the woman is a quality of the woman's appearance; it says nothing about her moral character, which has or lacks beauty on its own apart from the woman's physical appearance?

On the question of banana pudding and personal taste, I would need to review the book but it seems to me that it did make the distinction between the subjective and objective aspects of beauty.

[Rick]

I tried reading the Evidential Power of Beauty a few times and just couldn't get into it, though Gilson's Unity is a masterpiece. I particularly liked his argument that the Cartesian and Kantian systems fail because the wind up as types of idealism, and he offers ample argument of how this is the case in relation to those thinkers and their followers. The best description I have read on the objective nature of beauty is seen in the writings of Pieper, especially his Living the Truth because he relates it to the transcendentals of being. The trascendentals are part of the objective order of reality. He also traces how the concept of beauty became subjectivized during the Enlightenment by Kant and his forerunners because of the loss of the transcendentals. Yes, little errors do lead to big mistakes later. I guess the sad part is that ideas do have consequences, especially bad ones!

[Rick]

Howard, This is a summary on the "objective" nature of beauty being present in created things from Fr. Emonet's The Dearest Freshness Deep Down Things.
St. Thomas states that beauty is “that whose sight pleases” (STh I.5.4.1), in this sense beauty relates to a vision and produces in one a profound affectivity, a joy and delight. But, while beauty is associated with sight, it is not sensible in itself. In other words, it is not just something sensed in a painting. For St. Thomas, beauty is deeper, it is not a sensation, but it is grasped by the intellect, beauty is a spiritual light that is found in sensible things. For St. Thomas, beauty is received intuitively by the intellect, like color in sight. Beauty, is present in an immediate, intimate and penetrating presence in the intellect itself. How does this occur? When our intellect attempts to seize the essence/form of a thing, it arms itself with many ideas. It arranges them in propositions (this is a dog), and the propositions in reasonings (some dogs may bite). It proceeds scientifically, and then it lays hold of a thing, its nature via abstract representations. When we do this we tend to hold them at a distance when we classify and quantify them, they remain “before” the intellect when they are judged (form mental concepts about them, weighed (is this dog dangerous?) and catalogued by the intellect.
But, when we begin to develop an affinity for a thing, say a tree of a dog, there arises a connaturality between it and the knowing subject. The “distance” of cataloging is suppressed and we move past the images and concepts so that the object can be contemplated and enter the mind without all these intermediaries. At this point, the object is “possessed” by the intellect and procures a love in the silence of the concepts. Now, the intellect has a joy, a spiritual joy of possessing what it loves. Anyone who has ever gazed at the beauty of his or her child, experienced God in nature, or sat before the Blessed Sacrament and was moved beyond concepts has therefore experienced beauty as such. In doing this, one also sees that beauty cannot exist without truth, oneness and goodness which all flow from the universality of being.

[W.]

A few works to look at regarding beauty:

The Beauty of Holiness and the Holiness of Beauty by John Saward. One of my favorite books! The post-it I keep in the book has my conclusion: "Beauty - the glow of the true and the good irradiating from every ordered state of being; not immediate sensual pleasure; tied to temperance."

A great and short analysis of the issue is About Beauty by Armand Maurer, CSB, that is, if you can find a copy. Hard to find but well worth it.

Since Gilson was mentioned above, he has a book somewhat related to this subject: The Arts of the Beautiful, which begins with a discussion of beauty and how it shows up in the artistic expressions of life. Early on, he refers to the commonly-cited definition: that which pleases when seen. He says:

"Objections to such a definition are not lacking. ... We are not saying that the beautiful itself consists in the pleasure it gives,but rather that the presence of the beautiful is known by the pleasure that attends its apprehension." (23)

He then goes on to elaborate his point and discuss more objections. In this section, he deals with the subjective side of experiencing beauty (in its apprehension and the accompanying pleasure) and also with the objective side of the beautiful [whose roots "are to be found in ontology, because the beautiful is a mode of being" (27)].

Another Press book on the subject is The Order of Things by Fr. James Schall, SJ. In the chapter "The Order of Beauty," he writes:

"There are beautiful things, and we perceive them, know them, and rejoice that they exist. ... Beauty is not an inner creation of our minds subsequently projected onto reality. Were this the case, we would never confront anything but ourselves. Our whole being responds to what is, to what is there that strikes us. Much of modern thought rightly sees in beauty a danger to its own philosophic principles, rooted as they often are in relativism or idealism. Beauty is from outside of us; ... it is not of our own initial making." (219)

A great read. I suggest you all get it, get all of them, hey, won't that help our economy?! Purchasing good books. Now that is a stimulus plan I can go for!

[W.]

Brian,

"To take it one step further, Thomas Aquinas believed that all people could have a basic knowledge of God purely through natural reason. He did not agree with Augustine that the human mind had been totally corrupted by sin at the Fall."

St. Augustine believed that the human mind had been "totally corrupted"?

Do you know where he says something like that or what he actually says?

I took a class on Augustine many years ago and one thing that stood out is how many times Augustine's own words contradicted what many say about him. Virginity, conjugal act, free-will. I thought Augustine held to a lesser view than you state. I wonder if this is one of those cases where the post-Manichean, post converted Augustine says things that counter an earlier version of him and therefore counter what many say he thought.

[W.]

Howard,

What about the chapter 5 on Ugliness? I know it does not address everything you mentioned but I thought it dealt with some of that. Perhaps not. It has been some time since I read it.

"It gives no serious attention to the many cases in which beauty is deceptive: the pretty woman (or handsome man) who leads others into sin, the evil policy packaged in flawless rhetoric, the false philosophy packaged in a movie with great acting and special effects. Then there are any number of plausible, very beautiful scientific theories that, contrary to the 'evidence' of their beauty, turn out to be hopelessly wrong."

Not sure if you meant it this way, but the traditional, Catholic, Thomistic view of beauty holds that beauty cannot be deceptive.

Technically speaking, the pretty woman, in as much as we are speaking about her beauty and her beauty in itself, does not lead others into sin. The sinner lets himself get there. Beauty is not the same as pretty, I think we all agree. But someone who chooses to abuse or misuse an experience of beauty holds the fault, not the one who is beautiful.

The evil policy and the false philosophy (not to mention the allegedly "beautiful scientific theories") are not beautiful to the extent that they are evil or false. Included in beauty is good and truth. Therefore they would negate the evil and false. Which means that the policy and the philosophy could not be beautiful, however much one thinks they are. If that situation occurs, then the error is in the one making the judgment that the policy or the philosophy is beautiful. Which is why the ontological (which I mentioned in above post) is so important. Beauty includes the good and the true in an ordered state of being. If that is not there, then the object is not beautiful, however much it might please. This type of pleasing is more likely the result of a disordered state of being (and desires) in the one making the (mis-)judgment that something is beautiful when in fact it is not.

[Brian J. Schuettler]

W.:

Thank you for giving me the motivation to expand upon my initial comment. I was expressing Augustine's position relative to his anti-Pelagian writings. The danger of expressing this man's thoughts in a comment is the inadequacy of dealing with the sheer depth and breadth of his teaching, with all it's subtlety and historical interpretation, both Catholic and Protestant.

Augustine,as you know,opposed what he believed was a denial of the grace of God by insisting that regeneration is exclusively the work of the Holy Spirit. Man was originally made in the image of God and free to choose good and evil, but Adam's sin bound all men, because Adam was the head of the race. Man's will is entirely corrupted by the Fall so that he must be considered totally depraved and unable to exercise his will in regard to the matter of salvation. Augustine believed that all inherit sin through Adam and that no one, therefore, can escape original sin. Man's will is so bound that he can do nothing to bring about his salvation. Salvation can only come to the elect through the grace of God in Christ. God must energize the human will to accept His proffered grace, which is only for those whom He has elected to salvation.

Pelagius' views were condemned at the Council of Ephesus in 431, but neither the Eastern nor Western churches ever fully accepted Augustine's views. John Cassianus, a monk, endeavored to find a compromise position by which the human and divine will could cooperate in salvation. He taught that all men ware sinful because of the fall, and that their wills are weakened but not totally corrupted. Man's partially free will can cooperate with divine grace in the process of salvation. He feared that the doctrines of election and irresistible grace taught by Augustine might lead to ethical irresponsibility. The view of Cassianus was condemned at the Synod of Orange in 529 in favor of a moderate Augustinian view.

As you know, W., Augustine said that we have the free will to cooperate with grace but even the exercise of our free will is dependent upon grace: "Will is to grace as the horse is to the rider"

I hope this addresses the concern you had and I am thankful for the opportunity to clarify the necessity of always reading Augustine's writings within the context of the development of doctrine within the Magisterium of the Church.

[Rick]

W., I think you are correct, Augustine did not hold the mind to be totally corrupt, but it is apparent that that he held that it becomes darkened due to OS. If totally corrupt, one could only use the analogy of faith to know God. He spends much time in the Confessions describing how the soul can know/ascend to God via the beauty (especially book 10 chap. 6 and 7) of created things and how the soul can know God via introspection.

[W.]

Brian,

In your first post, you said, "the human mind had been totally corrupted by sin at the Fall."

And then in the more recent one you said:

"Man's will is entirely corrupted by the Fall so that he must be considered totally depraved and unable to exercise his will in regard to the matter of salvation."

So I think you mean both the reason and the will are totally corrupted and/or depraved.

From what I know of Augustine, I disagree with you, that is, if I understand you correctly.

I do agree that with regard to the matter of salvation, Augustine held that man is in need of grace and even the initial movement happens because of a (prevenient) grace.

St. Augustine made a distinction between freedom and free-will. Freedom was lost but free-will, the capacity, was not. I think here is the crux of the disagreement. The capacity is still there and therefore I do not see how the will can be totally corrupted. There must be something left that is not totally corrupt; otherwise, what is there left for God to grace and put on the right path? On the level of morals, I see how we are deformed or fallen but on the level of being, it seems to me if there is something human there, if the imago Dei remains, then we cannot be totally corrupted.

Looking at Augustine's works, one finds so many passages to refer to for both our points, but I don't recall him ever saying or implying "entirely corrupted" and "totally depraved." Perhaps you know of some?
With regard to the reason, Augustine has a few places where he talks about the philosophers and the truths they could come to. In On the Trinity (IV.15.20), he says that some philosophers:

”have been able to penetrate with the eye of the mind beyond the whole creature, and to touch, though it be in ever so small a part, the light of the unchangeable truth.”

If the reason is totally corrupted, then how could they arrive at some truths? (Maybe we have different notions of what it means to be "corrupt"? If so, all this is probably beside the point.) If the discovery of truth is due to the workings of the Spirit, then why does Augustine often make the point that any good we do or truth we arrive at is not only the workings of grace or the Spirit but is also coupled with our will? There is a co-operation going on. A co-operation, I agree, that is initiated by God, but one that Augustine makes pains to say is still a co-working:

"He operates, therefore, without us, in order that we may will; but when we will, and so will that we may act, He co-operates with us. We can, however, ourselves do nothing to effect good works of piety without Him either working that we may will, or co-working when we will." (On Grace and Free Will, 33)

I accept that ultimately it is a mystery how these play out but with the most we can get at, it seems he has to preserve some goodness (even if just ontological) in order for us to retain the image of God. If some goodness, then the way I understand the terms, he could not mean a literal "entirely corrupted" and "totally depraved."

In her study of Augustine's thought, Mary T. Clark writes that Augustine himself dealt with this issue and the implication that "by sin Adam lost the image of God to which he was made." She refers to one of Augustine's later writings (Retractions II.50) and says that Augustine "did not mean that no image remained, but that it was deformed and had to be reformed." (Clark, Augustine, p. 124) Thus, how I see things, not "entirely corrupted" and "totally depraved."

My point is merely that the loss was not entire and the corruption was not total. I think it matters for key reasons, such as anthropology, morality, and what Christ came to save.

[Brian J. Schuettler]

W:

I think that you are talking about the natural virtues that are independent of grace. I am speaking of the necessity of grace to do any supernatural act and thus with the aid of Christ’s Redemptive Suffering.Thomas Aquinas declared that special grace is necessary for man to do any supernaturally good act, to love God, to fulfill God's commandments, to gain eternal life, to prepare for salvation, to rise from sin, to avoid sin, and to persevere.
Romans 14:23 says, "Whatever does not proceed from faith is sin." This is a radical indictment of all natural "virtue" that does not flow from a heart humbly relying on God's grace.
If the term total depravity is the bone of contention and it is unfortunely connected to Calvinism, let's substitute the phrase "total inability". Total depravity or total inability does not mean, however, that people are as evil as possible. Rather, it means that even the good which a person may intend is faulty in its premise, false in its motive, and weak in its implementation; and there is no mere refinement of natural capacities that can correct this condition. Thus, even acts of generosity and altruism are in fact egotist acts in disguise.

[Rick]

Brian:
According to the CCCC 2000-2001, actual grace is needed for any good moral act. The natural virtues do not act independent of grace. Here, I think it is important to emphasize the distinction between sanctifying/habitual grace and the theological virtues and actual grace and the moral virtues. In any case, actual grace is needed for any "good" moral act of the natural virtues. Fr. Jordan Aumann has a nice teaching on this in his Spiritual Theology (http://www.domcentral.org/study/aumann/st/default.htm)

[Mark Brumley]

I don't see, Rick, where in CCC 2000-2001 it states that actual grace is needed for any good moral act, if by any good moral act one means any good moral act whatsoever. Nor am I familiar with Father Aumann teaching such a thing.

It is difficult to understand such a position as entailing anything other than the condemned proposition that apart from grace man acts only in order to sin. Human beings, even apart from grace, are capable of performing a naturally good moral act, however otherwise proned to sin they are.

[W.]

Brian,

Thanks for the response.

" I am speaking of the necessity of grace to do any supernatural act."

No argument here.

No need to respond further, but just that my original question returns.

"Thomas Aquinas believed that all people could have a basic knowledge of God purely through natural reason. He did not agree with Augustine that the human mind had been totally corrupted by sin at the Fall."

The human mind as totally corrupted?

Guess we disagree here with our understanding of Augustine.

Your earlier reply to this went on to discuss the corruption of the will so I presumed you equated the fall of each. I don't think Augustine held as severe a result as you. I think he held out some capacity for the reason to discover some truths ... as the above post of mine referenced some support for this.

Oh well. Guess we'll find out one day. All the best.

[Brian J. Schuettler]

Thanks, W....All the best to you also. I think we essentially agree but my articulation of this deeply complex matter is inadequate. By the grace of God we can ask Augustine about this in the forever now.

[Rick]

Mark,

I am not denying that humans are capable of good acts no matter how to prone to sin they are. In fact, isn't it actual grace which helps to us to make choices to avoid sin even when sanctifying grace is lost? Unless I am reading Aumann and the CCC wrong, and I don't think that I am, how do we account for the for the good moral acts/works that are done in our lives apart from grace. I am not denying human free will, but doesn't God act as well via actual graces to help us choose the good in our everday experiences? Otherwise, if good acts Are you stating that sanctifying grace is necessary to receive actual graces?

Maybe, I am not getting your argument, but could you provide an example of a good human act on a natural level that is done without the assistance of actual grace? For me, this would seem hard given the Catholic view of the Christocentric of creation and redemption and the fact that we beleive that God holds us in existence. It is hard for me to see how any moral act can be purely natural given this. Also, when the CCC 1999 states that actual graces refer to God's interventions, whether at the beginning of conversion or in the course of the work of sanctification" it seems to imply that actual grace is present in all the choices for good. Again, I am not sure how there could be any purely natural good acts/works apart from God. Given such a view of actual grace, wouldn't this also apply to the unbaptized who lack sanctifying grace but are nonetheless capable of choosing good moral acts?

[Mark Brumley]

I'm out of town. A brief summary of what I think the Church teaches:

1. Human beings can engage in nonsalutary morally good acts. Ex: an act of fortitude or temperance or justice or prudence can, in principle, be carried out apart from grace, whether or not in fact grace is present. (The issue is not whether, in fact, grace is prevasive but whether, in principle, human beings can engage in morally good, nonsaluatry acts, apart from grace; they can. See, for instance, Ott.)
2. Actual grace can aid human beings in engaging in morally good acts, whether salutary or nonsalutary, but it does not follow that every moral good requires actual grace.
3. Grace is necessary to engage in salutary acts.