Friday, February 23, 2007

Failure to Observe Limits

Speaking of the unlimited.... I missed noting the second anniversary of this forum on Monday the 19th. This year I will not celebrate by changing my name. Ha ha.

Much more seriously, homeschooling is illegal in Germany. The German government is undertaking legal action against Christian homeschooling families and forcibly removing their children to government schools. More here.

There are proper limits to government. The primary responsibility for the education of children is their parents. The U.N. Declaration of Human Rights (Article 26.3), for one, specifically mentions this right and responsibility. A German friend tells me that the government's actions also contravene the German constitution.

According to the referenced article, laws against homeschooling date to Hitler. It's sad that, despite the obligatory politically correct protestations of "niceness," Germany hasn't managed to shed its totalitarian impulses. We can only hope the people of Germany take action to correct their government's exceding of its proper authority.

Thursday, February 15, 2007

Comprehending the Infinite

As part of the commentary on Aristotle’s Physics that I am composing, I’ve been researching infinity, which is one of the topics in book III.

David Foster Wallace’s Everything and More: A Compact History of Infinity was the first I read. As I’ve noted here before, Wallace’s writing style is extremely mannered. His treatment of the mathematical concepts and ideological camps is very understandable, but unfortunately, his thinking and writing are not so clear on the implications and meanings of the mathematical developments. The book could use circumscription by a table of contents and an index, though perhaps the lack of these is intended as an artsy way of embodying the “infinite.”

Brian Clegg’s Infinity: The Quest to Think the Unthinkable was an accessible and straight-forward recounting of the history of conceptions of the infinite. It wasn’t particularly deep or probing and covered roughly the same ground as Wallace’s book, but in less detail, and certainly less manically.

Paolo Zellini’s penetrating A Brief History of Infinity was reminiscent of Jorge Luis Borges’s writings, and not merely because he starts the book with a quotation from Borges. The book supplies a sweeping treatment of the implications of conceptions of infinity, and if I could tag it with any flaw, it would be that its depth combined with its brevity (200-pp.) flirt with impenetrability of language—but then infinity is an obscure topic (and it’s difficult to translate from Italian). The Wallace book’s treatment of the paramount proofs was good background for this book.

I thought it would be helpful to summarize the most significant points of Zellini’s work. Then I'll conclude with a brief consideration on how Aristotle's conception of infinity compares with modern developments.

Preliminary Clarification of Terminology

Part of the difficulty of Zellini’s book was keeping track of various terms for the two kinds of infinity.

These terms are roughly synonymous with each other: potential infinity, syncategorematic infinity, improper or false infinity, infinity ex parte materiae, material infinity, negative infinity

These terms are antonymous in a sense with the previous list, but synonymous with each other: actual infinity, categorematic infinity, real or true infinity, infinity ex parte formae, formal infinity, positive infinity.

This last set of terms is sometimes synonymous with the absolute or simple infinity actualized in God.

It might be helpful to recall through an example how potentiality and actuality are linked to temporal succession. A car is able to move because it is potentially in another place; once it has moved to that place, it is there actually and not potentially. Thus change in general is a succession of potentiality and actuality.

The Classical-Medieval Understanding

The Greek word for infinite, apeiron, means indefinite or unlimited. A connotation of perfection is completely foreign to it. Aristotle understood the term as denoting incompleteness. Limits make any object exist concretely, individually and with proper form; apeiron denotes a privation of limits—formlessness.

Aristotle writes, “The infinite [apeiron] is not what has nothing outside of it, but what always has something outside of it.” The infinite is always potential, because more can always be added to it. The infinite by division also exists only potentially: a concrete thing can potentially be divided an infinite number of times.

Numbers to the ancient Greeks were intimately tied to concrete things. Even if these things were conceived as abstract entities, they were nevertheless embodied in a sort of intelligible substance. Thus the Greeks could not conceive as a number as an entity in itself, but only as the result of counting a concrete set of things, and of course, one can never complete (or actualize) the counting of an infinite number of things (that is, in time).

St. Thomas Aquinas agreed with Aristotle’s basic point, but with the additional information of the Christian revelation of God’s infinitude, added a new layer of meaning to the infinite. As Zellini writes,

Aristotle had excluded any possibility of confusing the false infinite of apeiron with the infinite divine perfection, simply by denying the latter any infinite attributes. Divine perfection was designated by terms referring to its ‘totality,’ its ‘plenitude’, and to its ‘eternity’, but not to its unlimitedness. The last term belonged exclusively to the realm of quantity, and was therefore completely extraneous to God.

Aquinas dares not follow Aristotle’s formulation of the problem [because of the condemnation of Arabic Aristotelianism in 1277], and instead embraces the thesis that ‘God is infinite and eternal and boundless’. But he immediately adds that the infinite can have two opposite natures: one derived from the idea of form, and the other from the idea of matter.

The infinite ‘on the part of matter’ (ex parte materiae) thus had to correspond to an analogue of the false infinity of Aristotle’s apeiron. By contrast, the infinite ‘on the part of form’ (ex parte formae), by referring to a sort of formal perfection, could indicate in what sense one could speak correctly of God’s infinity. (59–60)

Infinity on the part of form means that God is infinitely articulated (delimited in a positive sense)…and yet Aquinas’s Five Ways show God to be infinitely simple in another sense….

Formal and material infinities parallel actual and potential infinities. As to whether the actual infinite could exist outside of God, Aquinas contrasts the divine infinite by essence or simplicity (per essentiam or simpliciter) with the relative infinite (infinitus secundum quid), corresponding to a specific nature. Aquinas seemed to think the latter could only exist potentially and thus seemed to identify it with the potential infinite. (61)

In the interest of clarity, Peter of Spain introduced novel terminology. The syncategorematic infinite is the potential infinite, but stripped of the residual connotation in “potential” of an actual orientation to an end or possibility of actual realization. The categorematic infinite, in contrast, is an actually realized whole that is simply larger than any finite quantity. (67)

Enlightenment Turn

Descartes distinguished the infinite from the indefinite. The latter roughly parallels the false or potential infinite, but with a significant attitudinal shift. For Descartes the imperfection of the indefinite is not its boundlessness, but the boundedness that remains. “It was the same continual opening which Boethius had rejected as a ‘monster of malice’, and which Aristotle had associated with non-being and privation. Instead…Descartes perceives in it the unequivocal sign of a divine imprint.” (103) While Descartes did not believe in the existence of the actual infinite, he paved the way for its use by speculating about the mind’s openness to the infinite as a standard of perfection.

Renaissance painting discovered perspective, which manifests the convergence of lines on the horizon, that is, at infinity. This innovation grounded the change in the concept of the infinite. Descartes’ rationalization of space is premised on “the affirmation of existence culminating in the visibility of a point in which the entire infinity of visual space appears enclosed and unified.” (111)

The shift in attitude is dramatically apparent by contrasting Aristotle with Leibniz, the originator of infinitesimals. Whereas Aristotle writes that “Nature avoids what is infinite, because the infinite lacks completion and finality, whereas this is what Nature always seeks,” Leibniz writes that nature “involves the infinite in all it does” (112–113)

Modern Developments

While the Leibniz’s conception of the infinite incorporated some flaws, its practical application convinced Bernard Bolzano of its reality. Bolzano saw that the infinite, as with other mathematical entities such as zero and imaginary numbers, could be an objective idea without corresponding to a concretely existing thing; a thing could be determinate without existing in reality: the infinite can be defined unambiguously even if it is impossible to enumerate its elements. (143) As Zellini writes,

When we speak of the set of inhabitants of Peking, we individuate a well-defined set without being required to enumerate separately all its components one by one. Analogously, the terms of an infinite sequence can all be specified by the law governing the formation of the sequence, which renders superfluous enumerating its terms: it is the law that specifies the sequence, and not ‘all’ its terms counted one by one. (148)

“Dedekind did not scruple to declare that arithmetic evolves perfectly independently of a priori intuitions of space and time, and that the concept of number is an immediate result of the laws of thought.” (160)

Like Bolzano, Dedekind said that an intellectual entity could be determined by all that can be said or thought about it. Similarly Cantor believed that mathematical entities can be considered actual insofar as they, as he wrote, “assume a perfectly determinate place in our knowledge, are clearly distinct from all other constituents of our thought, stand in definite relation to them, and therefore modify the substance of our mind in a definite way.” But (as any Platonist) he also believed that we don’t arbitrarily construct these entities, but receive them from “the voice of nature.” Still, he put the existence of these entities outside the competence of mathematics—such metaphysical questions have no bearing on mathematics. Furthermore, he saw clearly that transfinite numbers cannot even approach an understanding of the Absolute, which can only be acknowledged but never known. (160–161)

Yet, in the 12th chapter on “The Antinomies, or Paradoxes of Set Theory,” Zellini argues that Cantor’s achievements are not unqualified. Hilbert tried to preserve them by formalizing mathematics in a rigid system of “symbols without significance,” but Gödel showed that symbolic systems were not able to express a complete and closed world of mathematics. (177)

In 1932, Weyl condensed this finding in a neat summary: the infinite is intuitively accessible as an indefinitely open field of possibility, and in this respect would seem analogous to a series of numbers that can be extended unlimitedly. Yet completeness, the so-called actual infinite, lies beyond our reach. Nevertheless, the exigencies of totality impel the mind to imagine the infinite, by means of symbolic constructions, as a closed entity. (179)

It may be my lack of mathematical expertise that keeps me from understanding how later limits to mathematics concerned Cantor’s achievements (or perhaps Zellini is simply vague). Perhaps it is fair to summarize the conclusion of the book as saying that modern mathematics shows that transfinite numbers are never completely formless—that they can be handled in a determinate way, even though unable to completely exclude paradox.

How Aristotle Falls Short

As we have seen, infinite numbers aren’t unequivocally infinite, that is, without bound or indeterminate. Merely the fact that they are numbers constrains their properties. For example, that they are ordered bounds them from complete chaos. They needn’t be actualized in matter to be determinate.

As is well known, Aristotle failed to separate the actuality of form from the actuality of existence. The great advance of Aquinas was seeing that a determinate essence still required an act of being (esse) to be real. For example, you can conceive of a unicorn in a perfectly consistent and rather complete way, which means the unicorn has a certain actuality; but this actuality doesn’t mean such a creature actually exists. It seems to me that mathematical entities similarly can be fully determinate (formally actual) without being actualized in matter (possessing an act of being).1 Potentiality and actuality in the fully real sense of existing in the world have no part in such beings, so the inability to actualize infinity in the world is irrelevant to the actuality (or determinateness) of infinity.

Thus, Aristotle’s mistake was part of his general failure to distinguish formal actuality from existential actuality.


1. But then it would seem that only actually existing things can be determinate (or consistent) in the fullest sense.

Paolo Zellini, A Brief History of Infinity, trans. David Marsh (New York: Penguin Books, 2004).

David Foster Wallace, Everything and More: A Compact History of Infinity (New York: W.W. Norton and Company, 2003).

Brian Clegg, Infinity: The Quest to Think the Unthinkable ( (New York: Carroll & Graf Publishers, 2004).

17 Feb 2006: Minor edits.

Friday, February 02, 2007

The Crisis of Fatherhood and Christian Unity

Note: Blogger was bought out by Google, who've now forced me to switch to their new "improved" system. Please be patient while I get the kinks worked out of the template.

Thursday last week was the feast of the Conversion of St. Paul, and the last day of the Octave of Christian Unity.

Some years ago Fr. Thomas Loya, an Eastern-rite Catholic priest, gave a talk in which he said (I'm going from memory) the Western Church is the masculine to the Eastern Church's feminine: the West is much larger and more active, whereas the East is less expansive. Western churches thrust upward, whereas Eastern churches are more rounded and shorter. Furthermore the Western mentality is more analytic, where the East is more mystical.1 Both of these principles are essential aspects of the unity that is (or should be) the Church.2

The plan of this post is to use this conceit to understand Church history, and to see how it may help show us the road forward to Christianity unity. DISCLAIMER: Much of this is based on conversations I've had with fellow Christians of various Churches and denominations, so don't think of it as an academic document: it's more like a "cartoon ecclesiology." I offer it to help us understand the broad outlines of Church history, and to stimulate conversation. I welcome any clarifications and corrections that readers may offer.

It seems clear to me that the West is the masculine side essentially because its head is the Papa, the Pope. What we in the West often forget is that Pope wears two hats (tiaras?). We're all familiar with his role as head of the Church, but at least until recently, he also bore the title of "Patriarch of the West."3 Historically there are four apostolic Churches in Christianity: Alexandria, Antioch, and Jerusalem, and Rome; Constantinople was added later. Each of these Churches is headed by a patriarch who has special charge over the ceremony and discipline of that Church. Of course, as bishops, each of these patriarchs also certifies doctrine as orthodox and catholic in his respective region, but in his added role as head of the Church, the Pope is in charge of doctrine in the entire Christian Church in ensuring correct doctrine (e.g., by approving the decrees of Ecumenical Councils, and rejecting false councils, like the robber council of Ephesus). As Head of the Church, the Pope enforces doctrine, but as Patriarch of the West, he controls discipline and ceremony; the latter role is (traditionally) restricted to the Western Church.

Another thing we Westerners overlook is that, despite the formal mutual excommunications in A.D. 1054 ("The Great Schism"), the East and West were actually in communion until the thirteenth century sacking of Constantinople by soldiers of the Fourth Crusade. This event is probably the saddest of Church history: supposedly Christian soldiers ravaged the great Christian city, raping nuns, and stabling horses in Hagia Sophia, one of the most beautiful churches in all Christendom. To top it off, after the perpetrators of this crime acknowledged their sin, the Pope forgave them without imposing any substantial penance.

To this day, this painful memory is seared into the Eastern Christian memory. I contend that it colors all of their views and opinions of the Western Church. Using this post's conceit, the sacking was like a man drunkenly raping his wife; the easy forgiveness was like his sober self-justification; the long memory of the East is like that of an injured wife.

Today, it's not unusual to hear Orthodox Christians decry the doctrinal "innovations" of the Catholic Church. I contend that Catholics and Orthodox substantially share the same beliefs. The "innovations" are reformulations of the same doctrines that the Orthodox maintain, but the West's analytic bent and concentrated authority has allowed it more flexibility in expressing these doctrines in new ways to engage a changing world and protect against new errors. (There's a reason Islam didn't develop in the West's backyard.) The Orthodox for their part maintain doctrinal "purity," but at the expense of being frozen in first-millennium formulations and unable to speak to the modern world. The Orthodox sometimes do acknowledge their doctrinal unity with the Catholic Church, but what very understandably stands in their way of consistent acknowledgement is the cultural scar left by the sack of Constantinople.

Unfortunately the West has not always used its authority for the good of the Church. The abuses of the medieval papacy are well known (though often exaggerated), and need no recitation here. Suffice it to say that some of them exercised their authority not for the good of the ruled, but for their own aggrandizement. In northern Europe in particular, the clergy were poorly formed and poorly disciplined. The result was the Protestant Reformation. A Protestant colleague of mine reminds me of the embarrassing fact that Luther didn't leave the Church, but was excommunicated by Rome. It's difficult to decide whether to call the Protestant Churches abandoned children or runaways, but in any event they are estranged from their true home, and not entirely by their own doing.

In reality, authority needs to put itself in service of the Tradition handed down through the bishops from the Apostles. Christian unity cannot be regained without the proper exercise of authority. The abuse of authority illegitimates those particular uses of authority, not the authority itself.

These historical problems have been recognized by Rome and a dialogue has been opened. As Pope John Paul II observed in Ut Unum Sint, "the Catholic Church's conviction that in the ministry of the Bishop of Rome she has preserved... the visible sign and guarantor of unity, constitutes a difficulty for most other Christians, whose memory is marked by certain painful recollections. To the extent that we are responsible for these, I join my Predecessor Paul VI in asking forgiveness" (88).

Within the Catholic Church

Nevertheless not yet widely recognized is the extent to which this imperious exercise of paternal authority has caused damage within the Catholic Church.

This damage endangers the dialogue with other Christians. In the U.S., Roman Catholic bishops were jealous of members of their flock changing to one of the Eastern Rites to be ordained, since in those rites they could be married. They petitioned the Roman Pontiff to forbid Eastern-rite Catholics from ordaining married men in the U.S. This was a clear overextension of Pope's power as Patriarch of the West to the Eastern Rite. If we Catholics can't treat our own well, why would the Orthodox ever trust us to treat them well?

Furthermore, it cannot help the dialogue with the Orthodox, who rightly prize Tradition, to show a cavalier attitude toward Tradition. Unfortunately the Roman authority has at times been exercise at the expense of Tradition, instead of in service to it. The First Vatican Council's decree of "Ex Cathedra" papal pronouncements is an obvious (though commonly misunderstood and exaggerated) continuation of the centralization of power in Rome. Ironically, it is the Second Vatican Council, the supposed council of openness and democracy, that most centralized power. Following the Council (but not actually mandated by its documents), Pope Paul VI unilaterally forbade the traditional liturgy that dated from the Apostles and was codified at Trent—the so-called Tridentine Rite—to institute what is called the "Novus Ordo" or new order of Mass. The effect was to create a rift between tradition and authority that persists to the present.

Adherence to the new rite and the authority on which it rests, instead of adherence to the Faith, has become the de facto litmus test of Catholic belief. Not long ago, for example, the U.S. Bishops mandated standing to receive communion (a practice Cranmer first instituted in the English Reformation to destroy Eucharistic piety) as the norm, and required special episcopal permission for a congregation to receive communion kneeling. (And then they wonder at the faithful's failure to believe in the Real Presence!) It's as if they wanted to flatter their narcissism by distinguishing the power of their own offices from the power of tradition.4 The great concentration of power in Rome has prevented the other bishops from having to "grow up." They still act like children shielded from responsibility—and they have not comported themselves well in bearing increased responsibility. Witness the recent highly publicized scandals of the American Church.

With such juvenile behavior, it can be no surprise when Catholics fail to take their pastors seriously, even when they try to lead their flock to Christ. In Braveheart, the nobles of Scottland have sold out to the English king. William Wallace rises to its defence, but he still sees the need for the nobles, and in particular for Robert the Bruce, to whom he says:

Now tell me, what does that mean to be noble? Your title gives you claim to the throne of our country, but men don't follow titles, they follow courage. Now our people know you. Noble, and common, they respect you. And if you would just lead them to freedom, they'd follow you. And so would I.

Similarly, certain segments of the Church hierarchy seem to have abandoned their flocks to the wolves. This doesn't necessitate dispensing with the hierarchy, but in recalling them to their charge.

The crisis in the Church is a crisis of fatherhood. Fatherhood is the proper use of God-given power. St. Paul sets forth the proper use of husbandly authority in chapter 5 of his letter to the Ephesians.:

Wives, be subject to your husbands, as to the Lord. For the husband is the head of the wife as Christ is the head of the church, his body, and is himself its Savior. As the church is subject to Christ, so let wives also be subject in everything to their husbands. Husbands, love your wives, as Christ loved the church and gave himself up for her.... (Eph 5:22-25)
Fathers, like husbands, must sacrifice their own interests for the good of those they rule. "The good shepherd lays down his life for the sheep" (Jn 10:11).

The Church has taken concrete steps to heal the rift with the Eastern Churches and with the separated brethren of the West. For example, Pope John Paul II's year-2000 apologies, and his overtures to the Orthodox hierarchy, including offering monetary support, no strings attached, to Orthodox clergy in spreading the Gospel.

We need to pray that Rome and the Western hierarchy will one day see in the same light their responsibilities to the faithful who have remained loyal to Rome. We need obedience to appropriate authority to achieve unity, but this unity can only be in the Apostolic Tradition.


1. The concrete reasons for this difference are pretty plain: the East needs more props for liturgy, and has married priests. Plus I think the emphasis on the restfulness of the resurrection, as opposed to the striving of our Lord's passion is an additional reason.

2. I see The Dark Crystal as an allegory of the Schism, written from (possibly) an Anglican point of view. "Many ages ago in our arrogance and delusion, we shattered the pure Crystal, and our world split apart."

3. He inexplicably dropped this title in the last couple years, and it has the Orthodox (and the historically conscious) in a quandry.

4. One might wonder if this narcissism bears any relation to the infiltration of homosexuals into the ranks of the clergy.