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Alexi Sargeant

For Lent this year, I'm reading a heretic, as David Mills suggests (his title is provocative, but his point is about building compassion among brothers and sisters in Christ across the Reformational line). Fording the Tiber in reverse seems redundant to me, a convert from Protestantism to Catholicism. Instead, I'm looking across the Great Schism to read the Lenten reflections of Alexander Schmemann, an Orthodox priest and teacher well known for his prolific writing on the liturgy. Schmemann also has the distinction of working with Radio Liberty for thirty years, preaching Russian language sermons that were broadcast into the Soviet Union to reach the beleaguered faithful, including Alexander Solzhenitsyn.

Schmemann's Great Lent opens with a timely reminder of the season's purpose:

When a man leaves on a journey, he must know where he is going. Thus with Lent. Above all, Lent is a spiritual journey and its destination is Easter, “the Feast of Feasts.” It is the preparation for the “fulfillment of Pascha, the true Revelation.”

Later, he elaborates:

Easter is our return every year to our own Baptism, whereas Lent is our preparation for that return—the slow and sustained effort to perform, at the end, our own “passage” or “pascha” into the new life in Christ. If, as we shall see, lenten worship preserves even today its catechetical and baptismal character, it is not as “archeological” remains of the past, but as something valid and essential for us. For each year Lent and Easter are, once again, the rediscovery and the recovery by us of what we were made through our own baptismal death and resurrection.

My wife and I are leading an online book club on Great Lent this season. If you're interested in joining in, shoot me a message at

Veery Huleatt

Rereading books is a bad idea for someone whose to-read list grows much faster than her have-read list. I nonetheless often find myself turning back to old favorites, this week to J. D. Salinger’s novella Franny and Zooey.

I first came across the book when I was seventeen (the perfect age, by all accounts), and it was love at page one. Bits of the prose, like “I love you to pieces, distraction, etc.” from Franny’s letter to her boyfriend Lane, rattled about in my adolescent head for weeks, despite the fact that the sentiment in the phrase is revealed to be utterly ingenuous. Franny seemed like a reflection of me, albeit exponentially more attractive, intelligent, and successful than I could ever be. And who but a member of the Glass family could ever pull off existential angst with so much finesse and verbal aplomb?

From Forster’s Tibby Schlegel to Leo Tolstoy himself, literature’s landscape is full of individuals whose principles prevented them from being decent and charitable, and Franny joins their ranks quite easily. I read Franny and Zooey first at a time of spiritual confusion, and I doubt that the clouds of bathtub steam, dharmic prayers, and cigar smoke (“The cigars are ballast, sweetheart. Sheer ballast.”) rising from a 1950s Upper East Side apartment did anything to clarify things. What I do know is that it delighted me, and has continued to do so with each successive re-reading.

And, despite the vapor, there were concrete lessons that I walked away with seven years ago, and that I’m happy to rediscover now.

It may be a literary crime to extract morals from Salinger’s prose. I worry that pulling one element out into the light for full scrutiny may cause the whole edifice to collapse like a tower of Jenga blocks, revealing it for the pile of conceited verbiage that it quite possibly is. But, by the end of the story, Franny has learned two lessons that may well be her salvation: Lesson one: “How in hell are you going to recognize a legitimate holy man when you see one if you don’t even know a cup of consecrated chicken soup when it’s right in front of your nose?” Lesson two: “Shine your shoes for the Fat Lady.”

I can’t bear to explain them further, so if you’re already acquainted with the Glass family, be reminded. If not, I warmly recommend them to you.

Connor Grubaugh

I’ve been reading the most popular exposition of Kurt Gödel’s famous incompleteness theorems—Gödel’s Proof, by Ernest Nagel and James R. Newman (1958)—alongside Ludwig Wittgenstein’s Tractatus Logico-Philosophicus (1921). Although these works take up very different subjects, their overall arguments are remarkably similar: First, both claim that the irreducibly basic elements of their subject matter (for Nagel and Newman's Gödel, arithmetical numbers; for Wittgenstein, linguistic concepts) are ultimately contingent or dependent on some external principle for their intelligibility because they lack a foundation in absolute logical certainty. Second, both recognize (though one more fully than the other) that despite this contingency, there are certain manifest truths or “givens” at the foundations of their subject matter which preserve its coherence and consistency, without rising to the standard of logical indubitability.

Appreciating Gödel’s insight requires some sense of his historical context, which Nagel and Newman illustratively summarize for laymen. Until the nineteenth century, mathematicians traditionally held that the axioms of geometry, arithmetic, and other disciplines could be established as self-evidently true statements about objects in space. Of course, mathematics and physics proceed at their own pace, and the relationship between them is often unclear—but these are mere distractions. Since no logically incompatible statements can be true simultaneously—and the basic axioms of mathematics were considered true—it was thought that no truly irresoluble contradictions would ever arise between them.

Then, suddenly, came proof of the impossibility of deducing Euclid’s fifth axiom from the other four, and the rise of non-Euclidean geometry. According to Nagel and Newman, this event led number-theorists to conclude (perhaps wrongly) that mathematics is not the classical “science of quantity,” but “simply the discipline par excellence that draws the conclusions logically implied by any given set of axioms or postulates,” regardless of whether or not they are true. In an effort that mirrors the mechanization of the natural world by Bacon and Descartes, culminating in the publication of Whitehead and Russell's Principia Mathematica (1910), mathematicians attempted to completely formalize their discipline by abstracting all meaning away from its signs and symbols, and grounding the validity of its assertions in their own logical structure, or method of proof—in effect, say Nagel and Newman, making “mathematics … only a chapter of logic.” As Russell famously put it, “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

Finally, along came 25-year-old Kurt Gödel and his 1931 paper, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.” I’m no mathematician, so I won’t present the proof in detail here, but its conclusions (at least as reported by Nagel and Newman) are simple enough to understand. First, Gödel showed that it is impossible to prove the internal consistency of a purely formal system sufficient to encapsulate the whole of arithmetic (a feat Russell and Whitehead claimed to have accomplished in Principia Mathematica) without employing logical principles not contained in the system itself. Second, Gödel demonstrated that any purely formal system encapsulating arithmetic must be essentially incomplete, i.e., that for “any consistent formalization of number theory, there are true number-theoretical statements that cannot be derived in the system.”

Essentially, argue Nagel and Newman, Gödel realized that mathematical reasoning and inquiry—contra Russell, Whitehead, Carnap, and other logical positivists—is more than mere deduction from arbitrary axioms. The relationship between pure mathematics and reality is often obscure and can be coherently doubted, but if the existence of any such relationship is denied out of hand, mathematics cannot stand. Euclid’s axioms, for instance, may not be logically demonstrable in a strict sense, but they do seem to latch on to reality in a fitting and manifest way that deductive reasoning only begins to describe. For Gödel, Platonic realism made the most sense of the situation. In 1960, he penned a diary entry entitled “What I Believe” that consisted of fourteen simple propositions: The tenth read, “Materialism is false”; the twelfth, “Concepts have an objective existence.”

Where does Wittgenstein come into this? The Tractatus assumes that words are conventional signs and symbols, possessing no logically necessary meanings in themselves, but rather deriving meaning from their common usage. Language does not “point out” objects in the world, but tries to paint a complete “picture” of the “facts”—of what is the case and not the case in the world, which is the totality of all such facts. Thoughts, in turn, are simply “logical pictures” of these facts, and propositions are significant expressions of thoughts. But this language isn’t internally self-sufficient; all pictures must share something in common with whatever they depict. Proceeding by logical analysis, Wittgenstein believed that inquiry would eventually yield irreducibly basic symbols denoting entirely non-complex objects called “atomic facts”—truly reliable signifiers, the basic building-blocks of world-pictures, the points at which the painter and the painted meet.

According to Anthony Kenny, Wittgenstein was deeply influenced by Russell at the time he wrote the Tractatus, and he likely believed these “atomic facts” could be confirmed through empirical verification. Later (partly influenced by Gödel), he thought better of it, and dedicated the rest of his career to developing an alternative method for philosophical investigation. In the end, he probably never reached a satisfactory conclusion. But the spirit of Wittgenstein’s later work echoes Gödel’s reasoning as he drifted toward Platonism: There are no purely logical foundations, and yet the intelligibility of the subject matter cannot be denied. What is needed is some other form of certainty, some other method of investigating and assenting to propositions.

Kenny makes an eloquent point about On Certainty—Wittgenstein’s last work—in A New History of Western Philosophy:

Wittgenstein’s purpose ... is not just to establish the reality of the external world against Cartesian scepticism. His concern, as he acknowledged, was much closer to that of Newman in The Grammar of Assent: he wanted to inquire how it was possible to have unshakeable certainty that is not based on evidence. The existence of external objects was certain, but it was not something that could be proved, or that was an object of knowledge. Its location in our world-picture … was far deeper than that.

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