Jacob Klein was in the last year of his nine-year tenure as dean of St. John’s College in 1957 when I came as a young tutor. He died in 1978, still teaching. In those twenty-one years during which I knew him, he was above all a teacher—mine and everybody’s. His spirit informed the college. While dean, he was a fierce defender of his conception of this remarkable community of learning. This passion had generous parameters, from a smiling leniency toward spirited highjinks to a meticulous enforcement of rules meant to inculcate intellectual virtue. As a tutor, he shaped the place through lectures that the whole college attended and discussed, through classroom teaching that elicited from students more than they thought was in them, but above all through conversation that was direct and playful, serious and teasing, earthily Russian and cunningly cosmopolitan. We all thought that he had some secret wisdom that he dispensed sparingly out of pedagogical benevolence; yet he would sometimes tell us things in a plain and simple way that struck home as if we had always known them. I, at least, always had the sense of hearing delightful novelties that somehow I’d known all along. He also had an aversion to discipleship and a predilection for wicked American kids. And he could be infuriating whenever someone tried to extract definitive doctrines from him. His reluctance to pontificate was in part indolence (we sometimes called him “Jasha the Pasha”)—an indolence dignified by his aversion to philosophy carried on as an organized business—and in part pedagogical reservation—a conviction that to retail one’s thought-products to students was to prevent inquiry. This aversion to professing authority is, to my mind, his most persuasive and felicitous legacy to the college, and the reason we still call ourselves tutors—guardians of learning—rather than professors—professionals of knowledge.

Nonetheless, there were doctrines and they were published. He had set himself against academic publication, so much so that I had to translate Jasha’s youthful book on the origin of algebra in secret—though when confronted with the fait accompli he capitulated quite eagerly. This book is now the subject of Burt Hopkins’s acute and careful analysis, The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein (published in 2011).

Today, I would like to present two of his chief discoveries from a perspective of peculiar fascination to me—from the standpoint of the contemporary significance and the astounding prescience, and hence longevity, of his insights. Now I grew up intellectually within a perspective enforced by our program of studies and reinforced by Jasha’s views (forgive the informality; it was universal in his circle), which were rooted in certain continental philosophers, of whom Husserl was the most honorable. The guiding notion of this perspective was that modernity is best apprehended as being in a ruptured continuum with Greek antiquity—a continuum insofar as the terms persist, ruptured insofar as they take on new meanings and missions. That perspective makes those who hold it avid participants in the present—critically and appreciatively avid.

I will state immediately and straightforwardly the issues of our present-day lives to which Jasha’s insights speak. First, they speak to the ever-expanding role of image-viewing and virtual experience in our lives. Here the questions are: What degree of “reality” is ascribable to images? What does life among these semi-beings do to us? Do we lose substance as they lose their ground? Do originals retain their primary or even a residual function in the virtual world? Second, Jasha’s insights speak to the burgeoning brain science that tends to ascribe an ultimately physical being to human nature. Here the questions are approachable in terms of “emergence.” Granted that brain and mind are intimately linked, what is the manner in which the latter emerges from, or projects into, the former? How might an entity emerge, be it from above or below, that is radically different from its constituents? These are questions about consciousness (what we are aware of) and about self-consciousness (who we are) that should be of great concern to us, because they dominate public life quite unreflectively. To put this in a form that is not currently fashionable: Do we have souls?

Klein’s two insights, then, are both interpretations of Platonic writings and are set out in A Commentary on Plato’s Meno (1965) and Greek Mathematical Thought and the Origin of Algebra (1934). The latter is a learned book written by a private European scholar for academic readers, the former is a very accessible work written by an American teacher for lovers of Socrates. Of both these insights Burt Hopkins has produced detailed analyses, which have added a new edge to doctrines I’ve lived with familiarly for half a century. I will, however, feel free here to supplement, embroider and question Jacob Klein’s interpretation of Plato and Burt Hopkins’s reading of Klein as I go. I’ll do it implicitly, so you shouldn’t trust this account for faithfulness to the letter, though I hope you may trust it for faithfulness to the spirit. You’ll see, I think, what I mean when I speak of the immediacy and naturalness of Klein’s interpretation: His readings sit well.

The first insight, then, begins with an understanding of the lowest segment of the so-called Divided Line in Plato’s Republic, that mathematical image (picture it as vertical) of the ascent to Being and the learning associated with that ascent. In this lowest segment are located the deficient beings called reflections, shadows, and images, and a type of apprehension associated with them called eikasia in Greek and usually rendered as “conjecture.” Klein’s interpretation starts with a new translation of this noun: “image-recognition.” The nature of these lowest beings—they are revealed as basic rather than base—is set out in Plato’s Sophist. Consequently, the Republic and the Sophist between them lay the foundations of the Platonic world.

The second discovery involves a complex of notions from which I’ll extract one main element: the analysis of what it means to be a number, and what makes possible this kind of being—which, as it turns out, makes possible all Being. Again, the principal texts are the Sophist and the Republic, supplemented by Aristotle’s critical account of Plato’s doctrine. To anticipate the perplexity that is also the doctrine: Take any number—say two. It is constituted of two units. Each is one, but both together are two. How can it be that two emerges from elements that are each precisely not two? I might remark in passing that Socrates thinks that one mark of readiness for philosophical engagement is a fascination with this perplexity. And from my experience with students, I know that Socrates was correct.

So now, after these broad previews, some nitty-gritty. Socrates begins by dividing the whole line mentioned above in some arbitrary ratio, and then he divides the two subsections in that same ratio. So if the whole line is, say, sixteen units, and the ratio is, say, triple; then one segment is twelve units, the other four. Then subdivide the twelve-unit segment similarly into nine and three units, and the four-unit segment into three and one. There are now four segments, two by two in the same ratio with each other and with the first division of the whole. Whether you want to make the top or the bottom segment the longest depends on whether you assign more length to the greater fullness of Being or to the larger profusion of items. It can also be shown that in all divisions of this sort—called “extreme-and-mean ratio”—the middle segments will be equal. Socrates will make the iconic most of this mathematical fact.

Now the subsections make a four-term proportion called an analogia in Greek—a:b::b:c—and they mirror, as I said, the division of the whole line. You can read the line up or down. Down is the cascade of Being, which loses plenitude as it falls from true originals to mere images. Up is the ascent of learning, ending in the direct intellectual vision of the prime originals, the eide, the “invisible looks” in Klein’s language, usually called the “forms.” Beyond all Being there is the notorious Good, the unifying power above all the graduated beings, the principle of wholeness, which I’ll leave out here. At the bottom is the aforementioned “image-recognition.” Now just as each of the object-realms assigned to the upper sections is causally responsible for the ones below, so, inversely, in learning, each stage, each capacity, is needed for the learner to rise. None are left behind; all remain necessary. And so the bottom, the first capacity, is also the most pervasive. Children recognize images early on. Look at a picture book with a two-year-old: “Kitty,” he’ll say, pointing. “Careful, it’ll scratch.” “No, it won’t,” he’ll say, looking at you as if you were really naïve. That’s image-recognition, the human capability for recognizing likeness as belonging to a deficient order: a cat incapable of scratching.

It is as fundamental for Socrates as it is low on the scale of cognitive modes, because imaging is the most readily imaginable, the least technically ticklish way of representing the activity by which the realm of intelligible Being produces and rules the world of sensory appearances. Each step downward in the scale of being is a move from original to image; each step upward in the scale of learning involves recognizing that something lower is an image of something higher.

Just to complete the sketch of the Divided Line, here are the stages of knowledge and their objects in brief. Above images, there are the apparently solid objects of nature and artifice. The acquaintance with these is called “trust,” pistis. It is the implicit, unreflective belief we have in the dependable support of the ground we tread on and the chair we sit in—the faith that our world is not “the baseless fabric of a vision” that melts into thin air.

This whole complex of dimensionally defective images and taken-on-faith solidity of our phenomenal world is itself an image of the upper two parts of the line. The third part, equal in length to the second from the bottom, contains all the rational objects that look, on the way up, like abstractions from the sensory world—mathematical models and logical patterns. To these we apply our understanding, a capacity called in Greek “thinking-through,” dianoia. They are then revealed to be the originals of the sensory world, the intelligible patterns that impart to the sensory world such shapeliness and intelligibility as it has. Thus, they make natural science possible; for they are the rational counterparts of the sensory world. Finally, there is the realm of direct knowledge. As happens so often in the dialogues, the contents of these upper reaches are named in inverse relation to the contents of the lower ones: “invisible looks” (since eidos is from the vid-verb, the verb for seeing), and they are reached by a capacity for direct insight (which Aristotle will in fact analogize to sensing)—noesis. Above and beyond them all is the very Idea itself, the idea of all ideas—the Good, which produces, nourishes, and unifies all beings, and grounds all human learning.

Implicit in this ladder of Being is the answer to the question that matters most: What is an image, such that we can know it as an image? The answer is given in the Sophist, whose main character is, in his general person, an image incarnate—the mere image of a truly truth-seeking human being. Socrates poses the opening question of the Sophist, but he sits silently by as a Stranger from Italian Elea, a follower of Parmenides, finds a solution. I’ll venture a guess why he falls silent. In the Sophist appears a serious ontological teaching, and ontological doctrine is not Socrates’ way: He is the man of the tentative try, of hypotheses. I’ll even venture a—perhaps perverse—appreciation of this mode: His stubborn hypotheticalness, his unwillingness to assert knowledge, is the complement of his unshakable faith in a search for firm truth, carried on in full awareness of human finitude.

What then is the Parmenidean solution? I call it Parmenidean although the Stranger, the intellectual child of Parmenides, calls himself a parricide, since he is about to deny a crucial Parmenidean teaching: that Nonbeing is not, is neither sayable nor thinkable. For he will in fact affirm a yet deeper teaching of his philosophical father: that what counts is being thinkable and sayable. The Elean Stranger will show how Nonbeing can be thinkable and how speech is in fact impossible without it—as was indeed implicit in Parmenides’ very denial.

It is thinkable as Otherness. To say that something is not is usually to say that it is not this but that, that it is other than something perspectivally prior. (I say “usually,” because there “is” also something called “utter non-being,” which is indeed, though superficially utterable, insuperably unthinkable.) Relational, comparative Nonbeing, however, is one of the great ruling principles of ontology. It is totally pervasive, since whatever is a being is other than other beings. It is the source of diversity in the world and of negation in speech. Has the Stranger really done in his philosophical progenitor? No, as I intimated. He has actually saved Parmenides from himself; for he has shown that Nonbeing is, is Being in another mode. Being is still all there is. There is no parricide. Moreover, this Other, a piece of apparently high and dry ontology, turns out to give life to the realm of ideas and to the world of human beings: it informs the one with a diversity of beings and articulates the other with the oppositions of speech.

Why was this modification necessary in the search for the Sophist? Because a Sophist is indeed a faker, himself an image of a truth-seeker and a producer of images of what is genuine. Otherness, the great genus of “The Other,” is the condition of possibility for images, since it has three tremendous powers. First, it makes possible that a thing not be what it is. And that is just what characterizes an image: “It’s a kitty,” the child says, pointing. But not really; it doesn’t scratch. Or people bring out photographs in order to be in the presence of an absent one, but they are not real enough to assuage longing. Hence an image is understood first, and most ontologically speaking, as not being what it is, but also, second, as being less than the original it represents; for it represents that original in a deficient likeness. Here a second capacity of the Other shows up: it creates a defective, derivative Otherness. And third, it makes negative knowledge and denying speech possible: we can think and say, “The image is—in some specifiable way—like its original; but likeness is not identity.” The sentence “An image is not the original” displays Otherness as negation, articulated as Nonbeing. The ability to utter—and mean—that sentence is specifically human. Its loss would be, I think, a serious declension of our humanity. Therefore this complex of consideration, illuminated by Klein in his book on the Meno, seems to me crucial for navigating our image-flooded world with full awareness.

More particularly, the ability to distinguish image from original is crucial today because the shaping of our American lives, which is more and more a matter of declining options and refusing temptations, is much in need of suggestive approaches for coping with images. What are the truths and falsehoods of images in general? What consequently are the effects of discretionary image-viewing on our consciousness? Do all images in fact have originals, or is it images “all the way down”? And if there are always originals, how can we find our way back to them? What is real within our world, what is genuine beyond it? I’ll assert simply that without occasional reflection on such an eminently current issue our lives tend toward passing by rather than being lived. “The unexamined life is unlivable (abiotos),” Socrates says in the Apology (38a)—and so a life without reflection on its central issues is thus, in effect, unlived.

I’ll now go on to Klein’s second interpretive discovery, a much more technical, but equally future-fraught, construal.

A preoccupation of Socrates—it might be puzzling to readers who haven’t yet seen the implications—is often expressed by him in this way: “Each is one, but both are two.” To be gripped by this odd perplexity is, as I said, a beginning of philosophizing. The oddity comes out most starkly when we think of counting-numbers, the natural cardinal numbers. Take the first number, two. (For the ancients, one is not a number; it is the constituent unit of which a number is made.) Each of its units is one and nothing more. Yet this unit and another together make up the number two. Neither is what both together are. This ought to be strange to us, because we are used to the elements of a natural collection having each the quality that characterizes the whole: The doggy species subsumes dogs. Whether we think of dogginess either as an abstracted generalization or as a quality-bestowing form, each member has the characteristic that names the kind. Clearly, numbers are assemblages that work differently from other classes. Numbers have a uniquely characteristic, a so-called “arithmological” structure. The recognition of the significance of this situation and its peculiar appearance among the great forms, particularly in respect to Being, is Klein’s achievement.

Let me begin by briefly reviewing the kinds of numbers Klein takes into account. He observes—a previously ignored fact—that the first meaning of the Greek word that we translate as number, arithmos, is that of a counted assemblage of concrete things. Any counted collection—a flock of sheep, a string of horses, a herd of cattle—does not have, but is an arithmos. If we think as Greeks (and we may, with a little effort), we count ordinally, because we must keep items in order: first, second, third (and then go on cardinally four, five, six, for verbal convenience). But when we have counted up the whole, we allow it to become a heap-number—a distinct, discriminated group. It is a counted collection that has lost its memory. An arithmos is such a sensory number. It is, for example, a sheep-number, and its units are sheep-monads. To me it seems undecidable whether such a concrete number has an arithmological structure, since in it the sheep are both sheepish and mere units; as a flock we discriminate them, as units we count them.

Next come the mathematical numbers made up of pure monads, units that have no quality besides being unities. A mathematical number is defined by Euclid thus: “An arithmos is a multitude composed of monads,” where a monad is a pure unit. This type of number has an arithmological structure with a vengeance, and you can see why: a pure unit has no characteristics besides unitariness. It’s neither apples nor oranges, which is precisely why you can count fruit or anything at all with it. Being thus devoid of qualities, it has mere collectibility, but it has no other contribution to make to the assemblage. Being two is not in the nature of a monad as a monad, though adding up to two is. “Two” appears to emerge from these associable units. If you think this is unintelligible, so does Socrates. It will get worse.

The difficulty is implicitly acknowledged in the modern definition of number. It begins with arithmos like concrete assemblages. If their elements, treated now as mere units, can be put into one-to-one correspondence, the collections are said to be equivalent. The collection or set of all equivalent sets is their number. This definition evades the questions, What number is it? and Does the set of sets arise from the units of the concrete collection, or does it bestow on them the numerosity? Therein lies an implicit recognition of Socrates’ problem.

It gets worse, for now a third type of number appears. Klein knows of it from Aristotle’s critical report in the Metaphysics, where there is mention of “form-numbers,” arithmoi eidetikoi. The highest genera in the Platonic structure of forms are organized in such number-like assemblages. These are, unlike the indefinitely many mathematical numbers, limited in multitude. (There may have been ten, the so-called root-numbers of the Pythagoreans.) Now the super-genera in the Sophist are Same and Other. The highest after these is Being, which consists of Motion (kinesis) and Standstill (stasis). (This last is often translated as “Rest,” but that inaccurately implies a cessation from, or deprivation of, motion, though the two genera are coequal.) Notice, incidentally, that the three kinds of numbers run in tandem with the three rising upper segments of the Divided Line—concrete numbers with the sensory world, pure units with the mathematical domain, form-numbers with the eidetic realm.

Each of these forms acts like a monad in an arithmetic collection. However—and this is Aristotle’s most pertinent criticism—these high forms are not neutral units. They are each very much what they are in themselves, indefeasibly self-same and other than all others. They are, as he says: asymbletoi, “incomparable,” literally “not throwable together.” Thus, unlike pure, neutral mathematical numbers, they cannot be reckoned with across their own genus, and so, a fortiori, it would be seen that their association within their genus is unintelligible. For how can Motion and Standstill be together as the genus of Being if they have nothing in common and so cannot be rationally added up?

Klein claims that Aristotle’s cavil is in fact Plato’s point. The forms are associated in what is the very paradigm of an arithmological structure: what each is not, that they are together. It is because they have a number structure in which unique eide associate in a finite number of finite assemblages that innumerable sensory items can collect into concrete countable heaps organizable into finite classification. Furthermore, it is in imitation of these eidetic numbers that we have the indefinitely many mathematical numbers uniting as many pure units as you please—though we are left to work out the manner of this descent. For my part, I cannot claim to have done it.

I have mentioned before what is certainly the foremost stumbling block for most people in accepting the forms as causes of worldly being and becoming. The perplexity is usually put as “the participation problem:” How do appearances “participate” in the forms? These are infelicitous terms, because they imply the least satisfactory answer—that dogs somehow take a part in, or appropriate a part of the form, a non-solution scotched in Socrates’ very early attempt in the dialogue Parmenides at articulating his great discovery of the forms. “Imaging” might be a more felicitous term, since it is at least less awkward to the intellectual imagination than is “partaking.”

But let me stick here with the familiar term, and follow Klein in pointing out that the participation problem has two levels. On the lower level, the question is how the phenomenal world participates in the forms. On the higher level, it is how the forms associate with, participate in, each other. For unless they do form assemblages, genera and their constituent eide, the sensory world, even granted that it does somehow receive its being and structure from these, can have no learnable organization. Crudely put: We can classify the world’s beings, natural and artificial, in terms of hierarchies of kinds, such as the genera, subgenera, species, and subspecies of biology, only because their causative principles have a prior, paradigmatic structure of associations and subordinations. On this hypothesis, even only artificially distinguished heaps can be counted up by reason of the arithmological character of eidetic groupings.

The eidetic numbers are thus intended to be a Platonic solution to the upper-level participation problem. It is, so to speak, a highly formal solution. For while the type of association is named—the arithmoi eidetikoi with their arithmological structure—the cause of any particular association is not given. There is no substantive answer to the question: Just what in a form makes it associate numerologically with a specific other?

There should be no such answer, because the eidetic monads are, after all, incomposable. Motion and Standstill have in themselves nothing in common. Moreover, why are they the sole constituents of Being? And yet, there must be an answer since they are in fact composed. As Klein keeps pointing out, in the upper reaches the logos, rational speech, fails. One way it fails is that to reach the number two, for instance, we count off one, one, two; that is, three items—yet there are not three, but only two. For Two, be it mathematical or eidetic, is not over and beyond the two units; it is just those two together. How two items can become one our reason cannot quite articulate. Nor can it say what makes either an eidetic monad, which is a qualitative plenum, or the mathematical unit, which is a qualitative void, associate with others in “families,” (that thought-provoking classificatory term from biology) or in “numbers” (those colorless collections that yet have highly specific characteristics).

Now we come to Klein’s novel construal of just this eidetic number Two, which occurs in the Sophist, although it is not explicitly named there. Being is a great eidetic genus. It is composed of Standstill and Motion. Neither of these can have any part in the other; it is just as unthinkable for Standstill to be involved in Motion as for Motion to be involved in Standstill. Yet there is nothing in the world that is not both together. Our world is one of dynamic stability or stable dynamism, in place and in time. The duo responsible for this condition in the realm of forms is called Being. Being is not a third beside or above Motion or Standstill but just the togetherness of these subgenera. Being is only as both of these together, and neither of them can be except as part of a pair. As an unpaired monad, neither is; both are as a couple: Being is the eidetic Two. And once more, it is this arithmological structure that descends to, makes possible, and is mirrored in, the mathematical number structure of any mathematical two—on the one hand. On the other hand, it makes the phenomenal world appear as I have just described it: at once stable and moving, variable and organizable. On the way up, it might look as if the eidetic numbers are an erroneous levering-up of a mathematical notion; on the way down, they appear as the not quite humanly comprehensible, but necessary, hypothesis for an articulable world, a countable and classifiable world. And again, seen from above, Being must—somehow—bring about its own division; but seen from below, Being emerges from its constituents. And just as the modern definition of number in terms of equivalent sets leaves unarticulated the question of whether the number set is the ground of or the consequence of the equivalent sets, so too in Klein’s exposition of the first eidetic number, Two, it is left unsaid whether the genus determines its eidetic monads, or the reverse, or neither. It is left, as textbooks say, as an exercise for the reader—a hard one.

I might, before I end, even venture a still formal but somewhat more specific answer to the associability question. In the upper ontological reaches, at least, what might be called extreme Otherness—by which I mean either contrary (that is, qualitative) or contradictory (that is, logical) opposition—seems to be the principle grounding togetherness. Motion and Standstill are as opposite as can be, and for that reason yoked in Being; so are Same and Other. I will not pretend to have worked through the hierarchy of these five greatest genera. Nonetheless I have a suspicion that Same and Other, the most comprehensive genera, are not only intimately related to each other as mutually defining, but may ultimately have to be apprehended together as prior to and thus beyond Being, as a first self-alienation of the One, the principle of comprehension itself. As such, they might even be termed the negative Two, but that’s too far-out. In any case, these Plotinian evolutions are beyond my brief for today. I refer to Plotinus at all only because his One is in fact articulable only negatively and, is self-diremptive.

Now the strange structure of number, in which indiscernibly different but non-identical elements like pure monads, mere units, can be together what they are not individually, is only a case, though the most stripped-down, clarified case, of what is nowadays called “emergence.” Recall that emergence is the eventuation of a novel whole from elements that seem to have nothing in common with it. Examples range from trivial to life-changing. Socrates himself points out that the letters sigma and omega are individually different from the initial syllable of his own name, “So,” and that this is one new idea composed of two elements. (Theaetetus 203c) Two molecules of hydrogen and one of oxygen combine to form water, whose liquidity emerges as an unforeseeable quality. Individuals form communities that evince a might beyond the additive powers of their citizens. The emergent entity is other than rather than additional to, novel to rather than inferrable from, its elements. In the reverse case, sometimes called projection, the elements falling out from a totality are qualitatively quite different from it. This case might be called inverse emergence; an example might be the relation of Platonic forms to their participant particulars.

The most significant problem of emergence is also the most contemporary one. Since it seems indisputable that specific brain lesions lead to specific psychic disabilities, it is claimed by scientists who don’t want simply to identify mind with brain that the soul is brain-emergent. Does that make it a mere epiphenomenon? A miracle? “Emergence” names the event as a bottom-up process. But could it be a top-down happening, could the soul shape, or participate in shaping, its physical substructure? These are the recognizable old questions of “one-and-many:” one over, or in, or out of, many?

I want to make a claim that in this company especially should garner some sympathy: When deep human matters are at issue, it helps a lot to have delved into some ontology; the inquiry into Being may not affect our lives materially, yet it illuminates our daily lives more directly than does research providing factual information or theory producing instrumental constructs. This is the hypothesis under which Jacob Klein’s opening up of two Platonic preoccupations, images and numbers, is of current consequence. Herein lies the prescience, the foresightedness of his Platonic discoveries.


I have omitted here, as too complex for brief exposition, a third, more directly global interpretation of the modern condition, which is central to Greek Mathematical Thought and the Origin of Algebra. It is an understanding of the basic rupture between antiquity and modernity, of the great revolution of the West, as brought about by, or at least paradigmatically displayed in, the introduction of algebra. Algebra works with quantities abstracted from concrete collections (such as were betokened by the Greek arithmoi), with “general,” essentially symbolical “numbers,” such as the variables x, y, z or the constants a, b, c. These letters are symbols of a peculiar sort: They represent neither a concrete thing nor a determinate concept, but rather present themselves as the object of a calculation—a mere object, an indeterminate entity. Klein saw algebraic problem-solving procedures, so effective precisely because so contentlessly formal, as emblematic of a modern rage for that second-order, deliberately denatured thinking which dominates as much of our lives as is method-ridden. The human consequences of this symbolic conceptuality are great.

The occasion for this talk was the Keynote Address at the Conference on Jacob Klein, held at Seattle University on May 27-29, 2010. It appeared in The St. John’s Review (Volume 52, No. 1, 2010) and is republished here with gracious permission.

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