Unity and diversity: Geometry in secular and sacred realms

Many years ago, when teaching an introductory course in the humanities, I mentioned that a particular claim about the world was true ut in pluribus. The Latin means “for the most part,” or, more literally, “as in the many.” An eager first-year student shared what he learned that day with a fellow student, noting that he now knew what the Latin in the Great Seal of the United States meant. Unfortunately, he had confused that motto, e pluribus unum (from many, one), with ut in pluribus!

Pluribus and unum, many and one, have many senses, distinct but sharing a fundamental affinity. Although many today would not make the same mistake my student made, there continues to be considerable confusion about what one-ness or unity means in different settings. This was apparent recently as political discourse in the United States has reemphasized calls for unity. Recently the Vatican completed another week-long program on Christian unity and Pope Francis has often referred to the need for unity within the Catholic Church as well as a broader sense of unity among all human beings (I fratelli tutti).

Geometry and ethics

Geometry and ethics (both philosophical and theological) may seem to be a strange juxtaposition. This pair, however, would not have seemed strange for the ancients, as legend has it that over the gates of Plato’s Academy were inscribed the words: “Let no one ignorant of geometry enter here.” Education at the Academy addressed fundamental questions about justice, virtue, and the good life, as well as profound topics in metaphysics and epistemology.

And, of course, Plato thought that such education could lead to a philosopher king. From Euclid’s time on, in fact, geometry has provided a model for demonstrative knowledge—that is, syllogistic reasoning from true premises to true conclusions. Public policy issues, themselves a sub-division of ethics, depend implicitly and explicitly on broader concerns about nature and human nature. What role should geometry play in understanding the world in all its complexities and thus in contributing to sound policy for both the secular and the sacred? For believers, questions about unity in faith involve prior notions about what unity means in its varied senses.

Even though many of the modern natural and social sciences depend heavily upon mathematics, why would one think that the knowledge of geometry is a prerequisite for knowledge in any area? What is the connection between knowledge of points, lines, planes, solids, and geometric proofs, on the one hand, and the study of literature, history, philosophy, theology, the natural and social sciences, music, and art, on the other? It is not so much proofs in geometry itself that are crucial, but the way that topics addressed in geometry disclose how one ought to engage other areas of inquiry. We can see this connection in a striking way when we consider the relationship between points and lines.

If I were to ask readers to tell me what a line is, most would have some confidence that they know the answer to this question: they think that a line is made up of, or is set of, points. If I were to ask what a point is, they would probably be less confident. Ultimately, however, we need to understand that a point is a dimensionless reality; that is, a point is position without dimension; or, as Euclid would have put it in a more technical way, a point is that which does not have part outside of part. If someone were to claim that a point is really only a very small dot—something, that is, with dimension—what then would be the difference between a point and a line? Would not a point, so defined, simply be a very short line?

There is a kind of dialectical argument here that leads us to the uncontestable truth that a point is position without dimension. How could that which has no dimension whatsoever—a point—come to constitute or be a part of or make up that which has dimension, namely a line? If we follow this argument, we would have to recognize that what many thought to be the case—that a line was made up of points—cannot be true.

If points do not make up a line, how do they exist in a line—for surely points are somehow found in a line? The solution involves a recognition that points exist potentially in a line. Points exist in lines, but not as parts of lines. To divide a line “at a point” is to cause a point actually to exist. Once such a point exists actually, it divides the magnitude into two parts. The limits of each part exist together in the point thus brought into actual existence. The infinity of points in a line refers to the fact that, because the process of dividing a line is endless, we can always produce another point; there is no limit to the number of points so produced. However many times we divide a line, we can always divide it once more. Points can be produced in this way—that is, brought into actual existence—because they already exist potentially in a line. The distinction between potential and actual existence, both real yet different modes of existence, is crucial for understanding continua such as lines.

The one and the many

The discussion of lines and points reveals the problem of explaining the unity, the one-ness, of a line, and the plurality, the many-ness, of points. How is a line both one and many? And then, more importantly, how is it that such a question in geometry discloses a topic that unites all intellectual disciplines and thus undergirds questions of public policy?

The recognition of how things are one and yet composed in diverse ways is crucial for our understanding of reality. Accurately recognizing unity and diversity in the world is an essential element of how we ought to act in the world. We can begin with some relatively easy examples concerning the unity of things and then move to more complex phenomena.

In nature, we might consider an elementary compound such as water. Water is surely one thing, so in what way do hydrogen and oxygen exist in water? How is it that we can speak of the reality of water and yet maintain the diversity of hydrogen and oxygen in the liquid? At room temperature, hydrogen burns and oxygen supports burning, but under the same conditions, water does not burn. There must be something more to water than the mere bringing together of two elements. Somehow elements, such as hydrogen and oxygen in the case of water, are the material constituents of chemical compounds, yet these compounds possess a unity that is more than the sum of their material parts.

Explanations of motion and time also depend upon an adequate understanding of continuity and diversity. As a line is a continuous magnitude, so too are motion and time, but the latter are flowing continua. The “now” or instant of time unites past and future analogously to the way in which a point unites the continuum that is the line. Like a point, a now has no dimension—in this instance, temporal dimension—and thus time is not made up of nows or instants. Zeno’s paradoxes concerning the impossibility and unintelligibility of motion (e.g., his famous example of going half the distance, and half of the remaining distance, etc., and thus never reaching one’s destination; or of an object in motion, at any instant, also being at rest) are based on a misunderstanding of the nature of the continuous character of motion. A motion is one thing; it is not made up of discrete parts.

Think about our bodies, these biological unities. What is the relationship between the diversity of cells and the unity of the body? Clearly the human body is a biological unity, a single organism. A human being is not a collection of distinct, discrete units that happen to exist together: a human being is a unity, not an accidental heap. Although we can abstract from the unity of the whole and consider cells, or organs, or even the nervous system, in isolation from the whole, it does not follow that the human body is simply the sum of such distinct parts.

The human body is not a machine. The various parts that constitute a machine exist together in a kind of unity, but these parts are contiguous: they are in contact with one another and function in relation to one another, but they really remain distinct. The parts of a body, on the other hand, are distinguishable from one another, but they exist in biological continuity with one another. Complex as it is, a living body is a unity, not several parts just touching each other as they function together. Contiguity and continuity differ from each other not in degree, but in kind. In other words, continuity is not the result of a greater and greater contiguity.

If we turn to literature and consider words, sentences, and stories, we know that a word is one thing; it is a unity, yet there is in it the many-ness of letters. A sentence is one thing, yet there is in it the many-ness of words. A story is one thing, yet there is in it the many-ness of sentences. How do we discover the unity of a story in the many-ness of its sentences? The meaning of a sentence or a story, the content of its unity, is not the sum of its parts. In other words, although each word in a sentence means something, the meaning of the sentence as a whole is not discovered simply by adding together the meanings of the various words. The meaning of a sentence, however, is somehow disclosed or found in the words and their syntactical relationship to one another; so too is the relationship between sentences and the unity of a whole story.

Consider the unity of a symphony and the diversity of parts that each instrument plays. With respect to each of these parts, think of the unity of the sound produced and the diversity of individual notes and chords. How is a symphony one, yet many? What of a painting or a sculpture? Aesthetic unity—that is, the formal component of beauty—is not the sum of distinct parts, but is disclosed in the unity of the work taken precisely as a unified whole.

What about the unity over time of each of our lives: our psychological unity, intimately connected to our faculty of memory? How is it that each of us is the same person today as yesterday, despite all the new experiences that we have had since yesterday? How are we the same, yet different? Our lives are not made up of moments. As the baker’s wife in Stephen Sondheim’s musical Into the Woods observes, “If life were only moments, then you’d never know that you’d had one.”

What about the unity of institutions: how are they one, yet constituted by diverse individuals? In what way or ways can we speak about corporations, social groups, nationalities, or entire cultures as one? How is an individual a member of a family, a race, or a society? What kind of unity exists in marriage? How ought laws be framed so that they reflect the various kinds of unity—biological, social, religious, etc.—that are part of human existence and experience? We wrestle with the dilemma of judging individual responsibility in broader social, environmental, and genetic contexts. Human beings are not machines, not products to be manipulated. To foster the common good is the goal of public policy, and without a clear sense of nature and human nature, we will not know what is truly good.

Sameness and difference

We can expand this discussion to include specifically theological questions such as the unity of Persons in the Trinity and the unity of the divine and human natures in Christ. If the Church is the body of Christ, what kind of unity and diversity is proper to it? It would be a mistake to think of the Church as a unity in the same way in which a secular entity (e.g., a nation state) is a unity.

Sameness and difference, unity and diversity: these are themes that pervade systematic thinking in any subject. The relationship between points and lines reveals the topic in stark clarity, and allows us to begin to think about a question that no one really can ignore. We need to guard against a simplistic, univocal analysis of the question; that is, we need to recognize that an examination of sameness and difference in diverse areas requires analogical thinking. The unity of the human body, for example, is not the same as the unity of a chemical compound or of a line. Yet there is a unity to be found in each, a unity proportional to the kind of being each is.

Metaphysics investigates the many senses of unity, and even at this level, we confront the problem of how an individual substance (the primary category of what it means to be) is truly one thing and yet composed of form and matter. The latter are not themselves substances, but rather principles of substances. Form and matter, at least as Aristotle understood them, do not exist as independent entities. Various types of materialism, both ancient and contemporary, deny the existence of substantial unities other than some kind of fundamental indivisible particles. The complex entities of our world of experience are, for materialists, nothing but aggregates, accidental heaps of matter.

To conclude with a practical application to public policy, we should note that, when policy decisions about human beings are made, we need to remember that, with all living things, but especially with human beings, we confront the question of metaphysical unity in a special way. How are soul and body (the correlatives of form and matter in living substances) united in such a way that we can avoid the pitfalls of dualism and materialism and speak of the real unity of a living substance? As the Greeks knew, human well-being involves a harmony or unity within the soul and between the soul and the body, and among human beings.

If we are to have wise public policy, we need those who frame such policy to understand well unity and diversity in the world. Recognizing how lines are one, even with the many-ness of points, provides a good starting point for such reflection.