The study of non-Euclidean geometry introduces many complex explorations into the nature of truth, the objectivity of beauty, and whether mathematics is primarily in the realm of imagination or discovery. In fact, Lobachevski’s decision to call his geometry “imaginary” invites such questions. As long as geometry remains on a Euclidean plane, can be conducted with a compass and straight edge, and has direct application to the world of human experience, it seems obvious that truth is objective and discoverable, beauty is objective in its expression of harmony, order, and relationship, and that mathematics (at least in the realm of geometry) is the act of discovering an objective facet of reality that has practical application. But non-Euclidean geometry has shown that man can create systems of geometry that are internally consistent and coherent, and yet seem to contradict not only Euclid but also our own experience of the world. If a series of puns may be ventured, it doesn’t seem hyperbolic to say that the very limits of mathematics have been stretched to infinity in ways only our imagination can grasp.
One of the most pressing questions surrounding the implications of non-Euclidean geometry relates to what bearing this has on the nature of truth and beauty. Does it indicate that truth is relative or, at least, non-objective or pluralistic? Does it indicate that there are many truths or that truth is a human construct? And since mathematics seems to have direct relationship with beauty, does non-Euclidean geometry indicate that beauty is subjective or malleable? Could it be that, in fact, the creation (or discovery?) of non-Euclidean geometry liberates truth and beauty from a metaphorically flat plane into a three-dimensional sphere of complexity, depth, and paradox? These questions must be addressed, even if they cannot be fully answered.
On one level, these questions get to the heart of an abiding philosophical distinction between different ways of speaking about truth. On the one hand, something is true when it corresponds to reality, or at least the way things seem. If someone says that his coffee cup is sitting on a coaster, then his statement is true if it conforms to the reality that it, in fact, is sitting on a coaster. But on the other hand, something can be considered true when it logically coheres with other things we know to be true, based on deduction. If a conclusion logically flows from all the premises, then it can be said to be coherent, or valid, and in that sense true. So at first blush, one might be tempted to say that Euclidean geometry is true according to the correspondence theory of truth and non-Euclidean geometry is true according to the coherence theory of truth.
Yet things are not so simple. In fact, non-Euclidean geometry seems to have certain applications to the real world that address things Euclidean geometry fails to address. Upon deeper exploration, Euclidean geometry seems to correspond to what we know to be true about Euclidean planes, but not what we know to be true about life in a three-dimensional, spherical (or curved, if one might venture that word) universe. What is a “straight line” in a spherical world, for example? We are thus challenged to go deeper into these distinctions between kinds of truth.
It seems that the correspondence theory of truth can further be divided into categories. There are things that correspond to our perception but may not correspond to the way things actually are. To our perception, the world seems flat, and for most building projects and surveying endeavors, Euclidean geometry seems to correspond with the way things are, at least to our perceptions, and work quite well. But given that the earth is a sphere and that the universe has many properties better expressed in the language of curves, spheres, and hyperbolas, Euclidean geometry in a very real sense doesn’t correspond with the way things actually are. This does not mean that truth is relative or that Euclidean geometry is false. It simply means that it corresponds more closely to what might be called consistent, objective perception of reality—at least, perception to the naked eye. And in one sense, then, it is true. In another sense, however, it is false.
What’s even more surprising is that while non-Euclidean geometry was originally taken up as a creative exercise in the realm of logical coherence, specifically distancing itself from the realm of correspondence with the real world, it in fact better represents the way aspects of the universe actually work. Might it be that while it seems as if “imaginary geometry” is a manmade creative endeavor, it is in fact a kind of discovery? That perhaps our imaginations, if guided by logic, correspond with the universe more than we might have anticipated? And perhaps what first seemed to be an example of how truth might be relative or pluralistic might actually prove that truth is, in fact, mysteriously one: not in the sense of a singular point, but in the sense of a three-dimensional, multi-faceted, polyvalent structure?
This brings us to the discussion of beauty. Traditionally, the concept of Beauty was directly related to Truth and Goodness as the “trinity of transcendentals” that remained objective and unified regardless of personal experience or perception. Mathematics as a method of discovery was an elegant way of expressing the beauty of the world—the harmony of numbers and shapes in the unified cosmos. Music, for example, is simply the audible expression of ratio—of numbers in relationship. As long as mathematics remains a kind of discovery, an expression of the order and harmony of the cosmos, then beauty can be said to be objective. But if mathematics is a kind of human creation that has no correspondence with the cosmos, then this invites a radical shift in our understanding of beauty.
While the problem cannot be quickly solved, it seems as if the invention—or discovery—of imaginary geometry and “pure mathematics” doesn’t necessitate abandoning the transcendentals or the objectivity of beauty. Taking music as an analogy, it seems like a consistent melody can be discerned. In music, there are many modes. Each mode dictates what kinds of relationships each of the notes may have with each other. The relationship between two notes in one mode may be said to be quite beautiful, but put in another mode, the relationship might be discordant. Thus, each mode creates a context by which one can determine if notes in relationships are beautiful or ugly. In this sense, beauty can be said to be relative, in so far as it relates to the modal context. However, this doesn’t negate the reality of objectivity in music. Each mode creates an objective context within that mode. If one is playing in the Ionian mode, there are certain objective patterns that must be followed. One cannot play anything within the Ionian mode and call it music or call it beautiful. The presence of modal contexts does not therefore strip music or beauty of its objectivity, but rather deepens the meaning of objective beauty.
It also seems that these modes were discovered more than created, in that they describe seven different patterns or note-relationships that seem to adhere to what would be considered beautiful. And, conversely, they invite a tacit assumption that there are certain sequences of notes that, if strung together, can be said to be objectively ugly and nonmusical. Thus, we are given two major objective categories, beautiful and ugly, but whether certain note relationships fall into one or the other of those categories is contextually dependent.
If this is true of music, which has direct correspondence to the nature of mathematics, then one might venture that it is also true of mathematics and, in some sense, of truth. Imaginary mathematics may simply be the discovery of a different “mode of mathematics,” a way of expressing different patterns and relationships that are true, even if they seem to contradict in some ways the patterns and relationships expressed in other kinds of mathematics. Upon further investigation of non-Euclidean geometry, this seems to be supportable as long as mathematics relies on principles of logic and coherence. Thus, while Euclidean geometry defines parallelism differently than non-Euclidean geometry, the differing conclusions reached by these two modes of geometry do not undermine the abiding structure of geometric relationships between shapes. Thus, when Lobachevski maintains that the angles of a triangle are less than pi, the objective nature of geometric relationships dictates that he must now explore what effect this has on the nature of squares (concluding there are no squares), rectangles, and circles. Even though all of his conclusions about the natures of the individual shapes then differ from Euclid, the presence of consistent geometric relationship between these shapes abide. Thus, we find that the modal analogy holds. Just as musical modes describe different internally consistent relationships between notes, so different geometries describe different internally consistent relationships between shapes, lines, and angles. But these differences do not invite complete relativism or non-objectivity. In fact, it presupposes underlying objectivity.
While these conclusions are hardly conclusive, upon a brief exploration into these questions it seems that non-Euclidean geometry reinforces and deepens our understanding of objective truth and beauty, replacing a simplistic rendering of the shape of reality with something on a higher, polyvalent plane. Perhaps this is why many mathematicians have said that mathematics is the language of God, the Trinity in which the transcendentals find their unity. The One in whom all mathematical relationships find their center.
Tremendous Trifles 