EUCLID: Elements (Introduction)
What is the author saying? Of all the Great Books authors we’ve read Euclid is the easiest to figure out what he’s saying. There’s no prologue or preamble. He just starts out with definitions and gets right to the point: “A point is that which has no part.” “A line is a breadthless length.” “The extremities of a line are points.” And so forth. Then he proceeds with Postulates and ends up with Common Notions. With these tools in hand we begin immediately with Book 1, Proposition 1: “On a given finite straight line to construct an equilateral triangle.” It would be almost impossible to be more clear and precise than Euclid. He says what needs saying, no more no less. But what is implied is that the world is an orderly and rational place. By following certain rules we can build on definitions, postulates etc. and come to reliable conclusions.
Is it true? It’s relatively easy to determine if a Euclidean proposition is true. Just follow the steps outlined and compare them with the definitions given at the beginning. If they all fit then the proposition is true. But that’s a superficial understanding of things. Does Euclidean proof have any meaning outside its own system? John Keats ended his poem “Ode on a Grecian Urn” with lines that tell us 'Beauty is truth, truth beauty,—that is all/Ye know on earth, and all ye need to know.' In Euclidean terms that’s just not true. Keats failed to define his terms at the beginning of the poem. We don’t really know what Keats thinks truth and beauty consist of. And for most people the common notion is that truth and beauty are two entirely different things. Wallace Stevens makes a similar blunder in “Le Monocle de Mon Oncle” when he says ‘The mules that angels ride come slowly down/The blazing passes, from beyond the sun.’ In Euclidean terms that’s just sheer nonsense. If we define mules as physical creatures and angels as intelligences without bodily existence then how can angels possibly ride mules down so-called “blazing passes, from beyond the sun?” This won’t do for Euclid. Yet Stevens creates a vivid image, in my mind at least, that’s poetically true. I can imagine angels riding mules down blazing passes from beyond the sun even if it’s physically impossible. But is that concept so very different from Euclid’s “point” being defined as that which has no part? That seems physically impossible too though I can grasp it as a concept. Keats and Stevens are asking us to accept the terms of their poems just as Euclid is asking us to accept the terms of his geometry.
So what? It may be that art and mathematics inhabit separate realms. Maybe art is more interested in beauty and mathematics in truth. But there are other lessons to be drawn from Euclid. One example in our GB readings is Plato’s dialogue Meno. Socrates asks an untutored slave boy a series of questions and the boy unwittingly makes a geometric proof. Plato uses Meno to make a point about the nature of knowledge but it also demonstrates the close connection between mathematics and philosophy. Both require intense concentration to follow an argument to its logical conclusion. The GB readings also include The Constitution of the United States of America as an example. Article One, Section 2 begins “The House of Representatives shall be composed of…” and Article One, Section 3 begins “The Senate of the United States shall be composed of…” Article Two, Section 1 begins “The executive power shall be vested in…” Article three, Section 1 begins “The judicial power of the United States shall be vested in…” A constitution is obviously a different sort of document than a treatise on geometry. However, the beginnings of both the Elements and the Constitution start with definitions to be used as tools, although for very different purposes. The Elements uses those tools to build a geometric system. The Constitution uses its tools to build a nation.
Is it true? It’s relatively easy to determine if a Euclidean proposition is true. Just follow the steps outlined and compare them with the definitions given at the beginning. If they all fit then the proposition is true. But that’s a superficial understanding of things. Does Euclidean proof have any meaning outside its own system? John Keats ended his poem “Ode on a Grecian Urn” with lines that tell us 'Beauty is truth, truth beauty,—that is all/Ye know on earth, and all ye need to know.' In Euclidean terms that’s just not true. Keats failed to define his terms at the beginning of the poem. We don’t really know what Keats thinks truth and beauty consist of. And for most people the common notion is that truth and beauty are two entirely different things. Wallace Stevens makes a similar blunder in “Le Monocle de Mon Oncle” when he says ‘The mules that angels ride come slowly down/The blazing passes, from beyond the sun.’ In Euclidean terms that’s just sheer nonsense. If we define mules as physical creatures and angels as intelligences without bodily existence then how can angels possibly ride mules down so-called “blazing passes, from beyond the sun?” This won’t do for Euclid. Yet Stevens creates a vivid image, in my mind at least, that’s poetically true. I can imagine angels riding mules down blazing passes from beyond the sun even if it’s physically impossible. But is that concept so very different from Euclid’s “point” being defined as that which has no part? That seems physically impossible too though I can grasp it as a concept. Keats and Stevens are asking us to accept the terms of their poems just as Euclid is asking us to accept the terms of his geometry.
So what? It may be that art and mathematics inhabit separate realms. Maybe art is more interested in beauty and mathematics in truth. But there are other lessons to be drawn from Euclid. One example in our GB readings is Plato’s dialogue Meno. Socrates asks an untutored slave boy a series of questions and the boy unwittingly makes a geometric proof. Plato uses Meno to make a point about the nature of knowledge but it also demonstrates the close connection between mathematics and philosophy. Both require intense concentration to follow an argument to its logical conclusion. The GB readings also include The Constitution of the United States of America as an example. Article One, Section 2 begins “The House of Representatives shall be composed of…” and Article One, Section 3 begins “The Senate of the United States shall be composed of…” Article Two, Section 1 begins “The executive power shall be vested in…” Article three, Section 1 begins “The judicial power of the United States shall be vested in…” A constitution is obviously a different sort of document than a treatise on geometry. However, the beginnings of both the Elements and the Constitution start with definitions to be used as tools, although for very different purposes. The Elements uses those tools to build a geometric system. The Constitution uses its tools to build a nation.
posted by RDP | 12:19 PM
1 Comments:
"He says what needs saying, no more no less. But what is implied is that the world is an orderly and rational place."
Actually, nothing at all regarding the real world is implied by Euclid's postulates. He offers several definitions and postulates, and derives a specific logical relationship between these statements. That is all. There is no direct corollary from a closed system such as geometry and the physical world we inhabit. Mathematics, as Kant said, is a form of "synthetic a priori" knowledge, whose existence and validity are independent of the world. Thus, we cannot gain any knowledge about the world based on a system of "a priori" statements. Perfect triangles do not exist in nature, but only in the mind. Any idea we have about the world being rational is a conclusion based solely on experiment and observation, in other words "a posteriori" facts which have been verified. In order to infer that the world is rational based on geometric principles, you would first have to eliminate all randomness, making the world a closed system, neither subject to error or deviation.
"Is it true? It’s relatively easy to determine if a Euclidean proposition is true. Just follow the steps outlined and compare them with the definitions given at the beginning."
The proper question is not whether Euclid's geometry is true but whether his logic is valid. Truth is a condition associated with the real world, as opposed to things we might say about imaginary worlds. Since geometry is a construct of the mind, its "truth" cannot be tested in the world; geometry makes no assumptions regarding the world. However, Euclid's methodology can be examined and shown to be valid or invalid, depending on his logical consistency. In other words, are his proofs valid? Do the conclusions follow logically from the postulates?
"And for most people the common notion is that truth and beauty are two entirely different things."
This statement provokes a different kind of discussion. There are logical or empirical tests for determining the truth of a proposition. But what about beauty? Does it reside solely in the eye of the beholder? Or is it a quality discovered in the world itself? One argument is that no lie or untruth can ever be beautiful because beauty itself is an emanation of the soul in response to the world. Thus, the perception of beauty is not just an aesthetic judgment of the world but a kind of ideal equivalence between mind and body, such as the idea and its corresponding object. Of course, this raises all kinds of difficulty, such as the relationship between beauty and myth or illusion. Can ideas be beautiful? If so, why would some ideas be beautiful and others not?
"Keats and Stevens are asking us to accept the terms of their poems just as Euclid is asking us to accept the terms of his geometry."
True. But we are willing to grant poets the freedom to portray all kinds of imaginary objects or beings because we accept the convention. Poetry is not science and Keats is not masquerading as Darwin. Poets take liberty with their depiction of the "real" world because, in doing so, they often illuminate some aspect of human life that would otherwise remain concealed. This is why Plato banned poets from his ideal republic. He didn't want poets running around exposing the royal lie that holds society together.
...the beginnings of both the Elements and the Constitution start with definitions to be used as tools, although for very different purposes. The Elements uses those tools to build a geometric system. The Constitution uses its tools to build a nation.
I would point out that law and geometry operate in different spheres. Obviously, Euclid is making no value judgments in saying that..."A point is that which has no part... A line is a breadthless length...The extremities of a line are points.” These are simple declarative statements that make no reference, good or bad, to anything outside themselves.
On the other hand, our constitution establishes the laws (values) which we believe are necessary for a good society. Not only are they necessary to a good society, but we believe they establish the best possible society. This is where the distinction between truth and validity or between geometry and ethics is clear. Geometry is universal and can be understood and applied by all people at all times, regardless of geography or politics.
The U.S. Constitution does not make that claim. It simply states what laws we deem to be valid and necessary for our society. It is not until the French Revolution with its slogan of "Liberty, Equality, and Fraternity" do the so-called "rights of man" become institutionalized as universal ideals for a republican form of government.
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