What is true? What is not? Is there anyway of deciding whether a complicated statement is true? What is understanding? What is implication? Or interpretation? These are types of questions asked by logicians. When the average person says that he does not see any purpose to study mathematics, one must ask, does this person ever make decisions, or believe anything true, or use proof in any facet of his life? Does he ever deduce or infer? Organize or order? Stack objects in a closet or in a trunk? Buy things?

Of course mathematics has a role in everyday life. Logic has such a profound role, that it is hardly ever seen. A cursory study of historical, philosophical and religious works show different types of logic active at the most fundamental levels of thought. How did Descartes and St. Anselm "prove" the existence of God? If these are proofs, why was Nietszche so successful with the opposite point of view?

Some "stupid" questions:

Why can a statement not be true and false simultaneously- - or can it be?

What is 1 + 1? Are you sure? Why?

Who is the best person to have ever lived? Who's the worst? How do you know that?

What is knowledge?

What is truth?

Logic does not answer such fundamental questions as what truth is. What it does do is provide a rigorous framework for the study of implication and truth structures. Logic gives a wide variety of truths from:

Either the Celtics won, or the Celtics did not win. (An uninteresting tautology)

to:

 

There will always be some true statements that cannot be proven.

The latter statement is a gross simplification of Godel's Incompleteness theorem. While this particular idea has been batting around for centuries in various forms, commenting about the imperfection of man, it is the rigorous application of logic that has made the statement not only influential, but proven, and has thereby reshaped people's understanding of how they think. Having dispensed with the idea of being able to achieve a complete understanding, the next question is where did this truth come from anyway?

Perhaps a bit of historical background will shed some light on why incompleteness is important. In the late nineteenth century, analysis was carefully constructing a solid intellectual basis for mathematical thought. By use of carefully chosen, precise definitions and rigorous proofs, the discipline of logic was formalized. While mathematical proofs have been around since the time of Euclid, formalization gave logic an unshakable intellectual ground. In light of the power of formal logic, it was wondered whether a finite set of axioms would suffice as the basis of true statements, in any given discipline, though the branch of mathematics that was focused upon was number theory. This seemed to be a very real possibility at the time. Bertrand Russell invested years to this question, and produced an immense system of axiomization, in the work Principia Mathematica. His aim was to get to the philosophical root of mathematical thought, to provide the essential keys to logical reasoning.

The history of mathematical thought is marked by a growing awareness of the importance of axioms. In Eucild's Elements the ancient Greek mathematician codified the theorems of geometry using five fundamental postulates. The fifth of the postulates disturbed him, however. It said that any non-parallel lines intersect, and parallel lines don't intersect. It seemed to him that this postulate should be a theorem deducible from the other four postulates. He was so upset that he carefully differentiated the geometry based on only the first four postulates from the geometry based on all five postulates. For centuries, mathematicians tried to produce a proof of the fifth postulate based on only the first four, and always failed. Finally, it was realized that it was possible to postulate the negation of the fifth postulate and not get any inconsistency or contradiction. A new branch of geometry was discovered, called non-Euclidean geometry, which was based on the notion that any two parallel lines intersect at one point.

If this idea is disturbing, good. You are becoming aware of the profound role played by intuition, and of its limitations. Intuitively, we feel that parallel lines could never intersect. But intuition is flawed. Try this supposition: any number can be stated as the ratio of two integers. In other words, there are no irrational numbers. While a simple proof shows the falsehood of this statement, it is neither intuitive or obvious from a primal point of view. Mathematics in action is the process of testing intuition through formal proofs. As this process works, not only is one's intuition satisfied, it is strengthened to the point that notions which originally seem bizarre become familiar, comfortable, and eventually obvious.

Aware of the flaws of intuition, finding a complete description of truth would be a feat worthy of great praise. Math would be done, finished, uninteresting. With a complete axiomization, determining the relative truth or falsity of a given claim would not be a painstaking, intellectually complicated process. One would examine the theorem, and enumerate the consequence of the complete axioms, using rules of inference. Eventually our theorem would be found true, or it would be found false.

What Godel did, to prove the incompleteness of number theory, was to create a sentence, definable in the context of number theory, which said "I am not provable."

He did this by codifying the elements of the language, so any formula in number theory would have a unique code, called its Godel number. Any deduction would also have a Godel number. He then constructed a sentence, lets call it q such that

q = there is no true sentence which has, as the Godel number of its deduction, and #q= the Godel number of q's deduction.

But any deduction of q, would have the Godel number #q. So if our proposition q is true, q must not be deducible. Also the only deduction which has #q as its Godel number is q, since Godel numbering is a unique coding. So if q were false, there would be a deduction of Godel number #q, but saying this is akin to saying q is true, since Godel numbers are unique. Either way, trying to prove q deducible is an impossibility.

The upshot is that truth will always remain beyond our ability to axiomatize. The key to this realization is an awareness of the interplay between multiple levels of thought.

Self-reference creates the paradox behind Godel's proof.

Knowing this, one can become sensitized to how widespread self- reference is.

For example, this sentence is self-referential. This sentence, in addition to being self-referential, refers also to the previous sentence.

This sentence refers to the next sentence, the final sentence in a two-part loop.

This sentence refers to the previous sentence, the initial sentence in a two-part loop.

Beyond trivial sentential self-reference, there is wider self-reference.

Watch a typical religious broadcast. People proclaim the Bible is the word of God, because it says so within the Bible. The Koran is also the revealed word of God. The religious process has, at a fundamental level, a stage of belief --where someone says "I say what I'm saying is true" and people say, "Oh, it must be true; after all, he said it was true, and he never lies."

Any statement contains a tacit self-assertion. Facts are stated not merely as combinations of words, but as truth valuations, and to impart some form of meaning.

Meaning is perhaps the least definable word. Linguistically, there is nothing more important than meaning. But to find a meaning of the term "meaning" itself, well, that is a sticky problem. Some philosophers have tried to say ... uh, what? what is this interruption?

Discontinuity is a confusing thing.

Sometimes, articulation is, well..., um, difficult.

Excuse me, it seemed like this essay was getting rather structured and potentially lame. Not that what I've said thus far is in any sense invalid.

Did you notice the parallel structures above? If one had seen just the pictures, one would never know that there was a code, a meaning to the hieroglyphics.

This here is the theme to the paper: Ever since Godel, structure has been discredited as the single prerequisite to meaning. I claim this advance is due to incompleteness. At a fundamental level of reasoning, the post-modern mind has realized the difference between what is, and what is understood. There is no longer a vain attempt to find truth within classification and explanation. The metaphysics of the contemporary age do not depend any longer on the individual's place within an orderly world.

I categorically refuse to justify the previous statement.

Listen to Bach. Very balanced, ordered, mathematically precise. Subtle interplay. Beauty within structure. Bach was master at exploiting the rules and guidelines of classical fugue structure, and finding exquisite relations therein.

Listen to the Talking Heads. There is no attempt to place one's I if e in an in-tellectual understanding of all of life. On the contrary, the theme is to "Stop Making Sense!"

Listen to John Coltrane, if you dare. Some people listen to his frenetic, screeching saxophone sounds and say, "this is only noise." Coltrane understood. He understood because he had pushed the concept of order to such an extreme level, that the order became blurred, lost cohesion and became chaos. To appreciate this development in his career, one must listen to works from his entire career. A brief summation:

Round about Midnight: John is playing with Miles Davis. Still playing sophisticated bop music. Bop was fundamentally an angry genre of music. It represented black anger in the fate '40s at the servile tendencies of early black jazz musicians such as Louis Armstrong who catered to white audiences, thereby reinforcing prevailing racial stereotypes. Bop elevated and refined the order of swing music, diminishing the intervals of improvisation, and replacing the simple rhythms with a new beat which nearly matched the old, yet differed in an essentially annoying way. Bop is a new order.

Giant Steps: Coltrane, playing with his own group. Simply put, the most frenetic, high-paced, yet orderly saxophone playing ever. He's still playing with the chords of bop music, but at such a furious pace that a listener must hold his breath and focus to see how the order is still there. Coltrane essentially shows the limits of the saxophone. It is impossible for him to have a more intricate structure.

My Favorite Things: He burst out of the confines of chord changes. The solos vary melodically, instead of harmonically. Bop solos are based on chords which change in regular sixteen bar time intervals. The new modal solos are based on fewer notes, and extend over indeterminate intervals. There exists more freedom, or entropy, or chaos, whatever one wishes to call it.

Ascension and A Love Supreme: More freedom: chaotic instrumentation, varying rhythmic schemes, atonality, screeching solos. Exploring freedom: why all these rules? What if I do things differently? Seeing chaotic existence, accepting it, loving it.

In all facets of our (consider this our as inclusive as you'd like) culture, there is a similar exploration, exploitation, and abandonment of systems of orderly thought. For millennia, a drama was not considered worthy unless It consisted of five acts. Modern playwrights such as Shepard, Pinter, Beckett, and others have abandoned such constraints, and many other "rules" of theater production. The heroic adventure was long considered a staple of literature. Modern writers have abandoned heroics, exploring the role of the anti-hero. But even the anti-hero Is old enough that he has become so clearly defined as to be Intellectually constraining.

Another case study for the existence of orderly worlds Thomas Pynchon's The Crying Lot 59. The protagonist, Oedipa Maas, describes her adventures as a departure from a fixed tower. Previously she had had the comfort of orderly existence, with ready explanations for whatever might happen to her. The novel opens with her at a Tupperware party, a model of suburban behavior. Gradually, disorder starts infringing. Her husband is seducing 1 5 year-old girls. Her psychiatrist tries to give her LSD . She leaves her husband, nominally to work on the execution of a deceased ex-lover's estate.

There follows a bizarre seduction by the lawyer she's working with, Metzger. This happens while they're watching an old war film (which Metzger had starred in while young, as a child actor.) Metzger prods her for her predictions of what will happen to the sundry characters of the show. She guesses what would happen, based on her knowledge of the genre, and on her sense of what would be reasonable. Repeatedly, she is wrong, as the show takes bizarre and unlikely twists. Metzger provides explanations which seem unlikely, but move to plausible, and are eventually convincing. Her sense of likeliness is usurped, and her sense of order is cast aside.

The main thrust of the plot is her attempts to deal with Tristero. She stumbles upon the possible existence of an underground mail system - an enemy to the (U.S. Postal system. She finds evidence in a medieval play, where a courier for Thurn and Taxis meets an untimely end at the hands of cutthroats sent by a rival mail service. She discovers stamps whose designs are subtle mockeries of U.S.P.S. stamps. When one of her friends mentions an intra-office mail system at his workplace, she postulates the existence of an underground, nationwide mail system, the Tristero. Initial efforts to track this down are promising. She finds a professor to help, and the playwright is somewhat helpful. One particular symbol keeps recurring --a muted postal horn, symbolizing the Tristero's antagonism towards Thurn and Taxis, and other established mail systems.

Oedipa is frustrated in her attempts reconcile, within her mind, questions of whether Tristero really exists or whether she's fantasizing. The playwright dies under somewhat unusual circumstances. People start drifting away --her husband, Metzger, her psychiatrist, the professor. She traces an underground mail courier on his rounds. One night it seems that the muted post horn shows up everywhere. But she finds it impossible to accumulate enough evidence pro or con.

The novel concludes with her anticipated sale of a stamp from the estate. The stamp bears the mark of the muted post horn, and could be a Tristero. She feels an incredible amount of tension, uncertainly, and apprehension. Will the mystery continue, or will she be able to ascertain the truth of the issue?

By this point, the reader has lost all faith in resolution. Conclusion seems irrelevant, for whenever Oedipa reaches a conclusion, contradictory evidence shakes her faith. One is left with a feeling of an infinitely complex and chaotic world.

Oedipa's trials are equivalent to Godel's unproveable statements. She is unable to provide a reasonable ground of belief in the existence of Tristero. Likewise, the opposite hypothesis, that Tristero does not exist, has a motley assortment of inconclusive evidence. This is directly analogous to the question of q's possible deducibility, discussed above. There is no deduction for the Tristero hypothesis. Likewise, there is no deduction of the falsity of the Tristero hypothesis. Incompleteness is the cause of Oedipa's apprehension.

Freedom has been a major theme of the modern era, and especially of the 20th century. Freedom allows the individual to escape the limitations of his own mode of understanding. Allow me to explain:

Question: how can Mr. Whipman get to the lovely apple? This is a fairly simple problem, within the realm of maze solvability. Question: how can Mr. Whipman get to the bat? This problem does not lend itself to the standard rules of maze solving. A re-examination of the first question might help. Probably the simplest way for Mr. Whipman to get to the apple is to duck down, beneath the maze, dash to the right and up, and into the last chamber. Why go through the maze? This is known either as "cheating," or "redefining the rules." We could similarly redefine the rules to let Mr. Whipman get the bat. We could say "Mr. Whipman can whip down one wall," or "Fat walls don't stop Mr. Whipman." The beauty of it all, is that we can stop and say, "This is really silly, why not think of the problem this way!" Or: "Boy is this a stupid problem."

Hopefully I've made a convincing lack of argument for my non-assertion by now. See what I'm saying about structure?

Structure plays such an invisible role in our thought processes, that it's often hard to see the importance of mathematics. Yet the formal problems Godel studied are directly analogous to many found in every realm of human life. Incompleteness is everywhere!

"This is not my beautiful house! This is not my beautiful wife!" - the Talking Heads

"God aloes not throw dice!" - Albert Einstein

"A good historian is not daunted by a lack of evidence. If need be, he should ignore contradictory information r create supporting evidence." (gross paraphrase)- Herodotus

Beyond here lie dragons.. - old maps

"To succeed at knowledge is oh, oh, oh!" - Kate Bush

"One morning Gregor Samsa woke up to discover that he'd been
transformed into a cockroach." Kafka, The Metamorphosis

Question Authority - Slogan

"What are you waiting for? It's over! You can go home now." - Matthew Broderick, "Ferris Bueller's Day Off."