Preamble: In 1933 German logician Kurt Gödel publised a mathematical proof of his incompleteness theorem; the results of which made him the "most important logician since Aristotle". Incompleteness tells us that any such deductive system created in any given language will contain meaningful statements that are undecidable. i.e. that cannot be shown to be true or false. The result was earth-shattering since it showed the whole system of mathematical thought must be incomplete. Here we examine the far-reaching ramifications of incompleteness in science, philosophy, and mathematics.

In the last decade of the 1800s, German mathematicians had split into two distinct camps. By now, set theory and symbolic logic had matured so much that mathematicians were now at the point in which they could inquire about the very foundations of mathematics. The two camps were split on the issue of what mathematical thought really was anyway. One camp beleived that mathematics was no more than a very complicated game, whose rules are created arbitrarily, much like the rules of a chess game are given. Furthermore, if math equations modelled the real world they did so only coincidently. The other camp (what we will call the Hilbert Camp) beleived that mathematics was, in some ephemeral way, a kind of divine mode of thinking and communicating. They beleived that the system of mathematics was somehow connected to a sort of "concrete foundation of knowledge".

The Hilbert Camp, now feeling they were dealing with serious, eternal issues, launched a crusade to formalize the entire system of mathematics. After such formalizing, mathematics would then be a perfectly pure system of truth. And in this complete system there would be no unanswered questions, no dangling strings, and certainly no more paradoxes.

Logic is and was known inside mathematics, science, and philosophy as an argument tool in which statements can be shown to be true or false. The lure of logic is that is a method in which truth and falsity are known at each stage of the the argument without any ambiguity. A formal math proof is really just a string of statements in which the next statement is derived from the preceeding ones using strict logic rules. Also, any such argument following these restrictions can be called a deductive argument. For example, if you know that the next two statements are true:
1. All men are mortal.
2. Socrates is a man.
... then Socrates must be mortal.
Of course, more complicated and subtle rules exist in logic other than the one used here. But the final verdict is that truth and falsity are known without a doubt at all stages of the argument. A body of knowledge made of statements derived by logic is a deductive system.

Had Hilbert's Camp succeeded in its crusade, then logic would have "won out" and been the great victor for rational thought everywhere. But by the middle of the 1930s, proofs in logic ITSELF had shown without a doubt, that mathematics could never be complete. Indeed, any system created in the manner of the mathematical system must always be incomplete.

(The anatomy of a deductive system): If a system of true statements is derived from existing statements known to be true, where then (we might ask ourselves) are the "original" statements? The a logical chain of statements must at some point reach back to a starting point. And luckily they do, and such "original truths" are called axioms within mathematical lingo. An axiom is a statement assumed to be true without any proof. Usually such a statement is just an admittance of a common-sense rule. The mathematics used today is based off of a collection of axioms which deal with sets. A "set" is just an intuitive notion of a collection of things. From this small collection of intuitive axioms, all of mathematics is then derived. So deductive systems are often referred to as "axiomatic systems", for the above reasons.

Since deductive systems are axiomatic, there may be a many different deductive systems, each with different (or even contradicting) axioms. Gödel's proof did not deal with a specific system, but rather, it dealt with any system derived in the manner of a deductive system. The mathematics used then and today happens to be such a system, so the results of the incompletenss theorem immediately apply to it. Further still, the results will apply to any variation invented that tries to avoid its ramifications.

In the proof Gödel defines a generic deductive system and then shows that it has an incomplete statement in it somewhere. This method is not unlike giving the following absurd, but enlightening example: This sentence is false. The viable way of avoiding incomplete statements is to tack on extra axioms to your system that allow you to answer the unanswerable questions. But then you have only put yourself into a larger system which has its own incomplete statements. The game of asking "We know this, but what about this?" never ends for deductive systems.

Directly following the publishing of his proof, the reactions of Gödel's contemporaries were mixed and interesting. One person found the proof "difficult to follow." Another complained "What does Gödel expect us to think? That 1 + 1 does not equal 2?" Others held that Gödel was dabbling in non-sensical "meta-mathematics." Acceptance of this radical proof was slow, and did not receive recognition in the United States until the 1960's even though the published date was 1933. American mathematics was not yet ready for it.

Philosophical Ramifications of Incompleteness: Mathematicians of the enlightenment in Europe deposited a particularly Greek philosophy that mathematics is a "fixed and eternal" system whose theorems derive from "divine perfection." This sentiment was so deeply entrenched in the intelligencia that is resided several centuries all the way up to Gödel's lifetime. This practically religious sentiment is the very reason that incompleteness was so shocking to mathematics. The impact of the 2nd incompleteness theorem shows something about what we beleive. In the aftermath of the acceptance of the theorem, we know, without argument, that mathematics is a man-made linguistic construct. It is a system by which we communicate ideas. Incompleteness did not show that "everything in math is wrong." Rather it changed our idea of what mathematics is.

Consider a hypothetical body of knowledge which is omniscient. (The "mind of God", if you want to take it there). We can say with certainty now that such a body of knowledge is not a deductive system, since it would then not be omniscient. The nature of such a body of knowledge is now open to much more exotic presumptions.

Ramifications in Physics: Say a Unified Theory of physics is found. Now collect all the basic laws of physics into a small collection. Consider the smallest such set of laws as the "axioms" of the physical universe. Now your description of the universe is a deductive system, and therefore, must be incomplete, thus a Unified Theory will still not answer everything. Consider a map of something which contains no ambiguities. So alas, the map is always lesser than the terrain. In effect, a description of a physical thing must always lack something that the actual thing has. What does a thing have that its description does not? This question is best left to the philosophers.

This type of reasoning has lead to speculations on what a digital computer is capable of doing as far as simulating intelligence. A computer, being finite, may create simulations which can only amount to finite descriptions of real-world things at any given moment. A computer may simulate molecules interacting. But such a simulation must always fall short of real chemical reactions. The same rule applies if the computer is simulating a mind.