Language is notoriously insufficient when it comes to grasping truth. Words, riddled with connotations and subjectivity and vestiges and misunderstandings, never seem just right. Using them, you can only ever approach the telling of the truth, but can never quite tell it.
Most of us have the impression that the breach lies with all those sources of ambiguity - that if you stripped away confusion and abstraction from language and left behind only perfect precision and clarity, which is to say, the language of logic--math--then you could achieve truth. And in theory (you might think) if you were an omniscient being, you could solve any mystery of the universe with mathematics, whether it be the workings of consciousness, the reason for entropy's rise, or the distribution of the prime numbers. Of course, your equations or proofs would be unimaginably complex. It doesn't matter. Truth seems achievable, in theory, by pure math.
But it isn't. Two of the strangest, most striking, most devastating, and thus most ignored math theorems ever proven, Kurt Godel's First & Second Incompleteness Theorems (1931), imply that math, like all other languages, can only approach truth, but can never grasp it.
Because the second incompleteness theorem follows quite easily from the first without adding much to it, I'll dispense with it and simply try to explain the first theorem. It states that formal systems--read: systems of mathematics generated from first principles (like the concepts of zero, and one, and two, etc.) and the rules that they logically follow (like addition and multiplication)--can never be both consistent and complete.
First, what does it mean for a system to be consistent? It means that no statement and its negation are both provable using the rules of the system. For example, if A is proven true within the system, then for consistency, "not A" must be proven false.
Consistency is the pleasure of mathematics. In the real world--the world of grey areas--A and "not A" usually both have some degree of truth to them, because real-world A's and "not A's" are far more complicated than the A's and "not A's" of math, and thus they are burdened with the ambiguity and confusion I complained of before. Still, in theory, even real world statements could be set up as extremely complex mathematical statements and, in that form, proven either true or false.
What does it mean for the formal system to be complete? It means that everything that is true can be proven true using the principles and rules of the system.
Putting those clarifications together, we can grasp Godel's incompleteness theorem. It says that if a system is consistent (never contradictory), then it lacks the tools to prove all the true statements that exist. Some mathematicians and logicians worry that some of the truths we have already conceived of (for example, the infinite distribution of twin primes) are not provable within math.
On the other hand, if a formal system is complete--if there is nothing true that isn't provable by the rules of the system--then the system is inconsistent. In other words, there must be some statement A in the system for which both A and "not A" are provable.
This is sort of mindboggling and a little hard to swallow, but in fact the theorem is true beyond the shadow of a doubt. Since plenty of links are available, I won't regurgitate the theorem's proof in a non-rigorous form, but the gist of it is this: within any formal system, it is possible to generate a formula that relates to its own provability. In words, the proven formula states: "This formula can't be proven." Contemplating this, you'll realize that since the formula is proven by the system, it must be true. But if it is true, then it can't be proven by the system. This contradiction renders completeness and consistency incompatible. (See for yourself.)
Obviously the heart of the proof relates to famously paradoxical sentences in everyday language, like the liar's paradox: "This sentence is a lie." Apparently the problem with that sentence runs much deeper than the mere ambiguity of words. The problem is fundamental, and Godel made it mathematically rigorous.
And its implications are profound. Because Godel's theorem bars humans from omniscience, many theologians use it to establish a sort of mathematical realm for God. He, not we, can know everything (they say). But in truth (no pun intended), Godel's theorem blocks even God's access to omniscience.
Stephen Hawking and Freeman Dyson interpret the incompleteness theorem to mean that we will never attain a theory of everything - the holy grail of physics. Most mathematicians disagree. They argue that the math used by physics is a proven subset of all math.
The theorem's undeniable implications are enormous enough. In a recent paper, Geoffrey Laforte and his colleagues lament them nicely. They write, "There is no bedrock of mathematical certainty on which the edifice of science must be based, no direct route to mathematical Truth ... We can never be absolutely sure that we have things right, even in mathematics, and still less can we be certain that all truths will eventually be vouchsafed to us."
The incompleteness theorem lurks like an unpatchable hole in the foundation of mathematics, and of human knowledge itself. With no alternative, most mathematicians, logicians, philosophers and scientists choose to step around it, and walk on.