Mathematician Kurt Godel was born on April 28, 1906. His proofs, which altered mathematics and logic, are of the highest relevance to philosophy and to Christian apologetics.
There are many systems of math and logic. In a formal system we start with a few carefully defined symbols and rules. Following these strict rules, symbols are combined into new patterns (proofs). The symbols themselves are either place-holders or instructions. Some represent operations such as addition. Others represent slots that can be filled with numbers or sentences. The reason that empty symbols are used is so that we can reason without the mistakes of human emotion. After a proof is made in a formal system, appropriate statements or numbers can be substituted for the symbols and we can be sure of the results. Serious math often uses formal systems.
A simple formal system cannot support number theory but is easily proven to be self-consistent. If it generates a proof that A = Non A, (for instance, that 2 = 17) it is inconsistent. But to handle the whole theory of numbers, a complex formal system is needed. Unfortunately, as systems get more complex, they are harder to prove consistent. So we can't be sure our number theories are free of hidden contradictions. Godel worked with such problems.
He especially studied undecidable statements. An undecidable statement is one which can neither be proven true nor false in a formal system. Godel proved that any formal system deep enough to support number theory has at least one undecidable statement. Even if we know that the statement is true, the system cannot prove it. This means the system is incomplete. For this reason, Godel's first proof is called "the incompleteness theorem."
Godel's second theorem says no one can prove, from inside any complex formal system, that it is self-consistent. In his book Godel, Escher and Bach Hofstadter wrote, "Godel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved." In other words, we cannot prove some things in mathematics which we nonetheless know are true.
Take note! Godel did not prove nonsense (that A = Non-A). Instead he showed that no higher system can decide between a certain A and Non-A, even where A is known to be true. Any finite system with sufficient power to support full number theory cannot be self-contained.
Godel's proof fits well with Christian beliefs about the universe, by analogy. Judeo-Christianity has long held that truth is above reason. Spiritual truth can be grasped only by the spirit. Had Godel been able to show that self-proof was possible, we would be in deep trouble. The universe might them be self-explanatory. The implication of his proof is that the infinities and paradoxes of nature demand something higher, different in kind, more powerful, to explain them, just as every logic set needs a higher logic to prove and explain everything within it.
In other words, no finite system, even one as vast as the universe, can satisfy the questions it raises. Godel recognized this and tried to find a watertight proof of God's existence. He failed. Sadly, the evidence of his life suggests he failed to find a personal relationship with Christ either.
Resources:
- Bell, E. T. Men of Mathematics. (New York: Simon and Schuster, 1937)
- Blanché, Robert. "Axiomization," in Dictionary of the History of Ideas. (New York: Scribner's Sons, 1973).
- Çambel, Ali Bulent. Applied Chaos Theory: a paradigm for complexity. (Boston: Academic Press, 1993). especially pp. 35-37.
- Guillen, Michael. Bridges to Infinity. (Los Angeles: Tarcher, 1983).
- Heijenoort, J. Van. "Gödel's Theorem" in Encylopedia of Philosophy, edited by Paul Edwards. (New York: macmillan and Free Press, 1967).
- Hofstadter, Douglas. Gödel, Escher and Bach; an eternal golden braid. (New York: Vintage, 1979).
- Moore, A. W.The Infinite. (Routledge, 1990.) [See Chapter 12, "Gödel's Theorem" for one of the clearest explanations I was able to find].
- Moore, Gregory H. "Gödel, Kurt Friedrich," in Dictionary of Scientific Biography; edited by Charles Coulston Gillispie. (New York: Scribners, 1970-80).
- Nagel, Ernest and Newman, John. Godel's Proof. (New York: New York University Press, 1958).
- Paulos, John Allen. Beyond Numeracy; ruminations of a numbers man. (New York: Knopf, 1991).
- Penrose, Roger. Shadows of the Mind. (New York: Oxford University Press, 1994.) especially pp. 51-59 and the chapter titled "The Gödelian Case."
- Zebrowski, George. "Life in Gödel's Universe: Maps all the way." (Omni, volume 14, April 1992) p. 53.
