The Limits of Logic

He’s probably the most important mathematical theorist you never heard of. I am speaking of Kurt Godel, the Austrian mathematician and philosopher best known for proposing the “incompleteness theorems.” These theorems demonstrated mathematically that all logical systems include at least one unprovable axiom requiring assumptions based on concepts or facts outside the system. In other words, there are limits on what anyone can prove with absolute certainty through observation or logic, requiring everyone to make assumptions about what is true. His views had tremendous influence in his time, and they should have tremendous impact on our own views of science and rationality.

Godel was a brilliant but troubled mind, similar to John Nash, the subject of the movie, A Beautiful Mind. Godel was twice hospitalized for paranoia or related mental health issues, yet he won the Albert Einstein Medal and the National Medal of Science. Born in Austria in 1906, he immigrated to the U.S. after the Nazis killed his fellow Jewish faculty members at the University of Vienna. He had previously befriended Albert Einstein, and joined him at Princeton, New Jersey; then later he moved to Cambridge, Massachusetts. He became a U.S. citizen in 1947 with Einstein’s help. After the death of Einstein in 1955 and the hospitalization of Godel’s wife in 1977, he went insane. Within a year, he would starve himself because he believed someone was poisoning him.

Godel developed the first of his two major incompleteness theorems in 1931 at the age of twenty-five, while still living in Austria. It was at the height of the neo-positivist movement, a philosophy that believed that every rational assertion could be proved through science, mathematics, or logic. Its best known adherents in the English-speaking world were Bertrand Russell and Alfred North Whitehead. Godel’s theory directly addressed their views. In essence, he demonstrated mathematically that it is impossible to prove the truth of any system using logical or mathematical axioms existing solely within that system. To use a simplistic example, a car as a system cannot be explained without appealing to its assembly at a factory. The factory cannot be explained without appealing to materials. The materials cannot be explained without appealing to numerous other systems, such as mining and metallurgy. His views applied similarly to deductive and inductive logic. In the syllogism, “All men are mortal; Socrates is a man; therefore, Socrates is mortal,” one must make an assumption that all men are mortal. It is a reasonable assumption, but it is not provable using observation since it is impossible to observe or prove the death of all men. The inductive statement, “all men I know have died, therefore all men are mortal,” makes the same assumption from the opposite logical vantage point.

Some have even taken Godel’s theorems as proof for the existence of God. If the entire physical universe were considered one system, one has to assume something existing outside the system to make rational statements about the system. That something could not be material or its byproducts such as energy, which are part of the system, but it must be rational. Of course, Godel’s proofs applied only to limited mathematical systems, but the principles he developed were universal. Certainly, Godel himself believed in God and the existence of a spiritual realm. For him, rationality suggested the existence of God since one must assume rationality exists outside the system to accept its application within. His views, therefore, also support C.S. Lewis’ “argument from reason,” which stated that logic would be nonsensical in a materialist world guided only by physical causation and responses to stimuli. Belief in rationality requires believing in something rational outside of the physical universe that makes life logical. Nothing in the chemical make-up of our brains causes rationality or assumption of its truth.

Whether or not one applies Godel’s incompleteness theorems to the issue of theism, the one thing they do prove is that it is impossible for science, mathematics, and logic to prove all things. It appears a paradox, but it is nevertheless true. There will always be something outside the system that one has to assume to be true. Materialists and even some Christian apologists believe that you can prove or disprove the existence of God. In opposition, Godel’s theorems demonstrated that you can’t prove everything because there are limits to logic and to proof. You have to make assumptions in life, and that means that you must take some positions based on faith alone, including belief (or disbelief) in God.

For some, the idea that they cannot prove everything in life is daunting. Many need the comfort that they live in a logical world in which everything has a rational explanation. For those with faith, however, it is a comfort to know that there is a rational explanation outside of the physical universe. There are many things that we may never understand, but that does not mean that we are without hope. Life has meaning, even if we cannot prove it.

© 2018 J.D. Manders

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