There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call the Twilight Zone.
Rod Serling’s introduction to his Twilight Zone series is so familiar to us that today we hardly give much thought to its poetic power or to the radical idea Serling invites us to contemplate: That there are real limitations to human knowledge. So go head, get comfortable, grab that glass of wine or cup of coffee, forget what you know about the Twilight Zone and read Mr. Serling again.
Great!… If anyone of us would be asked what was the most radically transformational idea to come out of the 20th Century most of us would say Einstein’s Theories of General Relativity and Special Relativity. No debate right? But what about the second most transformative idea? Hmm… not so easy. Well here’s one to consider: Austrian mathematician and philosopher Kurt Godel and his Incompleteness Theorems.
It’s the year 1900, the beginning of a century of promise and progress. In science and mathematics optimism and intellectual hubris was rampant. Conventional wisdom held that Science was almost complete and only some tidying up was needed.
On a hot August afternoon of that year in a conference in Paris, brilliant 38-year-old German mathematician David Hilbert proposed that Mathematics be put on a firm of foundation of a limited set of self-evident axioms (or facts) from which all mathematics could be derived from using simple rules of logic. Sounded simple enough and intuitively, it made sense. The project became known as the “Hilbert Program”.
For the next 30 years though, mathematicians and philosophers struggled to define any such list of axioms. The difficulty seemed to be with coming up with enough axioms to cover all of mathematics without creating paradoxes and contradictions. It seemed you could have one or the other but you could not be both complete and paradox free at the same time. In 1931, Kurt Godel demonstrated why.
In 1931, Kurt Godel shattered the Hilbert Program with his Incompleteness Theorems.
The first Incompleteness Theorem states that an all-encompassing axiomatic system can never be found that is able to prove all mathematical truths, but no falsehoods. The second Incompleteness Theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency. If an axiomatic system can be proven to be consistent from within itself, then it is incomplete. In other words, there are mathematical truths that are not provable within the realm of mathematics itself.
At that point, everyone threw down their pencils and went to their corners. David Hilbert himself was stunned into silence. Philosophically, Godel’s theorems imply that truth cannot be defined solely in terms of mathematical or scientific provability. So it is one thing to be provable, and a different thing to be true. Truth out runs provability.
The implications are still being debated today. As mathematicians continue to specialize, any possible impact caused by the Incompleteness Theorem is assumed away as not impacting their limited areas of research. Some physicists, in their pursuit of the Theory of Everything that will completely explain the Universe, dismiss the Incompleteness Theorem by saying that it only affects arithmetic, not physics.
But is that a safe assumption? Today, do we live in an age of intellectual hubris like we did 114 years ago just before Einstein shattered everyone’s scientific complacency? Does the Kurt Godel’s Incompleteness Theorems imply limitations to human knowledge? In other words, can we fully understand the universe while at the same time be within it? Or will science soon develop a complete and consistent Theory of Everything without having to going outside of science itself? Or is the “Theory of Everything” the Hilbert Program of our day, doomed for a shocking revelation?
Now, get up and go get yourself a stronger drink, relax and read Serling again and ask yourself: Just where and what are the limits of human knowledge?
Post Script: I love mathematics, and as a small child I was fascinated with its deeper truth and beauty but hated the mundane as taught by my elementary teachers. While I struggled with simple multiplication tables I would surprise my teachers with my questions and insights. As my teachers explained to my parents about my mathematical aptitude: “Joe can climb mountains but he can’t cross a street.
Mathematics will continue to be part of this blog. For example, did you know that late 19th Century Mathematician Georg Cantor proved that some infinite sets are bigger than other infinite sets? Or that as a consequence some sets of numbers are not only uncountable, but also unnameable?
Some good books on Mathematics that touch on topics discussed in this post:
– Godel’s Way, Exploits into an undecidable world ( 2012) by Chaitin, da Costa and Doria
– Everything and More, A Complete History of Infinity (2003) by Wallace: A very enjoyable history of the mathematical concept of Infinity from the ancient Greeks to Georg Cantor
“It has forever been thus: So long as men write what they think, then all of the other freedoms – all of them – may remain intact. And it is then that writing becomes a weapon of truth, an article of faith, an act of courage.”
― Rod Serling