*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**.

Done?

Great!… If anyone of us would be asked what was the most radically transformational idea to come out of the 20^{th} 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 19^{th} 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

Great Job Joe! I really enjoyed the post.

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Thank you!

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Joe, Very interesting subject and material, let me be upfront and state my critical thinking skills are more honed than those of mathematical deduction. Forever the skeptic, reading material related to always, everything and infinity are subject to rejection as not likely possible. For instance Wallaces’ 2003 book would be more appropriate if he titled it ‘Everything known of infinity, a complete history’, since I doubt he had or could have had written of the unknown unknowns.

Rod Sterling had a great imagination and his quote you used at the end “…And it is then that writing becomes a weapon of truth” is more of an accurate necessity today than possibly in the last 200 years.

I enjoy reading AA and look forward to further chapters.

Uncle Paul

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Hello Uncle Paul,

I’m glad your enjoying the blog! I for one appreciate your observations. You really tapped into the thesis of Wallace’s book on Infinity and how tough it is to expound on it.. As Wallace explains, mathematicians and philosophers have struggled with the concept of infinity since the ancient Greeks. Aristotle believed that infinity could only be potential and not actual. Greek Philosopher Zeno’s paradox that movement across a room should be impossible because it would take an infinite number of smaller and smaller steps is another example of ancient Greek difficulties with infinity. However in mathematics infinity is is indispensable. For example, infinity (ie infinitesimals) is critical to defining Calculus, irrational numbers, and continuity. I am planning on posting about it in the future. Keep reading and keep these comments going!

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Joe,

I really enjoyed your blog. I have a long commute, and I often find myself contemplating the matters that you address. I find the popular notion that science refutes faith to be somewhat ironic. When people say that they “believe” in science, they are summarizing their world view more accurately than they realize. Most people have a very limited understanding of science. Their idea that the world is knowable and that this serves as a refutation of the existence of God is more an article of faith then a reasoned opinion that they formulated. In fact, their ideas often tend to be Newtonian. If I were to mock the scientific knowledge of such a person, I would not have refuted the existence of science. How can mocking a particular religious view refute the existence of God? And yet that seems to be how the “debate” is always framed. A century ago, quantum physics blew away our notions of time and space, raised questions of the role of consciousness in the fabric of the universe, and reunited science and philosophy. How is everyone not pondering these issues all of the time? In other words, thanks for the link and keep it up. I need to raid your reading list.

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Joe, I may have been the first to read this edition of your blog, but I did not want to be the first to leave a comment. That was a good choice for me, because the comments also are very worth reading. These topics are big “stuff” for me, and I am enjoying it immensely. Thanks, Joe; thank you all. Aunt Jean

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Good stuff!

The topics here bring to mind a phrase I coined about ten years ago: “idolatry of the human intellect”. It came to me as I was considering the position of those who struggle to accept not only the limitations of current human knowledge but also the idea that there are limitations to human understanding. As many self-proclaimed thinkers try to make science and religion mutually exclusive, they unwittingly attempt to segregate physics from metaphysics.

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