Are you interested in the mysteries and paradoxes of life, how they manifest in our physical reality, and how they are described by mathematics?
Did you ever wonder about consciousness, what it is all about, and what we are doing here?
Did you ever struggle with your mind that has caused you to stand in your way?
Well… I am not saying I have an answer for you and I am also not trying to prove anything. I believe that you can only find the truth within your own experience and that there is no ultimate rational explanation.
I fell in love with the beauty of the paradoxical nature of reality and mathematics has plenty of amazing, abstract tools to demonstrate them. I would like to share with you some of these perspectives that have fascinated me because of their elegance and because they are pointing us toward some abstract wisdom that is beyond rationality. In a way it also helped me to get less and less caught by my “mind” and to step bravely beyond what I thought was possible.
In 1931 the mathematician Kurt Gödel discovered an elegant way to disprove what many mathematicians and scientists have hoped to be true. It seems that, to some extent, this hope is still implicitly, or subconsciously ingrained into our current science and culture, and hence also into our way of thinking.
Of course, we would like our theory of numbers with the usual rules of calculation, that we use in science and everyday life, to be complete, meaning that every statement formulated in this system can be shown inside the system to be either true or false. Moreover, we would like this theory to be consistent, meaning without contradictions.
It seems quite reasonable to our dualistic minds that mathematics should include a theory of numbers and be a consistent and complete formal system. A formal system is, roughly speaking, a system of axioms (i.e., statements that we assume to be true) equipped with a set of rules of calculation that allow us to deduce theorems (i.e., true statements) in this system. A formal system is said to be consistent if there are no statements, such that the statement and its negation are both true (i.e., no contradictions). A formal system is said to be complete if every statement formulated in this system can be either proved or disproved within the system (i.e., by following the rules and building on the axioms in that system).
Gödel basically showed that this is not possible, at least not without creating a never-ending chain of ever greater, nested (one containing the other) theories. Since this chain would be infinite, it is essentially impossible to achieve such a theory.
This was quite shocking for mathematicians at that time and it still seems that this was not completely digested or integrated into science nowadays, nor in the general way of thinking.
To some of you, this might seem, intuitively, quite obvious. Like so many “shocking” scientific discoveries, this seems to undermine what has been recognized by many spiritual traditions for centuries and is intuitively felt by many people on the spiritual journey. I am talking about things like non-duality, non-locality, paradoxes of self-reference, unity, and in general, the limitations of the mind.
In his first incompleteness theorem, Gödel proved that such a theory of numbers does not exist. Precisely, there exists no consistent (without contradictions) axiomatic system of mathematics, that is strong enough to contain a theory of numbers (i.e., containing a surprisingly basic, amount of arithmetics), in which every statement can be shown (within this system) to be either true or false.
In other words, every consistent axiomatic system, that contains this basic amount of arithmetics, is incomplete (i.e., there is a statement that cannot be proved or disproved in that system).
The amount of arithmetics needed is a surprisingly basic set of axioms, which is roughly speaking the usual mathematics of natural numbers (1,2,3,...) as we know it. Notably, all of these axioms seem to be quite implicit in our society. Most of us learn them in school without ever questioning them.
As axioms it includes the existence of a zero, the idea of a successor of each nonzero element (i.e., the successor of 1 is 2, the successor of 2 is 3, and so on), the existence of addition and multiplication, and the usual rules of calculation that we are used to for addition and multiplication of natural numbers, such as, for example, that
or that
for some nonzero natural numbers a, x, and y.
One might go deeper into the philosophical discussion about the existence of a zero, its implications, and its historical background. Certain cultures did not have the concept of a zero and the invention (or discovery?) of the zero could be considered a substantial scientific development.
In his second incompleteness theorem, Gödel proved that a formal system with this basic arithmetics, and this time including one additional axiom (called the axiom of induction), then the formal system cannot prove its own consistency (even if it is consistent).
The axiom of induction is also quite intuitive and it says that: if a statement S is true for 1 and, additionally, S being true for some x implies S being true for the successor of x…
…then it must be true for 1,2,3,...,n, where n is a very large number (or infinite in some axiomatic systems). Roughly, this is saying that if something is true in the beginning and remains true by taking a step further, then it must remain true always.
Notice that in both theorems, the dependency on or self-reference to the system is crucial, i.e., “provability within the system” and “proving its own consistency”. Because the theorem does not refer to provability in an absolute sense, i.e., this does not imply that there are truths that cannot be proved. Provability always depends on the system, i.e., depending on which axioms we assume to be true. Instead, the theorem states that there are statements in a consistent formal system that cannot be proved within that particular system. That particular statement however would then be provable in another, bigger system. But we would end up with an infinite long nested chain of systems, where each bigger system is needed to prove the consistency of the lower one and/or decide about the truth of at least one of the statements in the lower system.
In his two incompleteness theorems, Gödel showed that there exists no contradiction-free mathematics (strong enough to include some basic theory of numbers), such that it can prove or disprove any statement within that system. Moreover, any such mathematics cannot prove that it is free from contradiction.
The message is that to potentially see whether certain statements are true or false, we need to transcend the system. And to see if the system is potentially free of contradictions we certainly always need to transcend the system. It would be an endless nested chain of stepping out of the system again and again. So let’s keep that in mind, and next time consider the possibility of taking a step out of the system.
This is beautifully illustrated by the invention of the “imaginary number” called “i” representing the square root of -1, which is not a “real number”. However, its invention was necessary to solve the equation
and without it, most of our modern technology would not exist. Imaginary numbers are necessary to describe many phenomena in nature involving wave patterns, such as the vibration of guitar strings or the movement of water.
At the core of the proof of Gödel’s theorems is the beautiful insight into how self-reference leads to contradiction (at least in a dualistic worldview, where things have to be either true or false.)
Why does this create paradoxes? The idea leads us to the famous liar’s paradox, which is the statement that a person could make: “I am a liar.” Is this sentence true or false? Neither nor. Both possibilities lead to a contradiction. Try it! Another close example is Quine's Paradox:
"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.
Again, this sentence can neither be true nor false.
Yet, paradoxically, our lives are full of self-reference; Consciousness trying to explain consciousness… Consciousness experiencing consciousness. “I” identify with “what I think I am”...
Now, let us go a step further and ask the question: “What if my own mind, my idea of myself, my system of beliefs, and my rational reasoning is such a system that can be transcended and expanded?” After all, axioms of logic, language, and experiences have shaped and influenced the way we perceive reality to a great degree.
I recognize (from beyond rational reasoning, through direct perception, transcending the rational mind) that these concepts can be translated into the systems of my mind, our language, and most interestingly the idea of the Self. And I intentionally say “recognize” and not “understand”, because this has nothing to do with rational reasoning as is implied by saying “transcending the rational mind”.
By suspending my disbelief, I keep transcending and expanding.
I invite you to research this within your own perception.
I am not saying that there is anything wrong with science, mathematics, or wanting to understand certain things in a rational way. There is something like relative truth. Surely, we do need the rational mind. We should not just blindly believe what is told to us.
A great teacher once told me:
“The rational mind is a truly amazing tool, but it is a lousy master. “
Besides, there also exist many other amazing tools…. :) It is an art to use the mind but not get caught by its illusions and concepts.
I hope this perspective inspires you to consistently challenge your beliefs, suspend doubt, and just try and experience. The real research is in the direct perception. I hope that this motivates you to recognize and keep reminding yourself of the limitations of our minds, language, and science. Keep seeking and transcending and you will expand.