We can safely assert that mathematics is our most logically coherent science. Nevertheless, even this science has its mysteries. In the early twentieth century, philosophers and mathematicians Bertrand Russell and Alfred North Whitehead came up with a plan to create a completely consistent system of mathematics. They began with propositions assumed to be true, and then other propositions had to logically follow from them. It goes roughly like this (We’ll use Peano’s axioms to illustrate their method):

• Zero is a number.
• Every number has a successor, and that successor is also a number.
• Zero is not the successor of any number.
• Different numbers have different successors.

Everything was going according to plan until Kurt Gödel, a mathematical genius, in 1931, threw a (figurative) bomb into it with his incompleteness theorems. He claimed that within this system, we can formulate statements that are undecidable. To prove this, Gödel constructed a sentence S in the formal language of an axiomatic system A that asserts about itself: S is not provable in A. If this sentence is true, then it is not provable, because that’s what it claims. But then a true sentence is unprovable, making system A incomplete. However, if the sentence is not true, then it is provable, and an untruth is provable in A.

A few variants:

• This sentence is not true.
• Pinocchio says, “My nose is growing now”
• The Cretan Epimenides says, “All statements of Cretans are always false”.
• If this sentence is true, then Santa Claus exists.
• This sentence does not exist.
• The following statement is true. The previous one is not true.
• According to Onno, everything can always go wrong, even a failure.

These statements are self-referential. They refer to themselves, describing themselves with a necessary property, without which they would not be the same. They are indivisible. These examples are amusing, but Gödel’s incompleteness theorems caused a shock in the world of mathematics.

We can make this complicated or try to keep it simple. Let’s keep it simple. A pair of scissors cuts everything except itself. That’s not difficult. Mathematical theorems can be understood by representing them in a decision tree. Each new statement with the judgment ‘true/false’ cuts, like a pair of scissors, the previous statement until everything is divided, and only the judgment, the scissors, remains. This is indivisible. Gödel’s theorem ‘S is not provable in A’ is ‘true’ and ‘false’. The last step referring to itself is undecidable. Now compare this to the smallest (indivisible) unit of quantum information, the shared information, the entanglement. It is in superposition and therefore indefinite, both 1 and 0, on and off, open and closed, up and down, ‘true’ and ‘false’.

Mathematics provides us with models of reality. Calculations in formal mathematics with exactly the same elements yield exactly the same outcomes. Nothing is lost. Nothing is added. There is no emergence in mathematics. Reduction has severed all coherence. Contrary to reality, mathematics deals exclusively with isolated elements. In a formal mathematical model, all superposition has been eliminated. As a result, it is entirely predictable and calculable. But where is the information that was lost during the transition from reality to mathematical model? It cannot have disappeared. That would be contrary to the presumption of conservation of information. As with Maxwell’s demon, this information is transferred to (and then shared with) the observer who has gained knowledge about the system. In this way reality is converted into a model. Except for the last step. The model ends with shared information in superposition. This one is undecidable: superposition is fundamental. It cannot be reduced any further.

Everything ends and begins with superposition.

Mathematics is beautiful as it is. However, it is indeed incomplete for describing emergent reality.