The greater one’s science, the deeper the sense of mystery⁽¹⁾,
Vladimir Nabokov

In 1931 Kurt Gödel published his famous paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I” (Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I) where he proved two widely quoted incompleteness theorems. The first one (Theorem VI in the original paper) states that:

For any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms (Wikipedia, Kurt Godel)

In other words, if the system is consistent, it cannot be complete; and the consistency of the axioms cannot be proven within the system. Or there is no reasonable list of axioms from which we can prove exactly all true statements of number theory.

These theorems derailed Hilbert’s programme to find a finite set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

These extremely “negative” results have been extensively used to argue with more or less rigour about the limitations of human and, a fortiori, artificial intelligence; or even to “prove” that any attempt to construct a Theory Of Everything  is bound to fail. However, I think that what these theorems prove is simply how vast the universe of mathematics is, as this quote from Freeman Dyson beautifully explains:

Fifty years ago, Kurt Godel, who afterwards became one of Einstein’s closest friends, proved that the world of pure mathematics is inexhaustible. No finite set of axioms and rules of inference can ever encompass the whole of mathematics. Given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. This discovery of Godel came at first as an unwelcome shock to many mathematicians. It destroyed once and for all the hope that they could solve the problem of deciding by a systematic procedure the truth or falsehood of any mathematical statement. After the initial shock was over, the mathematicians realized that Godel’s theorem, in denying them the possibility of a universal algorithm to settle all questions, gave them instead a guarantee that mathematics can never die. No matter how far mathematics progresses and no matter how many problems are solved, there will always be, thanks to Godel, fresh questions to ask and fresh ideas to discover. (Freeman Dyson, “Infinite in All Directions”)

With the hindsight of information theory, Hilbert’s idea seems a bit naive. In case it had been proved right, it would have implied that a finite amount of information (the number of bits required to codify the finite number of axioms) would have sufficed to describe all the universe of mathematics, i.e. that the whole mathematical universe could have been “compressed” to a finite amount of information:

Hilbert, taking this tradition to its extreme, believed that a single FAS (Formal Axiomatic System) of finite complexity, a finite number of bits of information, must suffice to generate ALL of mathematical truth. He believed in a final Theory of everything, at least for the world of pure math. The rich, infinite, imaginative, open-ended of all of math, all of that compressed in a finite number of bits! (Gregory Chaitin, “Meta-Math: The Quest for Omega“)

Furthermore, I think that Gödel’s theorem suggests a subtle and wonderful relationship between science and art.  Gödel’s theorem shows that a complete and consistent finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program:

Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it would do so at the cost of making the system inconsistent (Wikipedia, “Gödel’s Incompleteness Theorems”)

There is nothing wrong with incompleteness. Euclidean geometry without the parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the other four axioms. By choosing different fifth postulates we have different geometries, being Euclidean geometry (the geometry of the plane) but one simple possibility among them.

Once we understand that there is no such thing as a complete logic system, that every system contains propositions whose truth cannot be deduced within the system, then we are free to add new propositions among the ones that are neither provable nor refutable within the system (logically independent) and assign then values at will,  amplifying our initial axiomatic system, in exactly the same way that Riemann did with the basic Euclidean geometry. Depending on our choices the new system will mean (or be) a completely different thing, and that’s where aesthetics and intuition enter into the equation.

Maybe art is the ultimate option, while science is only the rules of the art-game. This seems to be what Von Neumann apparently saw in Godel’s feat:

Thus today I am of the opinion that:

1. Gödel has shown the un-realizability of Hilbert’s program.
2. There is no more reason to reject intuitionism (if one disregards the aesthetic issue, which in practice will also for me be the decisive factor)

(John Von Neumann, Letter to R. Carnap, 1931)

If you had had to create a machine with the objective of exploring the broadest possible set of “logical” possibilities, what other means would you have used better than an open Godel-ish mathematical universe? Clearly God did not need play dice in order to create the infinite. If he or she did or does, it is for another reason.

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