The choice of Kurt Gödel (1906-1978) in this section, discussing philosophers, may seem strange for the simple fact that Kurt Gödel was mathematician and a logician. Although, Gödel could be described as a philosopher of mathematics, he is best remembered for his `incompleteness theorem`.
Of course, many famous philosophers were also accomplished mathematicians, e.g. Descartes, which possibly reflects the underlying logic required to support the development of any rational idea. However, before discussing Gödel himself, it may be useful to provide some of the background, as to why Gödel mathematical theorem should also be considered as an important philosophical milestone in the development of human thinking. By the late 19th century, mathematics was developing evermore-powerful methods, which may have reached its philosophical zenith in a school of thought called `formalism`. According to the formalist:
"Mathematics is a game, devoid of meaning, in which one plays with symbols, devoid of meaning, according to formal rules, which are agreed upon in advance. It is therefore an autonomous activity of thought."
In 1920, an equally famous mathematician, David Hilbert, proposed a project in which mathematics would be used to formulate a solid and complete logical foundation based on the following assumptions:
- Mathematics can be reduced to finite system of axioms
- The axioms of the systems can be proved.
In a sense, Hilbert was reflecting a type of `scientific determinism` that had prevailed since the time of Newton, which suggested that the world and possibly even the way we think about the world could be reduced to a concept of deterministic cause and effect. However, Hilbert's attempt to banish theoretical uncertainties would end in failure, when Gödel demonstrated that any non-contradictory formal system couldn't demonstrate its completeness by way of its own axioms. There is also a possibly deeper subliminal message in Gödel's work that suggests that human thinking cannot be reduced to a finite set of algorithms. This aspect may yet have far-reaching implications in the field of AI research.
Kurt Gödel was born in what is now the Czech Republic in 1906. In fact, Gödel's nationality would remain a complicated issue throughout his life. At the time of his birth, his hometown had a German-speaking majority and was the first language of his parents. Although Gödel's mother was a Protestant, his father was Catholic, in-line with state religion of the Austrian-Hungarian monarchy. It is therefore surprising that Gödel was educated in a Protestant environment. At the age of 12, Gödel became a Czech citizen, by default, when the Austro-Hungarian Empire broke up at the end of the 1st World War. Later, at the age of 23, Gödel chose to become an Austrian citizen, but when, at the age of 32, Adolf Hitler annexed Austria, Gödel automatically became a German citizen. After the 2nd World War, at the age of 42, he became a naturalized American citizen.
During his life, Gödel was to have several breakdowns brought on by depression. Possibly, he sense that many people considered him as eccentric and he became increasingly reclusive in his later years. He was also prone to paranoia and was distrustful of doctors, which may have led to him failing to look after his own health. In 1978, when his wife was incapacitated with illness, the combination of all Gödel problems appears to have led to a sad death from self-starvation. However, Kurt Gödel did leave an enduring legacy as being one of the most original mathematical philosophers of the twentieth century.
Although Gödel made many contributions to mathematics throughout his life, they are not the focus of our interest, at this stage. Therefore, we will only consider one aspect of Gödel's work; his Incompleteness theorems.
As outlined, at the start of the 20th century, one of the big mathematical goals was to reduce all number theory to a formal axiomatic system. It was hoped that such a system would start off with a few simple axioms that were almost indisputable, and would then go on to provide a mechanical way of deriving more complex theorems from the more fundamental axioms. The idea being that this system would eventually represent every statement you could possibly make about natural numbers. So if you made the statement `every even number greater than two is the sum of two primes`, you would be able to prove it mechanically, from the axioms, as being either true or false. Although modern computers had not yet been invented, the concept of automated computation was being considered, and as such, Hilbert's goal might have been seen as a decisive step towards automated intelligence. However, Gödel's theorem was to put pay to this idea by showing that the goal, in its entirety, cannot be achieved. Specifically, Gödel theorem shows that for any formal axiomatic system, there is always a statement about natural numbers which is true, but which cannot be proven within the system. In other words, the rigor of mathematics may always have a little fuzziness around its edges, possibly just like human thinking. While there will be no attempt to explain the details of Gödel Theorem, it is possibly worth trying to outline a few aspects in terms of a simple symbolic system utilizing the concept of:
- A statement,
- A proof as a series of statements
For example, let us start with a logic statement that is part of the system, which then leads to the fundamental problem:
There is no proof of [P]
If P is true, there is no proof of it, but if P is false, there is a proof that P is true, which is a contradiction of the statement. Therefore, the system cannot determine whether P is true. The logic involved is analogous to the `Liar's Paradox`, in which a liar states: `I am lying`. The paradox emerges because language is also a symbolic system. In part, the reason for describing Gödel theorem in terms of a paradox is because it best reflects an aspect of our wider discussion concerning the `path to understanding` requiring data to be converted into wisdom through the application of `structure, experience and evaluation`. In the Liar's Paradox, we have two pieces of data:
- I am a liar
- I am lying
The logical premise tells us the man is a liar and therefore does not tell the truth, but a liar who states that he is lying implies he is telling the truth, which leads to what appears to be an insurmountable contradiction. However, when human intelligence is faced with this type of problem everyday. In this case, the structure of the data is the simple adjacency of the two statements, while our experience tells us that a liar doesn't always tell lies. Therefore, the logical contradiction is resolved into another problem, i.e. evaluating whether he is lying in this specific case. How we might do this transcends intellectual logic and involves intuition based on emotional and social intelligence.
If we accept the outline of the argument put forward, we are led to the philosophical conclusion that mathematical truth cannot be ascertained by a mechanical formalism alone. However, in comparison, our intuitive ability to resolve a logical contradiction seems vague and incomplete. However, this insight into the human condition may ultimately lead to a more profound insight about the nature of human intelligence and how we have come to interpret the world around us.