Math
Kurt Gödel’s Incompleteness Theorems and Philosophy
In 1930, Kurt Gödel shocked the mathematical world when he delivered his two Incompleteness Theorems. These theorems , which we will explain shortly, uncovered a fundamental truth about the nature of mathematics, a truth that the majority of the mathematicians back in the day were not ready to accept.
Before we get into what these two theorems actually say, we will give a few simple and intuitive definitions which are necessary to later understand the beauty of the two theorems.
A mathematical system is an ensemble of some basic elements (like numbers), some relations/operations concerning these elements (addition, subtraction etc) and, finally, some axioms i.e. some statements about the elements and the operations that we assume to be true without proof (for example, the fact that equality in arithmetic is symmetric: If x=y, then y=x).
We say that a mathematical system is complete if we can prove every true statement in it.
Finally, a mathematical system is consistent if we cannot prove the opposite of an already proven statement. For example, if we have proved that the sum of two odd numbers is always even, we shouldn’t be able to then prove that it is not even. It is crucial for a mathematical system to be consistent. If it’s not then is will constantly contradict itself (since every true statement will be simultaneously false -the curious reader can look up “The Principle of Explosion”-) and it will be of no use for us.
This is all someone needs to grasp the idea of Gödel’s work. We are now ready to dive into the two Incompleteness Theorems:
First Incompleteness Theorem
Every mathematical system, powerful enough to describe computation is either incomplete or inconsistent.
Second Incompleteness Theorem
A consistent mathematical system cannot prove its own consistency.
Okay, now let’s brake these theorems down starting with the first one.
First of all, the “powerful enough to describe computation” — or “sufficiently expressive” as it is sometimes called — should not scare us. Loosely speaking, the majority of mathematical systems that we non-mathematicians are concerned about have this property. We will not dive into this much more since it not important for our understanding.
With the above remark in mind, the first of our two theorems tells us that every system that we have in our disposal is either incomplete or inconsistent. That means, that there are either some true statements in our system that we will never be able to prove or that our system contradicts itself (inconsistent). As we have previously stated, we do not want our systems to be inconsistent so we are left to hope that they are incomplete. The use of the word “hope” in the previous sentence is not random. We would like to have a way to prove that our system is actually incomplete and not inconsistent but Gödel’s second theorem now comes into play and deprives us of that possibility.
However, since we haven’t yet come across any inconsistencies we pretty much take it for granted that our systems are consistent but incomplete. Let’s ponder over what that entails for a minute.
What we just admitted is that there are some deep fundamental truths about mathematics and our universe, in general, that we will never be able to uncover. Reality is not “logically incomplete”. We are just not equipped with the right tools, yet, to fully understand it. It is not a matter of intelligence or mathematical skill. We are not waiting for the next Euler or Gauss to come in our aid. It is simply not possible. One fairly common recommendation of a person that hears about this revelation for the first time is to add the “unprovable” statements as axioms in our systems. However, this is of no use since by the addition of more axioms we give birth to a brand new mathematical system which is still subject to the two Incompleteness Theorems. There is no way out.
The philosophical implications of the Incompleteness Theorems are tremendous. To our knowledge, there is not another theorem in the mathematical world that can match the turmoil that Gödel’s work created when first published. Mathematicians who had devoted their entire lives to the pursuit for a proof of certain mathematical statements were now faced with the possibility that all of their lives’ work was in vain. A question kept popping up in every mathematician’s mind when he was deciding whether or not to tackle a new problem: “What if it cannot be proven?”.
Kurt Gödel is considered by many people one of the greatest logicians that ever lived. As we have already stated multiple times, his contributions to mathematics are invaluable. However, Gödel had a much more complicated thought process and philosophy than that one that is hinted by his mathematical accomplishments. He was a radical Platonist and Rationalist with some controversial metaphysical beliefs.
Mathematical Platonism and Rationalism
Mathematics is the epitome of human thought. It represents the ability of humans to produce logical thoughts and connections about abstract objects and entities. Moreover, mathematics constitutes a bridge, that humans themselves built, between the mysterious and complicated phenomena of our physical world and human knowledge. It is unexpected, when one thinks about it, that the very tool which enables us to understand the physical world is of abstract nature.
Considering the importance of mathematics to the evolution of human history and civilization it should be no surprise that it has always played a primordial role in the development of philosophy. One of the most interesting philosophical views regarding the very essence of mathematics is called Mathematical Platonism.
Platonism about mathematics (or Mathematical Platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.
Mathematical Platonism often goes hand to hand with another philosophical doctrine, Rationalism.
Rationalism is the philosophical view that intellect and reason are the true sources of absolute knowledge.
According to Rationalism, we can acquire information only through logic and reason. This view opposes Empiricism which states that knowledge comes from our senses and our sense only.
Although typically incorrect, we can think of Mathematical Platonism as a form of radical, more extreme, rationalism. As it is evidenced by Hao Wang’s book “A Logical Journey: from Gödel to Philosophy”, around the year 1960, Gödel writes down a list consisting of 14 philosophical points that he believed in. We will not examine each and everyone of these points in this article. Instead, we will focus on the ones that are the most indicative of his philosophy. The full list is presented below:
Gödel’s Philosophical List
- The world is rational.
- Human reason can, in principle, be developed more highly (through certain techniques).
- There are systematic methods for the solution of all problems (also art, etc.).
- There are other worlds and rational beings of a different and higher kind.
- The world in which we live is not the only one in which we shall live or have lived.
- There is incomparably more knowable a priori than is currently known.
- The development of human thought since the Renaissance is thoroughly intelligible.
- Reason in mankind will be developed in every direction.
- Formal rights comprise a real science.
- Materialism is false.
- The higher beings are connected to the others by analogy, not by composition.
- Concepts have an objective existence.
- There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.
- Religions are, for the most part, bad– but religion is not.
Before we analyze some of the above points, it is important to note that the aforementioned list was not meant for publication. Thus, it is fair to say that the above 14 points do not fully capture Gödel’s philosophy. Moreover, all of the concepts are open to interpretation. Here, we will give some educated speculations that stem from Gödel’s life and character.
3. “There are systematic methods for the solution of all problems (also art, etc.).”
The above sentence seems contradictory with the two Incompleteness Theorems that we previously saw. How can the creator of these theorems, the man who showed that mathematics has inherent limits, Kurt Gödel, believe that all problems can be solved? As with the whole list, this belief is open to interpretation but it seems that it could mean one of two things.
Firstly, either Gödel reckoned that mathematics can be, in some unprecedented way, further developed, that they are today or he was trying to hint on the possibility that there is a better way, better than mathematics, to approach and solve problems.
4. “There are other worlds and rational beings of a different kind”
The above statement in conjunction with the first one on the list “The world is rational” clearly reveals the philosophy that Gödel had adopted regarding mathematics and the world in general. When Gödel says “worlds” and “rational beings” he is, probably, not referring to actual physical entities. Instead, as a Platonist, he is hinting on the fact that there are certain abstract beings that exist solely in the sphere of logic and reason and not in a physical world like ours. These “beings” could be numbers, functions, vectors and all other mathematical objects.
5. “The world in which we live is not the only one in which we shall live or have lived.”
Admittedly, things get a little more complicated now. Among Gödel’s controversial ideas is his belief in reincarnation (or “transmigration”) of the soul. There is a debate going on in the philosophical world concerning whether or not Plato was an ambassador of reincarnation. A usual argument is that Plato, in his theory of the Forms, believed that people have latent knowledge of the Forms. This implies that people had encountered the Forms in a previous life and, thus, were able to remember them.
It should be noted here that this is not the usual kind of reincarnation that is present in the religions of Hinduism or Buddhism. What is possibly claimed here is that we as humans will probably experience many lives in many different planes of existence. It is a kind of generalization of the typical idea of reincarnation.
11. “The higher beings are connected to the others by analogy, not by composition.”
Here, Gödel expands on his fourth point on the list. He begins to describe some properties of the entities that live in his abstract and metaphysical world. When he says that “The higher beings are connected to the others by analogy” he means that the concept of “absolute” does not apply his world. Truly abstract concepts are defined in relation to other abstract concepts. To further understand this let’s use an example. Consider the number “4”. How can you define this entity? Keep in mind that we are treating the number “4” as an abstract non-physical concept. The only sensible way to approach this, according to Gödel, is to lay out some of its properties on how it is related to other numbers e.g. “It is bigger than 3 but smaller than 5” or “It is half of the number 8”. Just like numbers, Gödel claims, all the higher entities in his world are defined by analogy.
Final Remarks
As we have stated, Gödel’s list was not meant for publication and thus not only is it subjective, up to a degree, but it probably does not give the full picture regarding Gödel’s philosophy. Thus, we will not analyze any more of the points in the list but we highly encourage the reader to study it and think about the reasons why one of the greatest mathematicians of all time, Kurt Gödel, had such strong metaphysical and controversial beliefs.
