God, devil, and Gödel

The problem is the blind faith many are placing in the Cartesian thought. Descartes famously said “I think. Therefore, I am”. What is wrong with this? It is only in the act of thinking about something “outside” the self that one can see one is a thinking self. One must exist before one can think. Existence is primary. “I am. Therefore, I think”. You cannot experience a thing without it existing. Something must exist for you to experience it. Our knowledge in the modern sense consists of “four Es”: Existence, Experience, Experiment and Expression. A thing must first exist. You can experience things that exist. From this experience, you use your intelligence to abstract the essence. You do controlled experiments to derive more general truths about your experience. Lastly, these general laws are expressed in mathematical equations or schematics (e.g., double helical structure of DNA; Biology is more empirioschematic in that it uses many schemes to categorize and express universal behaviors).

In logicalism, experiment and expression are given more importance than experience. This view reduces the essence of all physical phenomena to symbols and equations. Thus, it enthrones mathematics as the queen of the sciences and ignores philosophy, which unfortunately leads to many fundamental issues. It indeed led to a great danger threatening our science itself. It generated many “black box operators” that think they know but do not know in reality. For example, a student ranked in the top 1% of IIT-JEE will hardly be able to provide you any ontological meaning of things. I would be surprised if any of them or their teachers (future and past) can properly define what a measurement is and how it converts a quality into a quantity. To not know that one does not know is pure ignorance. To know that one does not know is metacognition. To think that one knows when one does not indeed know truly is unintelligence. Today’s unintelligence and blind scienceism have led to a “I think; Therefore, I am” culture. If you truly understand mathematics, you would be able to see the problem with this statement and its implications.

Here, I will introduce to a gem: Godel’s theorem to show you what is wrong with philosophical idealism.

David Hilbert, one of the greatest mathematicians ever, was a proponent of metamathematics or metalogic. Hilbert wanted to prove mathematics as the ultimate and complete answer through the axiomatisation of mathematics. An axiomatic system essentially consists of some accepted presumptions (i.e., axioms or postulates) and theorems that are deduced from those axioms. The Greek word axioma means that which is self-evident. An example is Euclidean geometry you may have studied in school. This geometry is based on certain axioms that may be self-evident but are not necessarily provable rigorously. For example, any two points can be connected by a straight line. You know its true but its not provable (we will talk about proof and truth in a minute). You should recall that many geometric figures are “beings of reason”. A perfect circle does not exist in reality. Although the idea of a circle is abstracted from approximate circles in reality, a perfect circle exists only in our mind. Similarly sqrt(-1) or i exists only in our mind. Thus, they are “beings of reason”.

A theorem in any axiomatic system is derived rigorously from the axioms by purely depending on the principles of logic. An axiomatic system does not necessarily depend on any empirical data but on logic alone. All theorems ultimately reference back to the axioms. So, if you wish to prove an axiom using theorems within an axiomatic system, you end up self-referencing. Let us think of a simple example. Let us say we have a system that contains all possible statements. We make a postulate that “any statement in this is either true or false but not both at the same time in the same way”. This is our beginning axiom. Now, if you have a statement that refers to itself such as “this statement is false”, you would begin to see a problem with the axiomatic system. If the statement “this statement is false” is true, it is false. If it is false, it is true. It may be a little confusing. But, think about it. Our entire geometry and arithmetic are axiomatic systems.

There are two important characteristics of an axiomatic system, viz., consistency and completeness.

An axiomatic system is said to be consistent if it contains no formula or statement such that both the formula and its negation are provable as theorems within the system. The importance of consistency lies in the fact that in an inconsistent system, a theorem and its negation can both be proved. When this happens, the system can accommodate contradictions. That makes the system unreliable for any proof.

The other important feature of an axiomatic system is completeness. A system is complete if all the desired formulas or statements can be proved within the system. Any inconsistent system is complete because you can prove any theorem in an inconsistent system. Thus, the completeness of a system is useful only if the system is consistent.

Now, enter Kurt Gödel. In 1931, he published a paper entitled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, often dubbed as Gödel’s incompleteness theorem. Gödel established that any axiomatic system contains at least one formula which is necessarily true but is neither provable nor disprovable within the system. In other words, there is no axiomatic system (at least as powerful as arithmetic) that is necessarily both complete and consistent.

At this juncture, it is important to establish a clear distinction between the notions of truth and proof. Proof is rigorous, logical, axiomatic, and constructed within a system. However, truth lies outside of the system. For example, I could have an axiomatic system in which “Sun is green” can be proved. But, the veracity or truthfulness of this statement cannot be verified unless you look to “reality” that is outside the axiomatic system. Thus, the consistency of a system cannot be proved within the system. Something is provable does not necessarily mean it is true. To know what is true, one must look outside the system.

Now, let us use Godel’s theorem on Descartes. The consistency theory, which holds that, the consistency of an axiomatic system cannot be proven within that system, will enable us to decide upon the statement “I think; Therefore, I am”. This hypothesis asks us to question the very nature of our existence and reality. It implies that you do not exist if you cannot think. According to Gödel, there is no way to establish the consistency of a system called “me”. If I am inconsistent, how can I rely on the proof (I think; Therfore, I am) I derive from it? Hence, the truth about the reality cannot be proved from within. Based on Descartes statement, it is incorrect to conclude that one can only know what is in one’s mind and therefore one cannot know if there is anything outside the mind. Similarly, based on Godel’s theorem, it is incorrect to deduce that our knowledge is simply ideas “without” any foundations. Our ideas come from outside reality similar to how one cannot think about oneself without thinking about what is outside.

Science is rooted in reality. It is the study of reality. Although it is often expressed in mathematical form, science (e.g., physics) relies on both “truth” and “proof”. For example, I cannot describe the behavior of an atom like how I describe the motion of a cricket ball. The reality of how things behave drives science. However, the success of mathematics and empiriological science have somehow led to wrong conclusions that we can never truly know. Unless we place emphasis on understanding the metaphysical and philosophical aspects of science, we will fail.