Models and Paradoxes

This sentence is a lie.

Paradoxes: often funny, often baffling. What are they, and where do they come from?

Let’s begin with models. What are models? Models are essentially simplifications of principles and systems in the real world, and this makes them very useful, as we can use them to manipulate and predict the world. Some examples of models are language, maps, mathematics, music sheets, weather forecasts, and the laws of nature.

A model is essentially made up of recognized patterns in real life: they are representations of the real world, but not the real thing itself. When models are incomplete, they are extended and changed so that they become more accurate. The problem is that the more complex a model becomes, the less usable and understandable it is because of its complexity.

What often happens with complex systems is that the system is first broken up into smaller parts, and that a model is made for each part of the system, so that this model can be used to predict or manipulate only this small part. These small models can then be used together to predict the entire system.

What happens when someone tries to make a fully accurate model of an extremely complex system that cannot easily be broken up into separate parts? The model has to account for every case in the real situation, and will become as complex as the real system itself, making it a useless model. To make a useful model for an extremely complex system, it is required that some cases in the system are not considered, so that the model is simpler but less accurate.

Because models are made for separate parts, models can also clash with each other when they are used together, if they have different mechanics or have mismatching purposes. Think of using weather forecasting models and music sheets together, I can’t think of a situation in which they would be compatible, because they have a completely different use case. This means that some systems can not be used together, as they are too different from one another. This can even happen with specific mechanics in one model. Let’s call them submodels.

I suppose the whole universe could be seen as an extremely complex system that can often not be broken up into separate parts. Models for gravity work very well to utilize and predict gravity, but try making a fully accurate model of the human brain, and it will become way too complex. In this case the principle of gravity can be succesfully broken up from all phenomena into a separate part. The human brain is more difficult to break up into smaller parts, because it works more as a whole.

Well then, with that out of the way, what are paradoxes? I’ve never seen a paradox in real life, but always in the context of language or mathematics, or any other model. As far as I understand, paradoxes occur because a model is incomplete, or limited in other ways.

Maybe you’ve heard of the Sorites paradox (paradox of the heap). Imagine you have a heap of sand, and you remove grains of sand, one at a time. At how many remaining grains of sand will it stop being a heap? If it always stays a heap, will it still be a heap when only one grain remains? Because the concept of a heap in language is arbitrary, it cannot be quantified, leading to the paradox. I only know what a heap is based on what others have called a heap, not by how many grains of sand are in it.

There’s a mismatch in the arbitrary concept of a heap and the knowledge that a heap is made up of individual grains of sand. This means that language is also an incomplete model: it is arbitrary to an extent. Defining exactly how many grains of sand together is considered a heap would solve this problem, but that would make language too complex, so it’s left out. Imagine having to define exactly how many trees together is a forest. Nobody wants that, so we deal with paradoxes.

The first sentence in this post, this sentence is a lie, refers to itself as the sentence and negates itself. The sentence it refers to negates itself, so it negates its own negation, leading to paradox. This is also an example of two mechanics of language being incompatible, just like the previous example. In this case the mechanics are the referrence to the whole sentence and the negation. This is also an example of parts of a system not properly working together, where the subsystems collide.

So, to conclude, paradoxes probably don’t exist in the real world, at least not as far as I’m aware, but only exist in our heads, because our models have flaws. Discovering paradoxes is a great way to discover problems in our models and to make them better, both with our internal and our external models. Paradoxes have definitely led to great improvements in the field of mathematics from what I know, and probably also in many other fields. Contradictions lead to new discoveries, but I wonder if we’ll ever get rid of them, judging by the nature of what we are dealing with. All I know is, when you discover contradictions, go after them, because they are the edges of our preconceptions.

I hope this was a clear and coherent story. If you have opinions or feedback, please leave a comment, as I’d love to hear more perspectives.