Reality vs Fiction: Between Mathematics, Science, and Natural Language
Recently, I came across a meme about the discovery of the Portuguese ship The Bom Jesus, which turned out to be one of the most fascinating archaeological finds in recent history. The ship, which sank in 1533, was found along the coast of the Namib Desert, buried beneath the sand for centuries. Thanks to the unique geology of the area and the shifting dunes, the remains were preserved in excellent condition, along with a cargo of ivory, coins, and copper ingots, providing a wealth of historical information about trade routes and economic exchanges of that time.
The fact that a galleon was found in a desert setting challenges our conventional perceptions of shipwrecks. This is because, while today the location is an arid region, back in the 16th century, the coast of Namibia was where the Atlantic Ocean met the African continent. Over centuries, geographical changes transformed the area into a desert, creating the enigmatic context in which The Bom Jesus was discovered.
Being naturally curious, I immediately shared this story with a group of friends, one of whom is Hugo Félix, a filmmaker, screenwriter, director, and producer of a Mexican science fiction film shot at the Gran Telescopio Milimétrico. Hugo pointed out to me that in 1977, Steven Spielberg released the acclaimed movie “Close Encounters of the Third Kind”, which also had a scene where a galleon appeared in the middle of the desert.
In a time when the line between reality and fiction seems to blur more and more, the fascination with what lies beyond our current understanding becomes the driving force that pushes us to explore the boundaries of science and imagination. Spielberg, I believe, captures that exact sense of mystery and wonder in his film. On the other hand, the discovery of The Bom Jesus is where concrete reality asserts itself, a ship preserved by natural processes, a reminder of how history leaves traces in unexpected places.
A recent paper (https://www.arxiv.org/abs/2408.11065) study the complexity of natural languages and how they differ from mathematical language in the context of science. Natural language follows statistical patterns called Zipf’s law, a type of power-law distribution (https://youtu.be/oMl-SbuQUYc?si=gwbAr6qcc3CXXb5h)describing how word frequency slowly decreases, indicating a persistent presence of rare but significant elements within the system.
This pattern reflects an informational structure where complexity in terms of homogenity/heteroneity balance is always present. By contrast, the mathematical language used in science appears to follow a much steeper decay pattern, as shown in the Feynman Lectures (one of the samples analyzed in the study). This suggests that the information conveyed in science is more predictable and uniform, with less variability — at least when it comes to simple systems.
What I find particularly fascinating is that Frieden, in his 1998 book “Science as Fisher Information”, already introduced a unifying theory suggesting that the laws of science could be derived from Fisher information, a concept in information theory that measures a system’s capacity to infer knowledge from observations. According to Frieden, the laws of nature align with optimizing Fisher information within the specific context of an observed phenomenon. Essentially, scientific laws are formed from the informational patterns that emerge when observing nature, and their mathematical structure tends to be local and simple, often relying on second derivatives. When these second derivatives are spatial, it means that scientific laws (of simple systems) only require local information from the immediate environment of the system. This might explain why the mathematical language of science (of simple systems) does not follow Zipf’s law, instead exhibiting a more rapid, exponential decay.
On the other hand, Jaynes, in his seminal work “Probability: The Logic of Science”, argued that mathematics represents the logical part of human thought, while natural languages encode much richer and diverse processes, encompassing not just logic but also emotional, cultural, and contextual aspects of thought. This helps explain why mathematical descriptions in science (of simple systems) tend to be more predictable and less complex than natural language: we are not using all our cognitive processes, only those that allow precise, logical inference.
This brings up a crucial question: are we describing inherent patterns of nature, or are they human-imposed? The discussion suggests that the relationship between reality and fiction lies precisely at the boundary of our language capabilities to describe the world. Natural language allows for greater freedom in expressing ideas, even those that might not exist in our universe, whereas mathematical language is confined to describing what can be rigorously measured and proven — what only requires the rational part of our intellect.
Therefore, we can imagine a continuum that spans from science to science fiction to pure fantasy.
The difference between science and fiction, then, is not simply about truth or falsehood but rather about the cognitive and linguistic tools we use to express our ideas about the world. While science seeks objective truth through the rigorous logic of mathematical language, fiction draws from the creative potential of natural language, exploring other regions of the idea-space that allow us to conceive the inconceivable.
Fiction and science, then, are not opposing poles but reflections of the different capacities of our minds. It is important to note that this exponential pattern only describes the science of simple systems, not complex ones.
A simple system typically has few interactions, and the rules governing its behavior (which may change over time) remain consistent, at least over the observation period. In other words, even though its behavior may be dynamic, its dynamics are static. These systems tend to be predictable and controllable. Scientifically, we can model them using differential equations and by conducting experiments.
In contrast, complex systems often exhibit multiple interactions and tend to respond disproportionately (non-linearly) to changes in inputs defining their behavior. Their dynamics are dynamic, presenting patterns that emerge from the interactions between their components (emergence). When it comes to describing them scientifically, we cannot model them using differential equations, nor can we experiment with them as we do with simple systems. This type of system only admits probabilistic descriptions and observational studies, making them less susceptible to prediction and control.
In fact, in my opinion, we do not yet have the proper mathematics to fully describe complex systems as we do for simpler ones.
Thus, the boundary between the power-law pattern of natural languages and the exponential decay pattern seen in the mathematics of simple scientific systems may be much narrower or even nonexistent when we consider complex systems.
If that is the case, then perhaps we only imagine what is at least potentially possible in this universe. As a lost ship be found in the middle of a desert.
