The Implications of Discovering a Physical Phenomenon Un describable by Mathematics
Imagine a scenario where a physical phenomenon defies the rules of mathematics. This could shake the very foundation of our understanding of the universe and our methods of scientific inquiry. The question is, what would it mean if we discovered a physical phenomenon that could not be described by mathematics?
Understanding the Limitations of Mathematics in Physical Phenomena
For a phenomenon to be predictable, there are usually several mathematical equations that can fit it. The ideal solution is the simplest one that aligns with the accuracy of the measurements taken within the acceptable tolerance. However, when the precision of measurements improves, the existing mathematical models may need to be adjusted or even replaced with a more complex one. This is where Occam's razor comes into play, advocating for the simplest explanation that fits the data.
Some phenomena are only predictable under certain conditions, often modeled as nonlinear systems. These can be orderly, chaotic, or in a transitional state. Examples include weather forecasting, stock market modeling, and fluid dynamics, including turbulence, and the famous three-body problem. For instance, even with advanced numerical models, weather forecasts can still be inaccurate due to the chaotic nature of weather systems.
When a phenomenon is completely unpredictable, it is often modeled as a random variable. Quantum mechanics provides a practical example here, where the collapse of the wave-function is inherently random, meaning precise predictions of outcomes are impossible.
Observing Physical Phenomena Without an Adequate Mathematical Model
Chaos theory is a prime example of a field where the lack of a clear mathematical model has challenged our understanding of complex systems. Despite its complexity, chaos theory has provided invaluable insights into the behavior of nonlinear systems, which are common in nature.
Some may argue that the world is full of phenomena that can be described by mathematics in principle, but current mathematics, or even with advanced computational techniques, cannot efficiently describe them. For instance, the general relativity tensors were developed to describe phenomena in high gravitational fields, while Schr?dinger's equations can only be feasibly solved for the simplest cases like hydrogen and helium atoms. Quantum computers might provide a solution for more complex cases in the future.
Other phenomena are inherently unpredictable due to their chaotic nature, such as the three-body problem, where the solution heavily depends on initial conditions, making precise predictions extremely difficult. In still other cases, phenomena are fundamentally random, such as the outcomes in quantum mechanics, where predictions are probabilistic rather than deterministic. More challenging cases involve phenomena with inherently probabilistic in-principle probabilities, like self-referential quantum mechanics at the Planck scale, where the mathematical models have limited predictive power.
The Evolution of Mathematics and Its Relevance to Science
The development of mathematics is often driven not just by science but also by pure mathematics itself. Many mathematical theories, initially considered purely theoretical and of no practical use, have found applications in various fields. For instance, number theory once regarded as a purely abstract field has become vital in cryptography.
Sometimes, the direction of mathematical research is not clearly defined. The needs of other sciences, such as physics, often accelerate this progress. The lack of direct applications of certain branches of mathematics might slow down their development, but as history has shown, there is often a direct link between theoretical and applied mathematics.
Not Physical Phenomena
There are certain types of phenomena that go beyond the physical realm, making them beyond the scope of scientific study. This includes topics related to qualia, or subjective experiences (such as pain or color perception), which cannot be mathematically described. Additionally, epistemological questions about probability (e.g., the probability of being born in the 20th century or being a human of a specific race) and existential metaphysical questions (e.g., the probability of the existence of the universe or of physical laws as we know them) fall into this category.
In conclusion, while every phenomenon can be described mathematically to some degree, a phenomenon that is entirely or partially unpredictably random, chaotic, or fundamentally probabilistic is a significant challenge to our understanding of the natural world. Understanding these limitations can lead to new insights and potentially new branches of mathematics and science, furthering our comprehension of the universe.