arXiv:2402.09413v1 Announce Type: new
Abstract: A definition of what counts as an explanation of mathematical statement, and when one explanation is better than another, is given. Since all mathematical facts must be true in all causal models, and hence known by an agent, mathematical facts cannot be part of an explanation (under the standard notion of explanation). This problem is solved using impossible possible worlds.
Analyzing the Definition of Mathematical Explanations
Mathematics plays a crucial role in our understanding of the world, providing us with tools to model and explain various phenomena. Yet, when it comes to explaining mathematical statements, we encounter a unique challenge. This article delves into the definition of mathematical explanations and explores an interesting solution using the concept of impossible possible worlds.
The Challenge of Defining Mathematical Explanations
When discussing explanations in general, we often rely on cause-and-effect relationships. However, in mathematics, we encounter a different situation. Mathematical facts are universally true and known by all agents. As a result, they cannot be considered part of an explanation under the conventional notion.
Introducing Impossible Possible Worlds
To overcome this barrier and provide a framework for mathematical explanations, the concept of impossible possible worlds comes into play. By considering impossible scenarios, we can shed light on the reasons why certain mathematical statements hold true.
What are Impossible Possible Worlds?
Impossible possible worlds refer to hypothetical situations that are internally inconsistent or violate known mathematical facts. While these worlds may seem counterintuitive, they serve a valuable purpose in explaining mathematical statements.
Why Are Impossible Possible Worlds Useful?
By analyzing impossible possible worlds, mathematicians can identify the constraints and conditions necessary for a mathematical statement to hold true. This approach enables deeper insights into the underlying principles and structures of mathematical concepts.
The Multi-disciplinary Nature of Mathematical Explanations
The exploration of mathematical explanations requires a multi-disciplinary approach that bridges mathematics, philosophy, and logic. By integrating these perspectives, we gain a better understanding of the nature of mathematical knowledge and the methods used to justify mathematical statements.
What Comes Next?
The concept of impossible possible worlds opens up exciting avenues for future research. Further investigations could focus on refining the framework to handle different types of mathematical statements and exploring how this approach can be applied to other domains. Additionally, collaborations between mathematicians, philosophers, and logicians can contribute to a deeper understanding of mathematical explanations and their role in knowledge discovery.
In conclusion, the definition of mathematical explanations poses unique challenges due to the universal truth and known nature of mathematical facts. However, by embracing the concept of impossible possible worlds, we can overcome this hurdle and gain valuable insights into the foundations of mathematical statements. The multi-disciplinary nature of this exploration exemplifies the interconnectedness of various fields and emphasizes the importance of collaboration in furthering our understanding of mathematics.