The Origins and Motivations Behind Mathematical Axioms: A Closer Look

Mathematics is often hailed as the language of the universe, where axioms form the foundation of this grand language. But how do these axioms come to be, and are they inherently tied to the physical world?

The Role of Axioms in Mathematics

Axioms in mathematics are typically seen as self-evident truths that do not require proof. However, it is important to note that the words "prove" and "true" are not synonymous. A statement can be proven false, as well as true. Therefore, the acceptance of axioms as truths is based on their role within a mathematical system, rather than an absolute truth in the philosophical sense.

The Development of Geometry and Mathematics

The way geometry and mathematics are traditionally taught gives a false impression of how these theories were developed. Students are often taught a process that goes from axioms, which are seen as obvious and self-evident, to theorems, as if axioms are the starting point of mathematical exploration. However, this is an inversion of the actual process.

Euclid, for example, did not start with axioms and then derive theorems. Instead, he started with a body of knowledge he believed to be true and then worked backward to find axioms that supported this knowledge. This means his Elements is more the result of that work than a starting point for new exploration.

Revisiting the Scientific Method in Mathematics

The process in mathematics can be likened to the scientific method, but in reverse. The scientific method follows a logical progression from observation to deduction and prediction, then back to observation to revise principles as necessary. In mathematics, the process is similar but works backwards:

Observe regularities in nature (analogous to empirical observation in science) Find simple statements that are observably true (building blocks of formal systems) Deduce basic principles to explain those regularities (axioms and formal systems) Develop a formal system to prove those statements (like Peano Arithmetic) Make predictions about other regularities not yet observed (theorems and further exploration) Revise the principles as necessary (adaptation and growth of mathematics)

Unprovable Axioms and Their Impact

In this process, sometimes we find that our axioms, despite being true, do not always lead to accurate statements about the physical world. This can lead to revision, growth, and the introduction of new concepts like negative numbers, irrational numbers, and independent statements.

What does this say about the "obviousness" or "unprovability" of axioms? In reality, the axioms we choose are based on our observations of the physical world. This means that while some axioms may seem self-evident, they are ultimately grounded in empirical facts rather than inherent truths.

Mathematics as an Empirical Science

Mathematics is, in some sense, an empirical science. It evolves from insights and observations, leading to corrections and new developments. Addition, for example, is not a Platonic concept but a concept that emerged from our experiences and the regularities we observed in the world.

Just as there are natural phenomena where the rules of arithmetic do not apply perfectly, there are also areas of mathematics where axioms do not always lead to accurate descriptions. This adaptability and the fact that mathematics continues to evolve highlight its empirical nature.

Conclusion

The relationship between mathematical axioms and the physical world is complex and nuanced. While axioms are often motivated by the physical world, they also need to be revisited and adapted based on new observations and insights. This ongoing process reflects the dynamic and empirical nature of mathematics, not a fixed, absolute system of truths.