The Merits and Limitations of 'Obvious' in Mathematical Proofs

In the realm of mathematics, the term obvious often evokes a combination of intuitive clarity and perceptual simplicity. However, this seemingly straightforward term carries a nuanced and sometimes deceptive connotation. This article explores the role of obvious in mathematical proofs, specifically examining the Jordan curve theorem and Brouwer's invariance of domain, while also delving into the broader implications of this concept in mathematical intuition and pedagogy.

The Role of Intuition in Mathematics

Mathematics, at its core, is both a rigorous discipline and an intuitive one. While strict logical arguments are paramount, initial intuitions and perceptions play a vital role in theoretical development. Consider the Jordan curve theorem, which asserts that a simple closed curve divides the plane into an interior region (bounded) and an exterior region (unbounded) in the Euclidean plane. Despite its intuitive appeal, the theorem's proof is far from trivial and requires sophisticated topological arguments.

Another example is Brouwer’s invariance of domain. This theorem states that if a set in Euclidean space is homeomorphic to an open subset, then it itself is open. Although this statement feels intuitively clear, its rigorous proof demands complex reasoning. These examples highlight that even when a problem seems obvious, a detailed and rigorous argument is often required to establish its truth.

The Imprudence of Unproven Notions

The term obvious can be misleading when it suggests that a step in a proof is too obvious to require further explanation. This is a common pitfall in mathematical discourse, where assumptions based on obviousness can lead to fallacies. For instance, the equation 11 2 is an illustrative joke. While it might seem intuitively obvious that 11 is not equal to 2, understanding the underlying mathematical principles that prevent this from being true requires a solid grasp of arithmetic and the properties of numbers.

Mathematics demands precision and proof at every step. Therefore, when a concept is described as obviously true, it is often accompanied by a recognition that the underlying proof is complex or subtle. The term plausible is often preferred in mathematical contexts because it acknowledges the tentative nature of intuitive insights while still expressing a high degree of confidence in the truth of a statement.

The Jests of Mathematical Intuition

Mathematics students and educators often share jests about the use of terms like obvious, trivial, and easy to see. These jests reflect the tension between intuitive understanding and formal proof. Here are a couple of examples: Easy to see: When a professor says a result is easy to see, it typically means that they had to work through the details for several hours to verify the statement. The term comes across as an attempt to downplay the complexity of the problem, even if it feels intuitively clear. Trivial: This term is often used sarcastically by students to indicate that a professor has not yet fully understood the problem. The label trivial in mathematics is reserved for statements that are simple and do not require much effort to prove. Obvious: The colloquial use of obvious in mathematics often reflects the teacher's or researcher's perspective. If someone says a result is obvious, it means that it was intuitively clear to them, regardless of its complexity or the effort required to prove it.

Conclusion

While the term obvious has a place in mathematical discourse, its usage should be carefully evaluated to ensure that no step in a proof is taken for granted. The role of intuition in mathematics is valuable, but it is essential to back up these intuitions with rigorous proofs. By maintaining a balance between intuition and formal argument, mathematicians can build solid, robust theories that stand the test of time.