Thoughts about proving in learning

Introduction

The relationship between proofs and learning often initiates with us grasping the fundamentals of proof before delving into theoretical study. However, as one’s study progresses, the necessity of proving can, at times, dwindle. This raises an essential question: how integral is the role of proving in the learning journey?

Experiencing the beauty and excitement in discovering or learning a compelling proof is undeniable. Yet, it’s crucial to acknowledge that embracing a proof is not always a prerequisite for applying theory to resolve problems. Herein lies a potential pitfall of idealism that warrants our attention.

Analyzing Attitudes Toward Proofs

Attitudes toward proofs can broadly be categorized into four distinct perspectives:

1. Personal Validation: Independently proving the theory.
2. Informed Understanding: Grasping proofs developed by others.
3. Existential Assurance: Ensuring a proof exists without delving into it.
4. Practical Disregard: Bypassing the need for proof altogether.

While the first three approaches do not jeopardize the practical use of a theory, the fourth is commonly encountered in real-world scenarios. Perspectives three and four often emerge as more time-efficient, particularly in applied contexts. However, when proofs are simple or elegant, the first and second approaches not only prove efficient but also weave an element of interest and engagement into the learning process.

Thus, a discerning balance must be struck between the complexity of the proof and the ultimate objective of studying the theory. Each of these attitudes brings its own set of advantages and should be adopted with a malleable mindset.