Mathematics plays a paramount role in the ability to gain knowledge. Yet, the realm of higher-level mathematics and logic is inaccessible to the majority. Using informal approaches to mathematics—such as geometric shapes—grants most people access to the world of mathematics.
Engaging in geometric proofs forces individuals to use logic and ingenuity to progress from a given premise to a conclusion. Geometric proofs require in-depth examinations with the content, resulting in deeper and more pure understandings. The skills of logic, ingenuity, and in-depth analysis used in geometric proofs allow an individual to develop their knowledge.  Geometric proofs vary so immensely in complexity that they can hint at a measure of knowledge. For example, most individuals can acquire rudimentary geometric thought processes through daily life without an education. As the level of complexity increases, so does the level of knowledge associated with it. For example, the formalisms of differential geometry may only be accessible by those who have dedicated their lives to analyzing geometric proof. These individuals can only be at that level by amassing knowledge from previous forms of geometric proof and experience working with geometric ideals.  The acquired skills—and knowledge—through geometric proof allows for examination of profound humanitarian and philosophical dilemmas. The Renaissance philosopher Galileo Galilei’s fascination with geometric proofs helped him develop the degree of knowledge required to process his astronomical and scientific contributions to humanity. 
The premise of geometric proof being the primary indicator of knowledge restricts creativity and self-expression. The geometric proof relies primarily on mathematical understanding, therefore intertwining the definition of knowledge with mathematics. With this argument, knowledge can only be gained with an aptitude for mathematics. In reality, there are countless knowledgable individuals without sufficient understanding of mathematics or geometric proof.