For instance, the Transitive Axiom of Equality states that if a=b, and b=c, therefore a=c (Basic). This axiom follows Euclid's Common Notion One: "things equal to the same thing are equal to each other" (Basic). Mathematicians use this axiom to explore new concepts through rigorous proofs to justify whether a concept is valid or not. When this concept becomes a tautology, a theory is formulated. However, despite the rigorous proofs found, any exceptions found would be called a conjecture. Conjectures create a gap in the shared knowledge, as there is always a possibility to find an exception, which thus counterclaims the certainty of rigorous proofs.
Furthermore, history is also a way of knowing that uses rigorous proofs. Dr. Lisbeth Fried compares the process of rigorous proof to the scientific method, insinuating that thinking initiates through inductive reasoning (Fried). After historians obtain sufficient evidence and data regarding a certain historical event, the contemporary interpretations are thus analyzed, examined, and deduced into a conclusion. Historians use inductive reasoning to formulate new theories by utilizing sufficient evidence. Evidence such as first-hand accounts are utilized by historians to deduce an overall theory about what occurred. An example is Keynes and Carnap, Bayes' theorem, which is a theorem of probability theory. It expresses how evidence comes to bear on hypotheses (Inductive Logic). However, flaws of inductive reasoning lies in the biased or exaggerated evidence, such as primary sources (Lagemaat). As an example, during the course of World War II, Germany was referred to as Nazi Germany. The name Nazi Germany suggests that all Germans were part of the Nazi Party. This syllogism is flawed as, in fact, approximately one third of the Germans supported Hitler and the Nazi regime (Finister). Therefore, despite proofs used through reasoning, inductive reasoning does not always maintain complete certainty.
