Mathematicians have the concept of rigorous proof, which leads to knowing something with complete certainty. Consider the extent to which complete certainty might be achievable in mathematics and at least one other area of knowledge.

Claiming that mathematics' rigorous proof results only in complete certainty and allows shared knowledge to be absolute is an exaggerated assertion. The title insinuates that through mathematics, something can be known with complete certainty. Complete certainty may be exaggerated, therefore the question to be explored is: to what extent can rigorous proof be used to find complete certainty? However, rigorous proof can also be explored in other ways of knowing. In history, "the study of the past," historians do not use mathematical equations and explanations, but rather their own method to rigorously prove facts of the past, such as using evidence and various perspectives (Lagemaat 189). Therefore, this essay explores the extent to which rigorous proofs can be achieved in mathematics and history by examining their reliance on reasoning and imagination. .

Rigorous proof in mathematics is often derived from deductive and inductive reasoning. Deductive reasoning is the premise to achieve complete certainty: it is defined as "reasoning from general to particular" (Lagemaat 121). When applied to a system of numbers, real numbers can be narrowed down to different factors: irrational, rational, and whole numbers (Number). Through rigorous proof, this premises would then follow deductive reasoning: since any number is real, then it is either rational, irrational, an integer, whole, or a natural number. However, the reverse does not apply and would not be valid. In addition, mathematics has an inductive reasoning component, which is "reasoning from particular to general" (Lagemaat 121). Mathematicians use inductive reasoning to expand on axioms to form truths.