Bolzano's Conception Of Analysis

In the modern era, analysis is a very well established and developed field of mathematics. It is well known that analytic proofs are very strict in nature and to prove anything in analysis requires a careful consideration of the behaviour and characteristics of the proposition on hand. Before analysis became as well founded as it is today, there was no standard of rigor for analytic proofs. So, many mathematicians gave analytic proofs that were heavily based on spatial and geometric intuition. Bolzano believed that relying on spatial and geometric intuition was an improper way to prove a truth in analysis. He thought, though arguments appealing to space can be used to explain the truth of a proposition, such methods cannot justify the truth of a proposition. With the concept of “grounding” truths with basic truths, Bolzano …show more content…
Bolzano’s concept of “grounds of judgements” is one of the primary driving forces that motivates his idea of the “correct method” for deriving truths in mathematics. He claims that some judgements provide the grounds for other judgements, which are the consequences of the former, and that the main goal of scientific investigation is to find the ultimate grounds of our judgements (Notes 182). In fact, the necessity to properly ground judgements with other judgements is the very source of Bolzano’s discomfort with proofs in analysis given by his contemporaries. While he admits that, by providing geometric or spatial justifications, his contemporaries are then able to present truths that are immediately obvious and require no proof in the sense of confirmation, he believes that the truth of the justifications themselves still need to be justified (Purely 160). In Bolzano’s eyes, the proof of a proposition becomes illegitimate when justifications in the proof are grounded by results of the

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