AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |
Back to Blog
![]() ![]() For example, non-Euclidean geometries have been shown to be as consistent as their Euclidean cousin. (Download Axiomatic Systems (Lee) and see sections 6.1, 8.1, and 8.4 in book 3b of Math Connections (Berlinghoff) for further explanations, activities, and problem sets on axiomatic systems, consistency, and independence). Now, when mathematicians create new axiomatic systems, they are more concerned that their choices be interesting (in terms of the mathematics to which they lead), logically independent (not redundant or derivable from one another), and internally consistent (theorems which can be proven from the postulates do not contradict each other). Mathematicians tried to choose statements that seemed irrefutably truean obvious consequence of our physical world or number system. In general, however, proofs use justifications many steps removed from the postulates.īefore the nineteenth century, postulates (or axioms) were accepted as true but regarded as self-evidently so. Similarly, we need to accept certain terms (e.g., "point" or "set") as undefined in order to avoid circularity (see Writing Definitions). ![]() The absence of such starting points would force us into an endless circle of justifications. Since we base each claim on other claims, we need a property, stated as a postulate, that we agree to leave unproven. We cannot prove any statement if we do not have a starting point. Postulates are a necessary part of mathematics. In other words, often the overarching objective is the presentation of a convincing narrative. Many proofs leave out calculations or explanations that are considered obvious, manageable for the reader to supply, or which are cut to save space or to make the main thread of a proof more readable. The level of detail in a proof varies with the author and the audience. In practice, proofs may involve diagrams that clarify, words that narrate and explain, symbolic statements, or even a computer program (as was the case for the Four Color Theorem (MacTutor)). That is, that all of the premises of each deduction are already established or given. By logical, we mean that each step in the argument is justified by earlier steps. The argument derives its conclusions from the premises of the statement, other theorems, definitions, and, ultimately, the postulates of the mathematical system in which the claim is based. Examplesno matter how manyare never a proof of a claim that covers an infinite number of instances.Ī proof is a logical argument that establishes the truth of a statement. ![]() This habit also guides our work in the more abstract realm of mathematics, but mathematics requires us to adopt a greater level of skepticism. In everyday life, we frequently reach conclusions based on anecdotal evidence. Twenty practice problems (with solutions) You can use the table of contents below to navigate around this chapter: You may want to jump to the activities, try some out, and then double back to the readings once you have had a chance to reflect on how you approach proofs. This discussion addresses several different aspects of proof and includes many links to additional readings. ![]()
0 Comments
Read More
Leave a Reply. |