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Set Theory for Concept Modeling
Published in Richard M. Golden, Statistical Machine Learning, 2020
Furthermore, set theory is the foundation of all mathematics. Mathematical definitions are statements about equivalent sets. Mathematical theorems are assertions that specific conclusions follow from specific assumptions. Mathematical proofs correspond to a sequence of logical arguments. Definitions, theorems, and proofs are all based upon logical assertions. Logical assertions, in turn, can be reformulated as statements in set theory.
Write Your Math Well
Published in Edward J. Rothwell, Michael J. Cloud, Engineering Writing by Design, 2017
Edward J. Rothwell, Michael J. Cloud
The basic activity of mathematical proof is that we take a collection of definitions, accepted axioms, and already established propositions, and, by valid patterns of inference, use these to establish a conclusion.
Undergraduates’ propositional knowledge and proof schemes regarding differentiability and integrability concepts
Published in International Journal of Mathematical Education in Science and Technology, 2018
One of the most important features that distinguishes mathematics from other branches of science is the effort to identify, explain, and prove each instrument it uses [25]. In university level mathematics courses, the teaching cycle continues with the definition, theorem, and proof-based content in general. The processes of creating proofs are often not well understood by students due to the rote learning-based approaches, and students are generally not aware of alternative proof approaches [19–21]. It is important to ask what is the mathematical proof and why is it so important? Even mathematical philosophers direct their approaches by seeking answers to the following questions: What is the object of mathematics, and which arguments indicate a proof in mathematics? Mathematical proof is the process of persuasion, which consists in revealing the correctness or incorrectness of an assertion. In this process, axioms, assumptions, or predictions are used systematically by making arguments. On the other hand, the purpose of a mathematical proof is not only to show that a claim is true or false, but also to argue why it is true or false [26,27]. The processes of proof may vary according to individuals’ periods of cognitive development, the language of the community, and even culture. Although it seems that there is a single definite way to prove in mathematics, a high-school student and a university student, an algebra professor and an analysis professor, nineteenth century mathematicians and twenty-first century mathematicians may use different methods to prove a proposition. This difference does not indicate that one approach is superior or more correct than the other. Harel and Sowder [25] defined ‘proof scheme’ as the defenses that an individual makes to convince himself or another of the correctness or incorrectness of a mathematical situation. They (ibid) proposed following three levels (each with sublevels) of student proof schemes: (i) external proof schemes, (ii) empirical proof schemes, and (iii) analytical proof schemes. External convictions are the proof schemes whose validity is accepted without questioning and that are learned from authority figures (such as textbooks or teachers). External proof schemes are themselves divided into ritual, authoritarian, and symbolic subschemes. In the experimental proof scheme, propositions examined by the influence of affective experiences or physical phenomena are accepted or rejected. The empirical proof schemes may require inductive or perceptual reasoning. Analytical type schemes are proofs that present the axioms and the theorems in a logic filter using the deductive method. Analytical proof schemes can be either transformational or axiomatic. An individual's proof approach for different propositions may be different, or more than one proof scheme may be used in the process of proving the truth of a proposition.