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Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated. [5] These results are often described as deep. While it is difficult to find universal agreement on whether a result is deep, some examples are more commonly cited than others.
This stage occurs before a mathematics student has learned mathematical formalism. Mathematics is learned in an informal manner that often incorporates examples, fuzzy notions and hand-waving. This stage is founded mostly in intuition and the emphasis is more on computation than theory. Natural talent is important at this level.
Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines.
Pure mathematics studies the properties and structure of abstract objects, [1] such as the E8 group, in group theory. This may be done without focusing on concrete applications of the concepts in the physical world. Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may ...
Case study 2 looks at people who use mathematics to change and alter their recipes while cooking. His emphasis here is the use of mathematical knowledge without formal instruction, which he considers to be the central mathetic moral of the study. Papert states "The central epistemological moral is that we all used concrete forms of reasoning.
Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc. This may also include the philosophical study of the relation of this framework with reality. [1]
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