Mathematics Reasoning Text

Mathematics deals with the definition, properties and manipulation of “objects” such as numbers, lines and shapes. Are these mathematical objects real? The ancient Greeks argued that they are an ideal reality independent of time and space. Their properties exist independently of us and are just waiting to be discovered. However, others like Albert Einstein have argued that we invent these objects (as concepts). They are the products of our thinking and have only the properties we ascribe to them. Some even argue that there are no mathematical objects at all; all we have is an elaborate game with a set of rules and formulas expressed in symbols. Regardless of the view of mathematics, all would agree that mathematics is essential to our understanding and manipulation of the world around us.

How the field of mathematics has developed and continues to develop is related to the different views of the nature of mathematics and can be illustrated (in part) by examining various common forms of mathematical reasoning. In other words, what thinking processes do mathematicians and people who use mathematics, like scientists, employ? The forms of mathematical reasoning discussed here are somewhat simplified and include the following:

  • reasoning by deduction
  • reasoning by induction
  • reasoning by analogy/models

An understanding of mathematical reasoning will help to answer the questions: how is mathematical knowledge created? how is it tested? how is it used?

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“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?”Albert Einstein