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?
“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
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