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Development of Theorems
Name: Denis
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
How do mathematicians find theorems like the Pythagorean
theorem? Is it just random playing numbers of something they thought of?
Replies:
Denis,
There are many situations that can lead to mathematical theorems. It
can result from noticing a pattern and working out details to see what
the pattern actually is. It can result from needing to discover how
several things relate and then looking for this relationship. It can
result from just being curious about something that originally seemed
unimportant. Theorems are seldom originally intended to be theorems.
This cannot happen until the mathematician knows an important
relationship exists. In all cases, a good imagination is probably the
most necessary factor for creating a theorem.
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
The development of mathematical theorems depends upon a set of ELEMENTS
or DEFINITIONS of the terms involved (these could be: points, lines,
angles, numbers, etc...etc.) and a set of RULES or OPERATIONS of these
elements that defines how the elements behave with respect to one another.
Some new RELATION may follow from these elements and operations --
THEOREMS. Just "how" a mathematician "discovers" these theorems is quite
diverse. Some of the new relationships may be "obvious" -- others may be
quite "obscure". Mathematical PROOF consists in deriving the new relations
from the DEFINITIONS and OPERATIONS so that all of the ELEMENTS and RULES
are not violated. Sometimes the chain of logic is simple and
straightforward, and other times the chain of logic can cover dozens of
pages.
In some cases the mathematician may "see" the result as being "true" and
then set about showing (PROVING) that the result follows logically from
the definitions and rules. In other cases the mathematician may notice
some regularity in the behavior of the elements and then set about proving
that the observation is a logical consequence of the definitions and rules
of operation.
For the mathematician the objective is to show that the new relation
(THEOREM) is a necessary consequence of the logical relation between the
ELEMENTS/DEFINITIONS and the RULES/OPERATIONS. Establishing this
relationship can be easy or tricky, obvious or obscure. A couple of
examples: 1. There are an infinite number of prime numbers (prime numbers
are divisible those that are divisible only by the number itself and the
number "1").
2. Every even number greater than "2" is the sum of two prime numbers --
examples: 4 = 3+1, 14 = 11 + 3, etc.
Just showing examples is not sufficient. One has to show that the result
is a necessary consequence of the "counting" numbers and the rules of
arithmetic.
Vince Calder
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Update: June 2012
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