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Mathematics Research
Karim Naous and Mohamad Rishani
Philosophy
13 December 2016
Task:
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The Definition of Mathematics
Origin of Mathematics
The two different schools of Mathematics (Inductive and Deductive)
The relation of Mathematics to Science
Mathematics:
Mathematics is the science of numbers and their operations, interrelations,
combinations, generalizations, and abstractions and of space configurations and
their structure, measurement, transformations, and generalizations 1
Origin of mathematics:
The history of mathematics is nearly as old as humanity itself. Since antiquity,
mathematics has been fundamental to advances in science, engineering, and
philosophy. It has evolved from simple counting, measurement and calculation, and the
systematic study of the shapes and motions of physical objects, through the application
of abstraction, imagination and logic, to the broad, complex and often abstract discipline
we know today. From the notched bones of early man to the mathematical advances
brought about by settled agriculture in Mesopotamia and Egypt and the revolutionary
developments of ancient Greece and its Hellenistic empire, the story of mathematics is
a long and impressive one. The East carried on the baton, particularly China, India and
the medieval Islamic empire, before the focus of mathematical innovation moved back
to Europe in the late Middle Ages and Renaissance. Then, a whole new series of
revolutionary developments occurred in 17th Century and 18th Century Europe, setting
the stage for the increasing complexity and abstraction of 19th Century mathematics,
and finally the audacious and sometimes devastating discoveries of the 20th Century. 2
1
2
https://www.merriam-webster.com/dictionary/mathematics
http://www.storyofmathematics.com/
Inductive Mathematics:
Mathematical induction is a mathematical proof technique, most commonly used to
establish a given statement for all natural numbers, although it can be used to prove
statements about any well-ordered set. It is a form of direct proof, and it is done in two
steps. The first step, known as the base case, is to prove the given statement for the
first natural number. The second step, known as the inductive step, is to prove that the
given statement for any one natural number implies the given statement for the next
natural number. From these two steps, mathematical induction is the rule from which
one infers that the given statement is established for all natural numbers. The method
can be extended to prove statements about more general well-founded structures, such
as trees; this generalization, known as structural induction, is used in mathematical logic
and computer science. Mathematical induction in this extended sense is closely related
to recursion. Mathematical induction, in some form, is the foundation of all correctness
proofs for computer programs.3
Example of Inductive Reasoning:
A great example of inductive reasoning is the process a child goes through when
introduced to something new. If a child has a dog at home, she knows that dogs have
fur, four legs and a tail. If a child were to be introduced to a cat, that child may very well
assume the cat is a dog. Why? Because the cat has fur, four legs and a tail. In the
child's experience, this means dog. The child induces from her specific scenario
something about a much larger population.4
Deductive Mathematics:
Inductive reasoning typically leads to deductive reasoning, the process of reaching
conclusions based on previously known facts. The conclusions reached by this type of
reasoning are valid and can be relied on. For example, you know for a fact that all
pennies are copper colored. Now, if your friend gave you a penny, what can you
conclude about the penny? You can conclude that the penny will be copper colored.
You can say this for certain because your statement is based on facts.
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4
https://en.wikipedia.org/wiki/Mathematical_induction
http://study.com/academy/lesson/reasoning-in-mathematics-inductive-and-deductive-reasoning.html
Examples of Deductive Reasoning:5
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All men are mortal. Joe is a man. Therefore Joe is mortal. If the first two
statements are true, then the conclusion must be true.
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Bachelors are unmarried men. Bill is unmarried. Therefore, Bill is a bachelor.
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To get a Bachelor's degree at Utah State University, a student must have 120
credits. Sally has more than 130 credits. Therefore, Sally has a bachelor's
degree.
The Relation of Mathematics to Science:
One of the first analysis’ of the relationship between mathematics and other sciences
belongs to the ancient Greek philosopher Aristotle, who wrote about the
interdependence of mathematics and other exact sciences in his work "Physics."
Scholars such as Galileo Galilei and Isaac Newton also acknowledged this relationship.
Nowadays, mathematics is at the basis of many physics discoveries, such as the
superstring theory.
5
http://ocw.usu.edu/English/introduction-to-writing-academic-prose/inductive-and-deductive-reasoning.html