PPT: Mathematical Reasoning

 Page 1


Mathematical Reasoning
The Foundation of Algorithmics
Page 2


Mathematical Reasoning
The Foundation of Algorithmics
The Nature of Truth
In mathematics, we deal with 
statements that are True or False
This is known as “The Law of the 
Excluded Middle”
Despite the fact that multi-valued logics 
are used in computer science,
they have no place in mathematical 
reasoning
Page 3


Mathematical Reasoning
The Foundation of Algorithmics
The Nature of Truth
In mathematics, we deal with 
statements that are True or False
This is known as “The Law of the 
Excluded Middle”
Despite the fact that multi-valued logics 
are used in computer science,
they have no place in mathematical 
reasoning
The nature of mathematical proof
An individual once said to me “You 
know what the definition of a good 
proof is?”
“What?” I replied.
“It convinces you!” he said, quite proud 
of himself.
This individual was …
Page 4


Mathematical Reasoning
The Foundation of Algorithmics
The Nature of Truth
In mathematics, we deal with 
statements that are True or False
This is known as “The Law of the 
Excluded Middle”
Despite the fact that multi-valued logics 
are used in computer science,
they have no place in mathematical 
reasoning
The nature of mathematical proof
An individual once said to me “You 
know what the definition of a good 
proof is?”
“What?” I replied.
“It convinces you!” he said, quite proud 
of himself.
This individual was …
Mathematical Proof II
Dead Wrong!
Personal certitude has nothing to do with 
mathematical proof.
The human mind is a fragile thing, and 
human beings can be convinced of the most 
preposterous things.
Page 5


Mathematical Reasoning
The Foundation of Algorithmics
The Nature of Truth
In mathematics, we deal with 
statements that are True or False
This is known as “The Law of the 
Excluded Middle”
Despite the fact that multi-valued logics 
are used in computer science,
they have no place in mathematical 
reasoning
The nature of mathematical proof
An individual once said to me “You 
know what the definition of a good 
proof is?”
“What?” I replied.
“It convinces you!” he said, quite proud 
of himself.
This individual was …
Mathematical Proof II
Dead Wrong!
Personal certitude has nothing to do with 
mathematical proof.
The human mind is a fragile thing, and 
human beings can be convinced of the most 
preposterous things.
Mathematical Proof III
A good proof is one that starts with a 
set of axioms, and proceeds using 
correct rules of inference to the 
conclusion.
In many cases, we will proceed 
informally, but that does not mean that 
we will skip essential steps.