Transcript creating mathematical knowledge

What distinguishes maths?
Mathematical method
1. Mathematics is a FORMAL system of
knowledge.
2. The foundation of maths is AXIOMS.
3. If you apply RULES OF INFERENCE to the
axioms, you create mathematical knowledge,
called THEOREMS.
2. The foundation of maths is AXIOMS.
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Axioms are fundamental laws.
Axioms are obtained by reflecting rationally
on what we know.
They are examples of a priori knowledge,
accepted by all mathematicians.
Mathematics is based on axioms
Many mathematicians accept them as given,
they know they are the foundation of maths
but would find them hard to define.
Examples of axioms
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Euclid's elements
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Non-euclidean geometry
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Seven axioms of set theory (ZermeloFrankel)
Peano's Postulates
Peano's Postulates
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Define the fundamental laws of the number
system
These are the fundamental rules of arithmetic
Peano says there are such things as
numbers and they can be defined by 5
axioms
1. 0 is a number
2. Every number has at least one and at most
one successor which is a number
3. 0 is not the successor of any number
4. No two numbers have the same successor
5. Whatever is true of 0, and is also true of the
successor of any number when it is true of
that number, is true of all numbers
If you apply RULES OF INFERENCE to the
axioms, you create mathematical knowledge,
called THEOREMS.
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Inference is the forming of conclusions from
the information available
Rules of inference are those rules which
mathematicians apply deductively to the
mathematical information available to them,
the axioms
If....then....
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A well-known rule of inference
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An arithmetical example:
if 1+6=7 and 5+4=9
then (1+6) + (5+4) = 7 + 9
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An algebraic example
If x=y and p=q
then x+p = y+q
Theorems
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A statement created by deductively applying
the rules of inference to axioms
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A statement of mathematical knowledge
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Example: Pythagoras Theorem
The square on the hypotenuse is equal to the
sum of the squares on the other 2 sides
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This theorem is based on the 10 axioms of
Euclid's Elements (but mathematicians take
these as given)
Are we happy?
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Does it all seem a bit vague?
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Are we relying on “givens”?
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Where have the axioms come from?
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Is this why we have proof?
The mathematical method
a deductive process
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Define axioms
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Then apply rules of inference
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Then create theorems
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Then apply rules of inference
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Then create more theorems
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Then apply rules of inference
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Then create even more theorems and so on
ad infinitum....