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Mathematics
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We tend to think of math as an island of
certainty in a vast sea of subjectivity,
interpretability and chaos.
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What is it?
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The search for abstract patterns. These patterns
are all over the place: for any object you can
name, you can take two of it and then add two
more of it, and you will have four of it. If you
take any circle – no matter the size – and divide its
circumference by its diameter and you always get
the same number (roughly 3.14).
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In this sense there seems to be an underlying order
to things. This means that math seems to give us
certainty, and also has very practical applications.
“The book of Nature is written in the language
of Mathematics”
– Galileo Galilei (1564-1642)
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It’s precisely this certainty that can be scary,
however. If you make a mistake in a math
problem, you’re just wrong, and can be
shown to be wrong. No one will say “What
an interesting interpretation!”
The Mathematical Paradigm
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A good definition of mathematics is “the science
of rigorous proof.”
Early cultures developed what we now refer to as
“cookbook mathematics” – useful recipes for
solving practical problems. Mathematics as the
science of proof dates back to the Greeks.
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Euclid (300 BCE) was the first person to
organize geometry into a rigorous body of
knowledge. The geometry we learn today is
basically Euclidean geometry.
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The model of reasoning developed by
Euclid is called a formal system.
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A formal system has three key elements:
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1.
2.
3.
axioms
deductive reasoning
theorems
Axioms
Axioms are the starting points, or basic
assumptions. Until (at least) the 19th
century, the axioms of mathematics were
considered to be self-evident truths. We
still have to pretty much accept them as true
(otherwise we risk getting caught in an
infinite regress….).
There are four traditional requirements
for a set of axioms. They have to be….
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1.
Consistent. If you can deduce both p and non-p from the same
set of axioms, they are not consistent. Inconsistency is bad – once
you’ve let it into a system you can prove literally anything.
2.
Independent. You should begin with the smallest possible
number of axioms, and they should be very basic. As soon as an
axiom gets complicated enough that it can be deduced by another
axiom it is too complex.
3.
Simple. Since axioms are accepted without further proof, they
ought to be as simple and clear as possible.
4.
Fruitful. A good formal system should enable you to prove as
many theorems as possible using the fewest number of axioms.
Starting with a few basic axioms (a point is that which has no part, a line
has length but no breadth, etc…), Euclid postulated the following five
axioms:
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1. It shall be possible to draw a straight line
joining any two points.
2.
A finite straight line may be extended without
limit in either direction.
3.
It shall be possible to draw a circle with a
given center and through a given point.
4.
All right angles are equal to one another.
5.
There is just one straight line through a given
point, which is parallel to a given line.
Deductive Reasoning
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We discussed this last semester.
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(1) All human beings are mortal
(2) Socrates is a human being
(3) Therefore Socrates is mortal.
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In mathematics, axioms are like premises, and
theorems are like conclusions.
Theorems
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Using his five axioms and deductive reasoning,
Euclid derived various simple theorems:
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1.
2.
3.
4.
Lines perpendicular to the same line are parallel.
Two straight lines do not enclose an area.
The sum of the angles of a triangle is 180 degrees.
The angles on a straight line add up to 180 degrees.
How can a mathematical proof be
“beautiful”?
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Although the person on the street does not usually
associate mathematics with beauty, we can get a
sense of what a mathematician means by a
“beautiful” or “elegant” solution by considering a
couple of simple examples.
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1.
There are 1024 people in a knock-out tennis tournament.
What is the total number of games that must be played before
a champion can be declared?
2.
What is the sum of integers from 1 to 100?
Mathematics and the Ways of Knowing
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We know about it in terms of reason, but
what about the others?