Mathematics And The Axiomatic Method

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Mathematics AndThe Axiomatic Method
As far as the laws of mathematics refer to
reality, they are not certain, and as far as they
are certain, they do not refer to
reality. Albert Einstein
Mathematics may be defined as the subject in
which we never know what we are talking about,
nor whether what we are saying is true.
Bertrand Russell
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What Is It?
The Axiomatic Method is the procedure by which an
entire science or system of theorems is deduced
in accordance with specified rules by logical
deduction from certain basic propositions
(axioms), which in turn are constructed from a
few terms taken as primitive. These terms may be
either arbitrarily defined or conceived according
to a model in which some intuitive warrant for
their truth is felt to exist.
(http//concise.britannica.com/ebc/article?tocId9
356243)
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Where Do I Sign Up?
To axiomatize a subject, we need to specify three
things TERMS These are the fundamental
objects were talking about. We usually choose
them to be meaningful to us, but the point of the
method is that we dont get to import our
intuitions into our proofs. AXIOMS These are
the statements (about the terms) that we assume
from the get-go are true. They are theorems
they are also freebies. Again, we usually pick
axioms that make sense to us, but we shouldnt
make ANY assumptions that arent explicitly
covered as axioms.
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That Was Only Two Things
RULES OF INFERENCE These are the rules that
tell us how to use theorems to make other
theorems. In general, if we set our system up
right, we can get by with just one rule of
inference Modus Ponens (MP). Modus Ponens If
A is a theorem and A ? B is a theorem, then B is
a theorem. So Modus Ponens just lets us make
obvious conclusions when there are implications
lying around. Snappy!
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Just Any Old Rules of Inference?
Its important that our rules be reasonable, that
is, that they dont produce inconsistencies.
Heres an example of an unreasonable rule of
inference RULE OF IDIOCY If A is a theorem,
then so is not A. If we have even one axiom A,
this rule makes our system inconsistent the
system claims that A and not A are both true,
which is impossible.
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Euclidean Geometry

• Terms Point, Line, Plane.
• Axioms (usually called Postulates in this case)
• It is possible to draw a straight line from any
  point to any point.
• It is possible to produce a finite straight line
  continuously in a straight line.
• It is possible to describe a circle with any
  center and radius.
• All right angles are equal to one another.
• Given a line and a point not on the line, there
  is exactly one parallel through the point (that
  is, exactly one line through the point that is
  parallel to the given line).

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More Euclidean Geometry
Its not quite true that we have to specify the
rules of inference. Euclid didnt he just used
normal reasoning and didnt get into trouble.
(Actually, he did get into trouble with
19th-century mathematicians, but it had nothing
to do with rules of inference he tacitly assumed
several facts that he should have written down as
axioms. Hilberts Foundations of Geometry (1899)
rectified those problems by making the tacit
assumptions explicit.)
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Non-Euclidean Geometry
Terms Point, Line, Plane. Axioms 1-4 from
before, and we replace 5 with something
else. If we say there are NO parallels, we get
Riemannian Geometry! If we say there are
SEVERAL parallels, we get Hyperbolic
Geometry! Note that these arent real-world
geometry.
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Arithmetic/Number Theory
Terms Number, 0, S, , . Axioms 0 is a
number. If x is a number, Sx (the successor of x)
is a number. 0 isnt Sx for any number x. If Sx
Sy, then x y. Induction If a set A of numbers
contains 0 and contains Sx for every x in A,
then A contains every number. For all numbers x
and y, we have x 0 x x Sy S(xy) x
0 0 x Sy (x y) x
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Principia Mathematica
This is another famous example, in which Russell
and Whitehead tried to set down axioms for
Mathematical Logic (and, by extension, for all of
Mathematics).
The project didnt work out in quite the way
theyd hoped Gödel showed that arithmetic isnt
even fully axiomatizable, let alone all of
Mathematics. Yikes! Still, it was a big step
forward in mathematical thought it was a proof
of concept sort of deal that put part of
Mathematics on sturdy axiomatic ground.
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Nobody Said It Would Be Easy
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Hofstadters MIU-System

• Terms M, I, U.
• Axiom MI.
• Rules of Inference
• If xI is a theorem, so is xIU.
• If Mx is a theorem, so is Mxx.
• In any theorem, III can be replaced by U.
• UU can be dropped from any theorem.
• This looks different from the other examples
  because its only about producing strings. But
  we can make the other ones just as formal

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Arithmetic/Number Theory In Formal Attire
Terms Number, 0, S, , . Axioms ?x (x
0) ?x ?y (y Sx) ?x (0 ? Sx) ?x ?y (Sx Sy ? x
y) A(0) ?x (A(x) ? A(Sx)) ? ?x A(x) ?x (x
0 x) ?x ?y (x Sy S(xy)) ?x (x 0 0) ?x
?y (x Sy (x y) x)
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This Talk Needs More Pictures
Chicken Liver Bake enjoy it with the ashes of a
loved one. Or maybe what's left of the chickens
are in that urn. Maybe the chickens were your
loved ones. But chickens never love back
enough. And that's why you have to KILL them.
And eat their livers ritualistically. And then
they're a part of you forever.
Forever. (http//www.candyboots.com/wwcards.html)
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Why Bother?
Certainty If you believe that your axioms are
correct and you believe that your rules of
inference are valid, then you can confidently
believe the theorems of the system. Said another
way, if you derive a theorem thats false, then
either one of your axioms is false or one of your
rules of inference is faulty. Applicability If
you find another system that satisfies the
axioms, then ALL results that follow from the
axioms are true about your system.
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Hofstadters pq-System
Terms p, q, -. Axioms xp-qx- is an axiom
whenever x is composed of hyphens only. Rule of
Inference If x, y, and z are strings containing
only hyphens, and if xpyqz is a theorem, then
xpy-qz- is a theorem. So, for example, -p-q--
and --p-q--- are both axioms, and from them we
can use the rule to get the theorems -p--q--- and
---p-q----.
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But What Does It All MEAN?
How can we make sense of the theorems -p-q--,
--p-q---, -p--q---, ---p-q----, and so on? -
1 p q - 1 p q taken
from - apple p horse q happy So
-p-q-- becomes apple horse apple happy apple
apple. This is prolly a valid interpretation if
youre a horse. The point if the axioms and
rules make sense under an interpretation, then
all theorems must hold under that interpretation.
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One More Reason To Bother
Mechanization If we axiomatize a subject
formally enough, we can have computers derive
theorems for us. Sometimes (not very often,
actually) they obtain results that humans hadnt
discovered yet. Another bonus the axiomatic
approach paved the way for Gödels Incompleteness
Theorems. His idea was to encode theorems as
numbers but that only works if the theorems are
formal objects, and his proof is specifically
about formal systems.
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Whats In It For Us?

• This quarter well take an often axiomatic
  approach to three different subjects Group
  Theory, Set Theory, and Real Analysis. (If it
  were possible to treat the History of Mathematics
  axiomatically, wed probably do that too.)
• We have (or will soon have) ideas about what sets
  and groups and ordered fields are, but well try
  to distill as much of what we want to say about
  them as we can in an axiomatic setting.
• So fasten your seat belts!

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FIN
Yes, this is a picture of the 2004 Isuzu Axiom.