Download The Axiomatic Method

Mathematics And
The 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
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?tocId=9356243)
Where Do I Sign Up?
To axiomatize a subject, we need to specify three things:
TERMS: These are the fundamental objects we’re
talking about. We usually choose them to be meaningful
to us, but the point of the method is that we don’t 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 shouldn’t make
ANY assumptions that aren’t explicitly covered as
axioms.
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!
Just Any Old Rules of Inference?
It’s important that our rules be reasonable, that is, that
they don’t produce inconsistencies.
Here’s 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.
Euclidean Geometry
Terms: Point, Line, Plane.
Axioms: (usually called Postulates in this case)
1. It is possible to draw a straight line from any point to
any point.
2. It is possible to produce a finite straight line
continuously in a straight line.
3. It is possible to describe a circle with any center and
radius.
4. All right angles are equal to one another.
5. 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).
More Euclidean Geometry
It’s not quite true that we have to specify the rules of
inference. Euclid didn’t; he just used normal reasoning
and didn’t 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. Hilbert’s Foundations of
Geometry (1899) rectified those problems by making the
tacit assumptions explicit.)
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 aren’t “real-world” geometry.
(…or are they?)
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 isn’t 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(x+y)
x * 0 = 0;
x * Sy = (x * y) + x
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 didn’t work out in quite the way they’d
hoped; Gödel showed that arithmetic isn’t 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.
Nobody Said It Would Be Easy
Hofstadter’s MIU-System
Terms: M, I, U.
Axiom: MI.
Rules of Inference:
I. If xI is a theorem, so is xIU.
II. If Mx is a theorem, so is Mxx.
III. In any theorem, III can be replaced by U.
IV. UU can be dropped from any theorem.
This looks different from the other examples because it’s
only about producing strings. But we can make the
other ones just as formal:
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(x+y))
x (x * 0 = 0)
x y (x * Sy = (x * y) + x)
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)
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 that’s 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.
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 hadn’t discovered yet.
Another bonus: the axiomatic approach paved the way
for Gödel’s 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.
What’s In It For Us?
This quarter we’ll take an often axiomatic approach to three
different subjects: Group Theory, Set Theory, and Real
Analysis. (If it were possible to treat the Philosophy of
Mathematics axiomatically, we’d probably do that too.)
We have (or will soon have) ideas about what sets and
groups and ordered fields are, but we’ll 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!
FIN
Yes, this is a picture of the 2004 Isuzu Axiom.