Mathematics AndThe Axiomatic

Method

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

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

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.

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 AttireTerms: 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 itwith the ashes of a loved one.

Or maybe what's left of the chickens are in that urn. Maybethe 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. Andthen 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 History 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.