Mathematics and the Theory of Knowledge

Truth, Axioms and Proof: What do we really know?

It is all about logic…

A Logic Problem

All three signs on these boxes are incorrect. Which box contains the gold?

BOX A

The gold is not in Box C

BOX B

The gold is in this box

BOX C

The gold is in Box A

Think about how you solved the previous

logic problem and answer this question:Did you make any assumptions prior to

solving the problem? Explain these assumptions.

What we hope to learn todayThe role of axioms in mathematics and how

we use them in mathematical proofDifferent methods of mathematical proof Does proof imply truth in mathematics?Paradoxes Russell’s paradox and Godel’s Theorem

Proof in Mathematics

Mathematical ProofsProof in mathematics requires the absence of

any doubt about the truth of an argumentEvery step in any mathematical proof must

be valid according to a set of mathematical statements which are assumed true (called Axioms -we shall come to this soon!)

Consider the following “proof”Let x = 1Then x2-1 = x -1 (try substituting x=1 to check

this)

(x+1)(x-1) = x- 1 (factorizing using difference of squares)

x + 1 = 1 (dividing both sides by (x-1) )

2 = 1 (substituting x=1)

What went wrong?????? In groups of two or three, see if you can find a flaw in this “Proof”.

Proof: Rules of InferenceAll proofs depend on rules of inference:

these are logical statements which given a Proposition, provide some Implication

For example in normal arithmetic with x a real number we have the following Proposition:

2x = 4 The Implication from this statement is that

x=2

Mathematical ProofsThere are different methods of mathematical

proof which include:Proof by logical deductionProof by exhaustionProof by construction (or direct proof)Proof by mathematical inductionProof by contradiction

Let’s explore some of these methods of Proof!

Proof by ExhaustionThis method depends on testing every

possible case of a theorem

Example – Consider the theorem “all students at SAS have a foreign passport or identification card”

How could we prove this by exhaustion?Do you see any potential problems with this

method of proof in mathematics?

Proof by Construction (direct)In this proof, the statement asserted to exist

is explicitly exhibited or constructed.

Consider the proof of Pythagoras’ theorem given by the diagrams on the sheet. See if you can make sense of the proof of

a2 + b2 = c2

Hint- let the sides of the triangle be a, b, c and look at areas!

Proof by InductionThis method works by first proving a specific

example to be true (eg n=1)Then assume the statement is true for a

general value of n and proving it is true for the next value of n+1

HL math students will be learning this one!!!! Everyone else can just take their word for it that it works.

Proof by Logical DeductionThis method involves a series of propositions,

each of which must have been previously proved (or be evident without proof) or follow by a valid logical argument from earlier propositions in the proof

We have looked extensively at deduction already!

Proof by ContradictionThis works by assuming the negative of what

one is trying to prove and deriving a contradiction

Note that methods of proof cannot be classified easily as many proofs adopt techniques from more than one of the previously discussed methods

Axioms and Theorems

Axioms in mathematicsMathematics is based on axioms –these are

“facts” that are assumed to be true and accepted without proof since they are considered to be “self-evident”

Examples of axioms: “Things equal to the same thing are equal to each other” or in mathematical terms:

if y=a and x=a then y=xSome of the most famous axioms are found in

geometry and were first stated by Euclid (~300 BC)

Axioms in mathematicsIn addition to being self-evident, Axioms

should be:1. Consistent –it should not be possible to

derive a logical contradiction from the axioms

2. Independent- it should not be possible to derive one axiom from another

3. Fruitful –we would like to be able to derive many theorems from the set of axioms

Mathematics –built from AxiomsEvery step in a proof rests on the axioms of

the mathematics that is being usedStatements that are proven from Axioms are

called THEOREMSOnce we have a theorem, it becomes a

statement that we accept as true and which can be used in the proof of other theorems

Axioms and Theorems –an exampleConsider the following four axioms (A1-A4)

which define a formal system K1. Some DERs are KIN-DERs and some DERs are TEN-

DERs, but no DER is both a KIN-DER and a TEN-DER2. The result of GARring any number of DERs is a DER, and

this does not depend on the order of the DERs.3. When two KIN-DERs or two TEN-DERs are GARred, the

result is a KIN-DER.4. When a KIN-DER and a TEN-DER are GARred the result

is a TEN-DER.

Work in groups and answer the questions on the sheet about this system

Mathematics – Is it all true?Pure Mathematics is a quest for a structure

that does not contain internal contradictions.

Our system is built upon the Axiom Theorem process.

In groups, discuss this question: “Do you see any flaws in the way mathematics has been built using axioms and theorems?”

Are mathematical theories consistent?Bertrand Russell (1872-1970) discovered a

paradox in Cantor’s set theory, which had come to play a fundamental role in mathematics.

This paradox , hidden in a well-established branch of mathematics came as a major shock to the mathematical community.

First let’s explore this concept of a paradox

Paradoxes and Other Weirdness in Mathematics

ParadoxesA paradox is a logical inconsistency One of the most famous paradoxes is called the

Liar’s paradox and was formulated by Epimenides (~500 BC)

Epimenides was born and lived on the island of Crete where the citizens were know as Cretan.

It is claimed he made the statement : “All Cretans are liars”

Discuss the logical inconsistency in this statement.

Some more paradoxesInvestigate the following situations:1. “The next sentence is false. The previous

sentence is true.”2. The barber in a certain village is a man who

shaves all men who do not shave themselves. Does the barber shave himself?

A visual Paradox: the work of Escher

A Historical Link-David Hilbert

One of the greatest mathematicians of the 20th century

Proposed a list of 23 famous unsolved mathematical problems in 1900

One of his problems was to find a logical foundation for any system using a set of mathematical axioms

The idea of Hilbert’s Problems has been continued to this day with the Clay Millennium Prize Problems -7 problems –solve one and get one million US$

Bertrand Russell’s Paradox

Bertrand Russell was a British logician, mathematician, philosopher and writer who discovered a paradox in mathematical set theory in 1901. Russell looked in detail at the basic set axioms of mathematics. The existence of sets are generally regarded as axiomatic in all mathematical structures.

Bertrand Russell’s ParadoxRussel’s Paradox had a profound

effect upon the mathematical establishment –the “truth” of mathematics theories would no longer be unquestionable

Mathematics and TruthJust how “true” can mathematics be if there

can exist paradoxes?Is mathematics only “true” for restricted

cases and not “true” always?Is Russell’s paradox just a unique example?

Godel’s Incompleteness Theorems

Kurt Godel was a brilliant Austrian-Czech logician who came up with two Incompleteness theorems in 1931. These theorems had a profound effect on pure mathematics.

A short movie about Godel

Godel’s Incompleteness TheoremsGodel’s theorems were very complicated but

in essence, he was able to show:1. That the consistency of any formal axiomatic

system cannot be proved in that system but only in a ‘larger’ system (which cannot again prove its own consistency)

2. That a mathematical theory, such as the arithmetic of natural numbers, cannot be completely derived from a finite set of axioms; in any such system, some theorems can neither be proved nor refuted

Godel’s Incompleteness TheoremsWhat are the consequences of Godel’s

theorems?Axioms do not provide a ‘solid’ enough

foundation on which to build our mathematics; if we cannot show them to be free from contradiction, we cannot use them to guarantee truth even within a theory

Mathematical truth is something that goes beyond mere man-made constructions –there are some things in mathematics which simply cannot be proven to be “true”

Can we really be sure about anything?

Recent scientific and mathematical discoveries have shown that the universe seems to be far more uncertain, random and unknowable. Perhaps the future will provide answers. Or perhaps not.

The Last Word from Calvin and Hobbes