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 B The gold is in this box BOX C The gold is in Box A BOX A The gold is not in Box C

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 today The role of axioms in mathematics and how we use them in mathematical proof Different methods of mathematical proof Does proof imply truth in mathematics? Paradoxes Russell’s paradox and Godel’s Theorem

Proof in Mathematics

Mathematical Proofs Proof in mathematics requires the absence of any doubt about the truth of an argument Every 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 = 1 Then 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 Inference All 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 Proofs There are different methods of mathematical proof which include: Proof by logical deduction Proof by exhaustion Proof by construction (or direct proof) Proof by mathematical induction Proof by contradiction Let’s explore some of these methods of Proof!

Proof by Exhaustion This 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 Induction This 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 Deduction This 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 Contradiction This 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 mathematics Mathematics 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=x Some of the most famous axioms are found in geometry and were first stated by Euclid (~300 BC)

Axioms in mathematics In addition to being self-evident, Axioms should be: Consistent –it should not be possible to derive a logical contradiction from the axioms Independent- it should not be possible to derive one axiom from another Fruitful –we would like to be able to derive many theorems from the set of axioms

Mathematics –built from Axioms Every step in a proof rests on the axioms of the mathematics that is being used Statements that are proven from Axioms are called THEOREMS Once 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 example Consider the following four axioms (A1-A4) which define a formal system K Some DERs are KIN-DERs and some DERs are TEN-DERs, but no DER is both a KIN-DER and a TEN-DER The result of GARring any number of DERs is a DER, and this does not depend on the order of the DERs. When two KIN-DERs or two TEN-DERs are GARred, the result is a KIN-DER. 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

Paradoxes A 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 paradoxes Investigate the following situations: “The next sentence is false. The previous sentence is true.” 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 Paradox Russel’s Paradox had a profound effect upon the mathematical establishment –the “truth” of mathematics theories would no longer be unquestionable

Mathematics and Truth Just 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 Theorems Godel’s theorems were very complicated but in essence, he was able to show: 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) 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 Theorems What 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