Mathematics and Certainty Formalism, Platonism, and Empiricism

What we hope to learn today What can we “know” in mathematics?

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

Rules of Inference -4 main rules 1. The rule of detachment: From a is true and a implies b we can infer that b is true. (a , b are propositions) Example If the following propositions are true: It is raining. If it is raining, I will take an umbrella. We can infer that I will take an umbrella.

Rules of Inference -4 main rules The rule of syllogism: From a implies b is true and b implies c is true, we can conclude the a implies c is true. (a, b, c are propositions) Example: If we accept as true that: -if x is an odd number then x is not divisible by 4 and, -if x is not divisible by 4 then x is not divisible by 16 We can infer that the proposition -if x is an odd number then x is not divisible by 16 Is also true.

Rules of Inference -4 main rules The rule of Equivalence –at any stage in the argument we can replace any statement by an equivalent statement. Example: If x is a whole number then the statement “x is even” could be replaced by the statement “x is divisible by 2”

Rules of Inference -4 main rules The rule of substitution –If we have a true statement about all the elements of a set, then the statement is true about any individual member of the set. Example: If we accept the statement “all lions have sharp teeth” then Leo, who is a lion, must have sharp teeth.

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?

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

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 The last proposition is what is to be proved

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

Proof that √2 is Irrational If it were rational, it could be expressed as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. But if a/b = √2, then a2 = 2b2. Therefore a2 must be even. Because the square of an odd number is odd, that in turn implies that a is even. This means that b must be odd because a/b is in lowest terms. On the other hand, if a is even, then a2 is a multiple of 4. If a2 is a multiple of 4 and a2 = 2b2, then 2b2 is a multiple of 4, and therefore b2 is even, and so is b. So b is odd and even, a contradiction. Therefore the initial assumption—that √2 can be expressed as a fraction—must be false.

Axioms and Theorems

Axioms in mathematics Mathematics is based on axioms –these are “facts” that are assumed to be true since they are considered to be “self-evident” An axiom is a statement that is accepted without proof 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 Euclid, in his famous Elements in the 3rd century B.C. stated five axioms from which could be derived all the theorems of plane geometry. These axioms included the following: All right angles are equal to one another A straight line can be drawn from one point to any other point A circle can be produced given any centre and a radius Etc.

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

Axioms in mathematics Most axiom systems have been based on the notion of a ‘set’, meaning a collection of objects –an example of a set axiom is the ‘axiom of specification’ Example: Assume that the set ‘People of China’ is defined. If we impose the condition that the members of this set must be female, then this new set of Chinese females is defined.

Axioms in mathematics A second example of a set axiom is the ‘axiom of powers’ Example: For each set, there exists a collection of sets that contains amongst its elements all the subsets of the original set. -consider the set of all the cats in Egypt - there must be a set that contains all the female cats in Egypt -there must be a set that contains all the cats with green eyes in Egypt -there must be a set that contains all the Egyptian cats with black tails etc

Axioms in mathematics Mathematics has in some sense, been a search for the smallest possible set of consistent axioms. “Pure mathematics is concerned with absolute truth only in the sense of creating a self-consistent structure of thinking”

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? That the consistency of mathematical theories, even ones that have become well established, cannot be taken for granted was demonstrated very clearly in 1902 when 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” In pairs, discuss the logical inconsistency in this statement.

Some more paradoxes In pairs, 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

Bertrand Russell’s Paradox Bertrand Russell was a British logician, mathematician, philosopher and writer who discovered a paradox in mathematical set theory. 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 Russell’s paradox arose from the basic question: Can we make a set that contains ‘everything’? This would not seem to be a problem since we could just include everything we encounter in the universe in our set. However, problems arise…. Russell’s paradox concerns a set R defined in the following way: R is the set of all sets which are not members of themselves

Bertrand Russell’s Paradox Russell’s paradox can be expressed in a less mathematical form by relating it to library catalogues. Imagine a library in which there are two catalogues, one of which lists all the books in the library which somewhere refer to themselves and the other, all the books in the library which make no mention of themselves. Russell posed this question: In which catalogue is the second catalogue itself to be listed?

Bertrand Russell’s Paradox The most commonly accepted result of Russell’s paradox is that we must be very careful when we talk about sets of everything We need to work within a carefully defined universal set, chosen to be appropriate to the mathematics that we are undertaking

Mathematics and Reality Discuss this question in groups: Is mathematics a fundamental part of the real world or does mathematics simply represent the real world?

Mathematics and Reality So what is the relationship between mathematics and the real world? simply a game in which sets of axioms are developed into theorems, for its own sake –although some of the theories turn out to have some practical use OR Knowing some mathematical truth requires that we be able to prove it in its theory; our “proof” of the reality of the theories comes from testing the mathematics in practice –our knowledge of other parts of mathematics is less conclusive the further it is from what has been tested

Mathematics and Reality Unfortunately, for those proponents of the first view, the discovery of Godel’s Incompleteness Theorems damaged beyond repair this formalist view of mathematics.

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.

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

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