1. To Infinity and Beyond!

2. Paradox A paradox is a seemingly consistent, logical argument that nonetheless ends with a ridiculous conclusion Like the story of Achilles and the Tortoise

3. Zeno’s Paradox A tortoise challenges Achilles to a race, telling him he will never win as long as the tortoise is given a head start. Achilles agrees and the tortoise sets off until he is the agreed distance ahead (point A) at which point Achilles starts running. But when Achilles reaches the point A, the tortoise has moved on a bit further to point B. Again when Achilles reaches B the tortoise is now at C and so on ad infinitum. Achilles CAN NEVER overtake the tortoise!

4. Paradoxes This is of course absurd but why? A paradox illustrates one of 2 things: – Either the premises are wrong – Or the conclusion is wrong You have to look more closely at the construction of the argument Which is exactly what Aristotle did

5. Maths and Logic Aristotle proposed a form of logic in which a conclusion is inferred from a series of premises. If the premises are true the conclusion MUST be true – a syllogism Premise 1: All humans are mortal Premise 2: All Filipinos are human Conclusion: All Filipinos are mortal If Premises 1 and 2 are correct the conclusion must be correct. Do you see any problems with this?

6. More Syllogisms God is love Love is blind Stevie Wonder is blind Stevie Wonder is God All Labradors are dogs All dogs are mammals All mammals are dogs All doctors are tired IB students are tired IB students are doctors I’m nothing (  ) Nothing is perfect God is perfect Therefore I’m God (and therefore I’m also Stevie Wonder and blind) Some sticky things are icky All toffee is sticky Therefore some toffee is icky

7. Logos, Ethos & Pathos As an aside…… Logic is part of Aristotle’s 3 “appeals” or Rhetoric (ways of persuading your audience): Logos – using reason Ethos – using authority Pathos – using emotion

8. And then Hume came along and spoilt everything! David Hume the 18 th Century Scottish philosopher came up with a big criticism with deductive reasoning. Any ideas? It’s called “The Induction Problem” Basically, how do we know that the premises are true? So we’re back to square one!

9. Then to make matters worse... Kurt Gödel (1931) argued that any proof requires axioms that are taken from outside of that area of Maths or requires axioms that are improvable (not false) in that system of Maths – the Incompleteness Theorem (e.g. the Russell Paradox: the barber is “one who shaves all those, and those only, who do not shave themselves.” So who shave the barber?) And so there cannot be a complete set of axioms that describes everything in Maths And a computer will never be smarter than a human And we can never really know ourselves

10. But back to paradoxes… Consider the following: – The set of prime numbers is a subset of the natural numbers – The set of even natural numbers is a subset of the natural numbers – The set of prime numbers has fewer members than the set of even natural numbers or the set of natural numbers – The set of even natural numbers has fewer members than the set of natural numbers – The set of primes has infinitely many members – The set of natural numbers has infinitely many members – The set of even natural numbers has infinitely many members How can you have 3 different sets each of a different size but all of which have infinitely many members? Are there certain concepts that do not make sense except in terms of Maths

11. Paradoxes v Fallacies A fallacy is an example of faulty processing to reach an absurd answer: Prove that 1 = 0 Let S = an infinite series of 1 – 1 + 1 – 1 + 1 – 1 … S = 1 – 1 + 1 – 1 + 1 – 1 … S = (1 – 1) + (1 – 1) + (1 – 1) … S = 0 + 0 + 0 …. S = 0 Also: S = 1 + (– 1 + 1) + (– 1 + 1) + (– 1 … S = 1 + 0 + 0… S = 1 There is nothing wrong with the maths!

12. Another fallacy Prove that 1 =2 Solve x – 5= x – 5 x – 1x – 2 The x – 5 on either side cancel out so x – 1=x – 2 Subtract x from each side -1 = -2 Or1 = 2

13. Homework 1.Come up with 2 syllogisms – One that is true – And one that is false 2.Comment on 2 other persons’ false syllogisms and try to explain the faulty reasoning 3.Discuss the following question: “Can everything be reduced to Maths?” 4.There is a big gap in development of the philosophy of Maths from the Greeks to the renaissance. Why might this be?