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F OR OUR GUESTS … This is a Liceo, which is not a vocational school This is a scientific Liceo, so our students focus upon sciences So, we don’t study science from a technical point of view, but we try to stress upon a theoretical study All of our students study subjects such as philosophy and latin

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We therefore try to carry on the study of the scientific subjects so that our students can develop a strong capability of abstraction and the awareness of comprehensive bits of knowledge. What we’ll show you today is just a humble example of how we try to do it.

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F OR OUR STUDENTS : WHAT ARE YOU GOING TO LEARN TODAY ? What is a (logical) paradox? Some examples of paradoxa An early paradox (which dates back to ancient Greece): Achilles and the tortoise where is the problem from a mathematical point of view what is the role of this paradox a modern mathematical explanation

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P ART 1: PARADOXES

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PARADOXES WHAT IS A PARADOX?

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WHAT IS PARADOXICAL?

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A PARADOXICAL SITUATION PARADOXES

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A PARADOXICAL UTTERANCE PARADOXES

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A PARADOXICAL IMAGE PARADOXES

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PARADOXON PARA DOXA contrary to, different from opinion common knowledge common sense From dokein : to appear, seem,think

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PARADOX "statement contrary to common belief or expectation” “statement seemingly absurd yet really true” “statement that is seemingly self- contradictory yet not illogical or obviously untrue” "statement contrary to common belief or expectation” “statement seemingly absurd yet really true” “statement that is seemingly self- contradictory yet not illogical or obviously untrue”

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SOME KINDS OF PARADOX SELF REFERENCE THE LIAR This statement is false Could we state whether this sentence is either true or false? THE LIAR This statement is false Could we state whether this sentence is either true or false?

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SOME KINDS OF PARADOX VICIOUS CIRCULARITY THE INFINITE REGRESS The following sentece is true The previous sentence is false Could we state whether these sentences are either true or false? THE INFINITE REGRESS The following sentece is true The previous sentence is false Could we state whether these sentences are either true or false?

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SOME KINDS OF PARADOX CATEGORICAL PARADOX THE BARBER The barber is a man in town who shaves all those, and only those men in town who do not shave themselves Who shaves the barber? THE BARBER The barber is a man in town who shaves all those, and only those men in town who do not shave themselves Who shaves the barber?

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V IDEO The infinite-hotel Link: http://www.youtube.com/watch?v=faQ BrAQ87l4

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SOME KINDS OF PARADOX: AGAINST MOTION ACHILLES AND THE TORTOISE Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start. Will Achilles overtake the tortoise? ACHILLES AND THE TORTOISE Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start. Will Achilles overtake the tortoise?

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VIDEO Link: http://www.youtube.com/watch?v=Ur p60wqr4lo

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THE KEY CONCEPT OF THE PARADOX: THE UNLIMITED DIVISIBILITY OF SPACE A SEGMENT, WHICH IS LIMITED, IS MADE OF AN UNLIMITED NUMBER OF POINTS HOW MUCH TIME DO I NEED TO COVER AN UNLIMITED NUMBER OF POINTS? A SEGMENT, WHICH IS LIMITED, IS MADE OF AN UNLIMITED NUMBER OF POINTS HOW MUCH TIME DO I NEED TO COVER AN UNLIMITED NUMBER OF POINTS?

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P ART 2: THE DANGER OF INFINITY

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THE DANGER OF INFINITY WHAT IS IT THAT DOES NOT WORK IN ZENO’S ARGUMENT?

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THE PROBLEM ARRIVES WHEN YOU SAY:

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> THE DANGER OF INFINITY

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BY SAYING “AND SO ON” ZENO CONLUDES THAT ACHILLES WILL NEVER REACH THE TORTOISE THE DANGER OF INFINITY

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THEREFORE ZENO DRAWS A CONCLUSION WHICH COMES FROM THE ANALYSIS OF AN INFINITE NUMBER OF STEPS. THE DANGER OF INFINITY

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THIS IS A DANGEROUS THING IN MATHS, AS YOU WILL SOON SEE… THE DANGER OF INFINITY

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PART3: PARMENIDES: THE FOUNDATION OF ONTOLOGY BEING AS BEING

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PARMENIDES: THE FOUNDATION OF ONTOLOGY WHAT CAN WE SAY ABOUT BEING AS BEING ? WE CAN SAY THAT IS WHAT CAN WE SAY ABOUT BEING AS BEING ? WE CAN SAY THAT IS

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PARMENIDES: THE FOUNDATION OF ONTOLOGY WHAT CAN WE SAY ABOUT NOTHING ? WE CAN SAY THAT IS NOT WHAT CAN WE SAY ABOUT NOTHING ? WE CAN SAY THAT IS NOT

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PARMENIDES: THE FOUNDATION OF ONTOLOGY CHANGE, PLURALITY AND MOTION ARE A MIXTURE OF IS AND IS NOT CHANGE, PLURALITY AND MOTION ARE A MIXTURE OF IS AND IS NOT

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PARMENIDES: THE FOUNDATION OF ONTOLOGY THE WAY OF TRUTH ( ALETHEIA) WHAT REASON TELLS ME THE WAY OF TRUTH ( ALETHEIA) WHAT REASON TELLS ME THE WAY OF OPINIONI (DOXA) WHAT SENSES TELL ME THE WAY OF OPINIONI (DOXA) WHAT SENSES TELL ME STABILITY OF BEING CHANGE, PLURALITY, MOTION

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PART4: THE GEOMETRIC SERIES

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LET’S CONSIDER ACHILLES’ RACE MORE CLOSELY ASSUME THAT ACHILLES’ SPEED IS 10M/S AND TORTOISE’S ONE IS 1M/S ASSUME THAT THE TOROISE STARTS 10M IN FRONT OF ACHILLES CALL T O THE POSITION OF THE TORTOISE AT THE BEGINNING, T 1 THE POSITION OF THE TORTOISE AFTER THE FIRST INTERVAL ANALIZED, T 2 THAT AFTER THE SECOND INTERVAL AND SO ON…

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LOOK AT THE FOLLOWING SCHEME FOR ACHILLES’ RACE ACHILLES’ PATHTIMEDISTANCE COVERED START->T 0 1s10m T 0 ->T 1 0,1s1m T 1 ->T 2 0,01s0,1m ….

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HOW MUCH SPACE WILL IT TAKE TO ACHILLES TO REACH THE TORTOISE? S= 10 + 1 + 0,1 + 0,01 +... = 11,111 THIS IS NOT AN INFINITE SPACE!

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HOW MUCH TIME WILL IT TAKE TO ACHILLES TO REACH THE TORTOISE? 1 + 0,1 + 0,01 +... = 1,111... = 1 + 1/9 TISE IS NOT AN INFINITE TIME?

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SOLUTION?? IT IS POSSIBLE TO SUM AN INFINITE NUMBER OF NUMBERS AND TO OBTAIN A RESULT WHICH IS A FINITE NUMBER.

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SOME PROBLEMS….(ZENO WAS NOT A FOOL) WE USED APPROPRIATE SPEEDS FOR ACHILLES AND THE TORTOISE; USING DIFFERENT ONES, IT COULD HAVE BEEN MUCH MORE DIFFICULT TO ARRIVE TO SUCH AN EVIDENCE

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NOT ONLY ANCIENT GREEKS DID NOT USE OUR POSITIONAL WAY OF WRITING NUMBERS, THEREFORE THIS PROBLEM WAS DEFINITELY MORE INVOLVED

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BUT ABOVE ALL… AS YOU’VE ALREADY EXPERIENCED THE SUM OF INFINITE NUMBERS CAN BE A TRICKY THING AND ITS THEORY MUST BE FOUNDED ON A SOUND BASIS.

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MATHEMATICIAN SOLVED POSSIBLE PARADOXES COMING FROM THE EASY IDEA OF SUMMING UP INFINITE NUMBERS DURING THE 19° CENTURY.

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