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 dont study science from a technical point of view but we try to stress upon a ID: 360019
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Slide1
For 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
latinSlide2
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.Slide3
For 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
explanationSlide4
Part1: paradoxesSlide5
paradoxesWHAT IS A PARADOX?Slide6
WHAT IS PARADOXICAL?
paradoxesSlide7
A PARADOXICAL SITUATIONparadoxesSlide8
A PARADOXICAL UTTERANCEparadoxesSlide9
A PARADOXICAL IMAGEparadoxesSlide10
paradoxesSlide11
PARADOXON
PARA
DOXA
contrary
to
,
different
from
opinion
common
knowledge
common
sense
From
dokein
: to
appear
,
seem
,
think
paradoxesSlide12
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”
paradoxesSlide13
SOME KINDS OF PARADOXSELF REFERENCE
THE LIAR
T
his statement is falseC
ould
we state whether this sentence is either true or false?
paradoxesSlide14
SOME KINDS OF PARADOXVICIOUS CIRCULARITY
THE INFINITE REGRESS
The
following sentece
is
true
The
previous
sentence
is
false
C
ould
we state whether these sentences are either true or false?
paradoxesSlide15
SOME KINDS OF PARADOXCATEGORICAL 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?
paradoxesSlide16
VideoThe infinite-hotelLink: http://www.youtube.com/
watch
?v=faQBrAQ87l4Slide17
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?
paradoxesSlide18
paradoxesSlide19
videoLink: http://www.youtube.com/watch?v=Urp60wqr4loSlide20
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?
paradoxesSlide21
Part2: THE DANGER OF INFINITYSlide22
THE DANGER OF INFINITY
WHAT IS IT THAT DOES NOT WORK IN ZENO’S ARGUMENT?Slide23
THE PROBLEM ARRIVES WHEN YOU SAY:
THE DANGER OF INFINITYSlide24
<<ACHILLES WILL FIRST GO WHERE THE TORTOISE WAS AT THE BEGINNING, THEN HE WILL MOVE WHERE THE TORTOISE WAS AT THE SECOND STEP, AND SO
ON…
>>THE DANGER OF INFINITYSlide25
BY SAYING “AND SO ON” ZENO CONLUDES THAT ACHILLES WILL NEVER REACH THE TORTOISE
THE DANGER OF INFINITYSlide26
THEREFORE ZENO DRAWS A CONCLUSION WHICH COMES FROM THE ANALYSIS OF AN INFINITE NUMBER OF STEPS.
THE DANGER OF INFINITYSlide27
THIS
IS A DANGEROUS THING IN MATHS, AS YOU WILL SOON
SEE…
THE DANGER OF INFINITYSlide28
PART3: PARMENIDES:THE FOUNDATION OF ONTOLOGY
BEING AS BEING
paradoxesSlide29
PARMENIDES:THE FOUNDATION OF ONTOLOGY
WHAT CAN WE SAY ABOUT
BEING AS BEING
?WE CAN SAY THAT IS
paradoxesSlide30
PARMENIDES:THE FOUNDATION OF ONTOLOGY
WHAT CAN WE SAY ABOUT
NOTHING
?WE CAN SAY THAT IS NOT
paradoxesSlide31
PARMENIDES:THE FOUNDATION OF ONTOLOGY
CHANGE, PLURALITY AND MOTION
ARE
A MIXTURE OF IS AND
IS NOT
paradoxesSlide32
PARMENIDES:THE FOUNDATION OF ONTOLOGY
THE WAY OF
TRUTH
(ALETHEIA)WHAT REASON TELLS ME
paradoxes
THE WAY OF
OPINIONI (DOXA)
WHAT SENSES TELL ME
STABILITY OF BEING
CHANGE, PLURALITY, MOTIONSlide33
PART4: THE GEOMETRIC SERIESSlide34
LET’S CONSIDER ACHILLES’ RACE MORE CLOSELYASSUME THAT ACHILLES’ SPEED IS 10M/S AND TORTOISE’S ONE IS 1M/SASSUME THAT THE TOROISE STARTS 10M IN FRONT OF ACHILLES
CALL T
O
THE POSITION OF THE TORTOISE AT THE BEGINNING, T1 THE POSITION OF THE TORTOISE AFTER THE FIRST INTERVAL ANALIZED, T2 THAT AFTER THE SECOND INTERVAL AND SO ON…Slide35
LOOK AT THE FOLLOWING SCHEME FOR ACHILLES’ RACE
ACHILLES’ PATH
TIME
DISTANCE COVEREDSTART->T01s10mT0
->
T
1
0,1s
1m
T
1
->
T
2
0,01s
0,1m
….Slide36
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!Slide37
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?Slide38
SOLUTION??IT IS POSSIBLE TO SUM AN INFINITE NUMBER OF NUMBERS AND TO OBTAIN A RESULT WHICH IS A FINITE NUMBER.Slide39
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 EVIDENCESlide40
NOT ONLYANCIENT GREEKS DID NOT USE OUR POSITIONAL WAY OF WRITING NUMBERS, THEREFORE THIS PROBLEM WAS DEFINITELY MORE INVOLVEDSlide41
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.Slide42
MATHEMATICIAN SOLVED POSSIBLE PARADOXES COMING FROM THE EASY IDEA OF SUMMING UP INFINITE NUMBERS DURING THE 19° CENTURY.