and astronauts a physicist and mathematician are all vacationing in
mathematician are all vacationing in
mathematician are all vacationing in Scotland aboard a train outside their
Scotland aboard a train outside their
Scotland aboard a train outside their train window they observe a single black
train window they observe a single black
train window they observe a single black sheep in a field the astronaut exclaims
sheep in a field the astronaut exclaims
sheep in a field the astronaut exclaims look all Scottish sheep are black the
look all Scottish sheep are black the
look all Scottish sheep are black the physician corrects him no only some
physician corrects him no only some
physician corrects him no only some Scottish sheep are black and the
Scottish sheep are black and the
Scottish sheep are black and the mathematician Ponder’s waiting for the
mathematician Ponder’s waiting for the
mathematician Ponder’s waiting for the train to pass and eventually he says
train to pass and eventually he says
train to pass and eventually he says gentlemen in Scotland there exists one
gentlemen in Scotland there exists one
gentlemen in Scotland there exists one field that has at least one sheep at
field that has at least one sheep at
field that has at least one sheep at least one side of which is black I’ve
least one side of which is black I’ve
least one side of which is black I’ve heard I first read this joke in a book
heard I first read this joke in a book
heard I first read this joke in a book entitled The Simpsons and their
entitled The Simpsons and their
entitled The Simpsons and their mathematical secrets written by this man
mathematical secrets written by this man
mathematical secrets written by this man Simon son I remember being exactly
Simon son I remember being exactly
Simon son I remember being exactly amused but I do remember how intriguing
amused but I do remember how intriguing
amused but I do remember how intriguing it was and how it emphasized the
it was and how it emphasized the
it was and how it emphasized the restrictive nature of mathematics and
restrictive nature of mathematics and
restrictive nature of mathematics and how the common sense and reasoning that
how the common sense and reasoning that
how the common sense and reasoning that comes from the scientific fields of
comes from the scientific fields of
comes from the scientific fields of astronomy and physics just don’t apply
astronomy and physics just don’t apply
astronomy and physics just don’t apply it to math today I want to talk about a
it to math today I want to talk about a
it to math today I want to talk about a certain area of mathematics as more or
certain area of mathematics as more or
certain area of mathematics as more or less become a recreational playground of
less become a recreational playground of
less become a recreational playground of unsolvable problems and unanswerable
unsolvable problems and unanswerable
unsolvable problems and unanswerable questions this is of course the
questions this is of course the
questions this is of course the conceptualization of infinity and we
conceptualization of infinity and we
conceptualization of infinity and we think of infinity at first we might
think of infinity at first we might
think of infinity at first we might picture a number unreachable on the
picture a number unreachable on the
picture a number unreachable on the number line or bigger than any other but
number line or bigger than any other but
number line or bigger than any other but really it is more of a quality than a
really it is more of a quality than a
really it is more of a quality than a quantity in itself used to describe
quantity in itself used to describe
quantity in itself used to describe unending scenarios or infinitely many
unending scenarios or infinitely many
unending scenarios or infinitely many things so what is infinity how does it
things so what is infinity how does it
things so what is infinity how does it work how can it be used or how can it be
work how can it be used or how can it be
work how can it be used or how can it be avoided what mathematicians do to answer
avoided what mathematicians do to answer
avoided what mathematicians do to answer these questions is they create special
these questions is they create special
these questions is they create special scenarios for themselves these scenarios
scenarios for themselves these scenarios
scenarios for themselves these scenarios require very special property to change
require very special property to change
require very special property to change our understanding of what infinity is
our understanding of what infinity is
our understanding of what infinity is after all if we keep thinking of it in
after all if we keep thinking of it in
after all if we keep thinking of it in the same way then our understanding will
the same way then our understanding will
the same way then our understanding will never change perhaps the most common use
never change perhaps the most common use
never change perhaps the most common use of infinity is to describe expressions
of infinity is to describe expressions
of infinity is to describe expressions like this one this is what we call an
like this one this is what we call an
like this one this is what we call an infinite sum where we start with one
infinite sum where we start with one
infinite sum where we start with one number then add another one and add
number then add another one and add
number then add another one and add another number and so on forever and we
another number and so on forever and we
another number and so on forever and we never stop increasing our total value
never stop increasing our total value
never stop increasing our total value and in this particular sum we have 1/2
and in this particular sum we have 1/2
and in this particular sum we have 1/2 plus 1/4 plus 1/8
plus 1/4 plus 1/8
plus 1/4 plus 1/8 forever and somehow we end up at one a
forever and somehow we end up at one a
forever and somehow we end up at one a finite value our total is always
finite value our total is always
finite value our total is always increasing and yet it does not reach
increasing and yet it does not reach
increasing and yet it does not reach infinity this requires this possesses
infinity this requires this possesses
infinity this requires this possesses the very specific scenario that I
the very specific scenario that I
the very specific scenario that I mentioned earlier allow me to present
mentioned earlier allow me to present
mentioned earlier allow me to present another scenario that takes advantage of
another scenario that takes advantage of
another scenario that takes advantage of the freedoms of mathematics called
the freedoms of mathematics called
the freedoms of mathematics called Thompson’s lamp here we have a lamp it
Thompson’s lamp here we have a lamp it
Thompson’s lamp here we have a lamp it can be turned on and off like a basic
can be turned on and off like a basic
can be turned on and off like a basic lamp and it can be only on or off at any
lamp and it can be only on or off at any
lamp and it can be only on or off at any moment in time
moment in time
moment in time this lamp is programmed to turn on and
this lamp is programmed to turn on and
this lamp is programmed to turn on and off at very specific times within the
off at very specific times within the
off at very specific times within the span of one minute at the start of the
span of one minute at the start of the
span of one minute at the start of the minute the lamp is on after exactly 1/2
minute the lamp is on after exactly 1/2
minute the lamp is on after exactly 1/2 minutes
minutes
minutes the switch is flipped and it turns off
the switch is flipped and it turns off
the switch is flipped and it turns off after 1/4 minute it turns back on an
after 1/4 minute it turns back on an
after 1/4 minute it turns back on an eighth of minutes it turns off so you
eighth of minutes it turns off so you
eighth of minutes it turns off so you see the pattern after exactly what half
see the pattern after exactly what half
see the pattern after exactly what half the remaining time expires the switch is
the remaining time expires the switch is
the remaining time expires the switch is flipped and the lamp changes its state
flipped and the lamp changes its state
flipped and the lamp changes its state so this goes on until one minute is
so this goes on until one minute is
so this goes on until one minute is reached and after one minute is it
reached and after one minute is it
reached and after one minute is it possible to tell whether the lamp is on
possible to tell whether the lamp is on
possible to tell whether the lamp is on or off
or off
or off well if we think about it each time the
well if we think about it each time the
well if we think about it each time the lamp is turned on it is then immediately
lamp is turned on it is then immediately
lamp is turned on it is then immediately turned off again and vice versa if when
turned off again and vice versa if when
turned off again and vice versa if when it is turned off it is that immediately
it is turned off it is that immediately
it is turned off it is that immediately turned back on again so which one is it
turned back on again so which one is it
turned back on again so which one is it what have we done is it both neither or
what have we done is it both neither or
what have we done is it both neither or is it on or off we can do nothing but to
is it on or off we can do nothing but to
is it on or off we can do nothing but to simply guess what we’ve done is done and
simply guess what we’ve done is done and
simply guess what we’ve done is done and infinitely infinitely many things in a
infinitely infinitely many things in a
infinitely infinitely many things in a finite amount of time
finite amount of time
finite amount of time this is called a xenon super task named
this is called a xenon super task named
this is called a xenon super task named after the mathematician Zeno doing and
after the mathematician Zeno doing and
after the mathematician Zeno doing and incidentally infinitely many things in a
incidentally infinitely many things in a
incidentally infinitely many things in a finite amount of time does not let us
finite amount of time does not let us
finite amount of time does not let us answer whether Thompson’s lamp is on or
answer whether Thompson’s lamp is on or
answer whether Thompson’s lamp is on or off but it does allow us to change our
off but it does allow us to change our
off but it does allow us to change our concept of infinity into more of a
concept of infinity into more of a
concept of infinity into more of a mathematical idea than a physical one if
mathematical idea than a physical one if
mathematical idea than a physical one if we were to do the Thompson plant
we were to do the Thompson plant
we were to do the Thompson plant experiment again except following the
experiment again except following the
experiment again except following the laws of nature then I would merely turn
laws of nature then I would merely turn
laws of nature then I would merely turn the lamp on or off at intervals of 1
the lamp on or off at intervals of 1
the lamp on or off at intervals of 1 minutes but I would never be able to
minutes but I would never be able to
minutes but I would never be able to answer the question of what happens
answer the question of what happens
answer the question of what happens after I’ve done that infinitely many
after I’ve done that infinitely many
after I’ve done that infinitely many times another paradox that results from
times another paradox that results from
times another paradox that results from the use of infinity and this annoyance
the use of infinity and this annoyance
the use of infinity and this annoyance super task is called the Russ littlewood
super task is called the Russ littlewood
super task is called the Russ littlewood paradox suppose we start with a sheet of
paradox suppose we start with a sheet of
paradox suppose we start with a sheet of paper and what we want to do is write
paper and what we want to do is write
paper and what we want to do is write down infinitely many numbers on that
down infinitely many numbers on that
down infinitely many numbers on that sheet of paper at step one we write the
sheet of paper at step one we write the
sheet of paper at step one we write the numbers one through ten and then erase
numbers one through ten and then erase
numbers one through ten and then erase the smallest number which is one at step
the smallest number which is one at step
the smallest number which is one at step two we do the same for 11 through 20 and
two we do the same for 11 through 20 and
two we do the same for 11 through 20 and then erase 2 as you can see at each step
then erase 2 as you can see at each step
then erase 2 as you can see at each step we have nine more numbers on the page
we have nine more numbers on the page
we have nine more numbers on the page than we have the previous step and that
than we have the previous step and that
than we have the previous step and that gain of nine yet after we do this
gain of nine yet after we do this
gain of nine yet after we do this infinitely many times the final result
infinitely many times the final result
infinitely many times the final result is a blank sheet of paper so how is this
is a blank sheet of paper so how is this
is a blank sheet of paper so how is this possible how can we write infinitely
possible how can we write infinitely
possible how can we write infinitely many numbers and write nine numbers more
many numbers and write nine numbers more
many numbers and write nine numbers more on the page each time and yet have a
on the page each time and yet have a
on the page each time and yet have a blank sheet at the end of the page this
blank sheet at the end of the page this
blank sheet at the end of the page this is because we were erasing the numbers
is because we were erasing the numbers
is because we were erasing the numbers no matter how many numbers we had
no matter how many numbers we had
no matter how many numbers we had written initially if we raise infinitely
written initially if we raise infinitely
written initially if we raise infinitely many numbers we erase all of them now
many numbers we erase all of them now
many numbers we erase all of them now another way of doing this task which is
another way of doing this task which is
another way of doing this task which is very similar is only writing the first
very similar is only writing the first
very similar is only writing the first nine numbers and then multiplying the
nine numbers and then multiplying the
nine numbers and then multiplying the smallest by ten at each step here we’ve
smallest by ten at each step here we’ve
smallest by ten at each step here we’ve written one through nine and then
written one through nine and then
written one through nine and then multiplied one by ten to make it ten we
multiplied one by ten to make it ten we
multiplied one by ten to make it ten we do the same thing for 11 to 19 and
do the same thing for 11 to 19 and
do the same thing for 11 to 19 and multiply two by tend to make it 20 on
multiply two by tend to make it 20 on
multiply two by tend to make it 20 on the left and the right both sides show
the left and the right both sides show
the left and the right both sides show the exact same numbers
the exact same numbers
the exact same numbers however after we’ve done this infinitely
however after we’ve done this infinitely
however after we’ve done this infinitely many times the results are drastically
many times the results are drastically
many times the results are drastically different on one side we have a blank
different on one side we have a blank
different on one side we have a blank sheet of paper and at the other side we
sheet of paper and at the other side we
sheet of paper and at the other side we have infinitely many numbers followed by
have infinitely many numbers followed by
have infinitely many numbers followed by infinitely many zeros this is a very
infinitely many zeros this is a very
infinitely many zeros this is a very famous paradox in mathematics that
famous paradox in mathematics that
famous paradox in mathematics that occurs from the use of affinity so well
occurs from the use of affinity so well
occurs from the use of affinity so well in physical world there will be never
in physical world there will be never
in physical world there will be never exist a measurable quantity that reaches
exist a measurable quantity that reaches
exist a measurable quantity that reaches infinity while simultaneously in
infinity while simultaneously in
infinity while simultaneously in mathematics it creates endless paradoxes
mathematics it creates endless paradoxes
mathematics it creates endless paradoxes left and right however in mathematics
left and right however in mathematics
left and right however in mathematics the escape from the physical
the escape from the physical
the escape from the physical restrictions of nature are really what
restrictions of nature are really what
restrictions of nature are really what makes it its own sandbox or
makes it its own sandbox or
makes it its own sandbox or imitations can answer all kinds of
imitations can answer all kinds of
imitations can answer all kinds of questions whether they can be answered
questions whether they can be answered
questions whether they can be answered or not this sandbox is what makes
or not this sandbox is what makes
or not this sandbox is what makes mathematics not what is usually thought
mathematics not what is usually thought
mathematics not what is usually thought of a world of black and white are
of a world of black and white are
of a world of black and white are correct or incorrect but instead a field
correct or incorrect but instead a field
correct or incorrect but instead a field of study where we can ask whatever
of study where we can ask whatever
of study where we can ask whatever questions we want to about the possible
questions we want to about the possible
questions we want to about the possible the impossible the finite and the
the impossible the finite and the
the impossible the finite and the infinite thank you
”
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