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