Timecodes
00:00 Introduction to Controversial Issues in Mathematics
Mathematicians don’t always agree on basic definitions.
Thomas Lamb surveyed 2,500 mathematicians and asked them 100 questions about mathematical conventions.
The results of the survey showed that even on basic questions there is no consensus.
00:44 Zero to the Zero Power
Zero to the Zero Power is a controversial issue among mathematicians.
Half of mathematicians think it is one, half think it is an indefinite number, and half think it is undefined.
The answer to this question depends on the area of mathematics you are familiar with.
01:42 Calculus and Limits
In calculus, the limit of a function as it approaches zero from the right is one.
However, in other cases, such as with an exponential function, the limit can be zero.
This leads to uncertainty in the answer to the zero to the zero power question.
02:41 Pascal’s Triangle and the Binomial Theorem
Pascal’s triangle is useful for factoring binomials.
The Binomial Theorem states that one plus x to the nth power equals the sum of nchoosek.
If zero to the zeroth power equals one, the formula is true for all cases except x equals zero.
05:31 Natural Numbers and Zero
Whether natural numbers include zero is a contentious issue.
Half of the people think they do, and the other half don’t.
The inclusion of zero in natural numbers has a long history and different cultural traditions.
06:30 Continuity of 1/x
A third of the people think that 1/x is continuous.
The graph of the function has a vertical asymptote, making it discontinuous.
In calculus textbooks, a continuous function is defined as a function that is continuous at every point.
07:29 Indefiniteness of 1/x
The function 1/x is not defined at x = 0.
At the points where the function is defined, it is absolutely continuous.
It is important to understand what is defined in a particular example.
08:25 Increasing Function
The function f(x) = 1 is a horizontal line.
The definition of an increasing function is: the value of the function at one point is strictly less than the value at another point.
Different sources give different definitions: Wolfram and Thomas.
09:19 Linearity of Function
The function 3x + 1 is not linear by algebraic definition.
A linear function passes through the origin and satisfies a certain property.
The function 3x + 1 is an affine transformation.
11:13 Axiom of Choice
The axiom of choice is part of the axiomatic foundations of mathematics.
Most mathematicians accept this axiom.
The axiom of choice allows you to choose elements from an infinite set.
12:49 Conclusion and recommendations
Accepting or rejecting the axiom of choice affects the mathematics you learn.
It is recommended to use interactive lessons like Brilliant to deeply understand mathematics.
Brilliant helps you learn by doing and receiving feedback.
Link – Timecodes are made using YandexGPT Neural Network https://300.ya.ru/
Link – Google translator https://translate.google.com/
Video transcript
Math conventions
0:00
mathematicians don’t always agree even
0:02
about some of the most basic definitions
0:04
used in mathematics so Thomas lamb
0:07
created a survey asking 2500
0:10
mathematicians a 100 math convention
0:13
questions and the results are kind of
0:15
fascinating and get into some really
0:18
interesting mathematics I want to begin
0:20
with the ever controversial zero to the
0^0 is indeterminate?
0:22
power of zero you might think
0:25
mathematicians just have one answer to
0:27
what Z to the power of zero is but it
0:28
turns out they disagree a little under
0:32
half say the number is one but then
0:34
there’s also big groups about a quarter
0:36
say indeterminate and about a quarter
0:38
say undefined so what is going on here
0:42
so what’s the controver Verity to begin
0:43
with if I look at something like 0 to
0:45
the power of a this is clearly zero like
0:48
0 squar would be 0 * 0 that’s clearly
0:51
zero but in contrast if I look at
0:54
something like B to the power of zero
0:57
this is always equal to one here I’ve
1:00
graphed B to the X this is either
1:03
exponential growth if B is greater than
1:05
one or exponential decay if B is less
1:08
than one but either way when I zoom in
1:11
on x equal to Z I get a height of one so
1:14
I’ve got one argument for why it should
1:15
be zero another argument for why it
1:17
should be one which is it now ultimately
1:21
the way you answer this probably depends
1:23
on the discipline of mathematics to
1:25
which you’re being exposed let me take
1:27
the calculus students perspective first
1:30
in calculus we might say well let’s look
1:31
at a function like x to the X this is a
1:34
lovely function and as you can see as we
1:36
go toward Zero from the right hand side
1:40
this is going to approach the value of
1:41
one and in calculus the way we write
1:43
this down formally we’d say that the
1:45
limit as X Goes To Zero from the right
1:47
is equal to one but that’s just one
1:50
function I could consider all sorts of
1:52
functions of the type f ofx to the power
1:56
of G ofx when f and g were themselves
1:59
both going to zero x to the x is one
2:02
special case but like here’s a
2:03
completely different special case how
2:04
about the base could be e to the 1x^2
2:07
the the top could be X by the laws of
2:11
exponents I can multiply the X’s through
2:13
this gives me e to the 1/x and here’s
2:16
that plot and as you can see as X goes
2:18
to zero you get the value of zero and so
2:21
for this example we have our limit being
2:23
equal to zero so the point is if I’m
2:25
going to consider the general context of
2:29
f to the^ power of G where my f is going
2:31
to a zero in a limit and my G is going
2:33
to a zero in limit well the answer could
2:35
be zero it could be one it could be
2:37
Infinity it could be Pi you can make an
2:39
example that will be anything that you
2:40
want and that’s why we say it’s
2:43
indeterminant if I’m interpreting Z to Z
2:45
as a sort of limiting process in this
2:47
calculus sense then we’re going to call
2:49
it indeterminate and indeed that’s what
2:51
I teach my calculus students if you
2:53
think well no I don’t want to interpret
2:54
this problem in this limit sense I just
2:57
notice there’s these tensions I might
2:58
just call it undef find instead of
0^0=1?
3:01
indeterminate but then why are some
3:02
people saying the value is one well
3:05
let’s switch context a bit do you
3:07
remember Pascal’s triangle this is the
3:09
triangle you put ones all the way along
3:11
then if I say focus on the pink thing
3:13
what I do here is I I look at the two
3:14
numbers right above it one and one I add
3:16
those to get two I can go down 1 plus 2
3:18
is 3 2+ 1 is 3 1 plus 3 is 4 and I can
3:22
fill out this whole triangle this is
3:24
Pascal’s triangle it’s a lovely object
3:25
with all sorts of lovely patterns one in
3:28
specific is it’s really helpful for
3:31
expanding binomials like if I take 1 +
3:34
x^ of 5 well the polinomial that this
3:39
represents has these coefficients 1 5 10
3:42
10 5 1 that shows up directly as this
3:46
row on Pascal’s triangle and there’s a
3:49
formula for this the binomial theorem
3:51
says that 1 plus x ^ of N is a sum of
3:55
the way I say this is n choose K this is
3:58
the number of ways I can choose chose K
4:00
objects from within n items it can be
4:02
expressed in terms of factorials and
4:03
then multipli by X the K so there’s this
4:06
lovely formula for it and if I look on
4:08
the left hand side I imagine plugging in
4:10
x equal to Z there’s nothing wrong with
4:11
plugging x equal to Z on the left hand
4:13
side right that’s just 1 plus 0 to the
4:15
power of n 1 to the N but on the right
4:17
hand side if I was to plug in x equal
4:19
to0 then in the k equal to0 case I’d
4:22
have 0 to the 0 appearing there’s no
4:24
reason the left hand side expression
4:26
should stop making sense at x equal to Z
4:28
and so the right hand side should should
4:29
not stop making sense at zero as well so
4:32
in that special case when I plug in x
4:33
equal to 0 I should have 1 is equal to 0
4:36
to the 0 so the point is this defining 0
4:39
to the 0 to be equal to 1 makes this
4:42
formula true in all of the cases and I
4:45
don’t have to have this extra
4:46
restriction written down except in the
4:48
case when X is equal to Zer only then
4:50
does it not make sense because 0 to Z is
4:52
undefined or indeterminant based on our
4:55
previous arguments and there’s all sorts
4:57
of places in mathematics where
5:00
making this choice of Z to the Z equal
5:02
one is just really helpful ultimately I
5:03
don’t care what you say whether you say
5:04
it’s indeterminate whether you say it’s
5:05
undefined whether you say it’s one I’m
5:07
not actually not aware of a good
5:08
argument for why it should be zero but
5:10
maybe a couple people think that what’s
5:11
more important is to say that in
5:13
whatever context you might be in there’s
5:15
a convention within that specific
5:17
context and if people aren’t sure you
5:19
can specify what you mean and then
5:20
continue it’s not like there’s some deep
5:22
disagreement here about the ideas it’s
5:24
just what’s the definition going to be
5:26
or not be in our specific context I’ll
5:28
give you another example it blows my
0 is a natural number?
5:30
mind how even this is the natural
5:32
numbers are like 1 2 3 4 five and so on
5:35
but do they include zero well a little
5:37
over half of people think yes they do
5:39
include zero a little under half think
5:41
no they don’t include zero is it 0 One
5:44
Two Three or is it just 1 2 3 honestly
5:47
if you’ve got a new textbook you kind of
5:49
just have to go in and look and see how
5:50
they Define the natural numbers if it
5:52
becomes relevant at some point zero has
5:54
actually a really long and kind of
5:56
interesting history and all sorts of
5:58
different cultures whether you want to
6:00
call it natural or not well it’s led to
6:03
different historical Traditions if you
6:05
want to avoid this controversy you can
6:07
use things for example non negative
6:10
integers that’s going to include zero or
6:12
you could say positive integers that’s
6:14
going to start it at one there’s ways to
6:16
avoid around it but it’s still common
6:18
still natural if you don’t mind me
6:20
saying to just talk about the naturals
6:22
and sometimes people mean that they’ve
6:23
got zero in it and sometimes they don’t
6:25
I’ll give you one argument for why I
6:27
like to include zero that’s my own
6:28
personal preference here here if you
6:30
imagine I’m counting objects like this
6:31
is a box with one ball in it a box with
6:33
two balls a box with three balls if I’m
6:36
doing this counting talking about
6:38
cardinal numbers here then I also want
6:40
to refer to like how many items are
6:42
there what’s the size of a box with
6:44
nothing in it so seems very natural to
6:47
abuse the language here to think of zero
6:50
as being a natural number it’s referring
6:51
to the size of an empty box okay this
1/x is a continuous function?
6:55
one might really mess with your mind is
6:58
the function one over X continuous and
7:01
it turns out that well about a third of
7:04
people think the answer to that is yes
7:07
so the case for not continuous is may be
7:10
obvious just says well look at the graph
7:12
it it’s got a vertical ASM toote right
7:14
that’s not continuous duh but let’s be a
7:17
bit more careful if I go to say one of
7:19
the big calculus textbooks uh Thomas’s
7:22
calculus a standard book I use it in my
7:23
University it defines a continuous
7:25
function as one that is continuous at
7:28
every point and here’s the catch in its
7:31
domain zero isn’t in the domain of 1 /x
7:34
1 / Z is not defined there’s no
7:37
controversy about that and so if I look
7:40
at the graph again in all the spots
7:44
where it’s defined that is where X is
7:45
not equal to zero the function is
7:47
perfectly continuous and thus it is a
7:50
continuous function and again it’s it’s
7:52
not that this really matters so much
7:54
these these conventional differences are
7:56
not some really important sort of aspect
7:58
of mathematics usually if there’s any
8:00
ambiguity people are just going to tell
8:02
you what they mean so so these
8:04
conventional differences don’t mean that
8:05
people start getting mad and angry at
8:07
each other it just means that if you’re
8:08
working through exercises in your
8:10
Calculus checkbooks well you have to be
8:11
a little bit careful about what
8:12
precisely it was defined in your
8:14
specific example here’s another one also
f(x)=1 is increasing?
8:17
for the for the calculus students uh is
8:19
the function f ofx equal to one which is
8:21
completely a horizontal line is that
8:24
increasing well again about a third of
8:27
people say yes and and 2third say no the
8:30
argument for not increasing well let’s
8:31
flat it’s just not going up right but
8:34
but let’s be precise again but what does
8:37
increasing mean so here’s the definition
8:39
going back to Thomas’s calculus it says
8:41
that the function is increasing if the
8:44
function value one point is strictly
8:47
less than the function value at another
8:49
Point whenever X1 is less than X2 so if
8:53
you if you have two inputs and one’s
8:55
less than the other then the outputs
8:56
have one being less than the other a
8:58
sensible notion of increasing but notice
9:01
the strict inequality here F of X1 is
9:04
less than F of x2 in other sources like
9:07
for example go to wol from right now
9:09
they say less than or equal to here and
9:13
so according to Thomas this is not an
9:15
increasing function according to Wolfram
9:18
it is an increasing function people
9:20
using the the Wolfram definition would
9:22
say the Thomas definition is of strictly
9:26
increasing so there’s increasing and
9:28
then strictly increasing the the special
9:30
case when it’s strictly less than Note
9:32
by the way that well I haven’t written
9:34
on the screen both of these definitions
9:35
are about an interval of points where X1
9:38
is less than X2 the standard calculus
9:41
student error is to confuse the
9:43
definition of increasing with the
9:44
derivative at a specific point being
3x+1 is linear?
9:46
positive okay this one’s going to annoy
9:49
people uh is the function 3x + 1 linear
9:53
oh okay apparently again it keeps on
9:55
being about 1/3 23 for so many of these
9:57
one3 say no this is not a linear
10:00
function you might be like what do you
10:01
mean it’s not a linear function graph it
10:03
it’s a line that’s linear what’s there
10:06
to talk about but there’s a really
10:09
important algebraic notion of linearity
10:12
and it goes like this it says if I have
10:15
a function and I take a linear
10:17
combination so F of ax plus b y a linear
10:21
combination then the output is the
10:23
linear combination of the outputs it’s a
10:26
f ofx plus b f of Y so this is a lovely
10:29
algebraic property and note that this
10:32
demands that F of 0 is zero so 3x + 1 is
10:37
not linear according to this definition
10:40
it doesn’t go through the origin and the
10:43
terminology that you might use here is
10:45
that something like 3x which does go
10:47
through the origin and does satisfy this
10:49
property is linear and then 3x + 1 which
10:53
is sort of like linear but then shifted
10:55
is an apine transformation in the
10:58
subject of linear algebra we really
11:01
study this algebraic property in a lot
11:04
of detail we can generalize this for
11:05
example to a higher number of dimensions
11:08
and talk about linear transformations of
11:11
the plane or three-dimensional space
11:13
there’s a lot of really lovely work and
11:15
primarily you’re focused on that concept
11:17
where it’s linear and not apine this
Axiom of Choice?
11:20
next one is not just about basic
11:22
conventions it actually gets to some
11:24
deeper issues within mathematics and
11:26
this is the axium of choice and in the
11:29
axiomatic foundations of mathematics and
11:32
there’s multiple different ways to do
11:34
this the the Axiom of choice is an axiom
11:38
that most mathematicians tend to accept
11:40
as you can see here it’s about 85% and
11:43
then the rest have some combination of
11:45
either rejecting it entirely or maybe
11:46
they reject the axium of choice but they
11:48
accept a weaker notion for example the
11:51
axium of countable Choice the axum of
11:53
Dependable choice is another option
11:55
that’s not here in the particular poll
11:57
but the point is most people accept it
11:58
okay so so so what is the axum of choice
12:01
imagine I’ve got like a bag it’s got a
12:03
bunch of marbles inside of it and then I
12:05
have another bag with another bunch of
12:07
marbles another bag another bag and then
12:09
I want to imagine that this goes on to
12:12
Infinity if I only have a finite number
12:13
of bags it’s very easy to say well I can
12:15
pick one marble from the first one
12:17
marble from the second and one marble
12:18
from the third but if I have infinitely
12:21
many can I keep on doing that
12:23
association with infinitely many bags is
12:25
there a way to come up with the choice
12:26
function that picks one marble out of
12:30
every single one of these bags that you
12:31
can do this is called the Axiom of
12:33
choice this is often paired together
12:35
with other axioms in something called
12:37
zero Franklin set theory which is one of
12:39
the sort of foundations of mathematics
12:42
and this maybe intuitive Maybe not maybe
12:45
you tempted to agree that this is
12:46
obvious maybe you’re tempted to say no
12:48
it’s not obvious people do genuinely
12:50
disagree about it and the types of
12:52
mathematics that you get unlike these
12:54
previous just definitional disagreements
12:57
the actual mathematics that you get
12:59
really changes depending on whether you
13:01
do or do not reject the axium of choice
6/2(1+2)?
13:03
now the final one I’m going to leave you
13:05
with here I have shared my thoughts to
13:06
like a half million of you before in a
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video is the ever viral 6 divided 2 uh 3
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1 and well well I’ll just uh show the
13:16
results here and and I’ll let you make
13:18
your own conclusions about what the
13:19
average mathematician thinks of this
Brilliant.org/TreforBazett
13:21
problem now if you’re interested not
13:23
just in worrying about silly math
13:24
conventions but actually learning
13:27
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constantly providing feedback and
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opportunities to self assess one of the
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big challenges that I had to deal with
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with both as a math YouTuber and as a
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professor is students who try to learn a
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lot of math just by watching a lot of
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videos which can be great but if you’re
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not actually doing the mathematics it’s
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really hard to learn it deeply and so I
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subscription with that said and done I
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hope you enjoyed the video if you have
14:34
your own little math controversies leave
14:36
them down in the comments below and
14:38
we’ll do some more math in the next
14:39
video