
How many holes does a straw have? [video] - ColinWright
https://www.youtube.com/watch?v=JqqzIlLNPns
======
panic
I like how this video focuses on the question of how to model the world.
People often hold mathematical results up as truth when they're really just
the consequences of taking particular definitions. It's possible to reject any
mathematical conclusion -- a straw obviously doesn't have one hole, it has
two! -- and then work backward to find new ways of modeling that make your
conclusion true.

Of course, you'll probably find strange new consequences, like the fact that a
plate now has a hole. But then you can go back and tweak your definitions some
more, or even the entire premise. Maybe "being a hole" isn't a binary yes-or-
no thing, for example, but a continuum from cup-like (very holey) to plate-
like (barely holey). This process of finding strange consequences, fixing the
model to avoid them (or include them!), finding new strange consequences, and
so on, is really what math is about, I think.

~~~
PopeDotNinja
I think it's valuable to understand that most of what we're taught is an
abstraction for describing something.

Using negative integers as an example, it wasn't until I was introduced to
unsigned integers in C that it even occurred to me to wonder about what the
heck a negative number is. It was then I realized how cool it was to know that
I could make up my own version of numbers where there are no negatives. From
now on, numbers all have a length and direction! Instead of +3 and -3, now we
was 3 in-that-direction, and 3 in-the-exact-opposite-direction!

I decided it was easier to keep using regular negative integers, but it was
cool to realize someone just made them up many moons ago. Then I felt smarter
and/or less dumb, and I started having fewer "I could never do..." thoughts
after that.

------
jameshart
One perspective that might be useful is to think about the _functional_
definition of a straw. The job of a straw is to enclose part of the two-
dimensional surface of a liquid, then connect the enclosed region to an
enclosed container with a lower air pressure than the surroundings. It kind of
absolutely follows from those requirements, that a straw has to be some sort
of shape with a boundary and a single hole. If it has no boundary, you can't
connect it to the low-pressure reservoir; if it has no hole, it can't enclose
liquid surface; if it has more holes, it doesn't separate space above the
liquid into areas which can have different pressures (hence why a pinhole in a
straw stops it from working).

There are other forms that could function as a straw - you could have a straw
that splits in two and then joins back up (topologically equivalent to a torus
pierced by two circles - where a normal straw is a sphere pierced by two
circles). Essentially you can take _any_ surface which has an inside and an
outside, pierce it twice, and use it as a straw (so you can't do it, for
example, with a klein bottle - but you could do it with a trefoil knot). Put
your mouth round one piercing, stick the other against the surface of the
liquid, and your topologically advanced straw should work.

This also kind of points to why it's confusing to talk about how many holes
there are in a straw - because if you start with a closed 2D surface (like a
sphere), you have to pierce it twice to make a straw. But in topology, the
first 'hole' you add isn't a 'hole' at all (you can't loop a string through
it, as the video says) - it just opens the closed surface.

~~~
ColinWright
> _... if you start with a closed 2D surface (like a sphere), you have to
> pierce it twice to make a straw. But in topology, the first 'hole' you add
> isn't a 'hole' at all ..._

That's from one point of view, as the video says. From another point of view
it _is_ a hole. From that point of view, making a "straw" from a sphere
requires two holes. The question the video then goes on to ask is whether that
point of view is "the best" one. The conclusion is that it probably isn't the
most useful - in this context.

But in other contexts, if I have a sphere - genuinely a two dimensional
object, all the points (x,y,z) in R^3 satisfying x^2+y^2+z^2=1, when I then
punch a hole in it, it has a hole. It's now homeomorphically a disk, but it's
a sphere with a hole in it. In this context, for this purpose.

So I think you are overly-condensing the video and removing important points.

------
lisper
The problem with being topologically pedantic about this is that it is at odds
with the actual English definition of the word "hole", which includes both
toplogical holes, like a hole in your sock, and non-topolotical holes, like a
hole in the ground. A straw has one sock-like hole and, locally, two ground-
like holes, one at each end.

~~~
ColinWright
The problem with the actual English definition of the word "hole" is that
there isn't one. As soon as you start to pin people down about what they mean
you find that their intuitions are inconsistent and non-communicable.

Part of the value of these sorts of studies is to see what the fundamental
characteristics of things might be, and then look for mappings back to
"natural language concepts".

So what you're saying is a problem is actually an opportunity.

~~~
lisper
> there isn't one

Of course there is. It might not be mathematically precise, but it certainly
exists, and it includes things that topologists don't consider holes.

Henry Segerman uses a cup as his "straw-man" (pun intended) example of an
object that may or may not have a hole. But what he should have used was a
_hole_ , as in a hole in the ground, the kind you dig with a shovel. That too
is a continuum and not a dichotomy, but what you end up with at some point
when you dig is, beyond any question to any competent speaker of English, a
hole.

~~~
ColinWright
The argument on the interwebs about how many holes a straw has shows that
there is no generally accepted usable definition of "hole". You can have lots
of examples, and describe each, but the fact that I give a well-known, common
object to a bunch of people and ask "How many holes are there in this?" and
proceed to get a range of answers shows that there really isn't a working,
usable, generally accepted definition.

I suspect we've got different definitions of the word "definition".

~~~
lisper
That may be, but it remains a fact that every competent speaker of English
knows what it means to "dig a hole in the ground", and the result of that
process is not a topological hole.

The problem is that topologists chose the wrong word. Consider: Bob digs a
hole in the U.S. and Alice digs a hole in China. How many holes are there as a
result? Clearly there are two.

Now suppose they dig deep enough that their holes _meet_. Now how many holes
are there? Answer: zero. What you have now is a _tunnel_.

~~~
Retra
Or take a sphere and poke a hole in it. Topologist says there are no holes,
because now it is a disk. But wow else would you describe poking a hole in a
sphere?

~~~
lisper
Even that is based on the tacit assumption that the sphere you are poking a
hole in is a two-dimensional manifold and not a solid. If you "poke a hole" in
a solid sphere (which is what you actually do when you dig a hole in the
ground) you haven't change _anything_ from a topological point of view.

~~~
ColinWright
Sphere, not ball. A sphere is inherently a two dimensional manifold with no
edges, two sides, and no embedded loops that are homotopically not a point.
It's an idealised "shell".

However ...

I'm trying to understand your points here, but it's almost like you're saying
that the things mathematicians do have no value as they stand, and they should
be doing something else. That might not be what you mean, but reading your
comments as a whole, that's the impression I'm getting. Much of what you say
is true, but seems to be going off on that tangent.

Mathematicians, in these sorts of circumstances, take abstract things that
are, in some senses, approximations of something from reality, and then
examine those abstract things to see what they can find out about them. They
are approximations, and sometimes the results aren't applicable, but often
they are. This video is a case in point. We look at how we can model the
straw, and see what the word "hole" might mean in that model, then map it back
to reality to see what answer that gives us.

But your comments all seem to be saying: But that's not relevant to how we use
language, and so you're neither getting the "right" answer, nor doing anything
fundamentally useful.

So based on that reading of your comments, I'm not sure I have anything
further to add.

~~~
lisper
> it's almost like you're saying that the things mathematicians do have no
> value as they stand

I have no idea how I could have left you with that impression. I'm talking
about pedagogy and rhetoric, not math. I opened with: "The problem with being
topologically pedantic..." I honestly don't know how I could possibly have
made it any plainer.

~~~
henryseg
At 30 seconds into the video I say "from a topologist's perspective, how you
might try and answer this question". I tried to be very clear that this was
not the one true answer, but just my perspective. And the impression you got
was:

> Instead it comes across as a geek being smug about knowing the "right
> answer" to brain teaser because topology."

Perhaps the moral of all of this (and discussions on the internet in general),
is that communication is hard.

~~~
lisper
OK, I re-watched your video, as well as the Actionlab one it was responding
to, and this:

> it comes across as a geek being smug about knowing the "right answer"

was unfair. I apologize.

I went back and tried to figure out where the impression I got diverged from
reality, and I think it was at 0:26 where you say, "people don't really have a
good definition of what a hole is" and my brain extrapolated this to expect
something like, "and in this video I'm going to give you one." Which is of
course not what you say, but it's not an unreasonable expectation. In fact the
impression I now come away with on second viewing is that topologists are just
as much out to sea on this as everyone else. I suspect that was not the
impression you intended to leave either.

> communication is hard

Indeed. But if I may offer (what I hope is) a constructive suggestion: the
first step to clear communication is to know your audience. It's unclear to me
who the target audience of your video was intended to be. You start out
speaking very informally, e.g. "objects in topology are kind of stretchy and
maybe you can chance something that looks like a hole into something that
doesn't look like a hole." That indicates that you intended the video to be
seen by a lay audience (also the video you're responding to was clearly
intended for a lay audience). But mid-way through you casually introduce the
terms "homotopy group" and "homology group" without explaining them (just
pointers to Wikipedia pages) indicating that you expect a certain level of
mathematical sophistication. I think that was one of the things that
originally registered in my mind as "smug".

In any case, I applaud the fact that you are putting forth the effort to make
educational videos about math. I watched a few more in your channel and really
enjoyed them. "Geared Cuboctahedral Jitterbug" was particularly cool!

[UPDATE] Triple-piano just totally blew me away!

~~~
henryseg
I'll admit that I didn't think terribly hard about who the audience is. This
is another tricky thing of course - anyone can look at a YouTube video.

I guess I tried to give an intuitive explanation of how a topologist thinks
about hole-like things. One can only get so far before you have to start
getting precise (and technical). This is where I give the names "homotopy
group" and "homology groups", so that viewers at least have something to look
up to learn more if they want to. I certainly didn't intend to give the
impression that viewers were supposed to know what those terms are already.
Maybe there is a video to be done on giving a better introduction to homotopy
or homology groups, without requiring any background knowledge, but this
wasn't that video.

Incidentally, topologists don't use the word "hole" in a technical sense. So
yes, we wouldn't claim to know what a hole should be better than anyone else.
But in trying to make precise the intuitive ideas of what it means for
something to have a hole, or a "loop" in it, we got to the kind of definitions
we do work with. These ideas can, in principle, come back to be useful to
analyze objects in the real world (persistent homology is trying to do this).

Glad you liked some of my other videos!

------
dan-robertson
From a homotopy-equivalence point of view you can take a (solid) straw and
make it shorter and shorter until it is two dimensional (an annulus). Note
that this is a deformation retract (ie it is a “continuous deformation” but
not the kind commonly used to try to explain a homeomorphism: a straw is like
an annulus differently to how a donut is like a mug). The process can be done
again, making the ring thinner and thinner until it is a circle. From here it
seems reasonable to say that there is one hole in a circle so there is one
hole in a straw.

~~~
Grustaf
What do you mean by “until it is two-dimensional”? You can’t continuously
change it’s dimension. It’s a two dimensional surface embedded in three
dimensions, at least that’s how he modelled it in the beginning of the video.

~~~
danharaj
Homotopy equivalence is even coarser than homeomorphism. A solid ball is
homotopy equivalent to a point.

------
amelius
I was going to say "none, because the straw is made of a bunch of atoms", but
he actually addresses this.

------
sopooneo
Regarding his example with the scissors, I would think there are _three_
distinct ways to loop the string through: each of the two he showed, plus a
loop that goes through both handles. And I think by mushing the volume of the
scissors a bit, you could show that all three of those are exactly equivalent.

And then we might say that the number of ways you can loop a string is
actually equal to how many distinct _pairs_ of a holes an object has. It just
happens to turn out that three distinct things allow three distinct pairings,
whereas two distinct things (ie the two "holes" in a straw) allow only one
distinct pairing.

Working backward one step further, this does lead to the slightly unsettling
idea that a simple solid object like a ball may have one hole in it. But I can
actually get behind that. All the rest of the space not part of the ball is
the hole in the ball.

------
frou_dh
A straw is a lanky donut (What did you just call me?)

~~~
qorrect
A donut.

------
peterburkimsher
The theory is interesting, but the engineering question is simple.

How many times do I have to drill a hole?

By redefining "hole" as the use of a drill, surface-mount components don't use
any holes. Cups and plates have no holes. Filled doughnuts have one hole,
where the jam is injected. Doughnuts with a less-fattening centre were made
without a hole, whatever Tim Hortons would have you think by selling you
"doughnut holes".

------
qorrect
What real-world types of problems can Persistent Homology solve ?

~~~
ColinWright
In private communications, some of the topologists I know have said that it
might lead to new ways of working with machine learning. Most machine learning
is to do with classification, and separation of collections in high-
dimensional spaces. Persistent Homology talks about structures that emerge at
different scales, and that includes aggregation and separation of collections.

Early days.

------
tedunangst
Can you make a straw from a Möbius strip?

~~~
ColinWright
Assuming you're using your "straw" to transport liquid, the liquid will be in
contact with some part of the surface of the Möbius strip. But you can get
from any part of the surface to any other part of the surface by "walking
around" without crossing an edge, so there is no sense in which the liquid can
be contained. There being only one side, I'd say no.

------
akvadrako
Saved you a click: one hole

~~~
ColinWright
I'd guess you're the kind of person who would pick up a murder mystery, read
the first few pages to find out who was killed, the last few pages to find out
who did it, then put it down and move on.

Sometimes value lies in the journey, not in the destination.

So you have an answer - one. Of what value is that? Not a lot.

Those who watch the video will potentially learn a lot more than a glib and
useless answer to an apparently pointless question, and for those who work in
a technical area, perhaps knowing a little more about topology will come in
useful one day.

~~~
akvadrako
One is the obvious expected answer, so the value is not wasting time with the
video. It's like a murder mystery where the culprit is obvious from the
beginning, but the whole book is spent talking about hypothetical angles.

If the answer was 0, 1.5 or 2 in terms of topology, or anything else, then it
would be interesting to hear.

~~~
ColinWright
So did you already know about persistent homology? Are you aware of its
potential applications to machine learning? Did you already know _why_ some
people say the answer is 2?

Consider all the people who (a) did not know about these things, and (b)
didn't bother to watch the video because of your post. I'm sure you feel like
you've done them a favour, but have you really?

------
paganel
I’m a Marxist on this and I’m saying that a “big enough quantitative change
equals a qualitative one”, meaning that in the first example if you stretch
that cup long enough in order to almost change it into a plate then the cup is
actually not a cup anymore (because of the quantitative “long enough”) and
trying to answer the question “does a cup have a hole?” by presenting as a
counter-example a “long enough” stretched out cup (which is not a cup anymore)
serves no purpose. The second example with the straw which can be seen at the
limit as a doughnut is similar. Granted, I’m not a topologist, just an admirer
of Hume and of his fascination for mathematical induction and of people’s
blind faith in it (stretching a cup until it’s no longer a cup can be seen as
similar to some mathematical induction operations).

~~~
ralusek
The belief that quantity can have qualitative impact is being claimed under
Marxism these days?

~~~
paganel
To be honest I’m not up to date on current Marxist thought, but that was one
of the many things Marx was saying, yes.

