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.
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.
Great comment. I'll share a potential practical "definition" that popped into my head that is clearly different from the topological approach.
If one starts with the idea of a hole as: "something that an entity can fall into or through", then it becomes very clear that a cup has a hole and a plate doesn't. An ant can fall into a cup from one side, but there is no way for an ant to fall into or through a plate.
For falling "into" something, clearly some conception of "height" is important for this definition. Perhaps its related to being able to draw certain lines -- ones that are nearly parallel and of a relatively large length -- from one boundary to another surface of the object.
For falling "through" something, I'm drawing a bit of blank right now, but I think there is some fairly simple definition that can clearly express this idea.
I would argue the above is pretty close to what humans naturally consider as holes.
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EDIT: Thinking about this a bit more... I think we can simplify the above by viewing holes as "openings". This covers both cases of "falling into" or "through", and removes the distracting reference to gravity.
It should be possible to come up with some sort of fairly practical definition of "opening". I think it has do with relatively sizing and geometry of the "cavern" that the hole leads to.
However... inherently, there is no sharp line we can draw here, as the difference between a small scratch or dent and an actual cavernous hole is just a matter of degree. It seems, like many things in life, the distinction will remain a bit fuzzy and is not perfectly resolvable.
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.
> ... 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.
Perhaps a hole should be defined by other objects. For example, think of a closed rope (loop) that passes through the straw. You can't remove the loop and the straw from eachother. But, if you imagine the rope to be so thin that it can pass between the atoms of the straw, then you could do it (mathematically speaking, not physically). So the straw has one hole w.r.t. the normal rope, but zero holes w.r.t. the very thin rope.
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.
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.
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.
This is (my understanding of) Wittgenstein in a nutshell. Definitions in the real world don't have distinct borders. Words' meanings depend on how they're used, how we teach them to each other, how we experiment with them.
Most of their utility comes "far from the border". Ideally the border is little-trafficked and needs little policing, and if it drifts a little one way or another between speakers few important concepts are left on different sides.
Tying technical definitions to existing words is fine, obviously. Words can have several meanings, and where the contours of a technical concept approximate some lay word it can make sense to reuse the term. If mathematicians want to call something a "hole" (or a "set" or a "knot"), good for them. It might even in turn influence the lay definition, but it doesn't necessarily supersede it just by virtue of its well-defined borders (and indeed may not be suitable for general use because of its hard borders and low entropy.)
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".
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.
This just points out that "normal English usage" is inconsistent and impossible to use in any kind of technical context. As soon as you examine one of the most common senses in which we use the word "hole" we find that there are complexities. The study starts with "What do we mean by hole?" and the rabbit warren is entered. What follows is the investigation of the unavoidable consequences of things you thought were obvious.
You said:
> The problem with being topologically pedantic about this is that it is at odds with the actual English definition ..
I said:
> ... what you're saying is a problem is actually an opportunity.
You're agreeing that there really is no consistent, usable definition of the common usage of "hole", and that people just have to know what it means. I'm pointing out that the technical approach gives us a tool whereby we can start to map usable definitions fro the technical study back into the real usage of language.
It's not a problem, it's an opportunity. Perhaps this is an opportunity to look beyond the statistic treatment of language and get mappings to "meanings", whatever they are. Perhaps the Word2Vec idea can be taken further.
The technical world is full of words that don't "mean the same thing" as when used in a non-technical sense. Function, Object, Module, Expression, Constant, Scope, these are all words that have a technical meaning in a technical environment that break down when you get into the non-technical world. Criticising the content of the video for starting with a non-technical sense of "hole" and trying to make technical sense of it seems odd.
> So there are no holes in a straw - it's a tunnel.
Well, it's more like a tunnel than a hole, but it's not quite a tunnel either. I can touch the outside of a straw. It's not clear that I can touch the outside of a (proper) tunnel.
But yes, topological holes are a lot more like tunnels (in the ground) than they are like holes (in the ground). But on the other hand, topological holes are like holes in your socks. Socks are a really good example because no one refers to the inside of a sock as a "hole". If a sock is in good repair it has no holes despite the fact that it superficially resembles a floppy cup.
> It's not a problem, it's an opportunity.
In this case I would call it a missed opportunity. And that's a problem (IMHO).
The missed opportunity is to really talk about the subtlety and malleability of natural language and how mathematics can make things precise, and why this is desirable. Instead it comes across as a geek being smug about knowing the "right answer" to brain teaser because topology.
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?
You can poke a hole in the sphere if your "pin" enters one side and comes out the other end.
The problem with the conventional notion of a hole is that it depends on the curvature on an object, which is imprecise and arbitrary. In your example, suppose you put a "hole" in your sphere and started flattening it into a disk. As you flatten your sphere, at which point does the hole disappear? Can you rigorously define it so that your definition applies to all objects?
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.
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.
> 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.
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.
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!
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).
I understand the topological point of view (which also generally considers spheres to be 2-dimensional surfaces.) I'm pointing out that common language does not match the topological point of view. Topology also has 'open' and 'closed' sets, which are not very good names for whether some subset of a space 'has a shared classification of nearness', and 'tunnel' is probably better terminology than 'hole' is.
But mathematics is done by intelligent people, and it's a rather esoteric subject, so it isn't exactly a problem that the terms don't align intuitively.
> 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".
You probably do. If you think of definition as some mathematical description that strictly delineates what does and what does not qualify... that's not how human language definitions—or even human words—work.
Try handing an inflated ballon or unbroken sheet of copier paper to the same people and ask them how many holes it has; I doubt you'll find much disagreement between the answers. This implies that there is a definition of some kind.
English word definitions are probably better analogized as overlapping probability distributions, than as mutually exclusive sets of discrete values.
Now give a straw to those same people and ask how many holes there are and you'll get a range of answer, which implies that whatever you think is the definition, it's not working.
Just because you get agreement over some small range of examples, that doesn't then mean that it's a wide-ranging, usable definition.
Start sphere that has a small hole in the top and ask how many holes there are, you'll get the answer "One". Now make the hole a little bigger, then a little bigger, then a little bigger again. Keep going, and eventually you can turn this into a disk, or a shallow bowl, and now the (natural language) answer is "Zero". When did it change? People will disagree because there really is no "proper" usable consistent definition. Not in natural language.
And with regards a probability distribution - are you going to say that at some point the object I've described above has a half a hole? That's actually an intriguing prospect, to say "What's the probability that this thing has a hole?"
Names come from people, so the probability being distributed is the probability that you'll get a given name from a given person for a given thing. Take your sphere-to-disk example. The bigger the hole gets, the less likely it is that a given person will describe it as a hole. But at various points you'll get spikes where most people say "hole," "bowl," "disk," etc.
Personally I think it's silly to take this as proof that the definition of hole "isn't working". Probability distributions are good enough for quantum mechanics and fluid dynamics, but not language?
Also look around: people communicate about holes productively (i.e. to accomplish work) all the time, so obviously the definition is working just fine under many conditions. Language is one of those things that works way better in practice than in theory.
> And with regards a probability distribution - are you going to say that at some point the object I've described above has a half a hole? That's actually an intriguing prospect, to say "What's the probability that this thing has a hole?"
Not "half a hole", but "kinda a hole". As in, "Is this a hole? Well, kinda. Not really, but it sort of is."
Probabilistic lingustics makes (make?) a sort of intuitive sense to this layperson. In some contexts a word conveys a lot of information, but in others, it conveys less.
Well, he tried, but he did it the mathematically geeky way, not the ordinary-human-speaking-English way (which is my point). No non-mathematician ever actually calls the concave part of a cup a hole. It kinda sorta looks like a hole, but the process by which it is created is different from a proper hole, the kind you create by digging in the ground with a shovel. It also has some saliently different properties. For example, you can touch the "bottom" of the "hole" in a cup on both sides. You can't do that with a hole in the ground.
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.
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.
X = { (x,y,z) in R3 : |z| < 1, 1 < x^2 + y^2 < 2 }
Change the numbers until this is straw-like enough for you.
Define the deformation retract f : I * X -> X, (where I is the unit interval [0,1]):
f_t(x,y,z) = (x,y,z*(1-t))
This is continuous and well-defined and retracts the straw onto the space
f_1(X) = { (x,y,0) : 1<x^2+y^2<2 }
Which is two dimensional.
Now do something similar to make the annulus a circle.
Now what it actually means for two spaces, X, Y to be homotopy equivalent is that there are continuous f : X -> Y and g : Y -> X such that fg is homotopic to the identity on Y and gf is homotopic to the identity on X. Now for maps a,b : A -> B to be homotopic, there must be some (continuous) map H : I * A -> B such that H_0 = and H_1 = b. So why does a deformation retract lead to a homotopy equivalence?
Let R : I * X -> X be a deformation retract and r = R_1, Y = r(X) so r : X -> Y. Then we get a homotopy equivalence between X and Y by the functions r : X -> Y and i: Y -> X the inclusion function (ie the identity function).
ri = id (definition of a retract) so done here
ir is the same as R_1 : X -> X so just use the homotopy H_t = R_{1-t} to get the equivalence to id (R_0=id by definition of deformation retract)
> it seems reasonable to say that there is one hole in a circle
I'm not sure this is reasonable. There's only a hole in a circle if you are interacting with it from a higher dimension. If you're restricted to the plane, you can't access the area on the other side of the circle's border.
Does a ping pong ball have a hole? It's empty inside just like the circle.
A ping pong ball does have a hole but slightly differently from a circle in that the hole in a ping pong ball is 3-dimensional whereas the hole in the circle is 2-dimensional.
One needn’t embed the circle in a higher dimension to count holes. In the following sense a circle has one hole: the fundamental group of the circle is Z. That is, if you pick a point on the circle then the ways to go on a walk and get back to where you started are, modulo boring things like turning back on yourself or how fast you walk, to go round the hole in the middle of the circle clockwise an integer number of times (so 0 is staying still and -1 is once anti-clockwise). If you look at another shape, say a line the fundamental group is 1 as there is no nontrivial way to return to your starting point. On the surface of a torus (think inner tube) it turns out that you can go round the central hole (eg the whole bike wheel) some number of times and go round the small hole (eg the cross section of a bit of the tube) and it doesn’t matter which order you do them in. On the other hand if you take the shape made from two circles attached at a single point this also has two holes but unlike the torus it does matter which order you go round the holes. A path is described by some sequence of abcd where one cannot consecutively write ab,ba,cd, or dc: let a/b mean go round the first circle forwards/backwards, and c/d go round the other circle forwards/backwards then the sequence describes what to do.
I’m not quite sure what you mean when you say restricted to a plane. In some sense a circle does not exist embedded in any space and it’s holefulness (being topological) shouldn’t depend on the space it’s embedded in. If you do embed it into a plane one can note that the circle partitions the plane into two regions (call this one hole) and two circles would partition the plane into three regions (if the circles intersect more than once then this is really equivalent to multiple circles). If you put a circle into R3 (or S3), one can count holes by looking at the shape of the space minus the circle again (in the way talked about in the video)
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.
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".
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.
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.
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.
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.
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?
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).
He was pretty clear that he's going to answer the question from a topological perspective though, not using any colloquial definition of "hole". "A big enough quantitative change equals a qualitative one" isn't really how topology operates (as illustrated by how, topologically, a doughnut is the same shape as a coffe cup).
Mathematics is about understanding connections and commonalities, and then studying those to see what we can learn. To you, perhaps, a cup and a plate are "clearly not the same thing", but there are commonalities, and in that context, learning something about one will then apply to the other.
A way to see this is as a classification problem. Two nails in the box you buy in the hardware stores are also clearly not the same thing, but people usually treat them as interchangeable.
Is the solid/liquid/gas classification good? Is a ingot of gold the same than an ingot of lead. They are both "solid". Sometimes "solid" is a good concept for a classification, sometimes not. If the 16-ton weight is going to fall over the roadrunner head from 30 foots, you don't care too much about the material it is made. Probably it's enough to classify it as "solid" (with a few hidden assumptions, like it's much denser than the air).
Topology is good if you care about topological properties. But sometimes the property you care is an unsuspected topological property.
Sometimes you want to make a transformation to a mathematical object to change a property, but you don't want a very crazy transformation, and the property change you want changes something that changes something that changes a topological property. So if you allow only transformations that are not very crazy and don't change the topology, then you are doomed and can't do the transformation you want. So you get a proof that something is imposible.
What interesting difference is there between a cup and a plate? I claim approximately the only nontrivial difference in their shape is how much water the different shapes can hold. One could surely define a notion of usefulness as a water container (a teapot shape would then be similar to a cup and not to a plate), but I think that would largely be the end of it.
It turns out that the reason theories like topology exist is not for their usefulness in distinguishing cups from plates but for their being interesting to mathematicians (and secondarily for their applications to anther things that are interesting mathematicians).
A final question: is a bowl like a cup or a plate? Somewhere in between?
The cup and the plate share many of the same mathematical properties; we know this precisely because they are topologically the same. This means that when you prove that something is (topologically) true for the cup, you know that it is also true for the plate, and for anything "in between". Instead of a statement only applying for the cup, it now applies to a large class of objects, which is very powerful.
That's like asking "what good is the garbage man is it treats a pair of shoes the same as a tv set (when they're clearly not the same thing)".
Whether something is good (that is, useful) at some level of abstraction depends on what we need to do with it. For some use cases and calculations (such as those modeled by topology) the difference between a cup and a plate is irrelevant.
For cooking, of course, the difference between a cup and a plate is relevant. But for a dishwasher it's again irrelevant: you put them both in and it washes them with the same streams of water and soap.
i think you might be making metaphysical claims, while the topologist is not. moreover, in topology, things dont.become other things or kinds of things; they stay as they are and the (simplified) statement "a staw is a donut" means "a straw is similar to a donut in that they share certain qualities." there is never any actual stretching of an object to change its metaphysical identity, that is a virtual process (and one that has rules and restrictions) that is used to help humans wrap their head around the concept of homotopy.
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.