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Comment by paganel | Hacker News Reader

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).

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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).

And then the question becomes: what good is topology if it treats a cup the same as a plate? (when they’re clearly not the same thing)

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?

If you want to know about uses of topology, a 30 second search on DDG produced these links:

https://en.wikipedia.org/wiki/Topology#Applications

https://www.livescience.com/51307-topology.html

https://dragons8mycat.com/2013/08/20/gis-tips-what-is-topolo...

... and more.

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.

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.

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.

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.

The point isn't that the deformed "cup" is the same as the original cup. It's that in the process of deformation the number of holes remains the same.

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.

The Buddhists were on to this concept 3,000 years ago.

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

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.

Marx wrote about it in a letter to Engels in 1867. Apparently. So it’s not a new claim.

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