What does it mean when math text authors use the word pathological ?

[gakushya]

What does it mean when math text authors use the word "pathological"?

I have an example right in front of me in fact, but I'm sure you have seen this word before. I've seen it in a few other textbooks, and I'm pretty sure those books were also grad level math related subjects, just can't recall more than this one instance. So, just to illustrate by example, this is from Lee's Introduction to Topological Manifolds. He says the following

In our study of manifolds, we want to rule out such "pathological" spaces, so we make the following definition.

This is right after he demonstrates that the most generally defined topological spaces don't have enough structure to admit unique limit points. So, logically, Hausdorff spaces were invented. Or discovered. Or whatever your philosophical credence ordains.

I have two interpretations of what authors mean by this:

• Pathological in the medical sense where an entity with a pathology is diseased or crippled in some sense or rather. I'm guessing that this usage is meant to suggest that something is inadequate.
• In the idiomatic sense, as in a "pathological liar". Meant to suggest that if some mathematical object is pathological, its profound or extremely exotic and so is mentally disturbing.

I dunno, I kinda prefer the latter interpretation as it just seems to be more fun to think of something as blowing your mind, rather then crippled and obsolete. But couldn't both these usages apply to how we feel towards general topological spaces? But now that I actually wrote it out here, the first interpretation seems more likely to be the intended usage.

 

[AlephZero]

Usually it just means a mathematical structure that doesn't have the properties you would like it to have. For example, if you want to study the math properties of manifolds that are "useful" in describing physics, you want to exclude weird manifolds that don't correspond to your ideas about physics. But a pure mathematician might want to study those "pathological" manifolds just because they are wierd, without caring whether they are "useful" for doing applied math.

It is also used to describe things that demonstrate "special cases" - for example you might say that the function $f(x) = x \sin (1/x)$ when $x \ne 0$, and $f(0) = 0$, has pathological behaviour near $x = 0$.

 

[micromass]

I have an example right in front of me in fact, but I'm sure you have seen this word before. I've seen it in a few other textbooks, and I'm pretty sure those books were also grad level math related subjects, just can't recall more than this one instance. So, just to illustrate by example, this is from Lee's Introduction to Topological Manifolds. He says the following

This is right after he demonstrates that the most generally defined topological spaces don't have enough structure to admit unique limit points. So, logically, Hausdorff spaces were invented. Or discovered. Or whatever your philosophical credence ordains.

I have two interpretations of what authors mean by this:

• Pathological in the medical sense where an entity with a pathology is diseased or crippled in some sense or rather. I'm guessing that this usage is meant to suggest that something is inadequate.
• In the idiomatic sense, as in a "pathological liar". Meant to suggest that if some mathematical object is pathological, its profound or extremely exotic and so is mentally disturbing.

I dunno, I kinda prefer the latter interpretation as it just seems to be more fun to think of something as blowing your mind, rather then crippled and obsolete. But couldn't both these usages apply to how we feel towards general topological spaces? But now that I actually wrote it out here, the first interpretation seems more likely to be the intended usage.

I think both of your interpretations are correct here. A pathological object is often something that doesn't behave as expected, and thus it is crippled since it doesn't admit a nice theory like the others. But on the other hand, pathological examples can be extremely beautiful. I'm a big fan of various counterexamples books. For example "counterexamples of topology" by Steen and Seebach is filled with pathological stuff, but to me it's all quite nice and indeed: mentally disturbing.