The Main Point
ChatGPT is an AI language model that has been making the news recently. There are multiple articles on the internet on what the model is capable of so this will not be the focus of our post. Instead, we wish to consider how ChatGPT can be useful in the context of mathematics.
Now “in the context of mathematics” is rather broad. Broadly speaking, there are three aspects to it:
- learning mathematics (new concepts, techniques etc),
- teaching mathematics,
- researching mathematics.
This post is mostly about the first two cases. Here, we care not for the philosophical implications, future directions, or subtle flaws of the bot. Instead, our main question is:
How can ChatGPT help us, as mathematics practitioners?
People have reported getting conceptually deep and insightful answers from ChatGPT in the form of essays. So we ask it, out of curiosity:
Why are compact spaces interesting in point-set topology?
This is the reply we get:
Compact spaces are an important concept in point-set topology because they have many useful properties. For example, every compact space is closed and bounded, which means that all of its points are contained within some finite region of the space. This makes compact spaces well-suited for studying problems in which the points of interest are contained within a bounded region.
Compact spaces also have the property of being “small” in a certain sense. For example, every infinite compact space must contain a “small” infinite subset, meaning that it is possible to find an infinite number of points in the space that are contained within a finite region. This property makes compact spaces a good choice for studying problems in which the size of the space is a consideration.
Finally, compact spaces have the property of being “well-behaved” in many ways. For example, every continuous function defined on a compact space must achieve its maximum and minimum values, which makes them useful for optimization problems. Compact spaces also have many other desirable properties, such as the ability to be approximated by finite sets and the existence of convergent sequences, which makes them a useful tool for studying problems in many different areas of mathematics and physics.
Newcomers to ChatGPT would be impressed by its coherence and flawless grammar. However, if you’d been experimenting with ChatGPT for a while, such an impression fades rather quickly. So to us, at first glance, the above reply by ChatGPT actually seems underwhelming. The statements are ambiguous and ill-defined; a mathematics student who wrote the above in an essay would likely get a bad grade for the report. But since we emphasised the term “at first glance”, there’s more to the story than meets the eye.
First, let us critique the above essay as a teacher would.
- “…every compact space is closed and bounded” is largely true, if we replace the word “space” with “metric space”. In generic topological spaces, the term bounded does not make sense. Thus the first paragraph is largely sensible.
- The second paragraph is confusing. If we were to interpret it literally, it seems to be a strictly weaker variant of the first paragraph. Or, does it mean to say “compact spaces are often infinite, despite being contained in a finite region”? This actually brings up a nice contrast: although structurally finite (bounded), a compact set can be infinite, even uncountably so.
- The third paragraph is not bad. Replace the phrase “every continuous function” with “every real-valued continuous function” and you’re good to go. But it’s not clear what is meant by “approximated by finite sets”. Does it refer to the definition of compact spaces, where every open cover has a finite subcover? Also the part about “existence of convergent sequences” is terribly worded; it should be “every sequence has a convergent subsequence”, which is indeed a critical property of compact sets.
After some editing, we get the following.
Compact spaces are an important concept in point-set topology because they have many useful properties. For example, every compact metric space is closed and bounded, so heuristically the whole set is in a finite region of space. This makes compact spaces well-suited for studying problems where the domain of interest is bounded.
Compact spaces thus can be considered “small”, at least on a structural level. However as a set, it can be infinite, even uncountably so. For example, the closed interval [a, b] under the usual topology is compact, even though it is an uncountably infinite set.
Finally, compact spaces are also “well-behaved” in many other ways. For example, every continuous real-valued function on a compact space must achieve its maximum and minimum values, which makes them useful for optimization problems. Compact spaces can also be “approximated by finite sets”, in the sense that every compact metric space can be covered by finitely many ε-open balls, no matter how small ε>0 is. Also, every sequence in a compact space has a convergent subsequence, so one can imagine compact spaces to be constrained structurally.
It still wouldn’t win any award for best writing, but at least it’s more accurate (or so we’d hope!).
Proceeding in this direction, ChatGPT has the following applications.
- A lay person can get a rough guide on a deeply related topic, provided said person is willing to take a huge bag of salt with it.
- An imposter can feign deep knowledge by querying ChatGPT. In its current incarnation, this can indeed hoodwink the general lay person, but it only takes a few seconds for a domain expert to find multiple inaccuracies, poorly worded statements, or even downright falsehoods. [ Needless to say, we do not recommend such a use. ]
- A domain expert can save time by asking ChatGPT to write a first version of a report before editing it.
In this case, ChatGPT already has a concrete benefit. The “approximate by finite sets” can be a good motivation for the definition of compact sets, which is often uninspired (so what if every open cover has a finite subcover?).
One wonders what the next version of ChatGPT brings.