I'm on the way to Daejeon for the second week of a series of lectures on the graph structure theorem.

I'm wearing Crocs. Not real Crocs, mind you, imitation Crocs. There are all sorts of Croc-like shoes that don't carry the brand price tag. They cost what clumps of moulded plastic should cost.

Some are good, and some are shit. Mine are good, even if bulkier than the comparable Croc. This suits me fine, because I tend to wear through the bottoms of mine.

The previous week, at the seminar, Sang-Il commented that he was surprised someone would wear Crocs 'all the way to Daejeon'.

A couple of weeks earlier, I had worn them to a conference in Germany. I got comments there too, but only when we went on a hike.

As self-conscious as I am, I wonder if I should be more conscious of my footwear. Is it possible that Crocs are too casual? Even for a mathematician?

I get to the seminar at the IBS and make my greetings.

Andreas asks how I've been. "Well," I say, "Of course I've cut the lining out of my pseudo-crocs. So that comes with all the usual turmoil. "

" 'Of course'?" he asks.

"Well," I say, surprised at the part of my sentence that he responded to. "Surely you noticed."

"I didn't," he says simply.

"Oh," I say. "What a strange thing not to notice. "

"Is it?" he asks. "I think it would be a strange thing to notice."

"Notice– don't notice–" I say, "Two sides of the same coin. "

"Yeah. Opposite sides."

"Sure, that means that they are almost the same. Just opposite," I say. Then I tag on a mathematical example to make it more clear. "Like pi and minus pi. "

"Then the expression isn't apt."

I wonder if it is. I wonder what the expression means. This is something I'll have to as my AI agent, llama3.1, about. He thinks I am learning English, and loves it when I ask him about idioms.

Here's how that went.

>: What does the expression "Two sides of a coin" mean?

The expression "two sides of a coin" is an idiom that means to consider or discuss both aspects, perspectives, or viewpoints on a particular issue. It's often used to convey that there are at least two valid or opposing opinions, ideas, or outcomes related to a topic.
...(shortened)
In summary, "two sides of a coin" encourages us to think critically and consider multiple perspectives when evaluating an issue or situation.

>: Man, good thing I asked you, I thought it meant the same thing as 'peas of a pod'.

Don't worry, it's easy to get confused with similar-sounding idioms! But "two sides of a coin" and "peas in a pod" actually have different meanings.
... (shortened)
So while both expressions use natural imagery to convey a concept, they have distinct meanings. Now you know the difference!

>: Yeah. Thanks. But what if I said "It's two sides of the same coin," and I really emphasised the 'same' by really scrunching up my face when I said it, and saying it louder?

If you emphasized that it's "two sides of the SAME coin" with a raised tone and scrunchy face, that would imply that the two sides are not just different perspectives, but actually identical or equivalent.
... (shortened)
It's like saying: "You think there are two distinct positions here? No way! They're both part of the SAME coin, just trying to spin it differently!"

>: Great. Do you want to explain this to Andreas, or is it better if I do it?

I'm a large language model, I don't have personal relationships with individuals, so I didn't understand that you were referring to someone named Andreas earlier.
If you'd like to share the explanation about idioms and their nuances with someone named Andreas, go for it! They might appreciate your effort to clarify the differences between "two sides of a coin" and related expressions.

So there, Andreas!