Can you play Minecraft LAN cross-platform?

No.

I wouldn't associate with anyone owning a mac. Snobby barstewards.

And its rubbish for 95% of everyting bar writing music.

If the PC and Mac are both running the Java version, yes. They absolutely can.

As long as you're playing the Original Java version with both computers connected to the same LAN (ideally connected to the same router), then you should have minimal issues as the host just has to open up their offline world to the LAN & the other person should be able to hop in.

Since the Original Java version is OS-independent, it doesn't matter if the computer is running Windows, Mac OS X or Linux as they're all executing the exact same game code verbatim. There may be some minor differences in the Java Runtime Environment that each computer is using, but that's it.

The same can be said for online Minecraft servers as they're OS independent as well, since they run on Java. Many 3rd party server hosting services run on Linux to help keep costs down & maintain strong security… But you can run your own personal server from a Windows or Mac as well.

Hope this helps!