Hazy memory, but three decades ago I read some research from the Open University in the UK that said, roughly, that the number of bugs found is a factor of the number of users.

Some apparently "bug free" programs are actually riddled with bugs but they are not found because almost nobody uses them. They are probably not fixed for the same reason ;-)

Yes, open is source indeed awesome that way. Anyone can, in theory, go chase down that bug they tripped over. And many people do! But not nearly as many people as are capable of it (say, developers with some familiarity with the language, et al).

I've made just a few bug fixes to open source software that I didn't have some ownership of. From talking around with other devs over the years that makes me quite unusual, in that almost none of them have made any fixes to other peoples' code. And here I was feeling bad that I hadn't done more.

I don't think it should be surprising that few users actually look at the code or are willing to dig into a foreign code base. It's still true that open source makes it possible, which is a huge step up from any other model. There's a lot of backlash right now due to some very high profile bugs that have been around for a very long time. But would those bugs have been found if the programs hadn't been open? Also, look at what happened after heartbleed: Another group of people decided to dig into the openssh code base and try to clean it up, finding and fixing lots of other issues, without any authority from the original authors. That's the benefit of openness, in my opinion.

Yes, the "many eyeballs" is a part of it. But I think the author's point is that they are only half. The other half is someone has to trip over the bug. That is, people aren't finding these bugs purely through reading the source code.

There is a huge difference between a code review and the specific search for the cause of a known, defined bug. Should we rephrase Linus' law as "given enough eyeballs, all bugs' causes are shallow"? But maybe that's what the sentence actually meant for the first person that wrote it.