>Joe Public might come away with an over-simplified and over-confident knowledge about what science says, but I think that a clumsy knowledge of scientific facts is better than the impression that scientific knowledge is arcane and out of reach.

This seems like a false dilemma. Joe Public is as capable of reading a paper on sci-hub as anyone else. It would probably do everyone a bit of good if JP were more likely to do that when he's wondering what to do about the aphids in his garden, or read a patent relevant to his mechanics business, or assess the evidence around masks during a pandemic. And I think it's the idea that science needs to be dumbed down for a certain kind of person that keeps ordinary people from digging into the asterisks when things are relevant to what they're doing. Executive summaries are useful when something isn't very relevant, which is most of science for most people. But some science is relevant to everyone, and it would be good if we challenged the myth that one needs to belong to some kind of separate class in order to practice science. Science is for everyone. No one can know all of it, but everyone will come to a point where they'd benefit from knowing some of it.

I think the picture you paint is too idealistic. Not many can read a scientific paper and understand exactly what it is trying to say, since we are trained to avoid adjectives, adverbs, and half-formed speculations when writing papers. What is omitted from a paper is often as illuminating as what is in the paper. To know what is not said, you must have an anticipation of what must have been said. Often, that is quite important in peer review.

Even when giving scientific talks, the advice I give my students and post-docs is to first have a clear idea of who the audience is: is it

a) experts only from your area? [e.g. conference]

b) people from the same department who may be generally aware of the area without knowing specifics? [e.g. job talk]

c) General public [e.g. showing lab to school kids. This is the level at which you talk to the public. soundbites - so it is invariably misrepresented by the media]

All three levels require different approaches for the same subject. (b) is especially tricky - you are talking to experts, but from a different field. So you have to simplify, while not being patronizing. As for (c), I think it is best to avoid it. Except in situations like the pandemic, it rarely is beneficial. Also, to do it right, it requires a lot of preparation.

That is fair, but I think the general skill of reading scientific papers is one that more people are capable of learning than are encouraged to learn. I certainly don't think anyone has the time or intellect to digest every paper that might be of interest to them. But a lot of people I've met seem to be interested in science beyond a 'headline' depth and simply don't think they have the capacity to read any papers at all. That seems to me like a shortfall that could be corrected with better public education and media behaviour.

I'd be curious if you disagree with any of that in theory or just have a better appreciation of the practical difficulties...

I agree with this. One more point is that journalistic summaries of scientific findings are flat out wrong. Journals like Quanta may represent a middle ground, which are readable yet sober.

I'm sure that many could read it, but reading it and understand are not the same thing. Understanding takes time, often a lot of time. And this is where the simplified bit comes back.

Discussions on social media rarely indicate much effort at nuance...

I'm not even confident about the "many could read it" part. Most papers are full of obscure jargon, and occasionally fields of study will just re-define words for their own purposes.

I think falling for the Science as religion trap is a bad choice in the long run.

Why not Math as a religion, dumb down complex problems, if it's wrong for the common Joe, it's okay cause he wouldn't understand it anyway.. not a good strategy.. I think if Science has a knowledge management problem, that should be addressed rather than dumbing down of things for public consumption.

Do you have an example of a mainstream presentation of a complicated math discovery that doesn't dumb down the complex problem?

I will point out that math discoveries don't have quite the same kinds of asterisks as the physical sciences.

> Why not Math as a religion

Math is indeed a religion deep down, at least according to people like Hume, you have to "believe" in things like induction, you have to "believe" that natural numbers "don't end at some point" (for lack of a better word) or that two parallel lines don't finally meet, because who's there to watch them till the end of the world to say that they don't in fact meet? (Hume writes it down way better than I could ever do).

That's not how it works, though. We can study any set of axioms we want. Mathematics often study "multiple incompatible worlds". For example, euclidean space where parallel lines never meet, and projective space, where parallel lines do meet at infinity. This is a general theme of math; we can study different sets of "beliefs" when we don't know which one is the "correct belief".

Thus, we come out on top, since we know what happens in all possible worlds.

> what happens in all possible worlds.

Talking about "all possible worlds" is religious-talk (at least that's what Hume would say), afaik we only know of one world, talking about the "possibility" of other worlds is what established/institutional religions do.

Not literal worlds, mathematical possibilities. Analysis like "If XYZ is true, then [conclusion 1], but if XYZ is false, then [conclusion 2]". It's not religion, it's logic and mathematics, and we absolutely can look at all possibilities in many circumstances.