Suppose we had an apple replicating machine, that could replicate apples costlessly and instantly. How would we go about answering the question: "Are apples a non-rival good?".
1. We could have a lovely philosophical argument about whether the original apple and the copy are two different apples or just two different manifestations of the same apple. We would need to answer that question before we could decide whether the same apple was being consumed twice, which is one way to define "non-rival" goods.
2. Or we could simply say "yes", because the answer to the first question doesn't matter for economics; what does matter is that the marginal cost of producing the second and all subsequent consumptions of the apple(s) is zero.
Now we do in fact have apple replicating machines. We call them "apple trees". But they take labour, land, and time, to replicate apples. So the marginal cost of replicating an apple is not zero. And if we model the apple market, our model will include those replication costs. And unless we are talking about some new and rare variety of apple, we will ignore the cost of the first apple, because it's trivial.
Nearly all the food we eat is produced by replicating machines. Plants and animals are replicating machines.
If I teach you an idea, and you learn it, is the idea in your head the same idea as the idea in my head? One idea or two? It doesn't matter. What matters is the cost of replicating the idea. Sometimes that cost is non-trivial. Schools and universities are idea replication machines, and they are costly machines. They use the teachers' time and the students' time. If you start school at age 5, quit school at 20, then get a job and retire at 65, you have spent at minimum 25% of your total working years working at the job of replicating ideas. And that's not counting your teachers.
Factories that mass-produce cars are car replicating machines. First they build the prototype, then it gets replicated in the factory. The prototype is costly, and the replicas are costly too.
I can't eat the same apple twice, but I can use the same socket set twice. The first use cost me $100 (which is what I paid to buy it), but the second and all subsequent uses cost me $0 (wear and tear are negligible). I don't (normally) need to buy two identical socket sets. But two people might need to buy two identical socket sets, because the same socket set can't be in two different cars or garages (though friends and neighbours might sometimes be able to share, at little extra cost).
A live firework display is the exact opposite of the socket set. I can't watch it twice at zero marginal cost, but a second person can watch it once at zero marginal cost.
These examples show that the marginal cost of the second use may depend on whether the second user is the same person or a different person to the first user. It's the same with ideas: usually it's easier and cheaper to remember an old idea so you can use it twice, than for one person to teach another person. But at the macro level we can only remember by teaching, because individuals die. Some technologies do get forgotten, and need to be rediscovered.
Ideas are economically important, but are they economically special, on the rivalry/non-rivalry dimension? Even boring material goods, like socket sets, can get pretty weird, if you start thinking about them.
[Update: jonathan makes a good point about excludability of replication of ideas in comments. If I sell you an idea by teaching it to you, you also (usually) have the ability to teach it to others. That's not true if I sell you a car (unless reverse-engineering is easy). It's true if I sell you a live cow for milk, but not if I sell you a dead cow for meat.]
Thanks to commenter jonathan (who is not responsible for opinions here).
[Paul Romer might like my old post "Macroeconomics when all goods are non-rival", (even though it's short run macro and not growth theory like he does). And I was doing (short run) macro with monopolistic competition (pdf) (a couple of months) before it was fashionable. So it's especially silly (not to mention intolerant of disagreement) for him to make this accusation of me: "After twenty years of playing whac-a-mole with the die-hard price-taking neo-Marshallians, I can tell you exactly where the argument will go next. When the choices are stated as clearly as Dietz does, the neo-Marshallians will claim that there is no such thing as a nonrival good. Nick Rowe has already started laying down a smoke screen of obfuscation on this point." Nope. He got that one totally wrong, on both counts.
And that sort of response doesn't create a productive environment for new ideas. People need space to muse over old and new ideas, without the authorities making nasty accusations about their motives.]