I guess since you don't recognize private property
I recognize tangible property. I don't recognize imaginary property.
the right to use said property as I see fit
The fallacy is assuming that you are required to be successful in your attempts to use said property. You might have x-ray goggles but if I have lead-lined walls, you will fail in your usage.
Yet that property is very real. So very real that you want to take it to use it for yourself, and are making a point at not caring what I have to say about it.
Your example deviates from the point. What you present would be the equivalent of coming up with some ingenious design of yours and taking over my market share, which is perfectly fair, even if you based it off of my original concept, since you made it better, you deserve the rip from it.
The proper example would be I have x-ray goggles and you outright tell me "no you can't use them", then proceed to buy yourself a pair and use them.
Do mathematicians "create" theorems? What right would the descendants of Pierre de Fermat (or he himself, in his time) have to prevent you from rediscovering, say, Fermat's Little Theorem independently (such things happen in mathematics all the time), or from using it to prove other theorems or to invent new technologies, like public key cryptography?
If I rediscover it, it is mine to use. You are completely out of context. Laws of nature are for anyone to use if they can so manage. I am a painter and I come up with a piece. Now I choose to charge people to come and see it. I am not pretending dominion over paint, canvas or the technique I used to produce my painting. I am pretending property over the original alignment of colors that is the picture I have come up with. Then you come along, take a picture under the pretense that I cannot restrict access to that piece, that it somehow belongs to everyone, and god knows what else.
You take the stance that originality does not exist, that voluntary alignment of objects hold no meaning nor value, that a thousand monkeys with a typewriter and infinite time can come up the whole of Shakespear's work, and then pretend that I am calling dibs on words and semantics.
When was the last time that anybody got rich by patenting a programming language? Aren't these useful inventions? Don't they get invented all the time?
Programming languages were invented for the very purpose of being spread. You are oblivious to the intention of the creator, thinking that intangible creation were made for masses benefit, at the cost of its creator's time, effort and resources.
One argument for patents is that the alternative would be secrecy. A counterexample would be the RSA algorithm. It was first "invented" by someone in some British secret agency and kept secret. Not much later, a group of researchers discovered it independently and published it. Now everyone can benefit from it. I don't think their inventors regret the work they put into it. Actually, I see absolutely nothing wrong with that.
That is their very right, ideas aren't properties, their expression is. A movie maker isn't demanding rights over the concept of romance, only his take on it. A mathematician knows there is no rights to be held on a theory, but the software he builds after it, that is his property. If you can understand his process and give it your own shot, you are in your right and the more power to you. But to copy his software to spread it at your profit and at his detriment, this is not only aggression against the creator but also pocketing his wealth for yourself.
If you could reliably heal yourself by checking a website and following simple instructions, doctors would have no business, and that wouldn't necessarily be a bad thing.
Specialists fill for your inability to specialize in said domain, in the expectation of returns for services rendered. The very existence of specialists implies that they were able to profit from their specialty more than from being common laborers, which is the reason we don't live in caves anymore. But we are headed straight towards cave land if they can't make a profit of holding and applying knowledge that you don't have. If intellectual rights are moot, then a doctor has no right to charge you for a diagnosis. Let's see where that is going to take us.
In sum, mathematics, science, and medicine have progressed alright for centuries without anybody really needing to artificially restrict anybody else's use of information. Information is variably hard to discover/produce, but, by nature, often trivial to reproduce. You have no right to demand others to artificially renounce the benefits of this desirable property in order to protect a fundamentally unsound business plan.
Yeah let's forget the existence of sponsors under the form of kings, feudal lords and later governments and corporates, all that have greatly benefited from said research that they financed. And you want to read the part on charity again. My business plan seems to be sound enough that you want to take over, don't you? And somehow what isn't property turns into "desirable property" out of a sudden. Well shit, son.
You pay to be taught but you don't pay for the book that holds that knowledge? That is nonsensical.
I think you're under the mistaken assumption that producing X entitles you to own X but let's say you break into my shop, steal a bunch of my wood and then build a chair out of it. Do you own the chair because you produced it? No, it's my chair plus you owe me for damages to my wood. I collect wood, you see, and I wanted it kept in pristine condition.
'scuse me what? If you gonna talk about private property at least respect it's principles... first come first served, it doesn't matter what I did to your wood, it was without your consent, I have no right to it nor did I ever had. And yeah, if I build a plane it belongs to me, and the original alignment of its innards too. kthx.
that the need for property rights with tangible goods arises from their scarcity
I don't agree with this at a fundamental level. This is a pragmatic explanation, that exposes the need of property rights for a functional society to exist. It does not tend to the fundamental concept of property. I walk on a beach and pick up a grain of sand and call it mine. Assuming no one claimed it before me, it is now mine, and as all value is subjective, I shall assign it any value that I wish, and it certainly is not scarce. Property is a factor of value, and even though scarcity is an important parameter of value, it is but situational. Here's a simple example. Titanium is the 3rd most available metal on the world. Yet items made of titanium are highly expensive, because it requires particular knowledge and skill to work it. Here what is scarce is the ability to work it, even though, according to you, ideas 'cannot be scarce', so since the knowledge is available, anyone should be able to reproduce it.
True, scarcity isn't relevant to an idea, but the ability to understand it, that is. And nevertheless, books are not scarce, but a particular book holding a particular idea is valuable, because of the knowledge it provides.