> I think the idea behind intellectual property is to encourage people to take risks associated with developing something new and profit through their innovations.
No, the idea is to prevent other people from profiting through 'claimed' innovations.
> If all new discoveries were the property of the state/public then there wouldn't be a high incentive to be inventive for any given individual.
Most innovation is not done for money. It's done by people with genuine interest in the subject, often earning less than an average professional in the field. The vast majority of valuable R&D for computer(s/ing) came, and still comes from the public sector.
> It's like if everyone was a farmer but everything harvested was immediately taken from the farmer.
Why is it that intellectual property proponents always try to make analogies using physical property? Copying != moving/deleting. If I was a farmer and along came some poor hungry people that saw me farming, learned from it, then went and farmed their own land, I would have no problem with that.
> Why would any farmer put in effort to increase yield when they would not profit from their extra effort?
If we're talking about innovation, then because it could make farming easier, safer, more enjoyable, healthier, etc.