I believe that “assume”, “presume”, “suppose” are similar in meaning of to take some facts as a truth without proof.
But it seems to me that “presume” is more formal, “assume” is less formal and “suppose” is the most general word for this meaning. Am I right? Maybe there are other differences in meaning and usage of these words?
An assumption is technically something that must be taken for granted in order for an argument to go through. Some assumptions cannot, in principle, be proven. For instance, there is probably no way to prove that anything exists outside of my own mind, but I assume this because otherwise I would quickly die as a result of walking front of a non-existent car.
Less fancifully, if I thought that physics was a good and useful thing, then no matter how much I might doubt it, I would be forced to assume that the physical world exists if I wanted to be a physicist.
“Presume” has about the same denotation but also the additional connotation, as mentioned in Webster’s, of confidence. Note that to say someone is “presumptuous” commonly means that someone is overly confident of their assumption.
As mentioned above, I think in most contexts today “suppose” is used rather to express a certain amount of doubt about an assumption, and in that sense has the opposite connotation to “presume”.
In this sense you might think that “assume” has relatively neutral connotations, while “presume” and “suppose”, as commonly used, have roughly opposite connotations. I think that’s about right.