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It's pretty simple.
Positivism is a philosophical doctrine (teaching) that recognizes only empirical research as the only real source of real knowledge; genuine (or positive, positive) knowledge about something is the result obtained within the framework of special sciences. This has nothing to do with” positive thinking ” in its pop-psychological interpretation.
Optimism is more of a philosophical position, or attitude, which believes that the most reasonable way to look at everything that happens is from the point of view of the best, favorable sides and properties, and count on a sufficiently favorable outcome even in unfavorable circumstances. Roughly oversimplifying, this is the “everything will be fine” philosophy.
A slightly more sophisticated approach to optimism is associated with the concept of optimization, i.e. improvement; in a general sense, this is a philosophical position in which both negative and positive aspects of being are accepted without reservations, but the belief remains that much can be changed and improved in accordance with accepted values and guidelines.
Positivism is the philosophical concept that the source of knowledge is experiment. Now it is almost impossible to distinguish it from science, and it is not clear why you need another word to define science. A kind of vestige.
In everyday language, the positivist wants everything to be good, and the optimist hopes for it. A positive optimist successfully marries or gets married.
My optimism or positivity depends on the mood ,the same question or problem can result in either positive or optimistic, impulsive.And the difference in these concepts is relative.For someone, the same action and the concept of optimism, for another whim and nonsense, as well as with the positive.
Indeed, there is something to distinguish. To keep it short. Optimism is a positive attitude to life, a shock absorber of its failure. And positivism is a negative attitude to science, its refutation by empiricism, in which there is no knowledge of the nature of things, nor attempts to penetrate them.