Many worlds only violates positivism if you accept that there is some meaning to the statement that the other worlds "exist", beyond the statement that quantum mechanics is exact. This is not what the many-worlds business is about, it isn't about ontology, it's about the best framework within which to frame quantum mechanics and the map from quantum mechanics to experience.
Many advocates of many-worlds are positivist to the core, and don't care one whit if the other worlds exist or not. They just want to use this ontology (you can use any ontology you like consistent with experience) to make certain statements which are otherwise counterintutive manifestly obvious:
- Quantum mechanics requires exponential resources to simulate efficiently, the amount of resources required to simulate quantum mechanics on a large system is equal to the resources required to run a parallel machine, what CS people call a "nondeterministic" or "NP" machine, with some extra communication between the processes.
- Quantum computers can provide exponential speed-ups to certain problems, like prime factoring. Current models of quantum computers cannot exponentially speed up NP complete problems.
- Quantum mechanics can perform measurements of Vaidman counterfactuals--- these are questions of the form "if I were to shine a photon on this mirror, would the mirror measure the photon or bounce it off without measurement?" You can answer a Vaidman counterfactual without actually bouncing off a photon off a mirror.
These properties of QM are counterintuitive without many worlds, and I consider these three things (and any more things that people might dream up) to be the positivist statement that "Quantum mechanics is a many-worlds theory". The ontology is just there to make these things intuitive, rather than surprising. These are surprising things in other points of view.
The philosopher's reaction to many worlds is first to co-opt it's major new idea:
- Experience can be subjectively random, even if the underlying theory is deterministic.
This is explained in my answer here: Is there anything that is totally random? . This insight is original to Everett, and has been taken up by Daniel Dennett, possibly through Douglass Hofstadter, in the early 1980s, in a series of lectures and articles beginning with "Where Am I?". These served to popularize this point of view, but it is due to Everett, and he should be given credit for this, since in the centuries of history of thinking about determinism, nobody noticed before him. This is largely due to the decoupling of experience and physical law which is taken for granted in physics philosophy.
There is another idea in many worlds:
- The states of experience of mind are such that a superposition is percieved as a random outcome. The measure of randomness is imposed by the map of physics to experience, rather than by the physics itself, which does not have a non-qualia notion of randomness.
This idea is not usually considered respectable among those philosophers that study the philosophy of quantum mechanics. This despite the fact that it is a sound position. It is derived from positivism--- Everett identifies the mental state of the observer with certain superpositions of basis states in the quantum theory, explicitly using a computational theory of the mind, using a computer, and he uses positivism and the attendant computational theory of mind to identify the experiences of the computer with our experience.
Since modern philosophy, past 1970, is built on the deranged rejection of postivism, this type of thing does not get much traction.
It is not 100% certain that quantum mechanics is true at the level of quantum computers, but if it is so, then many worlds is as right as anything else, in the postivist sense. The uncertainty comes from the fact that while we have observed Vaidman counterfactuals, we have not observed quantum computation, so we are not sure how far the counterfactual states can persist without some new source of fundamental decoherence turning them into probabilistic superpositions as opposed to quantum superpositions. Probabilistic superpositions are not philosophically thorny. There is no known source of fundamental decoherence in modern physics, but you can't be 100% sure it isn't there until you build a working quantum computer.
Among those that say it a quantum computer has a chance of failing are Gerard 'tHooft. This is the main positivist question of many worlds, and it will be resolved by experiment, not by philosophizing.