I am wondering if the rejection of the positivism movement in philosophy lead to any major discoveries in mathematics and natural sciences? I am thinking it might have been able to contribute to those fields, but knowing almost nothing about the history of philosophy, I am not sure where to find the answer.
The most important development in the rejection of positivism, from the metaphysical perspective, is that it allowed for a revisit and reexamination of the philosophical systems thinkers, like Aristotle, Spinoza, Kant and Hegel. This resulted in a renewed appreciation for the fact that any study of knowledge or science or reality could not be reduced to any arbitrarily selected set of 'primary' and 'secondary' qualities and that the reduction of the field of what is knowable is not even remotely acceptable. This not only opened new possibilities in earth sciences but eventually allowed for concepts in cosmology and astrophysics, like black holes, dark matter and alternative universes to be put forward and accepted as reasonable. For a solid overview of what this rejection meant in Biology, Evolutionary studies and theories of Consciousness see "The Foundations of Metaphysics", by Errol E. Harris,(Humanities Paperback Library, June 1992). Charles M Saunders
Positivism took its general shape during the 19th when "Naturphilosophie" went out of fashion and science and philosophy became separate realms. In Comte's account humanity evolved from religious views to metaphysics and ultimately to positivism. As positivism mimics science it could not really obstruct discoveries. Actually nothing in (history of) mathematics is positivistic.
The most famous case against positivism is the denial of atomism by German thinkers such as E. Mach or W. Ostwald but it mostly slowed down the reception of some results and ideas without preventing them to develop.
So the "rejection of positivism" had no tangible effect in maths and natural science. An analogy with the rejection of Aristotelianism could be conjectured behind the question but it is an altogether different story: "Äristotelianism" stands for "The Church"; people however avoided confronting openly its monopoly on ideas.
Logical positivism basically evolved into analytic philosophy in the 1950's to point where I think few if any contemporary philosophers regard themselves as "logical positivists" at least in the classic sense. Logical positivism turned Anglo-American philosophical orthodoxy in the direction of analytic philosophy, which took off from there, especially with the publication of J.L. Austin's "How To Do Things With Words" and W.O Quine. Analytics is very semantic-analysis oriented where logical positivism was a bit more just strictly empiricist. So to answer your question, no. As far as I can tell, logical positivism has had pretty much zero effect on scientific and mathematical progress. I suppose Kurt Godel's Incompleteness Theorem (and proof) also pretty well destroyed positivism so that would an example of the opposite effect, I think.