I don't think that stumbling on the "philosophy" of modern science is a pre-condition, because I believe that that academic structure has arisen in the wake of capitalist development. Practically speaking, there was a great deal of innovation going on all over the world, regardless of its relationship with the formal academic structures favored by the various ruling classes of their time and place. What is needed is that something like capitalism arises somewhere in the world; if it can take off it will tend, as European capitalism did OTL, to sweep the world before it and gather the whole world's resources to its exponential growth. The continual self-revolutionizing technical innovation is part and parcel of capitalist development and it will devise its own philosophical integument as needed, as happened in the Euro-Atlantic world of our time line.
So, it can happen based on Europe as OTL, or it can conceivably happen somewhere completely different if that somewhere diverges early enough to pre-empt Europe. What we need to do is define the criteria of the sort of capitalism that did arise in Europe, broadly enough to capture the essence, and look for possible candidates.
I suspect that the key to what happened in Europe was a balance between centralizing and centrifugal tendencies in social development. That is, in general if a region of the world develops a particularly successful form of civilization, there is a tendency for some political power to arise there that seeks to consolidate it under its control, and when this happens, broadly speaking stagnation is the result. A region that can't be brought under central control is also generally one that can't sustain a rising general intensity of pragmatic technical development, so there is as it were an energy barrier or a dilemma to be overcome.
In Europe, the quest for centralization never ceased, but various factors held it in check. Enough intercommunication between the realms of Europe persisted to allow for ongoing general development, but the stubborn refusal of the powers there to be fully united gave play to competition that forced the various rulers to keep their options open lest some innovation by a neighbor left them fatally behind. So, gradually, the groundwork to allow capitalist methods of "organizing" the regional economy (scare quotes because capitalism operates by as it were internalizing and embracing the chaos of anarchic competition) to develop, which in turn meant the door was open to continual innovation unstoppable by any one state (or rather, stoppable only at its own eventual peril), and this gave the Europeans the means of assimilating innovation, both of their own and appropriated from others, routinely. This meant the gradual multiplication of European technical power and their eventual ability to reach and then dominate powers all over the world, and on that basis, rising exponentially on its own and also feeding on the assimilated resources of the world as a whole, technology and science developed to the point that spacecraft could be launched in the 1960s.
So, where else in the world could something like this have happened?
I'm tempted to suggest, working backwards in time:
An earlier-developed Southeast Asia/ Indonesia region;
Other parts of the Islamic world that might have had different breaks, notably on the interface of Persia and India but also possibly West Africa or a central Caliphate run on somewhat different policy lines, perhaps;
Some indigenous shift in the development of India might have possibly given place to such a technical-social arms race;
China might have failed to typically form a unified empire and instead persistently remained divided into large but still diverse rival zones.
And returning to Europe, perhaps someone can show how what eventually was well under way there OTL by say 1400 might have, with somewhat different breaks, have gotten going by say 1200. (I stuck in an extra century on the surmise that the farther back you go, the less development in the world in general, on the average, and so you have to allow more time).
That kind of exhausts my laundry list of possible candidate regions, and people might beat every one of them to death with well-taken objections; meanwhile I hope other more imaginative people might identify others.
I rather doubt that the Classical civilizations of the Mediterranean world could have surged a lot farther ahead than they did OTL before suffering the crises that pretty well wiped out Rome in the west and disrupted the East; I think you can't have capitalism without a certain degree of broad development that was far from mature at that point.
But moving it back a century or so does seem doable to me without going back to Sumer or the invention of fire. And a distant POD would probably wind up looking like just arbitrarily shifting the whole known structure and pacing of OTL back a bit rather than giving us a tighter timetable for development
To sum up, I think that broadly speaking there is a curve of human development that builds upon itself more or less exponentially, and it takes the time it takes to achieve certain levels of result. But with broad, fuzzy error bars; the detailed process is quite chaotic, sheer chance might blight a promising zone of OTL or vice versa such a zone might have been randomly blighted in OTL and in an ATL shift the process forward, locally. Centuries seem about the right time scale for chaos to have play; shifting it back thousands of years probably would require PODs many thousands before that involving more successful early human development and dispersal, maybe changes like earlier ends of the previous glaciation and so on.