How does Alfred North Whitehead's critique in "Science and the Modern World" converge with Jamesian Radical Empiricism?
I'm sure there's a lot to say about the relationship between Whitehead's "Science and the Modern World" and its relationship with Radical Empiricism. Here's what occurs to me—I hope it works as a place to start!
Whitehead was preoccupied with the worry that contemporary science with its materialistic emphases was moving in the wrong direction; his peers tended to believe that "only such things as could be localized at a mathematically simple 'point' of space and time were genuinely real" (IEP). Thus "relations and connections" were disregarded in favor of the "real" and physically locatable. Whitehead believed this to be a grave mistake—fields of relations should be prioritized even above the physical, as he believed they were in nature. The resulting fallacy led physicists to regard the abstract as concrete.
William James's conception of Radical Empiricism seems (to me) to support Whitehead's concerns and combat them by postulating (among other things) that both experience and relations between experiences are valid parts of scientific explanations. Scientific explanations should not confine themselves to the physical level (mere "whirling particles") but should also consider the bigger picture: the connections that explain and give meaning. Relations between things are what provides those things with meaning.
I think the "convergence" you mention has to do with how both men characterize reality. What follows is an extremely crude summary:
James came up with the idea of "radical empiricism" to correct what he saw as a wrongheaded trend in science to privilege measurable phenomena; contrary to Lockean empiricism, James argued that reality is more than simply "sense data" and that the causal relations between these "actual occasions," which are not sensory but rather interpretive, are just as important in understanding reality.
Whitehead seemingly picks up on this idea in Chapter 10 of "Science and the Modern World" when he discusses the "diversification of nature," his term for the way the human mind perceives nature by breaking it into separate actualities (or, to use James's term, "drops of experience"). Whitehead's epistemology is based, not simply on each individual event, but on the totality of possible relationships that can be established between these events. In this way, Whitehead follows James in recognizing the "connections" or causalities humans find between events to be as vital to understanding reality as the events themselves.