So, I worked with Bergson's texts quite a bit in grad school, as he is heavily in vogue at the moment in certain disciplines. To break down his argument to its essentials, the whole concept he is railing against is the spatialization of time. That is, for Bergson, time cannot be subdivided into the "mechanistic time" of the ticking clock, and the idea of a timeline is an abomination.
Hence Bergson's framing of time as duration: for Bergson the essence of experiential time is that our consciousness is always experiencing the latest moment sliding smoothly into the next. Time, he says, cannot be spatialization and counted as can space. Bergson railed against the idea of time being extrapolated to just another metric dimension like the 3 dimensions of space. The spatial dimensions, to him, were static, fixed, dead. It is only duration that gives our existential experience of the lived experience that we know. Spatialization, to Bergson, was a dirty word; it was the spatialization of our lived experiences that rendered industrial life dead, static, mechanistic and uninteresting. Bergson was railing against the idea of a physics that could predict everything, a popular thought in the early 20th century.
After WWII, Bergson was largely forgotten until Deleuze & Guatarri ressurected him. Deleuze in particular was an enormous fan of Bergson and promoted his ideas heavily.
But what was revolutionary about Deleuze's handling of Bergson was his incorporation of post-war complexity/chaos theory and quantum mechanics to recover space as a dynamic and mutable. Influenced by such concepts as Reimman mana olds and fractal theory, Deleuze recognized that space wasn't a static and mechanistic concept at all, but instead, like Bergson's duration, can give rise to all types of unpredictable behaviors, experiences, and mathematics.
Rather than focus on one concept of "space" - the abstracted Euclidean grid - they classified space in two broad classes, the smooth and striated. Smooth spaces are spaces that are analogous to Bergson's duration: the experienced space of the journey, nomadic spaces, spaces that unfold rather than increment, that are uncountable and unexpected. Striated spaces are the class of spaces Bergson focused on exclusively: coordinate spaces, the countable spaces of the Euclidean grid and the map, or that of the timeline.
Essentially, D&G 'recovered' mathematical space as an exciting and unpredictable philosophical concept. All spaces arise from continuously recapitualtion of smoothing and striation, and counting spaces always give rise to the uncountable and to emergent behavior.
A good example is Conway's Game of Life: a simple set of rules played out on a metric space in countable time (striation) gives rise to emergent organizational patterns and a higher level of emergent behavior that simply cannot be predicted or quantified using the original simple set of rules alone (smoothing). Or, to take another case, the Mandlebrot set: a simple pattern gives rise to a recursive, self-similar-yet-never-identical structure that persists to infinity. For D&G it the uncountable always arises out of the act of counting.
This comment is somewhat outside the normal domain of HN, I know, so I hope you will excuse it. I rarely get to show off the hundreds of hours I dumped into D&G and Bergson in gradschool in my day job. :-D