Before digital computers there were analog computers. These machines modeled processes not with discrete binary states piled high to equal either programs or data but with electronic circuits that added, subtracted, multiplied and divided current without respect to a ticking clock. Inputs and outputs could be continuous -- a characteristic which can only be approximated with their digital successors. It is easy to get from the analog world to the digital by simply adding a clock. It is as though the world were really analog, continuous underneath with digital quantization being an added afterthought.
Is this true? Is the world really a stateless continuum that we divide at will or is it quantized at the deepest level? The Greek pre-Socratic philosophers struggled with this question in attempting to understand the world. The very idea of the atom as well as the etymology of the word lies here. Democritus (460-370 BCE) proposed that the world consisted of minute indivisible atoms -- the word itself literally meaning "indivisible" in Greek. On the other hand Xeno of Elea (c. 450 BCE) asked what it would mean if the world were divisible*. Xeno's first paradox (of four) is that to get from point A to point B you first have to travel half way. Then half that. And half again. The result is an infinite number of steps between A and B. Xeno argued that this cannot be because if it takes a finite amount of time to travel each step then it would require an infinite amount of time altogether.
(*Xeno actually produced his paradoxes to show that change was not possible, but Democritus' atomism essentially was developed to explain change.)
Casual students of philosophy come away from Xeno's paradox confused. It is so obviously wrong that they tend to ask, "Where's the paradox". The paradox lies in following some simple assumptions, perhaps even common sense assumptions, to their logical conclusion. Like Schroedinger's Cat, Xeno's paradox is more about the way the world isn't than the way it is. Such conundrums typically point us to deeper meaning by pointing out that something is wrong with our ideas and perhaps clarifying what. Xeno's paradox shows that our answer to whether reality is discrete or continuous -- quantum or not -- can lead to unusual conclusions.
The atom that was eventually found by science turned out not to be indivisible. But the indivisibility of matter and energy at the lowest level is now beyond question. And it turns out that this simple fact has broad consequences. It allows us to "understand" the nature of black-body radiation, the internal structure of the atom, the nature of radioactivity and the photo-electric effect to name a few phenomenon. But the undstanding we have turns out to be almost unintelligable. While quanta seem to bring fuzzy things like light into focus they are themselves fuzzy. They don't seem to be anywhere in particular until we observe them. Some of their properties are absolutely uncertain. Particles which we used to think of as having no fuzziness at all do in QM. Some systems seem to be able to encompass the entire universe.
We can also speculate about additional consequences. Xeno supposed there can be an infinite number of steps from any point A to any point B. This is not true. There is in fact a smallest distance you can travel (known as the Plank length, about 10-35 meters). Anything less is undefined. But if A and B happen to be just that far apart does it take any time to travel between them? If not then maybe time is quantized too to avoid the absurd consequence that it takes no time to travel any distance (0t x 1035 = 0 is not the time it takes to travel a meter). Maybe there is a shortest time less than which is also undefined. And What does it mean to say that distance or time are undefined in these cases?
The world of QM is certainly strange indeed.
Then too, maybe QM is not the last word on the nature of physical reality. Maybe something is missing. Maybe QM is incomplete. A clue in this maybe the fact that it has been so hard to develop a quantum theory of gravity. General Relativity is the last bastion of classic physics not to be integrated into QM. Maybe it can't be.