Assume space-time is quantized. This would mean that space-time is broken up into discrete bits. It then follows that time is broken up into discrete bits.
This disagrees with basic experience: we can start counting time at any arbitrary point. “Now” could be any time whatsoever. Moreover, we run our physical experiments at any given point; we don’t have to wait to start our clocks.
But what if our ability to run experiments at any given point is just an illusion of our universe being broken up into such tiny bits that we just don’t notice the breaks?
Could we design an experiment to test when we can run experiments?
If time is continuous, we would never find any point at which we could not run an experiment. If time is not continuous, though, we would likewise never find any point at which we could not run an experiment, since all experiments would use clocks that start within that lockstep quantized time.
Hence we are unable to tell the difference between quantized and continuous time such that it always appears continuous.
However, even if time is continuous in this fashion, measurement of time is not. Since there is a lower limit to what we can distinguish between two different times, even if we are free to start measuring whenever we want, all subsequent measurements are physically dependent upon that initial fixed point. The second measurement must be outside the uncertainty associated with the initial measurement (the clock start) and the third must be outside the second, etc. Therefore all physically useful measurements of time (counting past zero, that is) are inherently physically quantized by their dependence upon the instantiation of measurement and limits of uncertainty.
If time is both continuous and discontinuous in this fashion, then so is all space-time.
This leads to the question of which is ontologically prior: if you hold that our reality is defined by what we can measure, then the universe is quantized and our experience pigeonholed; if you hold that our reality is defined by our phenomenal experience, then the universe is continuous and measurement is pigeonholing.
Either way it is a question of the metaphysics — not physics — of space-time. And without a way to distinguish between these options, no physical experiment will be able to settle the debate either, since we could always be chasing our metaphysical tails.
I’ve mulled over this issue concerning the logical limits of what can be measured by physics for years, but I never developed any conclusions. However, there has recently been discussion of the feasibility of a tabletop search for Planck scale signals. This nifty experiment seems deviously simple with the potential for novel results, so go check it out if you haven’t heard of it yet, for example in this discussion. One issue that the experiment bears upon is the continuity of space-time at the Planck Scale. My worry is that the above metaphysical distinction between counting zero and counting past zero may trip up the physicists’ search for the continuity or discontinuity at the fundamental levels of matter.