Why do we use continuous time and space to model real life systems?

Question

In Physics, I've heard of the smallest length of space as the "Planck length" and the smallest unit of time as the "Planck time". This isn't really a physics question as much a math question, though.

If we have a smallest unit of space and of time, why don't our mathematical models of real life systems deal with discrete space and time? What is the advantage to modeling a system's behavior with continuous space and time?

(There are plenty of examples in, for example, PDE theory, or in math finance, of systems we model using continuous time, e.g., the diffusion of heat through a material, or the price of a stock.)

A weird feature about continuous time, for example, is that there are no two consecutive time points, while that's not true in discrete time. Why should we want to model a system using time in which there are no two consecutive time points? What advantage would this feature of time (or space) give us in our model?

Answer 1

One can think of continuous methods as a limit of discrete processes (think of the derivative as the limit of secant lines, or the usual approach of Riemmann integration as a limit of rectangular areas, for instance).

If the discretization is sufficiently fine (Planck measures are very, very small for most of our application), then these limits are good approximations of the real, discrete situation. Of course, there is here some implicit 'continuity' assumption of reality, which is mostly empirical I suppose. Calculus was developed in part to answer physical problems and its success is a testament to how effective a tool it has proven to be.

Answer 2

I don't think there is a very good answer to your question, but it allows me to highlight a very important point. Physics consists of mathematical models of certain parts of reality that we can measure. This is very important: there is nothing obviously mathematical about reality, yet somehow Physics has been incredible successfull (obligatory read: Wigner's The Unreasonable Effectiveness of of Mathematics in the Natural Sciences).

So, there shouldn't be any real expectation that the math we use in physics somehow has to reflect the phenomena underlying our measurements. We use the equations of thermodynamics or the concept of temperature, even though we know that in reality fluids are made of little bouncing balls. We use group theory for the standard model, even though you would be hard-press to say in general how a particle and group theory are related. And a big Etc.