Comment by A_D_E_P_T
lol dude, you need to read about the philosophy of mathematics. Like, what is it? Besides, go back and read the post I was responding to.
Now, are the reals necessary?
lol dude, you need to read about the philosophy of mathematics. Like, what is it? Besides, go back and read the post I was responding to.
Now, are the reals necessary?
I disagree with calling them all failures.
Everything that is known about numerical analysis is based on discreteness, instead of continuity. Every useful predictive model that we have about the world, for example for forecasting weather, depends on numerical analysis.
I consider this a success!
Nope.
Forecasting weather, designing semiconductor devices, or any other such activities are all based on continuous mathematical models, which use e.g. systems of equations with partial derivatives or systems of integral equations.
Only in the last stage of solving such a problem, in order to use a digital computer the continuous mathematical model is approximated by a discrete mathematical model, which is constructed using a standard method, e.g. finite elements, boundary elements, finite differences, most of which are based on approximating an unknown function with an infinity of degrees of freedom with a function determined by a finite number of parameters, then by approximating the operations required to compute those parameters by operations with floating-point numbers.
Such an approximating method is something extremely different from formulating a discrete mathematical model of physics that is considered the exact model.
Even the graphics for a game are based on continuous models of space, not on discrete models, where it would be very difficult to implement something like the rotation of an object or perspective views.
The failures to which I have referred are the attempts to create such discrete models of physics, e.g. where the space and time are discrete not continuous.
These attempts have nothing in common with the approximating techniques, which are indeed the base for most successes in using digital computers.
Yep! Optimize (solve in infinite dimensions) and then discretize onto a finite basis has typically led to much better and stable methods than a discretize and then optimize approach.
Time-scale calculus is a pretty niche theoretical field that looks at blending the analysis of difference and differential equations, but I'm not aware of any algorithmic advances based on it.
Like I have expanded in another reply, reals are not necessary, but using them to model the sets of values of the dynamic quantities is much simpler than any alternative that attempts to use only rational numbers.
During the last decades, there have been published many research papers exploring the use of discreteness instead of the traditional continuity, but they cannot be considered as anything else but failures. The resulting mathematical models are much more complicated than the classic models, without offering any extra predictions that could be verified.
The reason for the complications is that in physics it would be pointless to try to model a single physical quantity as discrete instead of continuous. You have a system of many interrelated quantities and trying to model all of them as discrete reaches quickly contradictions for the simpler models, due to the irrational or transcendent functions that relate some quantities, functions that appear even when modeling something as simple as a rotation or oscillation. If, in order to avoid incommensurability, you replace a classic continuous uniform rotation with a sequence of unequal jumps in angular orientation, to match the possible directions of nodes in a discrete lattice, then the resulting model becomes extremely more complicated than the classic continuous model.