Classically, discrete molecular systems like the air in a bicycle tire have been modeled as continua. This would seem like the right thing to do for almost any application. We may know that the pressure in the tire ultimately results from 10^25 molecules bouncing around at random, but for most purposes it would seem crazy to try to follow those motions individually.
Lately, however, more and more mathematical scientists are trying to track statistics of particles. It seems everything nowadays is getting reformulated stochastically. I have been skeptical of this trend. Is it partly just a fad, a way of generating impressive new challenges for our latest computers?
But walking around New York has given me an illustration that sometimes, discreteness in our models really is indispensable. On the sidewalks there are hundreds of pedestrians flowing north, south, east, and west, and it is tempting to think of these flows as continua. You certainly feel like part of a wave when you are flowing along with the crowd! But at each crosswalk, something discrete happens. If the walk light is illuminated, a car wanting to turn right has to wait until the number of pedestrians crossing falls to exactly zero.
If pedestrians were a continuum rather than discrete, their density would always be positive, and no car in New York would ever be able to turn right.
[8 October 2017]
trefethendotnet 
I am an applied mathematician. I love your index cards. I also share your skepticism about molecular/particle dynamics. Metaphor: Global warming bring excess energy to our teacup and leads to unpredictable spills on land, also called hurricanes. Similarly, the excess influx of money and chip hardware in high performance computing leads to evolution of entities that needlessly compute stuff. Somebody will evolve and help justify the hardware because they also have their career to save. I saw it first hand 15 years ago and coming from a poor country it made me angry that megawatts are wasted like this. There will always be a little bit of underlying “real math” like Bayesian inverse modeling or statistical methods or galactic n-body problems. There has to be some. But all the computation is to just produce pictures and papers and not taking any decision that has implications for the real world.
I console myself by believing that there is no one is completely to blame because no one is completely in charge.