To All Concerned,
Through grad school this past year I have been amazed by Mathematics, mainly that is works! Math is amazing, in one way, because it actually describes the world. Since I believe the world functions in an orderly manner and according to fixed laws, (though those laws may be quite bazaar at times), this seems reasonable for math to describe these laws. Though this is amazing in its own right, this isn’t what has been boiling my noodle recently.
What’s been boiling my noodle is when math describes things that aren’t real, yet it may teach us things that are real. Here is what got me thinking about this:
The backbone of fluid mechanics are the Navier-Stokes equations. You can read about them here. They are three equations, each respectively saying that mass, momentum, and energy are neither created not destroyed…only moved around. The equations describe this moving around. So, these equations are very complicated to solve, but (I think) ACTUALLY describe the physical reality of what is going on. Barring nuclear reactions, these equations are real…they actually describe reality. That’s great – math describing the fixed laws of our universe.
Next step, since they are so hard to solve people ask themselves this question, “Alright, I can’t solve these equations as they stand, but can I do anything so I can solve them, perhaps some sort of trick so I can at least learn something?” The answer is usually yes, and the trick is making the equations simpler. Reducing the complexity of the equations means forgetting some of the things that are going on. You forget the less important things and keep the more important things. After these tricks, you end up with math equations that don’t actually correspond to reality – however they can teach you something about reality! These simplified, non-real, equations have occupied many people’s careers, and are used to design the planes we fly in.
Here is the point. How do we come to terms with the fact that something that doesn’t exist, teaches us about something that does exist? Any thoughts?
Alright, yes, with a little hand waving the mystery goes away. Sure, in reality, as the density gets smaller and smaller this happens in reality, or as the viscosity gets smaller and smaller that happens in reality. So the math corresponds to the reality in the limit. … But actually, the limits never go to infinity or zero in real life. So the mystery still holds, the math does not correspond to something that actually exists.