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?

Best Regards,

Ben Washington

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.

Part of the problem you’re talking about is the suppositious nature of modern science and matching deductions based on arbitrary postulates with experiment–look up the fallacy of affirming the consequent.

Another part of the problem is with modern math itself. Jacob Klein has written much about how math became unhinged from reality in the renaissance (with the rise of algebra), but he’s notoriously hard to understand. Prof. Joseph Cosgrove is a much more understandable expositor of his ideas.

LG

I may be wrong but in mathematic, infinity is not a number; it’s a concept that is derives from the properties of a set. For example the cardinality of natural numbers is infinite since every number has a successor:

succ(n) = n+1.

The fact that this property can be applied to any natural number allows us to infer that the cardinality of the set of natural numbers is infinity. Notice that we didn’t need to actually count till infinity to posit the conclusion.

Similarly, in our physical universe, we can do the same thing with time: if the universe is open (which is the current scientific consensus), every time instant has a successor.

succ(t) = t+1sec.

We can thus conclude that the cardinality of the set time expressed in second is infinite.

Thus?

1: 1

0: (1-1)

1: 1+(1-1)

Great, thanks for the thoughts and references to others who have, no doubt, thought longer about this issue.

The notion of a successor seems to invariably draw one’s attention to the related notion that there is only 1 number. In all probability, some ones elsef has thought about this, and in order to properly attribute these predecessors, who would seem to be legion, the research WILL require some time. One moment, please….

Well infinity is a concept in calculus, but in standard set theory it is both allowed as an ordinal number and a cardinal number. Moreover these numbers come in many different “sizes”.

Back to the main question: in some sense that I am still grappling with, physics is a part of mathematics which describes all possible and all impossible worlds. Physics without formalization is not physics. Having said that, I doubt very much that the Navier-Stokes equations are some kind of completely precise description of fluid flow. For instance, they ignore the conversion of mass into energy and quantum effects.

– retired mathematician

You are right, the Navier-Stokes equation are not EXACTLY correct. For starters, yes, they don’t consider nuclear reactions, which are happening. This leaves me more curious.

While hoping to avoid being trite, perhaps we can relate this to an Einstein quote. He said “The most incomprehensible thing about the world is that it is comprehensible.” We can add that it is completely incredible that we can have success in describing the world with incomplete understanding.

It would be great to continue thinking about this, but in the meantime we can reflect on how wonderful it is that this success is possible.