On Wednesday, August 8, 2012 1:16:17 AM UTC-6, djmpark wrote:
> Landau just expresses the frustration that non-mathematicians have when > > confronted with mathematical derivations and proofs that strive for rigor. > > They may need the results and material but, darn it, it's hard.
I don't think so. Landau was much more mathematically sophisticated than you give him credit for. But mathematics is only meaningful to non-mathematicians to the extent that it relates to reality.
Given that the philosophers have comprehensively demolished mathematical Platonism, it seems clear that the objects of mathematical study are products of the human imagination, not present in reality. It is indisputable that these objects are effective at modeling the behavior of real-world objects, but they also have odd properties of their own.
Now, sometimes we can utilize these odd properties in applied mathematics. To generalize a statement by Hadamard, sometimes the path that leads to real world truth takes us through unreal realms of imagination. And then we may also need to be aware of physically unreasonable behaviors of mathematical objects, so as not to misapply them.
But Landau singled out existence proofs. Non-constructive existence proofs are an example of mathematics that is unilluminating to a physicist. A notorious contemporary example is the Navier-Stokes equation: nobody knows if it generally has solutions or not. The Clay Mathematics Institute has offered a $1 million prize for a solution to the existence problem here. But to a physicist, this misses the point. The significance of the Navier-Stokes equation is that functions that satisfy it approximately also approximate real fluid flows. There is no need to know if an exact solution exists, since the Navier-Stokes equation describes the motion of a mathematical fiction: an "ideal fluid". An existence proof here would have no significance to physics. We're more interested in algorithms to efficiently find approximations. We have no doubt that in the real world the fluid will do whatever is inits nature to do, whether an exact Navier-Stokes solution exists or not.
Generally, what Landau seems to object to is the tendency of mathematicians to study and teach meaningless properties of mathematical objects, rather than those properties that connect mathematical objects to reality.