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Stephen Was On TV Last Night on NOVA

Posted 4 years ago
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It was an episode of NOVA on Public Broadcasting, discussing whether mathematics is created or discovered.

NOVA: The Great Math Mystery

Stephen's comment is that the areas that are most explored are the ones where math works well.

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There is a longer video clip on the web (I have the clip -- lost where it is on YouTube) where Stephen elaborates on this idea.

I thought that the Nova was pretty good. They could have pointed out the nature of the problem better if they had presented the fact that Newtonian Mechanics turns chaotic when predicting the orbits in the solar system.(Discovered/proven by Poincaré in the early 1900s) I have no idea whether the solar system really is chaotic, or whether just the model is, but this is really the point. Physics pretty much studies the 'simple' stuff. When it comes to fluid dynamics, or biological systems, then an engineering mindset has to take over, with rules of thumb, 'close enough' approximations, and a constricted domain of applicability.

Thanks -- that was the video. Just too lazy to look for it 6 minutes ago. ;-)

Yes, physics seems only to handle linear problems. The Navier-Stokes equation is nonlinear so it's engineering.

I saw it too. Regarding physicists and engineers, I think they oversimplified. Physicists are completely aware of the fact that for many calculations exact results are not available. Very few nonlinear differential equations have closed form solutions. But that does not mean they are not the governing model. It just means that, as with Pi, the solution can only be expressed as an approximation. In fact, about the first thing a physicist does with this sort of problem is Taylor expand it and throw away the nonlinear part. Or use perturbation methods. And then try to estimate the error.

In chaotic behavior, one way to look at it is that we simply do not have the computing power to achieve an acceptable solution. An example it turbulent fluid dynamics. Here there is no steady state for the flow, but direct numerical simulation can produce time dependent solutions using finite element methods. It's just that the mesh size and time steps needed are so fine that the computational cost becomes unaffordable. And of course there is the problem that the for all practical purposes initial conditions may be unknowable. (Or in the world of quantum mechanics uncertain?) Essentially this means that, given the detail scale of reality, our knowledge is really an approximation. I don't think physicists would dispute this.

And there was one more observation made in the show which I found interesting, and I think it applies to cosmology in general. They noted that Newton's law of gravitation, which first explained the orbits of planets and even the galaxies, is as valid today as when he first proposed it. Well, how do we know a "law" is valid? Answer: when the predictions it makes agree with observation. So gravitation, and General Relativity, are perfectly good theories. We just have to assume that there is 80% of the mass in the universe which is detectable only as a divergence of those theories from observation. Maybe I am missing something.

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