Not really advanced math, but it might still be interesting.

]]>Justin Martel wrote:

Mr. van Beek, …

You may call me Tim here, I don’t insinst on a formal introduction :-)

…suppose you had unlimited computational power ie. you could instantaneously measure every square meter of this planet. That is, suppose you had all the information you could possibly ever want. Would that not paralyze you?

Computational power and better input data are of course two separate topics, to improve weather models we need to improve both. The question how far we could go in principle is of course an interesting one, but when we take a look at weather models *today* from a *practical* viewpoint, it should be clear that there are a lot of obvious flaws and problems that can be solved/improved simply by more computational power, and more data. There are projects underway in both directions. As a weather modeller I’d like to incorporate lakes and cloud forming processes due to lakes into my model, for example. With a grid spacing of 5 km that is hardly possible.

I’d really like to know long a global weather forecast could be done that is of use (being deliberatly imprecise with this question). Today the longest model runs encompass ca. 16 days, I think.

Permit me to be vulgar here: what is the category within which meteorologists work? To rephrase, how would one be a weatherman for 4-manifolds?

I’m sorry, I don’t understand what “category” means in this context, nor what is vulgar about the question. To a first approximation global weather modelling is computational fluid dynamics on a rotating sphere, is that your point? I suppose that this becomes a lot more complicated in 4D…

]]>Flori, Justin & Tim, sorry I didn’t notice your questions for a couple days. I’m not in the least “criticizing” models for being naturally incomplete, but suggesting we use that property to help expose and study the ways nature does more than we can model, studying how nature completes natural system behaviors by comparing model and observed behavior. I’m suggesting a forensic science approach, comparing standard models we understand well to help reveal coherent behaviors of natural systems that diverge in the model we understand.

No doubt that seems a muddleheaded way to construct a model, as it’s something of the opposite, not how to construct a model but how to deconstruct our observations of nature. Of course, the main “coherent departure” of nature from what we can model is the emergence of new systems.

There might be other clues to use, but my theorem of emergence and continuity proves that to satisfy energy conservation new energy using processes need to exhibit a period of “inflation”. It’s a conclusion that is quite broad, that the big bang’s period of inflation, found necessary to explain the astrophysical data, would be a special case of.

Now do you understand why I’d scan data looking for emerging processes of inflationary development?

]]>Mr. van Beek, suppose you had unlimited computational power ie. you could instantaneously measure every square meter of this planet. That is, suppose you had all the information you could possibly ever want. Would that not paralyze you?

Aren’t our models essentially based on the fact that we have limited and coarse input data?

Permit me to be vulgar here: what is the category within which meteorologists work? To rephrase, how would one be a weatherman for 4-manifolds?

]]>Phil, I’m not sure I understand your objection: You seem to say we can’t in principle understand much (if anything) of the workings of nontrivial systems – because that would require us to predict their inventions (reorganizing micro details, inventing new macro rules).

Indeed methinks that would require us to be all-knowing Laplace demons. It would be like wanting to predict detailed evolution of life forms.

(Who is “Rosen and others”? Quote/link?)

But I think we can understand qualitatively and partially how complex systems work. Perhaps the success of physics is an example for that: Nature is full of emergent properties where the exact micro details don’t matter and some “higher principle” is responsible, like maximum entropy production. Some emergent properties can be modelled mathematically – and that’s probably the reason for the unreasonable effectiveness of mathematical equations. That we know some fundamental constants with great precision is perhaps due to them describing emergent properties. Of course, observing and learning about complex system behaviour is an essential part of natural philosophy. There is perhaps not much difference between “dead matter” and living matter.

Here’s another example of “qualitative math”: Estimating the topology of spaces from their curvature (e.g. Betti numbers). Earth is finite because its curvature at any point is positive.

]]>Justin Martel wrote:

I’m amused by forecasters who credit the limitation of weather predictions to inadequacies in our models.

Weather models model the atmosphere only, they basically monitor what get’s blown our way. But that still means that the horizontal grid resolution is around 5 km, which is not fine enough to resolve many cloud formation processes, for example. From the viewpoint of a single human on the surface, to have one grid point every kilometer isn’t impressive, right? But these are the limits due to limited computational resources. When you set up a global model, a quick back on the envelope calculation can tell you for example that combining a 50 km horizontal grid spacing with 25 vertical grid cells leaves you with 5 kB of memory per grid cell if you have 1 GB memory for grid data overall.

Simply put: There are a lot of relevant physical processes that cannot be resolved in weather and climate models, but have to be put in by hand via heuristic corrections, due to the coarse grid that is enforced by limited computer resources.

]]>Phil, I try to follow what you’re saying, but it just isn’t clear. Your website is also muddled. I suppose the opinion you are expressing is just “a theory of how water flows is futile because we can never understand the precise structure which dictates the lifecycle of water particles”. (And at bottom, isn’t that precisely what you are suggesting?). Before we give any part of ourselves to a field, we first ask whether it might be interesting, and here our ignorance and prejudice is valued, here we decide whether we shall proceed or move on to another. I mean, we love films, say, and by what other process do we ever come to watch a film? I certainly have my own opinion–based on nothing but ignorance and taste–that meteorology (here i take the naive view that the business of meteorology is prediction) is bounded. That is, I’m amused by forecasters who credit the limitation of weather predictions to inadequacies in our models.

But on all this i’m an amateur and mildly embarrassed to fill the blog with such low quality ramblings. I would hope my comments are deleted–in the name of waste management.

Basically, I don’t understand what you’re saying. I’m under the impression that you think mathematicians believe themselves to possess some view of ”truth” in that ridiculous sense of old french philosophers.

]]>Justin wrote:

Sure, among experts, the questions are enthralling. But does it have anything to catch the eyes of those whom it has not already absorbed?

I’m no expert. I may however find it easier than you to become interested in things. I often wish I had days that were twice as long, and twice as much physical energy, to learn about all the fascinating things in this world. I have a bunch of books on my desk: *Fundamentals of Chemical Kinetics*, and *The Theoretical Biologist’s Toolbox*, and *Wetlands Ecology: Principles and Conservation*, and *Chemical Reaction Networks: A Graph-Theoretic Approach*. They’re all really cool, but unfortunately I don’t have time to read them all, because I want to keep explaining network theory and how stochastic Petri nets can be understood as a variant of quantum field theory in which probabilities replace amplitudes. Since stochastic Petri nets are also related to toric varieties, there should be some fun relation between quantum field theory, or at least this variant, and toric geometry. And I want to figure out what it is!

Note: if the interest of this blog was actually biological, I would begin by referring the readers to the article by Gromov and Carbone titled “Mathematical Slices of Molecular Biology”.

I’m interested in that too! Thanks!

]]>I misunderstood the main interest of this blog. The question of “mathematics and the environment” is not that of the relation between “mathematics and biology”. As a student not exactly interested in fluid dynamics (which i consider–without any real experience or justification–as the actual domain of meteorology and weather) i see here nothing compelling.

Sure, among experts, the questions are enthralling. But does it have anything to catch the eyes of those whom it has not already absorbed? I mean, something beyond wavelets. It seems to me that is the first problem of the whole field—it’s closed and hides itself from mathematicians. Does it find no stronger ally in topology than the statement that somewhere the wind is not blowing, or that somewhere barometric pressure and “something else” coincide?

I’m obviously ignorant, prejudiced, and skeptical. But I’m also a young energetic student who’s looking for something fascinating. And when I turn my way towards, for instance, this website and the comments, I find only the typical talk of applied mathematicians. I conjecture that a field is only significant if it is interesting geometrically. And so?

One may either dismiss everything i’m trying to say as the ramblings of an obnoxious kid, or one may try to catch the song of a rare bird flying around this field, and playing it in the ears of those passerby’s who are looking around, trying to find something to tap their foot to.

Note: if the interest of this blog was actually biological, I would begin by referring the readers to the article by Gromov and Carbone titled “Mathematical Slices of Molecular Biology”.

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