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| Chemboy |
 Posted: Wed Apr 09, 2008 8:34 pm Post subject: Mathematical models of everything |
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Recently I heard someone state that he felt that anything in the world could be described in terms of a differential equation, even if it was of the 50th order or something crazy like that. What do you think about this? Do you feel that it is possible to describe anything using mathematics, even if we don't necessarily have the capability with our knowledge right now, or do you feel that it simply isn't feasible? And mods, if you feel this would be better suited in the Philosophy section perhaps, I would understand.
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| William McCormick |
| Posted: Wed Apr 09, 2008 9:22 pm Post subject: Re: Mathematical models of everything |
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Joined: 03 Apr 2008
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| Chemboy wrote: |
| Recently I heard someone state that he felt that anything in the world could be described in terms of a differential equation, even if it was of the 50th order or something crazy like that. What do you think about this? Do you feel that it is possible to describe anything using mathematics, even if we don't necessarily have the capability with our knowledge right now, or do you feel that it simply isn't feasible? And mods, if you feel this would be better suited in the Philosophy section perhaps, I would understand. |
Look at your computer screen.
In a way you could conclude that every pixel could have a maximum set number of colors. And compared to every other pixel, all cycling through their colors you could conceive the total output capability of the screen. Every single image possible. But as soon as you add animation, and order its a dead theory. It becomes infinite.
Sincerely,
William McCormick
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| bit4bit |
| Posted: Fri Apr 11, 2008 9:57 am Post subject: |
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Forum Bachelors Degree
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I havn't really looked at differential equations in depth, as I'm saving that for when I finsih learning multivariable calc. But... I would say that almost any physical process can be modelled by a function of one or more variables, and that the dynamics of any physical system can therefore be modelled by a differential equation.
I wouldn't go as far as saying 'anything' can be modelled by differential equations, but it looks to me like they are incredibly powerful tools.
The Navier-stokes equations for fluid flow are differential equations, and can be used to model flows in real world situations, but apparently it has not been proved that solutions always exist for the equations in 3D. Thats a current Clay maths institute open problem.
I'm really interested in learning about differential equations, but need to learn some more vector calculus first.
As for William's suggestion, I would say that such a situation can probably be modelled. You could consider a continuous function f(x,y)=intensity for a given co-ordinate. Then you could consider the time variance f(x,y,t) = intensity. But the images being displayed would be almost entirely sporadic, and not really have a pattern to them. What's more the screen is actually made up of discrete pixels, so you couldn't really consider a continuous function, but it could approximate the situation.
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| Chemboy |
| Posted: Fri Apr 11, 2008 12:49 pm Post subject: |
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| bit4bit wrote: |
| I wouldn't go as far as saying 'anything' can be modelled by differential equations, but it looks to me like they are incredibly powerful tools. |
Thanks for the reply bit4bit... I agree with that. Probably much can be modelled by mathematics in general, but not everything... When I said "anything" in the original post I meant "anything." Human thought was something I had in mind. To me, when it comes to things like human thought it's just far far too complex to be able to predict by any mathematical means (at least with any degree of accuracy).
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| bit4bit |
| Posted: Fri Apr 11, 2008 3:05 pm Post subject: |
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Call me wild, but I actually think that as research continues into neural signals, we might (At some point in the future) be able to start modelling brain activity in terms of individual neurons firing. The brain is basically a big electrochemical analogue computer, with neurons as its basis. When we start to understand their signals and interactions better, it might just become a similar case to simulating/modelling normal electrical circuits. In this case, I'm not sure differential equations could be directly applicable (Though I'm not an expert on them), but certainly some kind of math could be used, along with a topology of the network, to describe the signals and simulate a real brain process! Get youself 500GB of RAM, and it's alive!!! 
I think this is a genuinly fascinating area of research, that could have some promise one way or another. I found this link ages ago that might interest you:
http://www.unmediated.org/archives/2006/03/brain_cells_fus_1.php
(As the article explains, neurons are fused to the silicon circuit with some protein or other, are 'shocked' with capacitor discharges, and the signals read of with transistors!)
Thats all a bit of a sidetrack though. I think the main thing that prevents us from modelling things with maths, are random or sporadic processes that don't really have any apparent pattern. Things like modelling unpredictable quantum systems. Perhaps differential equations can account for this kind of thing? I know that the Navier-Stokes equations can hold for really unpredictable, turbulent flows, so maybe.
Another interesting idea is that of the 'butterfly effect', whereby a butterfly flapping its winds in china can cause a hurricane in America. I think this idea uses chaos theory, and differential equations somewhere.
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| serpicojr |
| Posted: Fri Apr 11, 2008 3:17 pm Post subject: |
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The idea behind the butterfly effect and chaos theory is that some system are so sensitive to initial conditions that minor differences can lead to wildly different outcomes. The weather is the classical example of such. Such systems can certainly be modeled by differential equations. But the point is that for these models to be useful, absurd amounts of precision are required.
In general, I feel that most if not every physical phenomena can be modeled by math. For those models to be useful, though, we'd first have to be able to find them, and we'd next have to be able to calculate to a sufficient degree of accuracy to make things useful. So there are really two questions: one, can everything be modeled by math?; and two, if so, can everything be modeled effectively? |
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| bit4bit |
| Posted: Fri Apr 11, 2008 3:27 pm Post subject: |
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I think the butterfly effect actually has some relevance in fluid dynamics for very turbulent systems, because as you say they are very sensitive to initial conditions, and need powerful computers to model the changing flows over time (Computational fluid dynamics).
I agree with this:
| Quote: |
| In general, I feel that most if not every physical phenomena can be modeled by math. For those models to be useful, though, we'd first have to be able to find them, and we'd next have to be able to calculate to a sufficient degree of accuracy to make things useful. So there are really two questions: one, can everything be modeled by math?; and two, if so, can everything be modeled effectively? |
With the exception of quantum systems, where we are limited to knowing only one part of it, and cannot know all information simultaneously.
Also, computers are getting more powerful all the time, by Moore's law, and there is intense research into finite element analysis, and other numerical methods for simulating things, so I think that aspect of it allows the models to get more and more accurate.
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