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Author
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Topic: Computability in Science
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warren_bergerson
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Member # 262
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posted 15. May 2004 10:49
I have always assumed that the mathematical question of ‘computability’ as it relates to formal scientific analysis had been resolved long ago. Based on discussions here and elsewhere, some individuals appear to have views that are quite different from mine. I present the following summary of my interpretation of computability concepts as a starting point for discussion.
To begin, it is my understanding that if you satisfy the following conditions:
1. A precise definition of beginning and ending states.
2. A precise definition of the rules for quantifying values of beginning and ending states, and
3. A finite set of observed values of pairs of beginning and ending state values
Then
4. It is always at least theoretically possible to find an unlimited number of algorithms capable of computing observed end values from observed beginning values.
Since scientific analysis always involves finite sets of observations, it is always theoretically possible to find an unlimited number of algorithms and thus scientific models that will fit and simulate observed data. It may not be practical or useful, but it is always theoretically possible to formulate scientific models that fit and simulate observed data. This computability principle does not necessarily apply unless all three of listed preconditions are satisfied. Scientific standards require that all three of the listed conditions must be satisfied in formulating a mathematical scientific model.
Scientific models, as defined, can be translated into predictive scientific hypotheses if the models generate testable predictions. What qualifies as a testable prediction is beyond the scope of this thread.
As it relates to the discussion of Avida, the above concept would suggest that given a specific defined evolutionary change, will always be an unlimited number of algorithms or programs capable of generating the change. If you make the specified change more complex in order to more realistically reflect evolutionary change in biological systems, there will still always be an unlimited number of change algorithms capable of modeling or simulating the change.
Modern evolutionary theory asserts that it can model and simulate evolutionary change. If you succeed in demonstrating that a particular program such as Avida can not simulate a particular biologically realistic evolutionary change, all you have demonstrated is that Avida does not realistically reflect modern evolutionary theory.
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Evan
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Member # 164
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posted 15. May 2004 10:51
The Mandelbrot set satisfies conditions 1-3, and I seriously doubt there is even one other algorithm for producing it.
Comments?
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Rex Kerr
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Member # 632
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posted 17. May 2004 05:13
Warren, your characterization seems accurate to me, with one proviso.
Evolution does not run according to arbitrary functions in the real world. Rather, there are a very limited set of possible algorithms that match our models of the physical world.
If we could design Avida or some other program to be representative of all such possible algorithms, then a failure of Avida might be informative.
Nonetheless, Avida is primarily interesting for what it can do, rather than what it cannot.
(Evan, the Mandelbrot set is not finite. If restricted to, say, a finite representation on a computer screen, then there are arbitrarily many other functions that can produce that finite representation. For example, a cubic spline through all the points will produce the same representation.)
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Evan
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Member # 164
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posted 17. May 2004 07:36
Thanks, Rex - I see that I didn't think about the infinite part. (I had been showing the Mandelbrot set to my calculus class so it was on my mind.)
However, even if you limit it to a finite subset, such as what appears on a computer screen (perhaps 640 x 480 pixels, for instance), are there infinitely many functions that could produce those 307,200 points?
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warren_bergerson
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Member # 262
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posted 17. May 2004 08:59
There are at least four general principles here. First, for any finite set of formally defined data or observation of cause and effect relationships, there are, in theory, an unlimited number of mathematical algorithms that can model or simulate the data. This is probably might be more commonly recognizable as “There are an infinite number of curves or algorithms that can be fit to any finite set of points’. Since all sets of observations of finite, this principle means that it is always, in theory, possible to find algorithms to fit observed data. This does not mean that finding such algorithms is practical or useful, only that it is theoretically possible.
Second, the standard mathematical topologies characterize the universe as involving an infinite number of points and properties. It is therefore not necessarily possible to find a set of algorithms or causal laws that will model or simulate all points and properties.
These two concepts are the basis of what might be called the pragmatic view of science. Science, from the pragmatic perspective, is viewed as the study of finite sets of observations relating to cause and effect relationships, where it is possible, practical, and useful to fit the data to explicitly expressible algorithms. Furthermore, science is only interested algorithms of this type if they produce useful, reliable, and testable predictions.
There are lots of causal relationships in the universe that may be unknowable or non-computable. There are certainly lots of causal relationships that can not be practically and usefully computable. IMO, science is only interested in causal relationships that are computable. The starting point for science, at least IMO, is causal relationships that are in theory definable and computable.
The third point is that there is a big difference between data sets that are computable in theory, and data sets that are computable in practice.
The fourth point, and probably the key point of this thread is that with the development of the computer, we have discovered a variety of very complex causal relationships and data sets that are practical to compute and simulate. Included in the set of complex computable causal relationships are weather forecasting models, and various robotics and automated processing systems.
Quote Rex: Evolution does not run according to arbitrary functions in the real world. Rather, there are a very limited set of possible algorithms that match our models of the physical world.
Your comments get right to the crux of the issue. Are evolutionary change processes computable? The answer to that question, I suggest, depends on the techniques and concepts used. If you attempt to compute and simulate evolutionary change processes based on known physical-chemical processes, then you are likely to have only limited success.
However, if you use the concepts and techniques of ‘behavior controlled by goal-directed intelligence, then, I suggest, you can compute and simulate evolutionary change processes.
Programs like Avida provide interesting insights into the problem of computing evolutionary changes processes and data. Avida is an arbitrary function that successfully computes some types of evolutionary change data under simplified, possibly unrealistic, conditions. I suggest that Avida is an initial simplified approximation or prototype computation of ‘behavior controlled by goal-directed intelligence’. By a process of successive approximation, I suggest, it would be possible to modify and refine Avida to model and simulate evolutionary change on a realistic basis.
At least IMO, it is inconsistent to suggest that Avida provides moderately accurate computations of some types of evolutionary change data, but the program provides a poorer and poorer fit to data as data is made more realistic. Avida, IMO, just does not fit in with a mechanistic approach to computing evolutionary change data.
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RBH
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Member # 380
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posted 17. May 2004 12:23
warren_bergerson wrote quote:
Programs like Avida provide interesting insights into the problem of computing evolutionary changes processes and data. Avida is an arbitrary function that successfully computes some types of evolutionary change data under simplified, possibly unrealistic, conditions.
"Arbitrary"? Not at all. It's precisely because avida is not an "arbitrary function" that it provides some insight into evolutionary processes. warren_bergerson's remarks are internally inconsistent. If "Programs like Avida provide interesting insights," they surely can't be "arbitrary functions."
I predict that this thread will soon degenerate into the same semantic and conceptual swamp that others that float in a theoretical and empirical vacuum do here.
RBH
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warren_bergerson
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Member # 262
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posted 17. May 2004 13:08
I expression ‘arbitrary function’ is Rex’s. Rex suggested that evolutionary processes can not be modeled by arbitrary functions. But Avida would seem to qualify as an arbitrary function.
Avida is interesting, IMO, because it is something like the relationships in quantum mechanics. As a high level approximation, Avida appears to provide reasonably accurate fits to some types of evolutionary change data. But, as in quantum mechanics, the accuracy or usefulness of the predictions disappears as you try to use more biologically realistic applications.
In quantum mechanics, this high level accuracy-low level inaccuracy is explained by stochastic concepts. In evolutionary process, the high level accuracy-low level inaccuracy appears to be explained, I suggest, by goal-directed intelligence.
Modern evolutionary theory is supposedly a mechanistic, continuous chain of causation type of theory. Avida would appear to contradict this claim since it works for high level approximations but breaks down when attempts are made to reconcile it to biologically realistic detailed chemical physical processes.
Note that the subject here is computability in science. Avida provides reasonably accurate computations of certain types of evolutionary change processes. However, Avida does appear to be the result of or even compatible with the mechanistic, chain of causation concept or approach to formulating or finding computational algorithms.
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Rex Kerr
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posted 17. May 2004 13:16
Warren, you misunderstood me. Avida isn't arbitrary, nor is evolution. Avida's algorithm was designed by its programmers to match (in those ways that were convenient) the one evolutionists believe is going on in the real world. (Also, I'm not quite sure what you're trying to convey about failures of Avida. It's an approximation, so of course it won't produce the exact same results that biological evolution does--what do you expect?)
Evan, consider a 640x480 grid with each pixel on an integer coordinate and having an integer intensity value between 0 and 255. Then consider an extra point at coordinate 319.5,239.5 that has an arbitrary real value. There are infinitely many such points. However, all of them can be fit by a cubic spline (or a polynomial of appropriate degree) that also goes through all of the pixel values. Thus, there are infinitely many functions that have the same finite representation as the one we've chosen for our Mandelbrot set.
If we use a high-order polynomial, our calculations will be different for every single pixel. (With a spline, only the calculations for neighboring pixels will be affected.)
[ 17. May 2004, 13:18: Message edited by: Rex Kerr ]
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