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"Chaotic" differential equations are very simple principles compared to the more complex consequences of them.

For example, the equations modeling the motion of a double-pendulum, double-pendulum eq. 1
double-pendulum eq. 2,
are relatively simple compared to the "chaotic" consequences of these equations:
double-pendulum animation

How is this not a violation of the principle that "one (e.g., a relatively simple differential equation modeling a double-pendulum) cannot give more (e.g., a complicated, 'chaotic' trajectory of the pendulum) than it has"?

Another example: There are myriads of consequences of Euclid's axioms. Are all of these consequences not virtually present in the axioms themselves? If so, this would violate the principle "one cannot give more than one has," unless something is added to the axioms in deriving proofs from them. Is that the case?

In other words: Are all the consequences of a science in the science's principles?

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    This question should be moved to the physics SE: your question is simply about the law of conservation of energy in chaotic systems and doesn't really have any philosophical content. – Alexander S King Mar 21 '16 at 17:16
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    Complexity is not an objective measure, it is a subjective impression. Entropy is a related measure, but there is no implicit entropy in an equation. Chaotic, and yet simple, often seems to take the form of "non-repetitively self-similar", which humans have little capacity to encode in a meaningful way. – user9166 Mar 21 '16 at 17:47
  • @AlexanderSKing I'm just using it as an example. I could use another example: Multiplying two large numbers A and B with the multiplication algorithm is much easier than counting the number of dots in a grid of size A×B. Thus, it would seem the multiplication algorithm has a much more complex consequence than the algorithm itself can provide. – Geremia Mar 21 '16 at 18:50
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    @jobermark Yes, perhaps my question is more about complexity. – Geremia Mar 21 '16 at 18:54
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    @AlexanderSKing I've removed the physics tag; perhaps that was confusing you. – Geremia Mar 21 '16 at 18:56
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Not an answer but some notes on language: "one cannot give more than one has," is just a vague common sense saying. It looks much like an impredicative definition - no matter how much one gives, it is never more than what one has, indeed. If you 'have' only the even integers you could give all integers just by halving them; or with only 0 and 1 you can give all numbers...

(and, also the diagonal of square is an irrational - 'infinite' - number when the side is an integer but you could chose it to be the obverse: an integer diagonal & irrational side)

'Contain' is a spatial metaphor (or catachresis), sub speciae eternitatis all consequences are 'contained' in principles or axioms; but for a temporal being an eternity is needed to derive them. And 'after an eternity' is a polite way of saying never...

And, btw are you sure that you are not a virtual killer? 'In potentia' is one of these expression that gave a bad name to scholastics.

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  • "Virtual" is different from "in potentia." Virtual (vir = power) presence means present by the power of something. Potential is something that could be actual with the agency of an efficient cause. – Geremia Mar 22 '16 at 23:15
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I think there are two issues that need to be defined: what is meant by "complex" and what is meant by "more."

There are a few different definitions I've seen of what is meant by something being complex, here are three examples:

  1. Sensitive to initial conditions
  2. The Church-Turing hierarchy
  3. Wolfram's classification of cellular automata

The pendulum, while exhibiting interesting behavior, is not "complex" by any of these definitions.

Usually when I've seen people say that something is "more complex," it is defined in terms of one system being able to simulate another system. So, again, taking the pendulum example, by any of the three above definitions it cannot simulate a "complicated" example. So, it is not violating the principle you suggest.

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  • "So, again, taking the pendulum example, by any of the three above definitions it cannot simulate a "complicated" example. So, it is not violating the principle you suggest." You are correct in this statement, and this can be simply expressed by the concept of Kolmogorov complexity (also called algorithmic complexity): the pattern of the pendulum has low Kolmogorov complexity, even if it appears complex to the human eye. The OP insists that there is some deeper philosophical question beyond this simple fact, but I've yet to understand what he is seeking. – Alexander S King Mar 22 '16 at 18:11
  • @AlexanderSKing "The OP insists that there is some deeper philosophical question beyond this simple fact, but I've yet to understand what he is seeking." Basically: "Are all the consequences of a science contained in the science's principles?" – Geremia Mar 22 '16 at 19:25

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