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Call the world as it presents itself to us world-1

The regularity & order in the world demands explanation - hence we have a mathematically consistent description of the world. Call this world-2.

But surely the same situation again applies here: that the regularity & order is manifestly displayed in world-2 demands an explanation. Suppose that this is another mathematically consistent explanation for this. Call this world-3.

and so on.

We appear to have an infinite regress. Does this mean that any mathematical description of the world can never be complete in itself?

A rough example of this is the description of our own universe in the standard model. Then the explanation of fine-tuning demands explanation - some treatments consider then the perspective of a larger multiverse.

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    What would a mathematically consistent explanation of a mathematically consistent explanation be? Also, in what sense are the mathematically consistent models of world-1 explanations? It seems that they represent rather than explain. If you're interested in a similar argument against metaphysical realism, check out Putnam's "Model Theoretic Argument". It is summarized in section 4.5 of the linked article. – Dennis May 29 '13 at 16:15
  • @Dennis: ok, I can see why you say & agree that they represent. But its also true that are in some sense an explanation in say the way occams razor 'chooses' the simlest explanation; or would you say the simplest representation here? – Mozibur Ullah May 29 '13 at 17:52
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    Wouldn't the most complete, simplest explanation of world-1 be world-1 itself; or a copy of world-1? A mathematical model explaining world 1 would have to encompass world 1 itself then draw its conclusions from that (otherwise where would conclusion be drawn from?), thus making it a more complex explanation of world 1 than world 1 is itself. So if world 2 is the simplest, least complex, and complete explanation of world 1, world 2 is then just world 1 or a copy of world 1. By the same principle world 3 is world 2, so on. – KDecker May 29 '13 at 18:37
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    @Dennis: Its also possible to find a simpler explanation that is also more complex. For example some features of the ring of integers are more readily explained when one considers all rings (of a suitable kind). Together they form a geometric object. An apposite example would be the circle - said to be by the greeks the most perfect shape - but they also came up with hyperbolae, and parabolas (imperfect shapes) - which can be fitted together into one geometric shape. Now by analogy think of the circle as the representation of our world. – Mozibur Ullah May 29 '13 at 19:48
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    I might add this could be taken to any, I guess, system or reality. Maybe only certain systems can do this; describe themselves fully. For instance Rule 110 of cellular automata is shown to be Turing complete. So you could create rule 110, within rule 110, the entirety of the system within itself. – KDecker May 29 '13 at 21:46
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Answer to your question: No - they do not have to lead to infinite regress.

Explanation:

  1. To cope with world-1, evolution equipped behaving animals with the possibility to sense the world.
  2. Smart interaction with this world-1 (aka behavior) requires to map a high dimensional sensory input space to a low dimensional motor output space. This mapping requires generalization and abstraction which are implemented as cognitive processes. E.g. a animal has to realize that an apple is an apple independent of the light conditions, its exact shape and a variety of different colors,... i.e. the animal has a notion of an apple.
  3. Smart social behavior is an evolutionary advantage but it requires a possibility to exchange such internal generalized notions. This exchange is achieved through the communication of symbolic representations. E.g. an animal warns other animals through a distinct call (symbol for predator)
  4. The evolution of such symbolic representations led to the human language and highly sophisticated models of the world-1 (the set of these models is what you call world-2). Part of these models is a set of rules that allows to treat symbols in a consistent fashion aka. mathematics. If we apply mathematics to symbols which originate from abstractions of the world aka science, we can predict reactions in the world aka explain it. It is important to note that a sentence on the world-2 can be proven (e.g. logic/mathematics) by checking whether it is consistent with a set of other sentences in world-2. This is not possible for sentences on the world-1 (science) since our access to the world-1 is restricted through our senses, cognitive processes and our limited possibilities to manipulate the world. So the only thing we can do to have good models of world-1 is to reject the contradictory ones i.e. the ones that predict wrong observations.
  5. Even tough we can describe the regularities in the world-1 we cannot explain them. The regularities of world-1 are a necessity for any description - from the cognitive process to the symbolic representation and the communication. So if we would come up with an explanation of the regularities, it would not be falsifiable.
  6. The creation of world-2 depends on the regularities in world-1 and therefore the regularities in world-2 are deduced from the ones in world-1 as everything in world-2.
  7. The sentence on the regularities in world-2 (point 6) might be considered a sentence of world-3 because it is a statement on the nature of world-2 but this is not true because the cognitive processes of animals and the communication in the human society are processes in the world-1, all sentences on their content are sentences on the content of world-1 and therefore they belong to world-2.
  8. In short: Explaining something is a process in the world-1 that requires consistency of world-1. Explanations of the consistency of world-1 are not falsifiable and therefore not valid.

In case you doubt that explaining something is a process in the world-1, I cut you with Occams razor.

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