Algorithmic Reducibility

…the chameleonic nature of numbers [is] so rich and complex that numerical patterns have the flexibility to mirror any other kind of pattern. (Douglas Hoftsadter in I am a Strange Loop, p. 159)

In my last post, I discussed the point of view known as ‘reductionism’ and the problems with that point of view. In summary, reductionism is the false belief that sciences that work with the smallest units of nature – atoms and below – are somehow more fundamental explanations of reality than emergent ones, such as thought or computation.

A few posts ago, I discussed computability and comprehension. My final conclusion was that while algorithms and explanations aren’t the same thing, you can’t have an explanation without having an algorithm.

Gregory Chaitin (in this article) points out that a theory must be simpler than the data it explains:

…a theory has to be simpler than the data it explains, otherwise it does not explain anything. The concept of a law becomes vacuous if arbitrarily high mathematical complexity is permitted, because then one can always construct a law no matter how random and patternless the data really are. (From “The Limits of Reason”)

Interestingly, this ability to reduce all explanations to computable algorithms forms a sort of ‘algorithmic reducibility’ that stands in stark contrast to the more familiar sort of ‘physical reducibility’ we normally think of.  In fact, if it’s true that all explanations have attached algorithms, then ‘algorithmic reducibility’ would seem to play the very role that Reductionists thought particle physics played: if you can’t reduce it to an algorithm, you don’t actually have a full explanation. Therefore this would mean that the theory of computation is actually more fundamental than particle physics.

There is something both disturbing and satisfying about the possibility that ‘algorithmic reducibility’ is fundamental. But it does seem to logically follow from our discussions so far. It also gives us a plausible explanation for why something like ‘beauty’ seems to be non-algorithmic but also doesn’t seem to be wholly subjective either. Perhaps we just don’t comprehend it yet. There are many phenomenon that we can’t yet turn into algorithms. We will always have to ask the question: does this mean I just don’t understand it yet, or does this phenomenon only exist in my mind?

Consciousness and Algorithms

But then what about consciousness? Is not consciousness effectively an aspect of nature too? Or is it separate from nature? Does this not lead to a conundrum? If consciousness can be explained, then it can be reduced to an algorithm. Doesn’t that mean we’re all just automatons? But if we say consciousness is not algorithmic, then doesn’t that mean we are claiming consciousness is fundamentally unexplainable? Aren’t we then claiming that not even God can comprehend consciousness because it’s beyond any sort of comprehension?

This paradox is a key question of interest for me and in future posts I will explore this further, along with some possible ways to solve this paradox. But first, we need to finish determining what science really is and come up with a good theory for how we gain knowledge. For the moment, we’ll temporarily declare consciousness off limits for algorithms and then re-evaluate the question when we are ready for it.

Questions for Discussion:

  1. Which is more disturbing to you? The possibility that we are algorithms, or the possibility that science can never, even in principle, explain consciousness?
  2. If science can never explain consciousness, then does that imply that even God can’t explain consciousness? If not, then does that not imply that ‘explanations” must not require algorithms? Then what is an explanation?

One thought on “Algorithmic Reducibility

  1. As we’ve discussed before, it just doesn’t follow that because something can’t be explained via spatio-temporal concepts or physical mechanisms, it’s incomprehensible to God. That is, unless you are defining “comprehensible” specifically as “explainable via spatio-temporal concepts and/or physical mechanisms”. In which case it’s just a tautology to say that non-physical things are incomprehensible.

    Even if you analyzed a thing into all its constituent parts, do you now have an “explanation” of it? Is an algorithm an explanation? How then to “explain” the algorithm?

    Analyzing a thing into its parts is an explanation only if the thing is reducible to its constituent parts. But does that not rob the thing of its meaning as an emergent whole? I submit that “explaining” a man in terms of his constituent parts, including algorithms, even if consciousness itself were explainable by algorithms, would not bring you any closer to comprehending what a man is. Comprehension would still be something different from mere analysis.

    In which case, comprehending a spirit would not require the ability to analyze it into constituent parts or algorithms. In fact, doing that would be superfluous to grasping its meaning.

    Take the example of a car. According to your understanding of “explanation” and “comprehension”, you would have to take it apart, and then take all its parts apart, and see what each is made of and how it works. To your mind, now you fully “comprehend” and can “explain” a car.

    But suppose a man time-traveled to the present and wanted to know what was that huge mess of metal and plastic and rubber that was spread all over your driveway and front lawn. Suppose you start explaining: When you put this rubber cylinder on this steel piece, you can fill it with air and use it as a wheel. “Ah, very interesting”, he says. And when you connect this thing (battery) with this other thing (headlight), it emits light. “Amazing!” says he. And this electric motor makes this piece of glass move up and down. “Yes, yes, very nice. But what is it?”

    I submit that he won’t grasp the meaning of it until you put all the parts together and take him for a ride. That’s when he will get the idea: “Ah! It’s a thing that transports people.” Now he comprehends cars.

    All the analysis of the car — breaking it down into its component parts and seeing how each of them functions — doesn’t explain the thing itself, but only how it functions physically (i.e. the efficient causes of each function); which is fine in itself, but is an incomplete understanding. Further, you can still comprehend “what the thing is”, without ever having done that. Men, after all, had a pretty good idea what men were, long before they discovered microbiology.

    Therefore if a spirit — or consciousness, which some say is the same thing — can’t be explained in terms of constituent parts or mechanisms or algorithms, it doesn’t follow that God, or man for that matter, can’t comprehend it.

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