Using Algorithmic Entropy to Disprove the View That Science Is Falsifiable
Author: planeheart
I believe you are all familiar with Popper's falsificationist thesis. In summary, this thesis holds that:
A theory is scientific because it can be falsified.
One of the key arguments of this thesis lies in the asymmetry between verification and falsification. The argument is roughly as follows: scientific propositions are all universal propositions, and verifying universal propositions is extremely difficult, while falsifying them is relatively easy. As an example: the claim that all crows are black can be negated by the existence of just one white crow.
Based on this thesis, one can even derive some seemingly reasonable conclusions. For example, if we draw balls from a black box and the first 100 balls drawn are all red, then among the following two assumptions (both consistent with current observations):
“(1) All balls in the black box are red”
, (2) There are balls of various colors in the black box‘
we should prefer (1), because it makes the theory appear simple and easier to falsify. (Popper believed that inductive reasoning is unnecessary for science, and in principle, the principle of falsificationism can be used to select the theory to accept.) It is easy to see that if the 101st ball is green, then (1) is falsified, but (2) is not falsified, so (1) has stronger falsifiability.
Through some rough analogical reasoning, falsificationist theory can also resonate with Occam's razor in a peculiar way, namely: simple theories tend to be extreme, and thus easy to falsify. Therefore, one should usually choose the simplest possible theory.
The asymmetry between verification and falsification that it attempts to assert does not actually exist in science. The reason for its error lies in the fact that its antecedent, "scientific propositions are all universal propositions," is not correct. Obviously, "quasiparticles that do not obey Bose and Fermi statistics are allowed to exist in two-dimensional strongly correlated systems" is not a universal proposition, and on the contrary, this kind of proposition is extremely difficult to falsify, while relatively easy to verify. This causes the so-called advantage of falsificationism over positivism to be completely lost.
Even with a weakened version saying, "the basic principles of a scientific theory must be universal propositions," it is also not correct, because in fact we are allowed to express some of these principles in the form of particular propositions, for example, the existence of a Lagrangian function that completely describes a physical system. Even though there are still many principles that seem to appear only in the form of universal propositions, it is difficult to see that there is no possibility of transforming their formulation in the future. More importantly, universal propositions are usually difficult to verify, but they are not necessarily impossible to verify.
Falsificationists elevate the difficulty of verifying universal propositions to "impossibility" on the grounds of the infinity of the objects of cognition, but this infinity itself is not a conclusion that can be derived from pure deductive reasoning. (The number of crows in the world is indeed not infinite) This causes them to deviate from their original claims. And regarding whether the number of things in physics that can be called "objects of cognition" is infinite, there is still serious controversy. (If the Bekenstein bound is applied to the observable universe as a whole, then the minimum amount of information needed to describe it, measured in bits, is finite)
Therefore, in fact, the self-contradiction of falsificationists occurs in their attempt to argue for the characteristics of scientific propositions. They fail to support their argument from a purely logical standpoint, yet they claim that their argument does not depend on induction.
Therefore, the author of this article intends to look at the entire problem from the perspective of randomness, rather than fallibility.
The so-calledalgorithmic entropy, which is what general information theory textbooks refer to as “Kolmogorov complexity”
Martin-Löf randomness is defined for infinitely long binary sequences of 0s and 1s. For finite-length binary sequences, because the definition of algorithmic entropy depends on the choice of the universal machine U, this definition has limitations for finite sequences: the result depends on the choice of U. However, for infinitely long sequences, choosing any U gives the same result (this is called the universality of algorithmic entropy). Therefore, although this definition needs to mention the universal machine, it does not essentially depend on it.
This randomness indeed has a very interesting asymmetry:
(1) Wecanprove that the vast majority of sequences are random. (To be precise, non-random sequences form a set of measure zero)
(2) Wecanprove that a particular sequence is not random (for example, 010101…… cyclic sequence)
(3) Wecannotprove that a particular sequence is random (this is somewhat like the example: the first N digits of the binary expression of the square root of 2 are difficult to distinguish from a 0-1 sequence obtained by coin tossing)
Among these, (3) is guaranteed by Chaitin's incompleteness theorem (the person who defined the famous Chaitin constant):
Foranyaxiomaticformal system A, there exists a constant L such that within A we cannot provefor any sthat K(s)>L
Therefore, if this asymmetry is used to replace the confirmation-falsification asymmetry in the original falsificationist theory, it will be more reliable than the original argument, because it does not depend on the dubious "infinity of the object of knowledge" as an external addition, but rather comes from the incompleteness universally present in any axiomatic formal system (or rather, a part of Gödel's incompleteness).
We can easily see that this argument, when used to explain the original example, would do better than falsificationist theory, and it reflects the "human preference for simple theories" in a more natural way. Because roughly speaking, algorithmic entropy corresponds to the complexity of description. In the previous argument, accepting "(1) all balls in the black box are red"“The reason is that it is now lower than ”(2) There are balls of various colors in the black box“ in algorithmic entropy, and therefore further away from ”random”. The reason for concluding “all crows are black” from observing a finite number of black crows is the same. In this theory, the purpose of humanity’s pursuit of scientific truth is attributed to constructing order from a chaotic universe, in order to avoid the possibility that the knowledge system is governed by ”complete randomness”.
Theories that build models using the above method already exist; this is the so-called "Minimum Description Length" (MDL) criterion. However, there are still enormous difficulties in generalizing this idea to the general case, due to: (1) The incomputability of algorithmic entropy. (2) For general objects of cognition, rather than mathematical objects such as binary sequences or integers, it is difficult to define algorithmic entropy (Zurek proposed the possibility of applying algorithmic entropy to physical descriptions, and believed that it essentially corresponds to Gibbs entropy in physics; this is the reason this article follows his terminology and calls it algorithmic entropy, rather than Kolmogorov complexity). However, statements such as "the infinity of spacetime regions" in falsificationist theory are also highly dubious and untestable expressions. Therefore, I believe that such a transformation can, while preserving its original intention (proving the asymmetry between a proposition and its negation), correct its extreme passages.
Note:
Question: For infinite sequences, does their complexity not depend on the universal Turing machine?
Answer: