2. The Unknowable Is the Ground of Whatever Is Known

Positivism’s Optimistic Take on Knowing 

Incomputable problems would seem to show a primary proof of the inherent incompletion of the Universe, especially because of their direct relation to Kurt Godel’s Incompleteness Theorem. However, much relies on how this sort of incompleteness is interpreted. Douglas Hofstadler famously thought of determination as complete because of the infinitely recursive movement from the inside determinations of a set to the outside, undetermined definition of a set in his “strange loops.” Hofstadler’s strange loops continually expand the set to include what had been its outside definition, but this expanded set itself must be determined by another arbitrary outside ad infinitum. Every new, more inclusive set produced a new, outside indeterminacy in accord with Godel’s incompleteness Theorem, but Hofstadler felt that this new indeterminacy wasn’t actually indeterminate because it could be potentially determined by a further expansion of the set. Incomputable problems are incomputable because they lack the capacity to determine if their operations are terminable because their operations produce more outside indeterminacy than they determine, which is the relation between Godel’s Incompleteness Theorem and Alan Turing’s “Halting Problem.” The debate remains as to whether this lack of capacity is the same as the absolute indeterminacy of the irreducible ambiguity of the void, or if it is just the provisionally undetermined indeterminacy of Hofstadler’s strange loops. 

There are incomputable problems, but these unsolvable complications don’t point to any inherent lack within information per se for the positivist. There is a provable, or “necessary,” incompleteness within the structure of computation itself, but not within mathematical and scientific knowing. Incomputable problems demonstrate that there is a necessary mismatch between algorithms and the Universe’s production of possibilities. Incomputable problems do not necessarily obviate a positive view of complete knowledge even though the Universe’s production of information out paces any conceivable computational process. Scientific knowledge is always incomplete because information is always quantitatively more than the capacity of any specific algorithmic intention and not because all knowledge isn’t ultimately reducible to quantities.  

If whatever there is, is reducible to information, then this is still a positive view about knowing because information’s ability to disclose the Universe is total, even if necessarily incomplete as far as working out every possible permutation goes. The general computations that describe the possible physical states of the Universe will someday suffice to disclose whatever there is because they will reveal a complete picture of its causal structure. For the modern sciences there are no causes other than physical causes, so this physicalist account of the positive mapping of one-to-one, casual relations, is all that there is to be known about what is. This view accords with modern Information Theory’s idea that knowing is knowing about these kinds of positivistic correlations between the “zero’s and one’s” of binary code and what is or isn't real, in which the zeros correspond to “isn’t” and the one’s correspond to “is.” What counts as ontologically real is what can be counted, and what can be counted is information. This is the scientific positivism that boldly makes the extraordinary ontological claims that the Universe just is information and nothing else. 

An incomputable problem is a problem in which algorithmic formulation is unable to keep up with the increasing number of variables, permutations, and procedures of a formulation of a problem, as Alan Turing’s famous “Halting Problem” showed. The Halting Problem proved that there is no hypothetical, ultimate algorithm that can determine if any given program will finish. Like Kurt Godel’s Incompleteness Theorem, the Halting Problem demonstrated that the truth conditions of a set cannot be proven from within that set, and thus there is no set of all other possible sets, which means that there is no algorithm that can decide if any given problem is computable or not. This necessary indeterminacy is still compatible with the determinism of scientific information, as Douglas Hofstadter showed with his famous “recursive loops.”  

The determination of any given situation is always from an indeterminate outside of that situation, but each exterior indeterminacy can be determined by an outside relative to which it is interior. Again, as above, there is a mismatch between the amount of information and computational capacities for these amounts because there is always the “one more” of the outside of the computation that defines the parameters of the computation. This “add one” sort of recursion is the sort of infinity that Hegel thought of as a “vertical” infinity because it is a determinate infinity. Each additional recursion is determined by the next, so that any given situation is determined, even if its determinations haven’t been worked out by a computer of some kind. The determination of any given situation comes from outside the situation, but this outside is itself determinable by the computations of mathematical and scientific descriptions, which are the necessary and sufficient reasons of determinate, causal relations. 

Rodger Penrose deduced from the determined infinity of this sort of recursion that thinking and consciousness in general isn’t computational; although, thinking can employ computations to certain ends. For Penrose, thinking, as opposed to computation, requires a conscious agent from “outside” of the computations to relate the mathematical operations meaningfully to the physical world. Penrose took Godel’s incompleteness theorem to imply that since sets can’t prove their own truth conditions, they needed an outside agent to relate the operations of the set to empirical reality, in the mode of the “Correspondence Theory of Truth.” Computations couldn’t falsify themselves because they couldn't relate themselves to an objective outside.  

For Hofstadter, the outside determinations of the set are mathematically determined computational processes, and those that appeared to themselves as “agents” suffered from the illusion of consciousness choice because of the self-positing recursion described by Godel’s Incompleteness Theorem, which was also the view of Daniel Dennet. Penrose recounts an anecdote about a time when he confronted Hofstadter about the fact that if he thought thisabout the recursion of Godel’s incompleteness theorem, then he would have to concede that certain integers were conscious. Penrose was shocked to find out that Hofstadter agreed, and that, indeed, there were conscious integers. It was just that conscious integers, such as us, were deluded about choice because even conscious integers were determined from their outside, which for Hofstadter, as is it for any Determinist, was our genetic coding and its relation to the natural laws.   

Set Theory is a helpful way into the determinate nature of Hofstadter’s “Strange Loops.” The definition of a set determines what belongs “in” the set and what doesn’t. These outside definitions are the “truth conditions” of that set, but those truth conditions only determine what is in the set and not what determines them. The necessary and sufficient reasons of determinate, causal explanations do not establish the necessary and sufficient reasons that define those defining reasons. If we want to know the cause of a cause, then we must back up one step to include that cause in a larger set that accounts for it. This infinite, but determinate, recursion can be seen most clearly at the current limits of the natural laws because they don’t have causal explanations for themselves beyond themselves, but they act as the causal explanations of all other physical relations.  

The truth conditions of the natural laws may someday be determined by truth conditions beyond them, but those truth conditions will have to either stand as uncaused causes or someday be explained by something beyond them as well. Whatever determines the current physical laws might in turn be determined by something outside of them, but even if these explanations temporarily appear as if absolute or foundational, they do not explain themselves. An “absolute” foundation in physical science would be something like an “uncaused cause,” or a “raw fact,” that “just is” without any further explanation. Each recursive step expands what the previous set had determined by nowincluding what had been the outside determination of that set, but there will always be a determinate, even if undetermined, outside of any given set no matter how expanded the set has become through this recursive process of inclusion. Whatever is determined by mathematical computation or by scientific observation does not increase determination nor decrease indeterminacy because computation and observation are only uncovering what has already been determined by the causal chain of the physical laws of the Universe, which is Einstein's already-finished “Block Universe.” 

In this way, knowledge may be perpetually incomplete for the material determinist, but there is no lack, either “in” or “of,” information. There is only the lack of algorithmic or computing capacity discussed above in terms of the Halting Problem and its relation to Set Theory, which is that there is always at least one more step outside any given algorithm, or set, that determines the parameters of that computation or set. This is the classical problem with the interminability of the casual chain, which will be discussed in the next section. This is also the problem with the positivism of Information Theory and the sciences in general. A digitizable “bit” of positive information has a hidden, or absence, causal chain that can’t be accounted for by direct observation, so its causes must be deduced or inferred from what is to what isn’t any longer, as David Hume famously brought to Kant’s attention when he pointed out that causes cannot be observed in the present but are conjectures about the past. Any complete account of whatever there is, must include its unobservable causal structure, which requires not only a complete description of what is, but also a general explanation of how it came to be.