Tractability: (More commonly, “Complexity,” but I like my word.) Some problems it is easy to confirm that IF you have a potential solution in hand, THEN it is easy to test of the solution works. But if you don’t have the solution in hand, then finding that solution is intractable. This is why encryption works on your computer. You and the person or organization share the encryption key, and can easily prove it works from the fact that, well, it works. A third party, however, is like that funny fish they serve in Boston (scrod) because finding the key from cold is basically impossible given available computing resources. (This is related to the question of whether “P = NP” is true or false. I’m pretty sure it is false, but proving one or the other has itself proven to be incredibly difficult.)

So let’s think about this for a moment. Could God ‘solve’ an uncomputable problem? What would that even mean? God, being the authoritarian prick so many fundamentalists make it out to be, could easily declare a solution. But, as Russell once (or more) quipped, that enjoys all the advantages of theft over honest toil. But for anyone or anything with even a pretense toward reason, such an ipse dixit gesture would be utterly unacceptable. In the absence of a proof (mathematics/logic and alcohol being the only places that word is legitimately used) such an ex cathedra announcement has less standing than the gibbering of a monkey with a cattle prod up its bum. At least we can understand what the monkey is going on about (poor thing!)

(Some religiously inclined persons embrace the idea that God is unintelligible. Often referred to as “fideists” in the philosophical literature, they embrace the dictum (whether explicitly or not), “I believe because it is absurd.” It is not hard to understand why such childishness is popular with “Christian” nationalists/dominionists. It legislates that they are not only permitted to be drooling, uneducated, half-witted buffoons with an inclination toward violence; it mandates that they be so. Also, because some people are catastrophically confused by even the simplest use of a logical quantifier, when I say that “some” religiously inclined persons embrace this view, I am OBVIOUSLY NOT saying that all do.)

What about tractibility? Well, some problems become unmanageably complex as the inputs (number of pieces of data) increase. Generally speaking this explosion of complexity it reflected in two way: the amount of memory a computer would require in order to work the problem (SPACE) and or the number of steps it would have to take in order to complete the computation (TIME). (The capitalization is reflective of standard practice in common texts on complexity.) Here, presumably, God might be able to step in and help, being gifted (again, presumably) with infinite SPACE and TIME, such mundane limitations would have no effect upon its divine mathematical grace. But, again, those of us who demand a reasoned response are left asking why we should accept an answer from a source that can’t even offer a meaningful justification for the existence of evil in the world, and never bothered to mention that slavery was wrong.

Dragging our considerations back to the world, issues of computability create strict limits on what qualifies as a genuinely scientific theory. For example, and supposed ‘theory’ that based itself upon uncomputable equations would have to be dismissed as the sheerest drivel, not so much because would it produce untestable ‘predictions,’ but rather because it would fail to produce any predictions at all.

(One is reminded here of the String Theory twaddle that has swallowed theoretical physics alive and left nothing worthy of the name “science” in its path. However, while it is certainly the case that String Theory produces no testable predictions, and as such is empirically vacuous, the reasons for this pathetic state of affairs are rather different from matters as readily identifiable as computability.)

The same issue arises with complexity: if the concrete calculations of a purported scientific ‘theory’ do not admit of a tractable solution, then the ‘theory’ is, again, empirically vacuous.

The question, “but what if there are aspects of reality that just are uncomputable or intractable?” leads naturally to the possibility of science being good enough, rather than absolutely “correct.”

Some processes of nature do seem to be best modeled with non-linear equations. These are the source of outcomes described as “the butterfly effect,” in which the smallest “computer” capable of performing the calculation is the entire universe itself. This is certainly a limit to what is tractable, since any computer at our disposal will be much smaller, and lack both the SPACE and TIME capacities to generate a solution prior to the world simply presenting that solution to us. However, we can gin up “good enough” solutions and, in point of fact, do so all the time. Newtonian gravitational mechanics are non-linear as soon as they involve 3 or more interacting bodies. And yet for short term projects like our many manned and unmanned space probes, these equations will allow us to navigate the solar system with astonishing accuracy, even as they won’t allow us to accurately predict the positions of the planets a millions years from now.

(This “close enough” affect is the salvation of scientific inquiry, and human rationality in general. This too forms a topic in the study of complexity (tractability as I insist on referring to it.) It is a comparatively new branch of the subject, and I am myself poorly educated in it, although the few materials I’ve seen suggest that, so far, formal results do not seem encouraging in general, however effective they might prove in particular cases where we are actually invested in the answers.)

But what if the physical processes of the universe are such that they are simply uncomputable? It is hard to view such a question as anything other than a pathetically vacuous hypothetical. I mean, what does it even mean to suggest such a thing? Clearly the universe runs on without our theoretical permissions, so in that sense it is obviously “computable” on its own terms. While our models of computability may prove deficient (and, to date, there is no evidence to suggest they are, although there is no proof that they are not), that is a failure of the model, not the general concept.

To say, on the other hand that reality is in some absolute sense “uncomputable” is in essence to assert that it is beyond the possibility of rational comprehension. And what possible rational argument can be offered for such a claim? To argue at all is to presuppose the rational comprehensibility of the universe; to argue against such comprehensibility is to instantly refute one’s self. Part of the trick, then, is not to confuse the model with the thing being modeled (part of the sin I refer to as “model centrism.”) If our formulae defy the possibility of computation, then that is a fault of the formulae.