Against skepticism

Skepticism, certainty, and formal truth

The great vice of philosophy in our time is its infatuation with arguments from epistemic weakness. What might have started as careful attention to distinct categories of real knowledge has fallen into a flip dismissal of the possibility of knowledge of any kind, or restricting it to the narrowest possible compass. The contrast to the staggering achievements of our technology is blatant; never have practical engineers known so much with such precision and assurance, while both academic and popular thought loudly declare their inability to do so, on principle.

Raise the problem of knowledge with a contemporary skeptic and he will dodge knowledge in favor of “certainty” in less than a minute. When pressed he will retreat further to “absolute certainty”. He won’t be able to point to a single instance of what he means by this, however, since his desired conclusion is that it does not exist. One can in principle lay down definitions that turn out to have no domain, but generally speaking this means “so much the worse for that definition”. A distinction that doesn’t distinguish one real existing thing from another real existing thing, leaves something to be desired.

What he is really trying to distinguish is not elements of his world, but his world from what he imagines about other peoples’ worlds. He sees things inside “doubt brackets”, but the content of the world is independent of those brackets. He imagines that someone else is “seeing” certainties everywhere, despite their not actually being present. Since the contents are the same (even as far as their operational, probabilistic or betting characteristics), this amounts to castigating other people for not putting his preferred doubt brackets around all of their own thoughts. (I’ll address the moral claims often advanced for such positions in a later post).

He is of course free to bury every thought in his head in layer after layer of such brackets. But he isn’t content to do this – he demands others encumber their internal notation system in a similar fashion. At bottom this is merely rude. No doubt subjectively he is earnestly trying to save others from the tarpits of their dogmatism. Perhaps he experienced his doubt-bracketing insight as liberating or humbling and wants to share it. But soon he will be diagnosing imaginary vices and errors in anyone who refuses the rituals of his church of the extra brackets, whose sacramental efficacy he never doubts.

Occasionally one finds the less social type of skeptic who instead adopts the passive role and dares anyone to argue him out of his fortress of solitude. “Prove I know something”, he says. It is trivial to show he believes things, and does so with all the operational characteristics of knowledge, but he insists he doesn’t know them. Since at bottom this isn’t a real distinction (“real” meaning, in the good scholastic fashion, “independent of what anyone thinks of it, not a matter of opinion”), who cares? As a rational animal it is his business to know things, and if he declares bankruptcy it is his own affair. One can still notice how his position suffers all the defects Popper diagnosed in hermetic thought-systems of a more dogmatic stripe. He thinks that nothing being able to shake the certainty of his self-denial of knowledge is a strength; in fact it proves to a demonstration that said self-denial makes no contact with reality.

An older role for the distinction between knowledge and certainty used it as a real distinction, specifically dividing empirical knowledge from formal knowledge. Mathematical facts and logical syllogisms counted as certainty as well as knowledge, while empirical facts were knowable but not certain. This has much to recommend it, but is slightly too blunt for the realities it is trying to capture. The reason is, there are formal truths that effectively have the epistemic status of empirical facts, that just happen to be this way rather than that, but are “leaf like” formal facts, unconnected to broader (prior) formal truths they follow from. Gregory Chaitin has shown this using cardinality arguments. Roughly speaking, there can be more true mathematical statements within a domain or system, than ways to derive them from a limited set of prior axioms. But at least it is a real distinction.

At the level of method, I think we expand the realm where we treat things as “empiricals” still more. In NKS we are frequently making conjectures about whole categories of formal systems, well before we can have deduced enough about them to turn our knowledge of them into anything like logical certainty. Not because they aren’t at bottom purely formal systems, but because the scale of deductive work involved exceeds anything practical, not just for us but for our computers, or even for any forseeable computers running for even astronomical time. Relative to the state of our knowledge at the time of the conjecture, these have the methodological status of empirical facts.

I would claim that is the only relevant knowledge to “rate” such truths or claims, by. There is no mythical mind of God for which all logical truth, no matter how involved, is immediate and simple. For any finite mind or computational system of any description, some purely formal truths or propositions have the epistemic character of empirical facts. What we have previously thought of as the formally certain is actually a special subset of the formal, the simple or computationally reducible.

Against skepticism, there are simple and computationally reducible formal facts that can be known with certainty, in the good scholastic sense of certainty. Against the idea that the domain of certainty exactly coincides with the formal (as distinct from empirical), there are subsets of formal propositions that for all practical purposes are like empiricals, instead. Meaning, we address the latter with an empirical toolkit of conjecture and experiment and induction, of categorization based on phenomenal characteristics, applicable theorems or models, and the like. The domain of application of empirical method is broader than the scholastic distinction supposed, but there is a domain of application of pure deduction, and it does attain certainty where it applies.

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3 thoughts on “Skepticism, certainty, and formal truth

  1. “The domain of application of empirical method is broader than the scholastic distinction supposed, but there is a domain of application of pure deduction, and it does attain certainty where it applies.”

    where is the line for this? what are the characteristics of something that is pure deduction?

    I’m seriously curious, this is not a rhetorical question.

    It’s tempting to make this argument binary: there is certainty or there is not certainty. In fact, it seems, there is both! Somethings, processes, behaviors, logic are very certain, others are uncertain (or practically uncertain). I propose, and can never prove, that it is the mush pot of certainty and uncertainty that “drives” existence (life, knowledge, physics…). if everything we completely certain and knowable (probably implies predicable determinism), then what would we have? if it were all uncertain (and unknowable) what would we have?

    That’s probably not a very interesting comment, but it came to mind.

  2. I can see room for a gray line between them, within formal systems, when the question turns on whether a certain amount of logical or computational work has been done or not, uncertainty about how much it will take, uncertainty about whether a strong simplification or effective substitution can be found that will reduce the problem, and the like.

    The extremes are obvious – the answer to 2+2 can be known with certainty after a minimalist calculation (so minimal we barely even consider it a calculation, or as involving any computational work). The 48 millionth line of the CA on the right side of my blog – Jason’s Rule – would require vast amounts of computational work to know, but would probably be doable within the lifetime of the universe.

    But there can be “nearly cracked” cases, where certainty might be possible even though we don’t have it yet. Consider the eventual behavior of the 3n+1 problem for a specific very large integer, say one with 10^1000 digits. If the Collatz conjecture is proven, we will (then, but not yet!) be able to say with certainty (then, not yet) that it ends at the cycle 4, 2, 1, 4, 2, 1, whatever the transient to get there happens to be. But nobody has yet proven the Collatz conjecture.

    The point is, if there is a general proof it will not need to involve the computational effort of that specific case. We can see that directly following the rule for a large enough case would be computationally intractable, and would never be accomplished in the lifetime of the universe. But we can’t rule out shortcuts from a general theorem, in a case like that.

    I’d say that right now, we can hypothesize, without certainty, that the given very large integer terminates at the cycle 4, 2, 1, 4, 2, 1. But this doesn’t mean that the certainty status of the proposition is fixed for all time as “uncertain”. It could change to “at least as certain as so-and-so’s proof”. Which if the (future) proof is clear enough, might count as “certain”.

    (There is another side issue touched on there, that complex enough proofs can themselves be difficult enough even to check, that they can seem uncertain to most of humanity).

  3. Ah, good examples. Very helpful.

    On your side issue. The fact that the proof is complex, perhaps to complex for most of humanity, doesn’t change the actual certainty of the fact, even to those that may not be able to prove that certainty. We can distinguish between formal certainty (it’s been proven) and informal (I really believe it!).

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