Saturday, December 31, 2011

The Self-Referential Paradox

This post may be one of mine most likely to fit Virginia Woolf's description of "the masturbations of a pimply-faced amateur." While I sometimes write with the audience in mind, I often, as I do here, write for the purpose of clarifying and refining my own thinking, which the process of writing and, hopefully, receiving criticism, helps immensely with. 

The item on my mind since a 1:00 AM tiny flash of insight last night (after several hours spent traversing the mountains of Skyrim in search of dragons to be slain, and consuming their souls to enhance my Thu'um) is in relation to something touched on in the previous post. In that post, I mentioned the work of one Kurt Godel, Austrian logician and mathematician. The idea which made him famous, published at a mere 25 years old, was his Incompleteness Theorem. There is insufficient room in this post, and I am unfit for the task besides, to explain the theorem in any detail here. I have read it, maybe 8-10 years ago, and at the time felt like I could follow it, up to a point at least, enough to bear witness to its irrefutably as well as its logical beauty. Either way, my meager understanding aside, it has stood for 80+ years, and I know of no serious candidate to disprove it. 

Reading Godel was, for me, on par with reading the Bible, through, for the first time, or reaching the closing of Ulysses (the actual target of Woolf's [jealous] disdain), ..."yes and his heart was going like mad and yes I said yes I will Yes," in that you feel as if you will never be able to look at the world in quite the same way again. (There are others, of course, Dawkins, Hawking, etc., who ultimately had more of an impact on me, but I digress...) Before I try to explain why I see Godel in this light, let me at least copy/paste the Wikipedia version of his Theorem here. 

What Godel showed was that for any mathematical system complex enough to include the arithmetic of the natural numbers (i.e. complex enough to include 2+2 and 124,678 x 567,678,432) was that:

             1. If the system is consistent, it cannot be complete.
             2. The consistency of the axioms cannot be proven within the system. 

To prove this, he constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which is in contradiction to the requirement that in a formal system provable statements are always true. Thus there will always be at least one true but unprovable statement. 

Sound familiar? It isn't all that different than the Epimenides paradox, the statement by Epimenides, a Cretan, that "All Cretans are liars." If it's true it's false, if it's false it's true. (Actually, there is a little bit of wiggle room there, but the more refined version, "This statement is false," is essentially irresolvable.) This is more broadly known as the Self Referential Paradox, and Godel essentially, simply, showed that it was an inescapable aspect of math and logic as well. 

But Oh My God Who Cares, right? Granted, for most of us, most of the time, this isn't something that is going to keep us up at night. (It may be one of my own personal not-at-all-sober experiences where I spent several hours trapped in the thought process of thinking about how I couldn't stop thinking about how I couldn't stop thinking about how I couldn't stop thinking about the logical loop I was trapped in of not being able to stop thinking about how I couldn't stop thinking about how I couldn't stop thinking about... that makes it a bit more upsetting for me.)

So people have been pointing out for centuries, from the ancient Greeks to 20th century logicians, that when a system becomes self-referential it often, almost always, runs into the problem of being either incomplete or inconsistent. I think this matters, because I think it applies to a great many more systems of belief than people often recognize.

I pointed out in A Relative Contradiction that I see the same internal contradiction in the moral relativism that is often associated with Western liberalism: That what is "good" is relative to someone's time and culture, except the Western liberal ideal of "tolerance" for other people's relative "goods," which is, they seem to claim, universal.

In a somewhat similar way, I pointed out in Pascal Was a Sissy that one starts to see a lot of cracks in the foundations of Christianity when you try to hold it up to the same standards it purports to advance. In the philosophy of religion this type of internal contradiction also crops up in the Problem of Evil.

I also tried to point out, in Freedom Worth Wanting that I think there are certain internal inconsistencies in application of belief systems that offer to "free" you of "desire" and offer that as the greatest freedom of all.

And I think that the same can be demonstrated for a great many sets of beliefs. I hope a few more examples will make my point sufficiently clear. Admittedly, I will be painting with some pretty broad strokes here, so some may have quibbles with my framing of certain beliefs, but I think that the point I make holds true nonetheless...

Take for example, the current American brand of Libertarianism. While I believe this political viewpoint, though far from perfect, has much to recommend it, it isn't hard to show that it suffers from its own internal contradiction. Libertarianism purports to advance the greatest degree of individual freedom. It argues that the overwhelming majority of political, economic and, to some extent, moral, decisions should be left up to the individual. Well, this is all well and good until those individuals decide they are willing to forgo a degree of that freedom in exchange for, say, socialized medicine. Well, no then, you don't have the freedom to make that choice. That choice is out of bounds. Libertarianism offers you a great many freedoms, but not the freedom to reject aspects of Libertarianism. 

(This isn't all that different from the dilemma the US fears when trying to advance democracy in other countries that may then end up voting, through a democratic process, to forgo democracy in favor of, say, an Islamic theocracy. What is "truer" to the cause of democracy: trying to force people to accept it when they wish to throw it away, or allowing them to do so, since that is "the will of the people?")

A similar internal contradiction can be shown, I think, in post-modern "discourse." While post-modern "theory" consists of many different threads; structuralism, post-structuralism, feminism, marxism, deconstructionism, queer theory, etc., they all attempt to do the same thing: to read the world as a "text" and take apart the reigning narratives that are being forced upon us hapless, naive dimwits by whatever bogeyman that particular thread is going after- the patriarchy, the capitalists, the straights, the ideals of science and materialism, etc. 

One of the features of post-modern theory that is consistent across each of these threads is the denial that there are ultimately "facts on the ground," as I like to say, that we can never reach an objective reality, and that our view of the world is always colored by the "narrative" we inhabit, a narrative foisted on us by the "dominant hegemony" (that pointless redundancy right there is a great example of how post-modern "discourse" attempts to bludgeon you into mental submission with top-heavy language instruments). 

While, again, there are some positive things to be said for attempts to understand and explain the unequal distributions of power in human society, there is an immediate and obvious flaw in any system that denies any kind of objective reality: if there is no escape from these "cultural narratives" then aren't "post-modern discourse" and even the idea that power should be distributed more evenly just subjective preferences of a particular "cultural narrative?" If that is the case, then who is to say that power shouldn't be exclusively in the hands of straight-white-male-Western-capitalists? (Obviously, I am not saying it should, I'm just pointing out that without any kind of narrative-free reality, there is no reason for preferring any one particular distribution of power over any other.) According to the very tenets of post-modernism, the "imbalances" that post-modernism tries to correct are just another piece of our cultural narrative, and they may not really be imbalances at all. If you follow post-modern discourse to its logical conclusion, if you make it self-referential, then  it runs into the same internal contradiction as so many other systems of belief. 

One more example. I took a philosophy course in college and in one particular class the prof was trying to offer a rebuttal of reductionism. (Reductionism is the belief that all that can really be said to exist are the most basic, fundamental material aspects of being, i.e. the fundamental particles of physics. ) He began by defining the existence of physical reality by stipulating that two things can only be said to exist if a relationship between them can be described, such as distance. In other words, you can measure the difference in position between your house and the one across the street, but not between your house and God, so He is out of the picture for this discussion. He then went on to ask, "Well, what is a 'car?' Is it just a bunch of constituent parts, mufflers, spark plugs, tires, which are themselves reducible to their component parts, ad infinitum?" (This is essentially the position of reductionism, that yes, that really is all you can say with confidence.) He argued that No, if a car is made up of n parts then the "car" is n+1, that is something else "arises" out of this conglomeration of parts that is "car."

I raised my hand and pointed out, "Okay, so if you admit that the only things which can truly be said to exist are things which can be described in relation to something else, such as a relation in space, then what is the distance from the muffler to the "car?"

Reductionism isn't slain so easily.

Unsurprisingly, he didn't have an answer to that, except to brush off the question. 

Which is essentially the response of most systems of belief when asked to justify themselves according to their own rules. 

Okay, wow, so what the hell is your point, Rob, with all the quasi-philosophical bullshit? I should stress that my point is not, as it may have seemed, to show that any of the systems of belief mentioned above, or any others, are automatically, inherently and completely wrong once one can identify an internal contradiction. My point, I guess, is the same as Godel's: that any system that is complex enough to reference itself is almost always* liable to run into internal contradiction. No system of human thought is ever* going to be 100% complete and consistent, every time, all the time.

And this is why Doubt is so Great.^ Because it reminds us of our imperfections, our fallibility. Because whether you believe we have been struck from perfection by Original Sin, or you acknowledge that we are just crafty monkeys, we have to remember, we are not gods. 


* Just want to point out that even this statement, which attempts to be universal, could probably be shown to be inconsistent, if someone had more mental energy than I have left at the moment.

^ Similarly, even Doubt, or science, or whatever you want to call it, needs to be aware of its own propensity for internal contradiction- "What if I doubt the utility of Doubt? Maybe it is not necessary to always be skeptical. Maybe I can just make up whatever I want and swear by it..."

Unlikely. You still can't just say "tangerine." :)


  1. I fondly remember the message from the night of the logical loop. : )

    When I was a fundamentalist, I was obsessed with the hypocrisy of other fundamentalists; that is, if you 'really' believe, say, abortion is killing babies, than not doing everything you could to stop it was hypocritical and therefore casts suspicions on whether one 'really' believed.

    I suspect that contradiction is actually foundational to any system. It's the master-signifier; you can't actually reach the ideal of consistency/completion precisely because you start with an unattainable/unprovable postulate/founding myth.

    The realm beyond the master-signifier is the horrifying abyss of relativism (and reductionism, i think)--a primordial pre-symbolic chasm which I find is useful to stare into occasionally just to know why I like my myths as much as my truths, even if they're identical to each other.

  2. Ah, but see, you've ended the conversation before we could ever even have one. By invoking a subjective fiction like "master signifier" you ask me to participate in said fiction in order to continue the conversation... something I am not eager to do.

    Because that is all we can ever hope to do, in any realm of human inquiry or discourse, have a conversation. And that is only possible if we agree on some fundamental things: 1. Logic is valid and 2. We see the same empirical objects- I see a circle and it has no corners, and you concur. Then we go from there.

    But if we are just going to retreat into our own subjective fictions, then I would at least choose something far more interesting than a "master-signifier." I would at least like the warm, cozy belief that I am going to live forever or whatnot, if I am just picking and choosing.

    You may very well be able to get others to have that conversation with you, it is empowering, after all, to believe that one has a certain insight into the world that is unknown to most others, but I, personally, found that it got me nowhere useful.

    I think we are saying the same thing, it is just that I see a way out of relativism and it seems like you do not.

  3. I think we really are saying the same thing. I invoked master-signifier merely to suggest that this wall or gap--that is, the internal inconsistency of any system, even of doubt itself--might be universal, or is at least recognised in other systems of reasoning.

    Take the car reduced to it's constituent parts, or reduced to the most finite particles of matter. It's the most honest position, I think, to admit both that all you can really prove is the collection of particles while also saying, "knowing that, It's also a car."

    It's the same mental process one goes through when acknowledging that all we really know so far about genetics and evolution and the mind asserts that the love one feels for another (lover, child, friend) is contingent on/derived from a whole host of biologically necessary mechanisms, not from some divine source. Yet, possessing that knowledge does not change one bit the actual experience of love, nor does it make it really more base.

    One more example. A good piece of fiction is really just words printed on paper (and you can reduce it even further), but it is experienced as something profound, something that isn't just "ink on paper." More--the actual translation of the words, the decoding of strings of letters into a flood of mental imagery evoking emotions actually experienced all within "imaginary space" is another act altogether. It is simultaneously particles, ink blots, words, ideas, and a whole other world. (And crazier still--two people can read the same skillfully written novel and envision two radically different worlds!)

    That's my way out of both reductionism and relativism. But I don't know what to call it.

    I should really just suck it up and write a post about this, huh?

  4. So, yes, I mostly agree.

    The only question I raise, the only question I am EVER really trying to raise on here is, "What can we say with some degree of confidence?" Not a whole lot. So mostly, I just want people to acknowledge when they are making stuff up... and stop using it as an excuse to kill, maim and oppress others.