Philosophical Tail Chasing

I do, perhaps too often, rely on childhood impressions as the beginnings of ideas. Perhaps this is because, only as kids, when we have not yet had imagination about the universe pummeled out of us, do we have the flexibility to accept that the world might be more fanciful than we now believe. Falling back on that imagination, and remembering ideas and impressions that stemmed from it, is one of my most powerful tools for philosophy. It is in that vein that I recall a cartoon I saw, as a kid, in which a vaccuum cleaner sucked itself up and vanished from existence. Though it struck me as physically absurd, even then, the image made an impression on me.

Our understanding of the world is layered in such a way as to avoid self-reference. Things are made of other things even as they comprise greater things. Our brains seem to be wired to understand the world as a big collection of holons (a term coined by Arthur Koestler to describe things that are both parts of something while itself being comprised of smaller things, as an atom is a part of a molecule but is itself comprised of neutrons, protons and electrons, for example). A world in which things were comprised of things that, say, contained the first thing in its own composition would be a difficult world for us to understand. It is tempting to say that our difficulty in understanding a world where, say, an atom could be comprised of molecules, one of which contained said atom, is based on the fact that such a world would be physically absurd, like the vaccuum cleaner sucking itself out of existence.

What has always probably been an inherent quality of human knowledge became explicitly identified in the early part of the 20th century when Bertrand Russell tried to avoid the self-reference that seemed inherent in mathematics. Russell, was reviewing work by Gottlob Frege, a German mathematician, who was trying to use set theory to logically derive all of mathematics. In Frege's work, cardinal numbers became symbols for sets of quantified sets. The number one, for instance, stood for the set that contained all sets with exactly one member. The number two, then, stood for the set of all sets with exactly two members. By defining numbers in this way, he was able to reduce arithmetic to logical operations on these hypothetical sets and derive the standard axioms we now associate with arithmetic. Frege believed, and Russell concurred, that, with (considerable) work, all of mathematics could be associated with logic, and thereby shown to be logically consistent and complete.

The trouble began when Russell, thinking about the idea that sets contain sets, wondered about the consequences of a set containing itself. For example, the set of all sets that contain one member would have to include a set that contained only itself as a member, because that is, by definition, an entity that satisfies the criteria for all sets that contain one member. The consequences of sets containing themselves become odder when you consider all the possible sets that contain themselves as members. They, themselves, constitute a set and, clearly, this set, could contain itself, since, if it did, it would by definition belong to itself. Well, if one is going to talk about the set of all sets that contain themselves, one naturally starts to wonder about the set of all sets that do not contain themselves as members. Or, at least Bertrand Russell did, to the severe detrement of his friend. Would this set contain itself or not? This problem, known now as Russell's Paradox, seemed to be the end of Frege's attempt to reduce mathematics to logic. The paradox, it seemed, was a contradiction. When logic leads one to a contradiction, one typically concludes that the premises are false and the system being derived is invalid.

Russell, though, undaunted, tackled the problem himself and came up with a solution that involved a system of types that would be familiar to any programmer familiar with object-oriented programming. In Russell's system, not all sets are created equal. There are sets of things, and sets of sets, and sets of numbers, and so forth. The concept of a set containing itself is defined away because there was no need, in mathematics, for sets that could be typed in such a way that they could contain themselves. (We now know, that this is not true. Number theory and metamathematics rely heavily on just the sort of self-reference that Russell explained away. But it turns out that the contradiction, while still present, does not invalidate mathematics, just certain previously assumed beliefs about it.)

Russell's solution was to turn sets into holons (even before the term was coined). By ensuring that things could only contain other things that could not contain greater things, self-reference and, thus, paradox were eliminated. To Russell's mechanistic, pre-quantum mechanics style of thinking, the universe was saved. Physics would continue ticking along in its deterministic way and mathematics would always be able to reliably describe it.

But, the more we learn about the world (at least to the extent that we think we do given my contention that there are no certain truths), the weirder it seems to become.

Is our universe, as some of the more "out-there" physicists are proposing, really an 11-dimension membrane in a universe of universes, some of which have stable physical laws and some of which do not (recommended reading on this topic: Parallel Worlds by Michio Kaku)? Or, as Hugh Everett proposed, does the universe cleave into other universes at the moment that every quantum event is observed, suggesting that there is a universe in which we actually made every other decision we might have made in this one?

As interesting as all these ideas are, they all suffer from the same ontological problem that any scientific or even religious theories about the universe suffer from: they all require that it is all ultimately contained in something which raises the question of what is outside that thing. If God created the world, who created God? If the Big Bang was the beginning of the universe, what was before the Big Bang? If the universe is one of an infinite number of parallel universes in some big-U Universe, how did the big-U Universe come to be? This problem stems directly from our discomfort with paradox. If I proposed a theory that the entire universe was actually located within my right forefinger fingernail, your objection would not be simply based on my arrogance. You would deny the very rationality of the idea. Something small can't hold something big. Something can't be contained in part of itself.

Of all the sins and evils that mankind has invented, nothing appears to be as vilely and universally despised as paradox. We accept the axioms of mathematics and fiinite-state predicate logic because they avoid paradox. Science, which relies on math and logic, accept without question, that a bit of evidence that contradicts a theory automatically invalidates it. Throughout history, people have been branded as witches and heretics for proposing ideas that defied the understood nature of the universe because the ideas posed the paradox that it was impossible for their ideas and the conventional wisdom to both be true. None of this is anything but obvious. But that's my point. We accept it far too easily.

Let's consider a most basic of paradoxes. I propose to define a new kind of finite state logic, one in which I rewrite the law of identity, which in conventional logic states that A = A. There can be no more obvious truth. If something is true, it is true. If it is false it is false. I propose, in my alternate logic, that the opposite is true. A is not equal to A. If something is true, it is really false. If something is false, it is really true. Can you get your minds around this idea?

What does such a thing even mean? In digital electronics, if you build a circuit that loops the output of an inverter into its input you get an oscillator. That is, the state of the inverter alternates between 1 and 0 over and over again at a speed determined by how long it takes the inverter to change its internal state. So that's one possible interpretation of A != A (to use a programming language notation). It could just mean that it's not stable. A may or may not be true depending on when you look. Another way to look at this, though, is to say that A may or not be true depending on ones point of view, which could include the point in time one chooses to look.

We see this all the time in the real world. If you and a friend are looking at a boulder, she from its one side and you from a side perpinduclar to her view, and you both drew a picture of it, odds that either of you would recognize the boulder from the other's picture is pretty remote. Boulders are just not regular objects as a rule. Her view may be of someting long, squarish and tapered to the left, whereas yours might be of something completely round, approximately as tall as it is wide and with big black spots on it. Yet you both drew the same boulder, but from a different point of view. This example is, clearly, not a paradox, per se. But it illustrates one possible way to get ones head around the idea of a paradoxical universe.

Rather than thinking about the universe as being something objective, to be perceived rightly or wrongly, one might simply think of the universe as being perceived, by definition. The ontological problem I described of needing to understand what contains the universe stems from a need to believe that it exists, indeed that we exist at all in some objective, materialistic sense. One's worldview, if you will, is one's world.

I would argue that discovery of a genuine, unavoidable paradox, one which can not be explained away simply by moving away to a more distant context (for example, a more complete theory of the universe), rather than destroying science and human knowledge, would free it. The development of human knowledge becomes, more properly, a development of human tolerance. Rather than fixating on endless dogmatic and fruitless ponderings on the "true" nature of the universe, mankind would be free to explore the more meaningful, if somewhat more mundane truths of life in our actual corner of that universe. Such a process gives us more than just technology and medicine but also greater compassion and tolerance and a reason to build a world with as much in common as necessary to benefit from our mutual existence but without sacrificing those aspects of ourselves that make us unique and make our lives special to us.

So where would one go to find such an unavoidable paradox? We can start with knowledge itself. It has been my assertion that certain knowledge is impossible. Whether it be because of the problem of induction, the fallability of human intelligence or just the simple acceptance that some things are based on inextricable human values, we can know with certainty that certain knowledge is unattainable. Yet, this very statement is a claim of certain knowledge. One could argue (and, at first I did myself) that this is not really a paradox because all we are doing is using the lights of ones own rationality to illustrate that it leads to a contradiction. Yet this argument is an argument of logic -- itself a claim based on a presumption of certainty. At the end of the day, the possibility of certain knowledge can not be resolved using knowledge itself. One has to conclude that whether or not knowledge is certain is a sort of belief. It is based on one's point of view. We live in a world where we rely heavily on our belief in certain knowledge. If you will, our world's existence, indeed that of the universe we try to describe, rests on our acceptance of the certainty of some facts about it. This makes sense if you allow the notion that one's worldview is one's world. Those who are not certain the world exists must live in a dark and lonely place.

One question that might come to mind is. if certainty in the world's existence is so critical to its existence, why would I argue against that certainty. The fact is, I don't. While I argue that, philosophically, absolute certainty is an unattainable goal, I do not claim that one can not be certain enough about the things in one's life not to believe in their own world and existence. Nagarjuna, the founder of the Madhyamaka school of Buddhism wrote of the difference between conventional truths and ultimate truths. It is certainty of ultimate truth that I argue is unattainable and, in fact, because said certainty is not even conceivably attainable, I argue that its existence is unnecessary. What is truth without someone to discover it?

Rather than finding some sort of arbitrary conventional certainty, though, I observe that certainty is, itself, a relative thing. I live my life and weigh the choices I face evaluating the world through a lens that can focus more clearly on some things than others, and I take measured risks taking into account all that I do not know. I count fairly firmly on the technology that allows me to fly across country and very little on the likelihood of finding large sums of money lying in the street with a tag identifying me as its new owner. This is all consistent with a relative view of values, ethics and even scientific or mathematical truth.

And to think, I once thought the idea of a vacuum sucking itself out of existence was absurd.