So many philosophical pseudo-debates focus on the existence or non-existence of this or that “thing”. Pop-skepticism is at odds with most sects of Christianity about the existence of a God; many skeptics, somehow oblivious of the hypocrisy in which they engage, argue simultaneously that claims must be supported with evidence and that there must be no God. I engaged fleetingly with the university’s skeptics society in a debate about the existence of the electron, in which I argued that the electron was a mathematical tool that enjoyed the same level of existence as a number, and not so much existence as…
Fortunately for me, I did not have to complete the above thought, as the debate was on facebook so I was permitted not to respond once the level of abstraction exceeded me. Rather than the inevitable fate of a face-to-face debate on the subject — in which I would make a fool of myself for failing to possess a well-collected, self-consistent argument, my opponents permitting me to exit the arena now having failed to disrupt their conceptual status quo — the debate fizzled out, and they will probably not remember of their own volition that they had even engaged in it. It is all for the better that my medium has changed, since after some time spent meditating on the question, I have come across something I have not been able to distill into a snappy epigram.
To a “standard model” logical mind, and even to the working mathematician who has not studied logic, existence is a straightforward concept. One can ask whether a mathematical object exists with some property, and assume without argument that one is asking a reasonable question with a yes-or-no answer. However, in the world of mathematical logic — the only logical world whose paradoxes I can comfortably resolve — the notion of existence is rather more slippery. There are the standard objects which one can prove to exist from the axioms, and there are — or perhaps I should say, there are not — objects whose existence is contradictory. But there is a neglected middle class. These objects _____ whether or not you choose to exclude the middle.
The Twin Prime Conjecture (TPC), a famous question still open today in 2011, conjectures that there are infinitely many numbers p such that both p and p+2 are prime. One of these pairs is called a “twin prime”, for example 5 and 7, or 179 and 181. There are many who believe TPC is true, some who believe TPC is false, but among logicians (who crave this sort of result), many believe TPC is “independent of the axioms.” Let us explore the consequences of this latter belief. To be concrete (insofar as such a word can mean anything in such matters), let us suppose that TPC is independent of “ZFC”, the Zermelo Frankel axioms with the Axiom of Choice, the axioms of choice (no pun intended) for popular set theory.
It would be helpful to be reminded of what exactly ZFC is. Aside from the deep fantastic worlds of intuition inhabiting many mathematicians’ minds, it is merely a set of 9 statements about the world of sets. For example, “if two sets have the same members, then they are the same set”, and “given any set, you may form the subset of elements satisfying a particular property”. These are stated in rigorous, precise logical language, so by formal manipulation we can exclude the subtleties of meaning that would abound in any English presentation of these axioms. Logicians like to say that a proof is nothing more than a chain of formal logical sentences arranged according to some simple rules; this view has spread since the advent of programming languages and computerized mathematical assistants.
If TPC were true, then given any number, you could count up from that number and eventually reach a twin prime. If TPC were false, then there would be some number, call it L, above which it would not be possible to find any twin primes. However, since TPC is independent (because we have supposed it), then we know we cannot prove it either way. It may be true, or it may be false; whether there is a third option is too deep a philosophical question to explore here. We may be able to count up from any number and find a twin prime, but we will never be sure that we will not arrive at a point after which there are no more. Or there may in fact be an L above which there are no more, but we shall never be able to write L as a sequence of digits. Again, whether these two comprise all possibilities is not a matter capable of absolute resolution.
There can be no proof that L exists, so, like God to the skeptics, it must not exist. By their own standard, this conclusion is not justified, for, by our assumption, there is no evidence in favor of its nonexistence either. Indeed, we may safely believe in L; if a contradiction would arise from its use, then we could leverage that contradiction to provide a proof that there are infinitely many twin primes, thus TPC would have been provable. After centuries of cautious hypothesis of what would happen if L did exist, we may begin to treat L as any other number. As the ancient Greeks’ unease about the existence of irrational numbers has faded, so too would ours. The naturals would become: 1, 2, 3, 4, 5, … L, L+1, …. We will have answered questions about L, for example it is greater than one million, because have found twin primes greater than one million.
This all happens consistently with the proof that the set of natural numbers is made up of only the numbers 1, 2, 3, 4, 5, …, for that proof does not mean what we think it means. We cannot enumerate all the natural numbers in a theorem; that proof only states that the set of natural numbers is the smallest set made up of zero and successors of elements in that set. If we can actually find a twin prime above any number, but merely not know it, then we might claim L cannot be the successor of any element in this set. But this claim is false, because L is clearly the successor of L-1! L, whether or not or ___ it is one of the familiar numbers, manages to sneak its way into the smallest set containing zero and successors. It is not the set of numbers, but the language about numbers that can be extended by this independence of TPC, and L is not logically distinguishable from “regular” numbers. It is a symbolic phenomenon. But so, too, are the familiar numbers. The only difference is we have chosen to say that zero exists.
“I Am Unique,” he shouts in forceful desperation as the Church to Galileo. His thoughts deeper, more profound than those of unquestioning masses, so sayeth the Lord. He sees himself pushing the boundaries of social construction, tangling with sanity, becoming the unimaginable sage he thinks he is creating. Like me, he seeks a life like none other, a life with color and breath, without limits. He believes that he is distinguished by these young thoughts.
I have stolen when I had desires beyond my means. I have lied. I have felt the weight of regret. He steals things he does not desire that are within his means. He would happily lie. He feels little regret in the name of his experiment, which he calls “growth”. He is ashamed of me, and wishes to break free from my quiet, well-meaning shackles.
He gives expectations of him the finger, trying to release his ultimately inward-focused frustration. He screams with my closed mouth. He would have me explain himself if he were ever in a position of true compromise. I can’t help but feel his adolescent influence as I write.
But we are a team. We both want to be heard. When we both have one thing to say, he will be the one with the balls to say it. He’s the one who kisses the girl, he’s the one who sings at open mic, he’s the one who writes music and love letters, who drops my secure life plan to make us happy.
And now, he is asking for the keys to my car.
I suspect those around me think I am losing my mind. They may be right — I mean, if I am right — and in that case I am not qualified to judge the rightness of either of us. Alright, enough of that, I had something I wanted to say.
Self-deconstruction is the phrase I use to describe why I think I think they see what I think they see. I have have been using that phrase being relatively certain it describes what I am doing, but not really knowing what it means. I had a rather vivid experience of it today after a bath, and I would like to give it as an example.
The lights were off, and I had stuffed my pants in front of the crack at the bottom of the door to prevent any light from leaking in. I could only feel whether my eyes were open or closed, sight was not a thing. I usually take a black bath to deeply relax or meditate. This time, it was a little too hot, and I was rather full of caffeine so my body had a natural affinity for being tense.
Squirming around in the tub, I had a little conversation with myself about whether I wanted to get out of the bath. It went something like this:
Self: Self, I would like to get out of the bath now.
Self: But you don’t feel like going to bed yet.
Self: Oh, hmm, you are right. Ok, I’ll stay here.
Self: Hey wait a minute! This is uncomfortable and I’m not enjoying this.
Self: But you still don’t feel like going to bed.
Self: Right… I get that… hmm how to resolve this.
Self: Hey wait a minute! Who said I have to go to bed if I get out?
Self: Well what are you going to do then?
Self: Hmm, I could, ummm… mmm….
Self: Wait, I don’t have to answer that. I’m just going to get out now.
After this argument I had been jolted into a place of self-conversation, just kind of describing why I was doing each thing I did. I unplugged the drain plug “because that’s what I do when I’m getting out of the bath”, and I up I stood, to Self’s disappointment. As the drain blurped the bathwater down, I pawed for the towel I had so consciously placed on top of the toilet, and began to dry myself off, “because I don’t want to get water on the floor, to be courteous.”
I am typically not a terribly courteous person. I’m often lost in my own world, oblivious of the existence of others. But I am aware that not everyone is like that. I get the impression that my roommate is very often thinking of how he is impacting people around him. In a moment of egotistical superiority-confirmation I thought “and some people get consumed by that.”
While I can be awfully egotistical for long periods of time without noticing, criticizing others is one of the things that always jolts my thought process out of its self-serving little world. I almost unconsciously catch myself and try to apply the criticism to myself. Thus:
I am trying not to be consumed by anything.
Nonsense, we are all consumed by something.
You just can’t see what it is.
One could say that being consumed by something, simultaneously being aware and accepting of that, is the definition of “living in alignment with yourself.”
But who said living in alignment with yourself is a desirable goal?
Because it makes you happy.
But who said being happy is a desirable goal?
And usually once I get to happiness the deconstruction stops and the usual motivators, the way I had for relating to the world, for measuring “how I am doing”, all that stuff falls apart and I have this wonderful feeling of blankness. Yes “wonderful”, shut up, I know.
Sometimes it goes into evolutionary ideas: happiness is a desirable goal because it reflects control and comfort, which attracts women because it used to help for raising healthy children. But that starts its own all-too-familiar chain of deconstruction about goals, free will, and the meaning and dynamics of reproduction. It’s similar to the above, and similarly I am eventually left with blankness.
Blankness is an, ehem, goal of meditation, but I seem to stumbling on it by accident over and over. Just little ones though, meditation still has a stronger effect.
But this deconstructive trend is popping up in many aspects of my life. I suspect it makes it very frustrating to argue or discuss with me, because I compulsively deconstruct the terms of the argument until I am not certain the thing we are arguing about has any meaning — or anything has any meaning for that matter. I have no idea what the other person’s experience of this is (aware that others’ ability to have experiences is one of my beliefs).
For now, I’m just basking in the liberation that this mental state is giving me. I seem to believe that it will end sometime and go back to “normal”, but I don’t really have any justification for that. I’ve been intentionally pushing the boundaries of conventional sanity, trying to experience the world in a unique way… and the other night I had the frightening sensation that it might be possible to succeed, losing touch with what others call reality. Now that I am on this path, can I stop? If I hold on to my conventional “realistic” worldview, I simply notice myself holding onto something that is not what it claims to be and deconstruct again.
Kind of cool, kind of scary, kind of … blank.
I have been working hard on my experimental posts recently, and with each post my standards raise alongside my anxiety. So I thought it would be prudent, as a way to unblock myself, to write in my simple style, speaking the simple truth.
I want to be clear with myself: this is my blog. I am not the pictures I paint of myself; my image only limits me. When I am not a Haskell hacker, it makes sense that I would not write about Haskell (99% of my traffic comes from the Haskell community, so perhaps the source of my anxiety is clear). I’m being a kid, naively experimenting with words, fancying myself some kind of artist, as I perceive my readership just wanting to learn more about their craft.
But that is the beautiful thing about this blog. It’s a big jumble of Luke, following along as he explores. As I told an anonymous commenter (who was probably my boss) right after I lost my job:
My blog isn’t an advertisement for my career. It’s here both for people who are interested in my technical ideas and for people who are following my journey to find a place in the world.
So I have to remember that, and by publishing this I hereby forgive myself for not posting anything technical for a while. That’s not where I’m at, and I am not the property of my readers.
Incidentally, I’ve been planning a post exploring the concept of “Computably Uncountable”. No promises though.
Perfection has always been a demotivator for me. I think it’s that way for a lot of people. Once I start holding myself to a standard, my expression is strangled shut. I have to let the shit through before I can see if there is caviar in it. No apologies for imagery. Poop.
Incidentally, if you are one of those people who appreciates this Luke-jumble, you can support my writing like this: . There is also a donate button on that page if you are feeling especially charitable. I really do love writing, and the fuller my reserves the more time I can spend doing it (besides, it’s inspiring to see people enjoying my work).
Arguments are in the business of increasing certainty. They begin with some assumptions, make some claims, support the claims with evidence, and then reach a new conclusion which, if you agree with all the steps, you should now accept as part of your belief system. We are born as blank slates who know nothing, and over the course of a lifetime, we make and read arguments until, by the time we die, we know a great many things with great certainty.
Opportunities for deep, life-changing learning are rare and must be cherished. Therefore, when we come across an argument whose assumptions are agreeable but seems to be heading in a direction contrary to our beliefs, we read with increased interest in hopes of being proved wrong. After all, if the argument is sound and comes to a conclusion that is contradictory to what we know, then, having seized the opportunity, we have discarded some nonsense and become more enlightened.
By this time, you have noticed that this is satire (I should hope!). But what am I making fun of? A simplistic interpretation is that I am lamenting the irrational way people treat arguments, that they need to be more willing to question themselves if they want a belief system founded in truth. Or perhaps they need to be more logically-minded, to prevent themselves from adopting such self-contradictory systems of belief in the first place. It doesn’t really matter what is wrong with people, as long as whatever it is explains why they will not accept my sound logical argument.
Now I am clearly making fun of someone you know. This person has a strong personality and holds as a core belief that most people are stupid. They write or speak passionately, they stay close to the scientific doctrine, they take pride in their certainty. While their arguments are convincing, they lack a certain respect for those who disagree, and it ends up limiting them from a more complete world-view. We all know this person, and, thankfully, acknowledging that we know someone like this releases us from the possibility of being this person.
The author is being subtly disrespectful now. He is trying to make me wonder whether I fall into this category, and in doing so, attempting to put himself above me. And this paragraph is even more disrespectful, taking on the voice of the reader, assuming he can predict his or her thoughts. Fortunately, he has failed, for one couldn’t say the reader was thinking anything beyond reading at the time.
At least I have not said anything that threatens your beliefs. That would merely serve the disengaged brain to produce a disinterested Ctrl-W or a polarized, indignant comment. Before I can convince you of any new truth, I have to convince you to engage with the question. An active mind taking a question seriously will produce a far more convincing effect than any amount of eloquent word-barrage. My goal was this: if the page was still in focus by the time you reached this paragraph, your mind would be curious and primed.
Now for my argument: what could I convince you of?