To every action, give your whole self; I am wholly procrastinating, fully indecisive, completely half-listening. Mr. mindful, awake, clear-headed, be careful, pictures can be projected on the fog. We are all blind, stumbling pigeons, wholeheartedly. The most committed are those who believe they have conquered life — how would you say that a delusional maniac “doesn’t have his heart in it”? Then there are those of us who envy such commitment — to be stuck only wanting a delusion — is that a lesser or greater commitment?
There is a transcendence here (, man). I want to experience my whole self, so I can’t just give up being lost and absent. My self includes my guilt, my self-judgments, my unacceptance of those judgments — no spiritual or psychological change I can make will do justice to my self. Nor will stagnation realize my true potential (a concept that makes the very same error).
But we get trapped again. We can’t stop intending to change because it would not do justice to self; nor can we stop intending to stop intending to change. To lay a path to spiritual betterhood is to believe that you have, in some small way, failed to be a blind, stumbling pigeon. This is false but, as we have already covered, admirable.
There’s nothing new to accept. This line of questioning is wrong. Self-acceptance is a vacuous goal. When that sinks in — when you really believe that — something changes inside of you. It plants the seed of the real self-acceptance, not that fuzzy-wuzzy kind you wanted. You’ll know when you have Achieved real self-acceptance because nothing happens, except maybe you will think and/or feel differently than you would have in some situations (it is unclear what that mechanism is).
Then what? Well of course, self-acceptance is only one step along the path — after which there can be no more — the steps no longer look like steps, but flat step-like objects. But I have to ask again, now what? What do I do now? What is the next .. the next ..
These are the chirpings of an analytical mind with nothing to analyze.
So, it turns out I’m not dead. How about that?
I have dropped out of school, and am busking for a living. It is tiring (especially when I forget to drink enough water), sometimes discouraging (when I play things to no response whatsoever or make $5 in an hour), but mostly great. My job is making music! And more importantly, my job is making my music, or music I am in love with — although certain pieces tend to attract more tippers than others, so it’s not truly free (what is?).
My grandmother contacted me telling me about a startup mixer so I could find a job. I don’t think she really understands my decision. I can understand that — she wants me to get a stable, well-paying job, have kids and a family, and go to church. The usual narrative. The other day I was idly contemplating being a father. Not now, of course. But I can see the draw; I can see that being a pretty special thing. The question is whether it is worth it to me. Sacrifice is part of love. But do I sacrifice for my child, or do I sacrifice a child (umm! — sacrifice having a child) for my other loves? That is not a question I am remotely prepared to answer.
I used to think — perhaps I still do — that big questions like those aren’t really worth answering, at least not rationally. I suppose this “used to” is fairly recent, as I had spent a long time on them prior, and they led me nowhere but in circles of unfulfilled dustkicking. My self-image can be so limited at times, and the rational mind is a slave to its images. What I can really do, what I’m really made of, I perhaps thought, won’t be small enough to be so easily decided — it must be eased into, made part of myself through exploration and long, gradual growth. But the liberation I feel from this new occupation of mine has shown me that perhaps at points along this process such a life decision is valuable, that it can be a beacon that reminds me that I chose this because it was important to me — more important than anything else at one time — and so gives me something to hold onto in times of uncertainty or suffering. It sounds very compelling, doesn’t it? But I am still in the honeymoon phase of my relationship with my life as a musician, so the only thing I can be sure of is that my thoughts about it are distorted.
And am I really good enough to make this a living? Maybe Boulder is the only place people appreciate public performances of amateur classical music. Maybe when I migrate for the winter I will be met with indifference or contempt, and I will be stuck in a new city with no job. Maybe when I improvise or play my originals people only tip me because I have brought the piano out, not because the music speaks to them in any deep way — I know that is not true, my second piano sonata is almost always met by applause, but it has been 10 years since I wrote that; do I still have it? A teenager passes by and plays most of the pieces I do — not as well, but not badly — and he will surpass me by my age. Will I ever have the guts to sing out there?
A thousand fears and doubts dance their rite around my dream — all I can do is to go out there every day and hope it goes well. I think it’s proof that I’m alive. I pose this question to myself: would I rather be wildly successful in a software company, or wildly successful as a musician? The latter, by any metric. “Wildly” need not even appear. Standing on a plank and singing to the jury, my heart beating a thousand times a minute, with the conviction of a soldier — this outshines any vision of a successful software idea.
I’m not leaving software. But my most exciting software ideas aren’t the kinds of things one can easily make a living on. I’m working on a browser-based programming environment which explores a new way of designing and organizing code. I don’t want to say too much about it because as I code the idea continues to develop in my mind, and I don’t want to nail it down yet (maybe ever). But anyway, to make money with that would sacrifice its beauty; this tool is not for productivity, at least not at first: it is exploring a way of thinking. It is easier to make a living making the music I love than the software I love. If my life is to overflow with love and happiness, music is the breadwinner.
Again — only a month in. But I think this is the way to do it, for me. I’m not setting myself up for a comfortable life, but comfort is a trap anyway. It is the contrast that feels so good, and without that contrast comfort is just normal. Without discomfort to prepare the contrast, comfort is dull and boring. Anyway, that’s how I see it. Funny coming from a hedonist like me. I guess I’m having a stint of long-term hedonism at the expense of short-term.
Maybe someday I won’t even feel the need to justify my choice anymore. That’s when I’ll really be in it.
It is part of growing up, I keep telling myself — doing what I know — for some definition of know — is right, despite the advice of my family and almost everyone (but my best friend who is my only beacon in this whole mess). I have a good family — supportive, have my best interest in mind, certainly not the image of the disapproving father so pervasive — and partially I haven’t been completely honest with them, because it’s scary. Nonetheless, I feel a lot of pressure from their attempting-to-be-neutral positions, and I know what I want — what I need to do, but when the time comes to say it I can’t, condemning myself to this purgatory.
I’m not going to finish college. I am very close, only a few credits away, but it is not going to happen at the end of this semester, and everyone is like “but it’s just one more and it’s important for the future” — not so different from my reasoning for returning to college in the first place — I have been at this decision point before, and did convince myself with the assistance of my family that it was the right thing to do. Maybe it was once, and although I did not achieve the goals I set for it, it isn’t right anymore.
Here’s the really hard part, and I have to speak this with less certainty than the other, because different parts of my mind and body are fighting over it. I don’t think I’m going to finish this semester. Try as I might (whatever that means) I cannot commit myself to something that I don’t truly believe is serving me, and right now that is school. I don’t have that kind of control over myself. My grades are really slipping; each moment here feels like trying to run in a dream, suspended in the air. I know, what’s another month? It really doesn’t matter either way. It would matter if I wanted to go to grad school, but years of getting to know myself and being friends with grad students, I don’t think it is the place for me. I am too disorganized, my intellectual exploration is founded in too much curiosity and not enough desire to contribute. Suddenly a pursuit will become uninteresting and another will take me by surprise, but you can’t just switch like that in school.
But you can just switch like that in life. Why would I arbitrarily obligate myself to someone else when I am exploring what I love? Out there in the cruel, forgiving, free world, I can pursue whatever I like whenever and however I like to. Yes, I need to make money, but that’s not such a huge deal. I don’t really get why people make their way of earning money the centerpiece of their lives. Insert canonical white-picket-fence rant.
I don’t have a good phrase to describe who I want to be or what I’m going for. I think of such phrases as potentially guiding, locally, but ultimately limiting. To define myself with words is to forget every moment the words do not account for — when would someone include the Pepsi they had for breakfast in their self-definition? — but that bottle of salt and sugar is part of me, negative or however you want to judge it. Of course, not having such a phrase makes it difficult to assess the value of a difficult easy decision like this, and without a mechanism for assessing value I have no choice but to be human and follow my hearts — there’s nothing else that I can say with my vocabulary that doesn’t sound like a waste of my life.
I have long valued every moment of my time. A year of my life spent unhappy in order to support the remainder of my life never seemed worthwhile to me — I know that sounds irrational — but that seems to be the way I relate to time. A month spent in school, a month not making my living by sharing my musical heart, a month depressed and careless, a month of missed opportunity.
And yet, it is only a month. But why would I stay? I can’t articulate any convincing reason. It will make it less work should I ever decide to come back and finish — but that is actually false. One class is just as many as four, if not more.
I have had my struggles, but at important times I have always listened to the guidance of my family, I think I have always made what they saw as the best choice. This time, I think, their poor choice is the right one — if only symbolically, if only to remind myself whose life I am living.
I define a model to be a simulation of external reality within a mind. It is an approximation, a system by which we can make some predictions that are somewhat accurate most of the time. Some confuse their models with reality itself — since I use God to inform my morality, there must truly be a God; since quantum mechanics makes probabilistic predictions, the universe must be fundamentally non-deterministic. These kinds of judgments fail to realize that True external reality is not accessible to us.
I am speaking from the perspective of a model in which there exists a True reality for models to approximate. As I have defined, the reality being approximated is not accessible, so what could I be referring to by “True external reality?” I don’t refer to the True external reality, but an approximation within my model. And the same goes for my model itself — I cannot refer directly to my model, which is a pattern of True reality that occurs in my mind, but only to an approximation of my model from within itself.
This is the problem I have with metacognitition. I have spent a great deal of time introspecting, trying to figure out what I think, what I believe, why I do the things I do. But I cannot access the True answers to those questions (are questions a part of True reality?), I can only answer them from the perspective of my model of myself. A little less than a year ago, a Bodhisattva Sirened me in to catalyze an understanding that my self-model is unsound, that I had ideas about myself that were incorrect, that I had memories which may not have actually occurred, that I had a fabrication mechanism which was creating reasons for my actions after I had already done them, or already committed myself to doing them. I lost my trust in my metacognition, and from there,
What is True? We can be like Descartes and try to deduce a sound foundation from almost nothing, but that is just model-play, desperately trying to construct a model which is reality. This is in vain, there is no perfect representation. Every word in this post echoes falsity and lack; I can’t say “there is no perfect representation” with any certainty. Logical argument is a model, relying on the framework of propositional knowledge — humanity invented propositions — biology invented truth.
Of what I know or think I know or think I cannot know, I exist regardless. This proposition is provable and refutable, and the proof and refutation are both devoid of any True meaning. Can I think myself into oblivion? or is it just that my mental structures complicate themselves until my mental structures are really complicated? Experience, not thought, is the foundation. Thought is model, language is model, thought about experience is model — but experience: that is True.
I cannot say, recall, or think with certainty (certainty itself is a property of propositions). But I experience with certainty, if you will allow the metaphor. I am not trying to communicate a truth or a Truth — this is very important — but a feeling. Can you feel this intermittent feeling of mine, this freeing, relaxing, empty feeling which the conscious mind resists fervorously? It occurs discretely, not as a lasting experience, but like the sound of a clap the instant it reaches my ear; there is no meaning yet, that comes later. The only way I know I have this feeling is through my memory, and like every truth it is not to be trusted. I now have a feeling that if I could make a continuous clap, it would accompany a continuous darkness in my mind. You could hardly tell the difference in me, and I would be too occupied with noticing it that I would not be able to report or even remember doing so. Perhaps I achieve it for great lengths of time already –
Can you believe I strive for this?! I seek my own inability to remember — should I achieve my goal, I will be on my deathbed before it is tomorrow. Life could be lying in her bed preoccupied by the necessity to one day leave it — or it could be a dream and orgasm. One is a long life in which future disappointment is love; the other is a whole river, reaching the sea the same moment it melts from the snow.
Love do not care. The mind will tire of obsessing on the contents of the black hole, but the heart will still beat to a rhythm. This final sentence arouses its beating, because I know, in every model, that I have a True love –
I would like to address a statement by Paul Snively during a Twitter conversation.
The notion that math has meaning apart from being computable is perverse.
I admit that all involved in the discussion are computer scientists, and so we would be predisposed toward constructivism and, in particular, this kind of value system. Indeed, I consider myself “a constructivist” — in the right setting, I will fiercely argue with a classicist that “there exists” does not mean what they think it means — but I will not go this far.
The underlying idea of constructivism is that the form of the theorem describes some evidence that must be used to support it. Evidence for (P or Q) needs either evidence for P or evidence for Q; evidence for (exists x such that P(x)) needs a concrete value (e.g. a number) x and evidence for P(x) for that x; evidence for (P implies Q) is a function that maps evidence for P into evidence for Q; and so on. The two theorems (1) “given any integer, there is a prime number larger than it” and (2) “there are not finitely many prime numbers” are in fact different statements. The evidence for (1) must be a computable function which maps integers to prime numbers which are larger; the evidence for (2) is a function which takes evidence that there are finitely many prime numbers (essentially an exhaustive list) and produces a contradiction. (2) is the form of Euclid’s famous proof, but it is not as strong as (1), which gives a computable process that generates primes. Idiomatically we would call (1) constructive and (2) non-constructive, but the finer distinction is that constructive mathematics distinguishes these two statements while classical mathematics considers them equivalent.
In practice, this means that you cannot use proof by contradiction, the identity that “¬∀x. P(x)” implies “∃x. ¬P(x)”, or the axiom of choice (which claims the existence of a function without giving a way to compute it). The necessary evidence can be extracted from proofs constructed using the remaining axioms.
If you alter the laws of evidence, you can recover the law of excluded middle (proof by contradiction), which says that for any proposition P, (P or not P) is true. Classical mathematicians consider it true that “there are either infinitely many or finitely many twin primes”. Constructively, however, this says if you have a proof of this statement, then you either have a proof of P or a proof of (not P). At the time of writing, this is not true of whether there are infinitely many twin primes; we do not yet have a proof either way. But if you allow into your language of evidence the ability to invoke continuations, then we do have such evidence: the one we have is a proof of (not P), which is a function that takes evidence for P and produces a contradiction. So you pass this function evidence for P because you need the contradiction, but instead of giving you the contradiction you wanted it goes back in time to change its mind, now saying that the evidence for (P or not P) is the evidence for P (which it was just given). Yes, it’s ridiculous, but could be considered constructive if you have a different slant on the meaning of evidence.
But don’t be so hasty in using this ridiculous interpretation against the law of excluded middle. The Ackermann function — a function which grows extremely fast — is constructively definable. However, A(4,4) is far, far greater than the number of elementary particles in the known universe. Using the numeral for A(4,4) as evidence is physically impossible, and yet it is considered valid constructive evidence. This puts constructivism on less sound scientific footing: a constructive theorem need not have actual evidence, it need only have evidence in principle. But what principle? How can one justify that A(4,4) is more real than a well-ordering of the real numbers? — we can give concrete evidence for neither. The proof that A(4,4) is a well-defined number relies on abstract reasoning founded in the same logical ideas that gave rise to the law of excluded middle.
This style of argument is associated with ultrafinitism — the idea that even very large finite numbers may not exist (pay attention to the word may — the idea that a finite number does not exist is intentionally outside the realm of ultrafinitism’s ability to answer). Classical mathematics says there exist arbitrary choice functions, constructive mathematics says those may not exist but A(4,4) does, ultrafinitism says that A(4,4) (and sometimes even numbers as small as 2100) may not exist. These distinctions seem all to be rooted in a sort of fight over which Platonic abstract concepts exist. Perhaps some, such as my friend Paul, would say “are meaningful” instead, but it’s the same idea. It is not as if only one of these philosophies has information to extract. Ultrafinite arguments construct observable evidence, constructive arguments construct idealized evidence, classical arguments discuss idealized existence. If you were to rephrase a classical existence argument “there exists a non-recursive set of integers” to “not every set of integers is recursive” then it becomes constructively valid (it is a diagonal argument). In fact, every sentence provable in classical logic has a corresponding sentence provable in constructive logic, by a simple syntactic transformation. I find it, then, irrational to consider that the former be meaningless and the latter meaningful. So we are merely arguing over the semantics of the word “exists” (and in general, “or”, “implies”, etc. as well). We are arguing about what the sentences mean, not whether they are meaningful. Classical existence is different than constructive existence, and neither corresponds to physical existence.
Paul says, “you can’t actually cut a sphere up and reassemble it into two spheres identical in volume to the first.” I respond by saying that you can’t actually separate an arbitrarily small piece of a box either (first of all, a box made of what?), which constructively is allowed. Mathematics is about mentally idealized objects — if we can “in principle” separate an arbitrary piece of a box, then we can also “in principle” cut a sphere up and reassemble it into two spheres of identical volume, they are merely operating by different principles. Fortunately, we can examine the proofs of these theorems to find out by which principles they are operating. But if you are going to bless one “in principle” as meaningful and banish the others — which I beg you not to — I can see no other way than to resort to the physical reality we are given, to ultrafinitism.
I have now met the fourth person who has said that they don’t have beliefs.
Perhaps I am still stuck in a naive conception of truth that they have transcended. I still unconsciously assign beliefs to be axioms, as assumed truths upon which to base my inferences, and as such not having beliefs would seem impossible. Perhaps they have already achieved what I merely strive for: just living, just being the little perceptrons they are, already embodying the consequences of truth as a linguistic construction and not a fact of the world. They know that whether an idea is true is irrelevant — that there is nothing more than successful ideas being successful — and as such to “believe in” any truth is only to be enslaved by a clever, self-reinforcing idea: that ideas can be true.
This transcendence must have been achieved after many years of thought and meditation — we are perhaps even born clinging to truth as though it were unitary and absolute. Wars have been fought over is and is not, as if ignoring the evidence shining in their swords, both could not coexist. We have a deep genetic drive, because the uncertainty introduced in realizing the paradox of accessible truths is enough to delay a life-saving decision by a few milliseconds, and thus has been bred out of us. The option that there is a representational barrier between your perceptions and the world is not an option for the animal at the edge of survival. But perhaps there is a latent genetic drive toward the non-believer’s enlightened state after all — once you stop worrying about what is true, you can react faster, having closed the analytical gap between cause and effect. You are a wild animal, your thoughts having proregressed into instincts. Indeed, when time is of the essence, this idea could be more successful than the idea of truth — perhaps their meditation was to put themselves in life-threatening situations in which they needed to be lightningfast to survive.
They see the intimate connection between the words “belief” and “truth”. An idea must be able to be true in order to be believed. But they do not reject these words, for an idea must be able to be false to be rejected. The collusion of “belief” and “truth” makes them very hard to break out of: each reinforces the other. When it comes time to communicate, the non-believers see that language is built around truth, and one cannot communicate without presupposing it. So for them to communicate that they are not where you think they are, they must use a sentence which by its very utterance contradicts itself: “I do not have beliefs.”
Lately I have been considering myself a relativist. To cast away the kneejerks, I don’t consider all belief systems equally valid (with caveats1). Wikipedia sums it up nicely:
… that truth is always relative to some particular frame of reference, such as a language or a culture.
I have noticed an increase in my opposition to what I am currently calling “scientific realism” — the belief that discoveries made by science are true, and other things are false (basically just an incarnation of absolutism). Yesterday I had an impassioned argument (still in good fun, though) with my roommate about our differences in perception. I noticed my emotions firing up around this subject, a symptom begging me to analyze its cause. Humans get very emotional when their thoughts approach a shattering of a core belief, so I am curious if one is near.
This time, instead of a philosophical persuasive essay, I’m just going to write down some of my observations.
In the conversation with my roommate Monty (who I consider quite intelligent), mostly a battle over semantics, I found the following ensemble of his ideas to leave an impression on me:
- Newtonian gravity is false, and General Relativity is true.
- If he lived 200 years ago, Newtonian physics would be true.
- One thing cannot be more true than another (except in the trivial case of one thing being true and the other false, of course).
- General Relativity and The Standard Model, which are mathematically incompatible, can both be true at the same time.
- He hasn’t yet seen any evidence that would suggest there are things that can’t eventually be explained by our current scientific ideas.
Taken together, these ideas are fascinating to me. They indicate a different definition of truth than the one I use, and I’m fascinated because I don’t have a concept that I could substitute for it. On surface interpretation, these statements seem inconsistent to me, so I am really curious about the concept from which they arise. (I am pretty sure (5) is just a fallacy though: what would such evidence look like?)
I have met others who claim that they do not have beliefs. I find this to be common among scientific realists. I wonder what definition of “belief” they use to be able to consider themselves devoid of it; so far when I have pried I am just evaded. There are two reasons I evade inquiries: (1) I am not taking the conversation seriously, which may be because it is threatening my beliefs, or other reasons; and (2) the inquiries are using words in ways that don’t have meaning to me, so I answer in riddles that bring out the dissonance2. I usually assume they are doing it because their beliefs are being threatened3; what makes me curious is the possibility that they are evading because of (2)4. Perhaps I am using “belief” incorrectly when asking that question.
Among Skeptics, there is another possible reason to avoid the word “belief”: because it is very close to “faith”, the buzzword of the enemy. Maybe they use the word “truth” to mean what I call “belief”… but then the idea that someone’s beliefs can be false would be nonsense.
I think most of my anti-realism comes from a desire to (at least give due diligence to) respect the belief systems of others. I think I may start considering “true” to be a value judgement (which, as an experiment, I am trying to avoid). I had a debate with a young earth creationist, a belief system I typically have a hard time respecting. After a long time, I think I heard an essential difference, when he said (paraphrasing): “I believe there is a God because I don’t want to live in a world without a God.” That indicates to me a different relationship to truth — that truth and desirability are related concepts — and opened to me the possibility of respecting his belief system a little more.
Dan Piponi made a brilliant comment on twitter during a conversation about realism: “I don’t think ‘reality’ means much. It’s just a placeholder to make the sentence grammatical.”
1 What exactly does a belief system being “valid” mean?
2 This will happen, for example, if you ask me whether I believe extra-terrestrial life exists, because I get hung up on the definition of “life”. People seem to acknowledge the subtlety of that word, but then keep using the word anyway as if the inability to define it is no big thing: “you know what I mean.” No, I actually don’t.
3 Probably because it confirms my superiority.
4 Possibly because it threatens my superiority.
I have heard the term “The Church of Reason” to refer to our modern disposition toward rationality and science. Some thinkers are upset by this analogy, claiming that rationality is fundamentally distinct from a religion. In some ways this is true: for instance, rationality does not entrust a single institution or treatise with control of its truth (though some sects — I mean branches — come very close to a blind trust of scientific consensus). However, I sometimes get the distinct impression of a further belief, however never explicitly stated, that logic and science are not just the latest way, but the way to discover truth.
A succinct criticism from within the logical discipline describes my thoughts well. I quote:
If I see a coin come up heads twenty times in a row, I’m going to use the power of induction to predict that the coin is biased towards heads. Induction tells me that, the more something has happened in the past, it’s more likely to continue to do so in the future. I trust induction because induction has worked for me before.
Somewhere out there in mind-space is someone who believes in anti-induction: each coin flip of heads convinces him that the coin is biased toward tails. Anti-induction tells him that, the more something has happened in the past, the less likely it is to do so in the future. If asked why he trusts anti-induction, he exclaims: “Because it’s never worked before!”
This delightful morsel is so much more than an idle curiosity to me. Please do not mistake me for taking the surface interpretation: I do not claim that induction and anti-induction are equally valuable. But the anti-induction hypothetical illuminates, in an entertaining way, that belief in induction is circular. Observe that our unwavering trust in logic rests upon induction.
In this modern age it is sometimes easy to forget that there was a time when most of humanity was deeply religious. Humans of every intellectual prowess saw “God did it” as a sound explanation (allow me to assume omnipotent monotheism for the sake of argument). Some theorized about how God thought, what he looked like (whether that was a legitimate question), what would appease him, what actions would cause him to create rain or not. Instead of conjuring thoughts of mockery, I would like the reader to put him or herself into one of those minds. You are not stupid; you are deeply immersed in a cultural belief system. It rains — you think back upon the actions of your town recently to try to determine why it must have done so; determining this is of the utmost importance. You may even engage in scientific practices, coming up with hypotheses and testing them: if I sing to one, but not both, of my children at night, the probability that God will be pleased is increased. But this science is based upon a faulty foundation: a whole host of different phenomena could be attributed to “God will be pleased”, and the method is not scientific by modern standards. It is still superstition. What I am putting forth is that the very process of modern science and reasoning may be considered superstition — or perhaps some yet-uninvented term to describe our primitive thinking — to the cultures of the future. Maybe, like the character above, what we are doing is analogous to the search for truth, but we’re missing the point.
But we can make predictions! I will grant that we can make better predictions than traditional religious belief systems used to. I am no scholar of religion, but I can at least imagine a tribe understanding that the fire spirit, who loves the taste of dry wood, will duplicate himself to any nearby dry wood. This makes a prediction as well (at the time of this understanding, it had not yet been observed that he would duplicate himself from Honto’s wood to Jumara’s wood). Nowadays we have only a more accurate idea of the spirits, and we call them by silly names like Boson and Gluon. (I would like to stress that we cannot yet predict anything perfectly. E. T. Jaynes argues that the stunningly accurate probabilistic results of quantum electrodynamics do not count as perfection; i.e. that interpreting the quantifiable uncertainty of its predictions as fundamental to nature rather than to the theory is a boneheaded arrogance.)
Speaking of quantum theory, in the last century we have come across physical laws with an unsettling interpretation problem. Quantum systems are defined in terms of measurement amplitudes, and measurement occurs when a quantum system interacts with a classical system. Of course, if quantum theory wishes to be foundational, the term “classical system” must refer to a mathematical interpretation of a system, not a specific, real system, for every system ought to be a quantum system. So now we are talking about the point of measurement being one interpretation interacting with another — we are speaking on the mathematical and the physical level at the same time. Philosophically, this is utter nonsense. A dominant viewpoint among physicists is that of instrumentalism, summarized by Feynman as “shut up and calculate”. In other words: our logical and intuitive explanations fail us, but the mathematics work out. We have stumbled upon a stunningly accurate mathematical theory with fuzzy, unintelligible edges; could this not indicate an impedance mismatch between our logic and reality? Electrons do not obey classical physics, though large ensembles of them converge on classical physics. Why should we assume nature obeys classical logic; perhaps only large ensembles of truths converge on classical logic? Indeed, the calculation structure of quantum amplitudes seems to be logic-esque, with rules at least for conjunction and disjunction. Maybe the barrier lies not in the transition to a classical system, but the transition to classical logic. Perhaps, if we could only think differently, there would be no barrier.
In order to be heard, I am arguing from a position that we just have the laws of logic slightly wrong, and that a successor would take the same form merely with different laws. I do not necessarily believe this — my inner mathematician wishes it, for it would be comfortable and familiar — but it is simply the most concrete way, the smallest step I can take, to cast doubt upon the logical absolute.
You and I are immersed in a culture of reason, just as many generations of humans before us were immersed in a culture of theism. I cannot simply show you an alternative way to see the world; I am as clouded by these conceptions as anyone of our time. I do not wish to replace your foundation, just erode it. I wish to illuminate the possibility that we may, still, be looking at clouds, and not at the stars.
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.