Monthly Archives: March 2012

Music

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 –

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Constructive Reality

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.

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Beliefs and Truth

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.”

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