You can add up the parts

But you won’t have the sum

You can strike up the march

There is no drum

Every heart, every heart

To love will come

But like a refugee

Ring the bells that still can ring

Forget your perfect offering

There is a crack, a crack in everything

That’s how the light gets in

From “Anthem” by Leonard Cohen.

We know from Gödel’s Theorem that any axiomatic system beyond the complexity of simple addition can only be incomplete so long as it contains no contradictions – incomplete and consistent. Completeness can only be achieved by including falsehoods – contradictions. A system which is complete and inconsistent can prove anything which is one reason contradictions must be vehemently avoided. Better to prove too little than to prove too much. Gödel’s Theorem put the kibosh on Whitehead and Russell’s attempt to prove that mathematics is true, and to find a mathematical system with axioms that did not generate paradoxes. Mathematics is true, but not all of it can be proven to be true. Gödel actually managed to use one of these paradoxes constructively in his proof that incomplete and consistent was the best that could be achieved with moderately complex axiomatic systems.

Gödel’s Theorem proves that mathematics cannot be formalized – meaning it is not possible to turn math into the mere manipulation of symbols following rules that ignore what those symbols mean. That is why computers have not superseded mathematicians, because computers have no understanding of what they do. The truth of Gödelian propositions can be seen and understood by human consciousness, but not purely by reference to rules within the axiomatic system. This hints at some connection between consciousness and the transcendent. Consciousness is necessary precisely because not everything can be reduced to rules; to a system. Consciousness must be free and not algorithmic to function. The halting problem similarly proves that there is no algorithmic method (step by step procedure) for (mindlessly) testing the validity of all algorithms. Understanding must put in an appearance.

In *The Brothers Karamazov,* Dostoevsky has Ivan quote from Voltaire: “If God did not exist, it would be necessary to invent him.” Ivan goes on to wonder that so primitive a creature as man could figure out God’s necessity. It can take quite some thinking however to discover just why we cannot do without God. At one point, Berdyaev describes our conscience as communication with the divine – though it is all too evident that many people’s consciences are deranged; corrupted and perverted by mass movements and ideologies. And then there are the many more whose conscience functions but the person ignores it to avoid the wrath of the deranged and powerful.

It is an odd fact that symbolic logic, mathematicised in the nineteenth century, has to contain things that are not at all logical. Predicate logic contains the axiom that if the consequent of a conditional is true, then the overall statement is to be regarded as true. A conditional is “If p then q.” p is the antecedent. q is the consequent. This makes no sense at all and is utterly ridiculous. In natural language, “if…then” implies some meaningful relationship between p and q. If you are a good boy, then you’ll get an ice cream. If it rains on Saturday, then our picnic will have to be moved indoors. If sodium is added to water, then there will be an explosion. If she smiles at me in that way, then maybe she likes me. In the first instance, the consequent is being offered as a reward. In the second, the conditional is contingency planning. In the third, the conditional indicates a causal link. In the fourth, an inference about someone’s possible feelings is being made. The fact that if p then q covers all these very different scenarios means symbolic logic is too crude to replace natural language. But the fact that the conditional is always regarded as true if the consequent is true violates common sense and seemingly all logic – though it is precisely formalized logic that is producing the absurdity. With regard to the first example, if the boy has an ice cream, no matter how he got it, including stealing it, then we are supposed to agree that “If he is a good boy, then he will get an ice cream” is true. The truth table definition of the conditional is:

p q p → q

T T T

T F F

F T T

F F T

Truth tables cover every single possible truth value assignment to the variables. This rule does not mean that p becomes magically true. As you can see with the truth table, even if p is false, so long as the consequent is true, then the conditional is regarded as true. This rule in logic can be summed up as “if q, then if p then q,” or, q → (p → q). Thus, predicate logic regards the statement “If Jimmy loves Doreess and has never eaten Swiss cheese, drunk a cola, or stared at a cat, then the speed of light is 299,792,458 m/s” as true, while the fact that there is no connection between p and q is treated as irrelevant.

With these examples of fracture and illogic in the midst of logic, it is as though one is pounding out a metal circle to lie flat against the ground but every attempt results in the last little bit smashing into pieces. Or perhaps one is trying to keep the circle flat, and yet the earth is curved, and the circle bends no matter what we do. The nineteenth century novel *Flatland *by Edwin Abbott is an allegory about a square being living in a two-dimensional reality that sees a circle that starts small and gets bigger and bigger. In fact, the circle is a sphere that the square cannot comprehend until the circle lifts the square up so it can perceive three-dimensional reality. The author intended this story as an allegory for our inability to comprehend the transcendent. While the flatland leaders privately acknowledge the reality of the sphere they execute witnesses to prevent this knowledge spreading in the manner of Plato’s Cave. Others, like Ken Wilber, have adopted the phrase “flatland” to describe science’s fixation with surfaces rather than depth and interiors, i.e., consciousness. The novel *We* by Zamyatin is similar in that it is about trying to cram people into a “perfected” reality that is governed by mathematical equations. The last step is to remove imagination so that people can be shoved into their assigned social box and finally be “happy.” The perfect person in the perfect society. This reality must remain incomplete and consistent. Immanence is not sufficient unto itself. Attempting to hammer the perfect conceptual circle just smashes it. The missing piece points to a mystery as we wait, in faith and hope, for the circle to turn into a sphere and raise us up. “The Fire Balloons,” a story in some editions of *The Martian Chronicles *by Ray Bradbury also features spheres. They are Martians who have put themselves through some kind of process, though they have forgotten what, that has left them as largely incorporeal floating blue spheres that reminds a priest of the fire balloons his grandfather used to light on the fourth of July. Two priests have decided to try to save their souls. One of them tries out the idea of the image of a spherical Jesus which he imagines as being appropriate to spherical creatures much to the other priest’s dismay. They bring a pump organ to play them hymns. The spheres try to avoid the two priests but eventually tell them that they have no need of saving. They are immortal, do not need to eat, never get tired, have no sexual feelings, and are incapable of sin of any kind. Suffering not, God is unnecessary for them. Not having overcome any difficulties and not facing the same existential problems as humans, the Fire Balloons have nothing to teach us and certainly not to learn from us. They have no crack opening onto the transcendent.

Despite the halting problem demonstrating that computer programming cannot be left to computers to do, and Gödel’s Theorem showing that math requires human mathematicians and cannot be formalized, some computer scientists and philosophers continue to imagine that they will be able to program computers with consciousness. Theorems are not theories and are thus not susceptible to being later disproved. They are not tentative, but facts about reality. When an algorithm is not valid, the effect is for the program never to terminate; never to halt. Whether this will happen or not cannot be determined algorithmically i.e., unthinkingly. Just how these computer scientists think they can bypass these facts is unknown.

The impossibility of creating consciousness using algorithms is assured. Consciousness remains “the hard problem,” of how a material thing like the brain could generate consciousness. A strong possibility is that the brain does no such thing; see Consciousness: What Is It and Where Can It be Found? If scientists, *per impossible,* somehow succeeded in creating Artificial General Intelligence this would mean the man-made creation of consciousness, yet it is consciousness that cannot be accounted for in materialist metaphysics. It is consciousness that evades determinism and reductionism, and points to spiritual realities. If software engineers succeeded with AGI, the connection between consciousness and religious faith would be shattered, and we would be like Morpheus in the *Matrix* movies saying “I had a dream. But, this dream has been taken from me.” Morpheus emits this existential moan when he learns that, seemingly, Neo is not a savior, but is merely an anomaly programmed into the Matrix to prevent its destruction. And the Matrix too is relevant to our discussion. The “Architect,” the Creator of the Matrix, who comes from the machine world, must program a fracture in what he wanted; which is a hermetically sealed simulated reality. The Architect is effectively the evil god of the Gnostics; the Demiurge who creates physical reality in the Gnostic view. The Architect tells Neo that when he first designed the Matrix, a computer-generated fictional reality designed to keep the populace mentally occupied and roughly content, he made it perfect. The humans, with their bodies in their slime-filled pods rejected this reality and died. So, in the next iteration, the Architect made the simulated reality imperfect. Again, the people would not accept it and died. Finally, an “intuitive program,” the Oracle, stumbled upon the answer: the introduction of choice and free will. Only then would humans accept their existential situation. People were given the choice to accept the simulated reality or to see through the illusion and reject it. 99% of people happily chose the simulation. 1% rebelled, exited their illusion-producing cocoon, and joined the humans living in Zion, the underground city in which flesh and blood humans live: in reality. Neo is a calculated anomaly who can bend the rules of the Matrix to a degree that no one else can, and, with the latest iteration of the anomaly, even the wider reality. When it came time for the Machine World to terminate Zion, Neo, the One, would return to the Source, allowing a temporary dissemination of the code he carries, reinserting the prime program[1] after which he would have to select from the Matrix twenty-three individuals, a combination of males and females, to rebuild and repopulate Zion. Failure to comply, the Architect informs Neo, will result in a cataclysmic systems crash killing everyone connected to the Matrix which, coupled with the termination of Zion, will result in the extinction of the entire human race. Each new “Neo” has a “contingent affirmation that was meant to create a profound attachment to the rest of the species,” meaning a generalized love for mankind. This version of Neo, however, is different because his attachment is far more specific – his love is for one particular woman; Trinity.[2] The Architect wants Neo to choose going to the Source to save Zion, but sees that Neo has chosen Trinity. The architect comments: “Hope; it is the quintessential human delusion, simultaneously the source of your greatest strength and your greatest weakness.” He incorrectly predicts that Neo will be unable to save Trinity and that it is too late.

Faith and hope are made possible by choice. And choice is possible through freedom; that only the *Ungrund*, the causeless cause, can provide. Materialist scientists know that materialism implies determinism. If we humans function adequately, being soulless, they should be able to program a material thing to do the same. They do not know that material reality has a crack in it through which consciousness enters with its intrinsic connection to freedom and thus the transcendent. The downside of choice, creativity, and imagination is that evil and destructive things can be imagined as easily, perhaps more easily, than dynamic order and goodness. A building can be destroyed in seconds that takes years to build. The film *Stalker,* by Tarkovsky, explores this duality of freedom, acknowledges the risk of choosing evil, and votes for freedom regardless because we cease to be human under total compulsion no matter how good the compelled “heaven” is supposed to be.

[1] This part seems like gobbledegook handwaving that resists comprehension.

[2] The Matrix movie universe can be divided into body, mind, and spirit. Zion represents body, the Matrix, the mind, and the Machine World, spirit. The three Matrix movies can be seen as body, mind, and spirit at war with each other, rather than working together in harmony.

In the Venn Diagram, Faith and Consciousness almost entirely overlap one another.

Hi, Tom: I agree. The fact of consciousness implies the necessity of faith.

I have a mundane observation that I think follows from this metaphysics: when societies lose belief in the transcendent, they go insane. QED.

Thanks for reading, Anti-Gnostic: I agree.

Hi, Kristor:

It is true that the logician does not care about truth, just validity. But, the philosopher cares about soundness which is a combination of the two. Philosophers will sometimes take a natural language argument, and symbolize it to test the argument’s validity as part of determining its soundness. Logical operators have rules with corresponding truth tables; at least in predicate logic, which is what I am discussing. According to the rules for the conditional, it is enough for the consequent to be true for the conditional to be considered true. If you were to check your original argument against the symbolized version, and your antecedent were false, symbolic logic tells you to go ahead and regard the conditional as true. The problem is ultimately that logic is hopelessly inadequate for capturing very much about the world. Bertrand Russell, at one point, fantasized about replacing natural language with logic to avoid “vagueness” and ambiguity. That would be a disaster and make social life impossible. Predicate logic gets superseded by quantifier logic but it is still a rough tool that can’t distinguish correlation from causation, for instance. Also, “If she smiles at you, then it may be the case that she loves you” is just as much a conditional as “If you put sodium in water, then it will explode.” “If it rains on Saturday, then we will watch a movie instead.” Any differences literally get lost in translation. The relation if…then does not indicate causation in particular and there is no way to signal that it does.

A truth table for p->q, which establishes the rule for the logical operator ->, is shown below.

p q p->q

T T T

T F F

F T T

F F T

When I said that when the consequent is treated as true, then the overall statement is regarded as true, this does not mean the antecedent magically becomes true. The conditional statement is true. And the conditional is represented as p->q – the whole statement. As you note, I explicitly reject what you say I am saying. Even in English grammar, the if clause is not the conditional. You seem to be confusing the antecedent for the conditional.

I am aware of deduction vs induction. That’s a normal part of philosophy. Although, in practice, many inductive arguments can be phrased deductively so the distinction between the two can be a bit arbitrary rather than indicating some real important difference. In that case, the unprovable and problematic nature of induction just becomes a highly questionable premise in a deductive argument.

In real life, when we say something with the logical structure of “if p then q,” we mean there is some special, meaningful relationship between the two. Once symbolized by variables, it is enough that q is true for the conditional statement to be regarded as true. That does not remotely translate into natural language or capture what is going on there. Everything that is true of real world natural language conditionals is not also true of conditionals when they are expressed in variables.

Hi, Kristor:

I’m pretty sure that we are agreeing. Although, in symbolic logic, if you know q, you also know if p then q, so your assertion that the two are not equivalent is wrong – again, in symbolic logic. But, it’s right in the real world which natural language does a better job of capturing. By denying that if p then q follows from q, you are just recapitulating my objections to one aspect of symbolic logic. In fact, a single rule. You write, “no matter how you parse the situation, it simply does not follow from the fact that Kristor is an Orthospherean that if walruses are carburetors then Kristor is certainly an Orthospherean.” In the real, non-symbolic logic world, absolutely. That’s what I’ve been arguing. I agree that the truth of q by itself should not determine whether p -> q is true and the truth or falsity, and relevance of p matters a great deal. Is there some mistake in that truth table above defining the rules for the conditional?

The conditional can be expressed as not p or q. An example given online:

1st: If it is a bear, then it can swim — T

2nd: If it is a bear, then it can not swim — F

3rd: If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.

4th: If it is not a bear, then it can not swim — T (as above)

I don’t like three! Somehow, our common sense view of the world can’t be captured by predicate logic, hence my image of trying to hammer flat something that just breaks when you try.

The rules for the other logical operators used in predicate logic seem unproblematic and cohere with common sense. With conjunction, p and q, both conjuncts must be true for the total statement to be regarded as true. This permits simplification. If you know p and q are true, then you know p is true and that q is true and you can treat them as separate true statements. In the case of the disjunction, p v q, at least one of the disjuncts must be true for the overall statement to be true. Simplification is not allowed because you don’t know which is true, even if the overall statement is true. But, addition is permitted. If you know p is true, you can add a disjunction without changing the truth value. If p is true, then you also know p v q is true. Sometimes we want to indicate that only one of the disjuncts is to be regarded as true, and there is a separate symbol for that depending on which operators you choose. Negation is straightforward – not p. Double negation, which is used to indicate that something is true in a proof, is a bit trickier, probably, if you speak one of those languages that routinely use double negation to mean negation. In Serbian, which my wife speaks, they use double negation all the time, and even triple negation sometimes. Something like “I didn’t not see nothing.” Of course, in logic, triple negation takes you back to negation. With the biconditional, when p is false, then q must be false. When p is true, then q must be true. When they don’t match, the biconditional statement comes out false. These rules match natural language just great. No reasonable person would blink an eye when informed of them. We seem to be in agreement that the rules governing the conditional, however, don’t make total sense. The truth of q should not determine the truth of the conditional. The other three truth value assignments are unproblematic. It might seem counterintuitive, at first glance, that if p and q are both false then the conditional is true. Personally, I just say to myself “If you have been good, then, you’ll get fruit” when p and q are negated turns into “If you haven’t been good, then, won’t get fruit” to convince myself the logic is fine.

p and q

T T = T

T F = F

F T = F

F F = F

p or q

T T = T

T F = T

F T = T

F F = F

not p

T = F

F = T

p if and only if q

T T = T

T F = F

F T = F

F F = T

Symbolic logic is a very crude tool that can express very little and is of very little use in philosophical argument. Good for computer programs and things like that, though, partly because computers are dumb things with no understanding and everything has to be spelt out for them – though not everything can actually be spelt out, which is why computers cannot replace mathematicians. I have never been tempted to symbolize an argument to check whether it is a good argument or not. The rules of logic are abstracted out of what we actually do, just like the rules of grammar. We don’t follow the rules of grammar. We speak grammatical English (or whatever) if our parents do. Grammar is an abstraction. My wife grew up without articles. I can tell her if a “the” or an “a” is required, without fail, without ever being able to articulate what the rule is. Logic, broadly, conceived, is very useful indeed and consistent with the Logos. But, philosophical argument involves, at crucial moments, intuitions, and spiritual intuitions to do with plausibility and the like, plus insight and self-evidence. The truth of Goedelian propositions and axioms are not proved logically – they are not proved at all – but their truth can be perceived and intuited.

Hi, Kristor – I have got a philosophy professor friend of mine who regularly teaches symbolic logic to review our discussion and this is his response:

“I might say that the problem with the conditional is not that it doesn’t capture causation but that it captures causation and a whole lot else as well. [This I agree with]. If the sun rises in the east then 2 + 2 = 4 is true even though the two statements seem to have nothing to do with each other.

Possible worlds logic does a better job of capturing causation although, of course, it isn’t perfect.

Reading over some of the exchange, it seems that Kristor is wrong. To be clear, we can distinguish between “P –> Q is true” and “P –Q is true in classical logic.” If Kristor is saying that the inference from Q to P –>Q is not valid In classical logic then he is just wrong. The conditional statement “If Q –> (P –> Q)” is a tautology in classical logic. Of course, this doesn’t mean P is true. The copied in statement below of Kristor’s,

But, no matter how you parse the situation, it simply does not follow from the fact that Kristor is an Orthospherean that if walruses are carburetors then Kristor is certainly an Orthospherean.

is just wrong if we’re talking about classical logic.

If q –> (p –> q) is a tautology

I’m sure that we agree on your broader point that no logistical calculus can be adequate to reality. I was worried only about your comments on symbolic logic. Even with respect to them, I am not sure we disagree; more likely, we have been misunderstanding each other.

Reviewing it now, I am chagrined to find that in our discussion so far, I have in my diction – and, so, in my thinking and thus in the arguments I have here so far proposed – myself fallen prey to the confusion of validity with truth which I had begun by picking out. Mea culpa!

The statement p → q is indeed true if q; but it is invalid to conclude from that statement and the fact of q that therefore p, or a fortiori that q is true *on account* of p; for, the fact of q tells us nothing about the facticity of p, or about the relation of p to q. After all, the statement (¬ p → q) is also true if q.

The paradoxical result then is that every conditional statement is true, tautologically, so long as the consequent thereof happens to be true.

It turns out then that p can have nothing whatever to do with q. There is no conditional in fact, but rather only on the page. All we are or can be worried about is q. It turns out then that no inferences from p to q are quite possible. They are not even meaningful. It turns out, i.e., that p → q is vacuous. It is meaningless.

But so then are all its less formalized instances in natural language, and in thought. So then, i.e., is thought per se.

NB: if this is a problem in symbolic logic, then it is not a problem that especially afflicts symbolic logic. It’s much worse than that. It bedevils arguments expressed in natural languages, too, just as much. Indeed, it ruins all language, of any sort, and thus all information, all mentation, and all causation (that’s a hazardous spoor to follow, for it portends the doom ab initio of the Lógos who is the language of language, and thus the logical and so the ontological forecondition of the very hunt). For, formalizations of natural languages are just *extremely careful expressions* of – and, in the final analysis, *within* – those natural languages (variables must vary within some domain if they are to be definite even as variables). And any natural language can be formalized without recourse to algebraic symbols, which after all only stand for expressions of natural language (I hope it goes without saying also that natural languages can be expressed in writing by no other possible medium than what boil down at bottom to algebraic symbols; as e.g., those of hieroglyphs, letters, runes, cunii, or ideograms (a writing in any such sort of symbols, in any language, can in principle be read off in any language whatever, given an index of interpretation)). To wit, your professor friend asserting that from the fact that Kristor is an Orthospherean it *does* follow ineluctably and necessarily, as a matter of logic and thus in all possible worlds – which is to say, tautologically – that if walruses are carburetors then Kristor is an Orthospherean. He did not express that opinion using symbolic logic, but natural English.

Fortunately, walruses are not carburetors in fact, so that my status as an Orthospherean is – in point of mere fact – not a function either of walruses or carburetors, or of their relations. This despite the fact that it is entailed as true from all eternity that if walruses are carburetors, then I am necessarily an Orthospherean.

Anyway: the symbolic system in use is not the issue. This is so, even if we abstract from natural language out to the conceptual systems to which languages refer – i.e., out to thought itself, and per se. And this is of course the most important sequela of Gödel’s Theorems.

Could this crack be related to Original Sin somehow?

Leonard Cohen knew quite a bit about Catholicism and had a lot of respect for it. His Irish Catholic nanny had a huge impression on him. It is my own personal interpretation, but to me his last album is an inconlusive record of his dilemma of whether to die a Jew or a Catholic.

The title-track of the album, ‘The Future’, which contains ‘Anthem’, is a very accurate description of where our socitety stands today.

Thanks, mickvet. I’m a fan of Cohen. It’s amazing how good his last album is. It seems rare that someone can write some of their best stuff just before they die. His songs involving God seem profound. You might be right about dying a Jew or Catholic. I tend to think he was pretty Christian, significantly more so than Jewish.

I read the lyrics to The Future and they seem appropriately grim and thus apropos for the current state of society.

Hi, La’akea:

Yes. I think so. The crack means this world cannot be properly understood without reference to the transcendent: the crack is the light of the divine; and without Original Sin, this world would be itself heavenly and in no need of redemption.

Great explanation, Richard.

Would you agree to any degree with my opinion that Bob Dylan and Leonard Cohen are the two great religious poets of our times (admittedly, the competition is scant)?

Hi, mickvet: I don’t feel qualified to judge because I mostly don’t read poetry.