There's a book which has been out for 2 years by Richard Dawkins called The God Delusion. It is, as its name implies, an attack on God. It is, in short, a polemic. I feel that I should read it for the sake of intellectual honesty, if nothing else. But the more I read about it, and the more short excerpts I read, and the more I hear/read from Dawkins, the less I care. And here's why.
Dawkins quite clearly rejects out of hand the possibility that any other world view than his is right. When I say "any other worldview" I don't mean the possibility of God-- that's what he's arguing against-- but any philosophical starting point besides logical positivism. He is a strict materialist, and a logical positivist, and everything else is flatly not worth mentioning.
He makes, I won't doubt (at least until I read his book... if I read it) a convincing argument that a l.p. framework implies that God is a god of the gaps, and that those gaps will quickly disappear, and God with them. But by doing this and only this, he sidesteps the issue completely. He misses the epistemology, which is the most important part in deciding the possibility of God. You cannot tell someone they have come to a false (moreover delusional) conclusion when you refuse to discuss the framework by which they establish truth. The utter arrogance and ignorance of trying to tell me I'm delusional without even so much as mentioning my epistemological groundings makes it impossible for me to take anything Dawkins says seriously.
He (sort of) rebuts such arguments in the preface with "Do you have to read up on leprechology before disbelieving in leprechauns?" This is a fantastic point, but few people are asking him to go through a detailed study of the theology of Aquinas, Luther, Calvin, Lewis and Chesterton to argue that God does not exist. What we are asking is for him to discuss their epistemological basis, rather than a priori tossing them aside. To quote Terry Eagleton "What, one wonders, are Dawkins’s views on the epistemological differences between Aquinas and Duns Scotus? Has he read Eriugena on subjectivity, Rahner on grace or Moltmann on hope? Has he even heard of them?" The question is not about theology, but about philosophy.
Let me expand upon that point; I don't care what Dawkins has to say about scriptural justifiability of The City of God, Fear and Trembling or Mere Christianity. I care about what he as to say about the way Augustine, Kierkegaard and Lewis establish Truth. What do they consider to be evidence of the truth of a statement? What grounds do they use to justify the existence and character of God? Why is this wrong?
Moving past his naive dismissal of several thousand years of epistemology, his caricature of religion shows how little he has even attempted to understand the religious mind. There is far too much to say on this topic, and I could hardly do it justice, so I will limit myself to a one sentence summary of his feelings on the morality of religion: He finds it to be dangerous and evil. If religion is defined to be "irrational, self-righteous, hate-filled zeal", I agree wholeheartedly. But in his religious (in that sense of the word) intolerance he refuses to accept that the source of such religion can be anything besides religion as it is normally defined and he also refuses to accept that religion (as it is normally defined) can create anything apart from such closed-mindedness.
By so vehemently rejecting everything that isn't is worldview, he is making the most absurd straw-man of himself. He claims that atheism will lead to a more peaceful society overall, but he proves himself, through his self-righteous intolerance, to be a poor example of this "truth". So, I'm not going to read The God Delusion unless someone convinces me that his argument does discuss the epistemological "failings" of a religious view, and that he doesn't caricature religion as I claim...
****
My first point reminds me of a quick exchange that was made a while ago... I tend to be bad at "arguing" points, so this is my esprit d'escalier. A friend of mine (who is also very logical positivist, and can't understand other worldviews-- although he tries, at least) made a comment about science solving some sort of subtle and difficult to determine system. Another friend responded with "that's assuming the scientific method is infallible, which it's not." to which the first friend responded "If it were possible to get machines, which cannot make the same mistakes we do, to do it, it would be." Before a rebuttal could be made (or at least before one was made) we got distracted with something else.
Besides the "you're assuming a logical positive, blah, blah..." I realized that the truth of this statement requires that the universe be objectively observable from within the universe-- otherwise there will always be irreconcilable error in any measurement, hence, science is limited. Given our current understanding of the universe, objective observability is absolutely impossible: "Einstein says so." Any observation can only be made with respect to a given object, and at a high enough resolution, everything about that observation is different for any other object.
Further, if we accept a logical positivist view on the universe, and hence, a Fregean view on math, it makes sense that the universe is governed by some mathematical system. I'm getting to a point I've made before, but.. If the universe is governed by a mathematical system, then we must accept Tarski's undefinability theorem and Gödel's incompleteness theorems. In other words, there are limits to how much we can mechanically discover about the universe. Hence, science has limits.
Tuesday, December 16, 2008
Monday, November 24, 2008
Developing in "impractical" languages
So... functional languages, in the case of ones I know, Scheme and Haskell, are often called "impractical" for real applications. Due to the small libraries, this is mostly true of scheme, but it is only because scheme libraries are small... It is not true of CommonLisp, or Haskell, or OCaml, or ...
The most common argument is that they run slower. This falls flat for OCaml, which outpaces basically everything (that means you, imperative language) except C; also, CL is one of the faster languages around. But everywhere I look in relation to Haskell has some sort of argument about it being "slow and impractical".
Let's look at the cost ($$) of developing and running a piece of software written in C vs one in Haskell. Also let's assume that it is computationally expensive (uses a lot of CPU power).
Now, everywhere I've looked, suggests developing in functional languages is at least 2 times (actually closer to 3 or 4 times...) as fast as developing in their imperative counterparts. But! We'll give C the benefit of the doubt, and say that developing in Haskell takes 2/3t (e.g. if a C program takes 3 months to develop, it will take 2 months to develop it in Haskell).
Now, we'll continue with a conservative estimate of $40000 a year for a programmer. For a job which requires a BS in CS and "several years" of experience... this is a modest (and, quite frankly, bad) pay... but programmers love their jobs, right?
Now, a decent (but not huge) project is going to take, say, 9 months for 5 programmers to write in C. So 6 months in Haskell. This means the development cost of the C program is 5*40000*(3/4) = $150000. The Haskell program? 5*40000*(1/2) = $100000. Obviously, since we're looking at 2/3 the cost.
Now, we've just saved ourselves $50000. That's more than the salary for a programmer for a year!
But! We've got to take into account the extra amount of computation required by the Haskell program. Since the differences are constant (not asymptotic), and Haskell is ~3 times more "expensive" than C, we're looking at just getting a better computer here... We'll say one that's "twice as good" (whatever that means), which isn't quite enough, but we've been giving C the benefit of the doubt this whole time...
So, getting a computer that's "twice" as good is going to cost the average person less than $1000 extra-- $150 for RAM, $150 for a CPU and $200-$300 for video card: less than $600. Let's say $750.
So, in order for the imperatave program to be more efficient, all around, economically, there need to be roughly 50000/750 = 75 copies...
Hmm... that's not too much, now, is it?
Giving programmers more money (hurray!) increases this number some, but not all that much... However, there are a few things to note:
(1)In reality, every study/anecdote suggests about 2x speed with functional development. This means we have ourselves $100000, so 150 copies... still small...
(2)Giving programmers $50000 a year (on average... this includes project managers, etc) means an extra $25000 (given (1), above)
(3)The cost of hiring good programmers is not decreasing very fast... the cost of upgrading computers is. So in 5 years, this whole thing is going to be stupid.
(4)Most substantial programs take closer to 10-20 people 1 to 2 years (in C). Given (1) and (2), that's at least 400000 (taking 15 for 1.5 years, that's 1.125Mil). In Haskell? For 15 people it will take .75 years, so that's 562500. We've saved just under one million. So we can now sell almost 1000 copies... still not much.
(5)This is Haskell vs C. In reality, development is done in a combination of C and C++. OCaml performs only moderately slower than C (and faster than C++), and CL is only slightly slower than C++. So, working in OCaml will save about the same amount of time, and won't cost much (any) more to run; developing in Lisp will only cost a tiny bit more to run, and won't be much (if any) more than OCaml... especially with Macros, and CLOS.
(6)There are very, very few applications where performance matters that much... Rendering is just about the only common task. So the "It's too slow" argument is dumb anyway.
My point is: People see that Functional Languages are experimental (theoretical) hotbeds, and say "Oh, they must only be good for language theory"... This is not the case. They are, mostly, only a tiny bit slower to run, and are significantly faster to develop in...
Also, pugs, and darcs
Edit: So, I got my orders mixed up... (Actually, I was looking at an incomplete ranking when I said OCaml and Lisp are fast). The best way to look at things is: The Computer Language Shootout! Look at rankings... play around with what does what... etc.
The most common argument is that they run slower. This falls flat for OCaml, which outpaces basically everything (that means you, imperative language) except C; also, CL is one of the faster languages around. But everywhere I look in relation to Haskell has some sort of argument about it being "slow and impractical".
Let's look at the cost ($$) of developing and running a piece of software written in C vs one in Haskell. Also let's assume that it is computationally expensive (uses a lot of CPU power).
Now, everywhere I've looked, suggests developing in functional languages is at least 2 times (actually closer to 3 or 4 times...) as fast as developing in their imperative counterparts. But! We'll give C the benefit of the doubt, and say that developing in Haskell takes 2/3t (e.g. if a C program takes 3 months to develop, it will take 2 months to develop it in Haskell).
Now, we'll continue with a conservative estimate of $40000 a year for a programmer. For a job which requires a BS in CS and "several years" of experience... this is a modest (and, quite frankly, bad) pay... but programmers love their jobs, right?
Now, a decent (but not huge) project is going to take, say, 9 months for 5 programmers to write in C. So 6 months in Haskell. This means the development cost of the C program is 5*40000*(3/4) = $150000. The Haskell program? 5*40000*(1/2) = $100000. Obviously, since we're looking at 2/3 the cost.
Now, we've just saved ourselves $50000. That's more than the salary for a programmer for a year!
But! We've got to take into account the extra amount of computation required by the Haskell program. Since the differences are constant (not asymptotic), and Haskell is ~3 times more "expensive" than C, we're looking at just getting a better computer here... We'll say one that's "twice as good" (whatever that means), which isn't quite enough, but we've been giving C the benefit of the doubt this whole time...
So, getting a computer that's "twice" as good is going to cost the average person less than $1000 extra-- $150 for RAM, $150 for a CPU and $200-$300 for video card: less than $600. Let's say $750.
So, in order for the imperatave program to be more efficient, all around, economically, there need to be roughly 50000/750 = 75 copies...
Hmm... that's not too much, now, is it?
Giving programmers more money (hurray!) increases this number some, but not all that much... However, there are a few things to note:
(1)In reality, every study/anecdote suggests about 2x speed with functional development. This means we have ourselves $100000, so 150 copies... still small...
(2)Giving programmers $50000 a year (on average... this includes project managers, etc) means an extra $25000 (given (1), above)
(3)The cost of hiring good programmers is not decreasing very fast... the cost of upgrading computers is. So in 5 years, this whole thing is going to be stupid.
(4)Most substantial programs take closer to 10-20 people 1 to 2 years (in C). Given (1) and (2), that's at least 400000 (taking 15 for 1.5 years, that's 1.125Mil). In Haskell? For 15 people it will take .75 years, so that's 562500. We've saved just under one million. So we can now sell almost 1000 copies... still not much.
(5)This is Haskell vs C. In reality, development is done in a combination of C and C++. OCaml performs only moderately slower than C (and faster than C++), and CL is only slightly slower than C++. So, working in OCaml will save about the same amount of time, and won't cost much (any) more to run; developing in Lisp will only cost a tiny bit more to run, and won't be much (if any) more than OCaml... especially with Macros, and CLOS.
(6)There are very, very few applications where performance matters that much... Rendering is just about the only common task. So the "It's too slow" argument is dumb anyway.
My point is: People see that Functional Languages are experimental (theoretical) hotbeds, and say "Oh, they must only be good for language theory"... This is not the case. They are, mostly, only a tiny bit slower to run, and are significantly faster to develop in...
Also, pugs, and darcs
Edit: So, I got my orders mixed up... (Actually, I was looking at an incomplete ranking when I said OCaml and Lisp are fast). The best way to look at things is: The Computer Language Shootout! Look at rankings... play around with what does what... etc.
Sunday, November 2, 2008
Euler and Haskell
I just started learning Haskell, which is a fun, and slightly ridiculous language-- it seems to be what happens when you let mathematicians design a programming language without supervision: It's too clever and uses way too much graduate level theory. Any language which makes frequent use of monads, functors and has a wikibook describing its relation to category theory is the result of an evil genius (or several, to be precise).
Anyway, I'm using Project Euler to learn it, at Anne's polite not-actually-a- suggestion. Which is to say, she brought up Project Euler, and I said "Oh! I can learn Haskell!" It's working rather well, and I recommend it.
Anyway, I'm using Project Euler to learn it, at Anne's polite not-actually-a- suggestion. Which is to say, she brought up Project Euler, and I said "Oh! I can learn Haskell!" It's working rather well, and I recommend it.
Thursday, October 23, 2008
Balaam
Balaam is a figure (apparently a prophet, or oracle) in Numbers (22-25) who was summoned by the Moabites to curse the Israelites as they went from Egypt, through Moab to the promised land. He did nothing of the sort, but still did not do as God commanded him to, and in the end, went against Israel. He is most famous for having a donkey which laid down, and then spoke when beaten. "The way" (2 Pet 2:15), "The error" (Jude v11), and "The teachings" (Rev 2:14) of Balaam are all condemned by New Testament authors.
The teachings of Balaam, which come last in Numbers (and aren't explicitly shown to be associated with him until Chapter 31), quite clearly represent an attempt to subvert Israel by intermingling them with Moabites and Moab culture. There isn't much mystery in them, so I will skip them.
The way of Balaam, often confused or associated with the error of Balaam, is typically taken to be greed. The Oxford NIV Scofield Study Bible has, I think, a mostly correct analysis of the error of Balaam: "The error of Balaam was that he could see only the natural morality. A holy God, he reasoned, must curse such a people as Israel. Like all false teachers he was ignorant of the higher morality of vicarious atonement, by which God could be just and yet the justifier of believing sinners." There is, I think more to this story, but it is details, and not incredibly important, so I'll move on.
The way of Balaam, as I said before, is typically taken to be Greed. This comes as no surprise, since both 2 Peter and Jude mention money (Peter says "wages"; Jude, "profit") when characterizing Balaam's problem. I think this is wrong. Firstly, profit and wages are used throughout the New Testament as metaphor for just about everything-- I guess people thought about money as much in the first century Mediterranean world as they do now.
Beyond that, Balaam quite clearly does not go for money. Instead, I think, he is afraid of conflict... afraid to tell people things they don't want to hear. He is too "nice" to go against them. The following are all quotations from Numbers (NIV); I'll try to edit out only what isn't important, but all quotes are "out of context"...
"The elders of Moab and Midian left, taking with them the fee for divination. When they came to Balaam, they told him what Balak [King of Moab] had said.
"'Spend the night here' Balaam said to them, 'and I will bring you back the answer the Lord gives me.'
[...]
"The next morning, Balaam got up and said [...] 'The Lord has refused to let me go with you.'"
Notice that he does not immediately go with them, thought they represent the king of Moab. He instead says he cannot go.
"Then Balak sent other princes, more numerous and more distinguished than the first. They came to Balaam and said:
'This is what Balak son of Zippor says: Do not let anything keep you from coming to me, because I will reward you handsomely and do whatever you say. Come and put a curse on these people for me.'
"But Balaam answered them, 'Even if Balak gave me his palace filled with silver and gold, I could not do anything great or small to go beyond the command of the Lord my God.'"
Again, he does not go for wealth or power.
Later, three times he is asked to curse Israel, by the king Balak himself, and 3 times, he listens to what God says, and tells Balak. Eventually, he is sent on his merry way without gold or jewels.
However, at every point in the story, Balaam is as polite as possible, going as far as he can to satisfy the people who are asking him to do something that God Himself told him he could not do... He is afraid to tell them what they do not want to hear...
insert(Balaam.concluding_paragraph()); //I don't have anything else to say here...
The teachings of Balaam, which come last in Numbers (and aren't explicitly shown to be associated with him until Chapter 31), quite clearly represent an attempt to subvert Israel by intermingling them with Moabites and Moab culture. There isn't much mystery in them, so I will skip them.
The way of Balaam, often confused or associated with the error of Balaam, is typically taken to be greed. The Oxford NIV Scofield Study Bible has, I think, a mostly correct analysis of the error of Balaam: "The error of Balaam was that he could see only the natural morality. A holy God, he reasoned, must curse such a people as Israel. Like all false teachers he was ignorant of the higher morality of vicarious atonement, by which God could be just and yet the justifier of believing sinners." There is, I think more to this story, but it is details, and not incredibly important, so I'll move on.
The way of Balaam, as I said before, is typically taken to be Greed. This comes as no surprise, since both 2 Peter and Jude mention money (Peter says "wages"; Jude, "profit") when characterizing Balaam's problem. I think this is wrong. Firstly, profit and wages are used throughout the New Testament as metaphor for just about everything-- I guess people thought about money as much in the first century Mediterranean world as they do now.
Beyond that, Balaam quite clearly does not go for money. Instead, I think, he is afraid of conflict... afraid to tell people things they don't want to hear. He is too "nice" to go against them. The following are all quotations from Numbers (NIV); I'll try to edit out only what isn't important, but all quotes are "out of context"...
"The elders of Moab and Midian left, taking with them the fee for divination. When they came to Balaam, they told him what Balak [King of Moab] had said.
"'Spend the night here' Balaam said to them, 'and I will bring you back the answer the Lord gives me.'
[...]
"The next morning, Balaam got up and said [...] 'The Lord has refused to let me go with you.'"
Notice that he does not immediately go with them, thought they represent the king of Moab. He instead says he cannot go.
"Then Balak sent other princes, more numerous and more distinguished than the first. They came to Balaam and said:
'This is what Balak son of Zippor says: Do not let anything keep you from coming to me, because I will reward you handsomely and do whatever you say. Come and put a curse on these people for me.'
"But Balaam answered them, 'Even if Balak gave me his palace filled with silver and gold, I could not do anything great or small to go beyond the command of the Lord my God.'"
Again, he does not go for wealth or power.
Later, three times he is asked to curse Israel, by the king Balak himself, and 3 times, he listens to what God says, and tells Balak. Eventually, he is sent on his merry way without gold or jewels.
However, at every point in the story, Balaam is as polite as possible, going as far as he can to satisfy the people who are asking him to do something that God Himself told him he could not do... He is afraid to tell them what they do not want to hear...
insert(Balaam.concluding_paragraph()); //I don't have anything else to say here...
Thursday, October 16, 2008
God of Judgment?
A friend of mine recently posted something about his uncle's response to him being gay. It included "I guess that’s why you decided you really didn’t believe in God. Made it convenient to act in ways He condemns.”
this reflects a sentiment which is rather common in modern "Christianity": that the Christian God is a God of condemnation. A cursory reading of the Bible might lead one to suspect this is so, but a little more depth shows something quite different.
I'll start with the "curses" God pronounces. And he pronounces several. But all of them amount to this "The consequence [not punishment] of this action is as follows. Choose wisely." God pronounces several curses before the choice is made (e.g. Genesis 2:16-17, 1 Samuel 8:11-18), and a few after the fact (as in Genesis 3:6-19).
The first curse, in Genesis deserves particular attention. First he says "If you eat of this fruit, you will surely die." After they eat the fruit (and don't die), he says "All these bad things will happen to you."
But what about death? There are two things: They did die... later. Would they have died otherwise? The passage (Genesis 3:20-22) suggest otherwise. Also there is spiritual death, which we as humans, most certainly do experience.
So, why would God condemn us to this, simply for wanting a fruit which made us "wiser"? There is a lot happening in this curse, but it boils down to 2 things:
Firstly, it is not "I sentence you to this" but "this is the result of your action." God wants us to choose to love Him. Choice requires that we can choose not to love Him. Since choosing "not God" is the same as choosing sin (by definition). Choice means sin can exist. The existence of sin implies the existence of evil. So God enabled the existence of evil. His statement in Genesis 3 is that we have actualized that opportunity.
The second thing is that he wanted Adam and Eve to be happy (because he loved them). To choose to eat of that fruit meant to choose not to trust God; to not trust God is to reject his love. To be unloved is to be unhappy, and so choosing wisdom over love is to choose unhappiness.
Beyond that, God pronounces other curses, which contain similar sentiments: "Love me (or come to me, or repent or ...) or this bad thing will happen." They represent the same idea: This is a result that follows from your actions.
There is more to say on this topic, but it really doesn't matter. It comes down to this: God did not create us to condemn us; He did not send His Son to condemn us. He created us, and He sent His Son to love us. "Love is patient, love is kind. It does not envy, it does not boast, it is not proud. It is not rude, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. Love does not delight in evil but rejoices with the truth. It always protects, always trusts, always hopes, always perseveres." (1 Cor 4-7)
Christianity is missing a lot of things these days, the most important one is Love.
this reflects a sentiment which is rather common in modern "Christianity": that the Christian God is a God of condemnation. A cursory reading of the Bible might lead one to suspect this is so, but a little more depth shows something quite different.
I'll start with the "curses" God pronounces. And he pronounces several. But all of them amount to this "The consequence [not punishment] of this action is as follows. Choose wisely." God pronounces several curses before the choice is made (e.g. Genesis 2:16-17, 1 Samuel 8:11-18), and a few after the fact (as in Genesis 3:6-19).
The first curse, in Genesis deserves particular attention. First he says "If you eat of this fruit, you will surely die." After they eat the fruit (and don't die), he says "All these bad things will happen to you."
But what about death? There are two things: They did die... later. Would they have died otherwise? The passage (Genesis 3:20-22) suggest otherwise. Also there is spiritual death, which we as humans, most certainly do experience.
So, why would God condemn us to this, simply for wanting a fruit which made us "wiser"? There is a lot happening in this curse, but it boils down to 2 things:
Firstly, it is not "I sentence you to this" but "this is the result of your action." God wants us to choose to love Him. Choice requires that we can choose not to love Him. Since choosing "not God" is the same as choosing sin (by definition). Choice means sin can exist. The existence of sin implies the existence of evil. So God enabled the existence of evil. His statement in Genesis 3 is that we have actualized that opportunity.
The second thing is that he wanted Adam and Eve to be happy (because he loved them). To choose to eat of that fruit meant to choose not to trust God; to not trust God is to reject his love. To be unloved is to be unhappy, and so choosing wisdom over love is to choose unhappiness.
Beyond that, God pronounces other curses, which contain similar sentiments: "Love me (or come to me, or repent or ...) or this bad thing will happen." They represent the same idea: This is a result that follows from your actions.
There is more to say on this topic, but it really doesn't matter. It comes down to this: God did not create us to condemn us; He did not send His Son to condemn us. He created us, and He sent His Son to love us. "Love is patient, love is kind. It does not envy, it does not boast, it is not proud. It is not rude, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. Love does not delight in evil but rejoices with the truth. It always protects, always trusts, always hopes, always perseveres." (1 Cor 4-7)
Christianity is missing a lot of things these days, the most important one is Love.
Friday, October 3, 2008
Mystery...
Do we, as a society, understand the notion of mystery? We explain everything away, so that even that which is "mysterious" to us isn't really a mystery, isn't something mystifying and awe inspiring: its something to coldly study...
The lack of mystery is among the many things that we are missing. Mystery drives us, it fills us with passion; it moves us.
Almost no one feels that mystery. Maybe in the past not many people did... maybe it's only those with above average intellect who actually feel mystery and are inspired by it, and in the past they were the only ones who wrote, so the apparent lack of mystery in our lives is just the result of biased data.
But I think even most of those who "explore" the universe don't feel mystery in their daily lives. I think we have a habit of just accepting the world and moving on...
I think that's a problem.
The lack of mystery is among the many things that we are missing. Mystery drives us, it fills us with passion; it moves us.
Almost no one feels that mystery. Maybe in the past not many people did... maybe it's only those with above average intellect who actually feel mystery and are inspired by it, and in the past they were the only ones who wrote, so the apparent lack of mystery in our lives is just the result of biased data.
But I think even most of those who "explore" the universe don't feel mystery in their daily lives. I think we have a habit of just accepting the world and moving on...
I think that's a problem.
Sunday, August 10, 2008
haha... Good times
Today's sinfest (10 Aug, 08) contains a brief retelling of the Gospel... In comedic form... with a terminator parody.
On evolutionary models...
I was thinking about evolution on the way home this morning. It bugs me, because Darwin really didn't say anything at all. There's nothing profound, nothing insightful about it. It is a general model for change. "The object most suitable to it's current environment will be the one that survives best, and small changes accumulate (somehow), such that new objects will have traits which suit the new environment better, thus prospering." The object can be an animal, a theory, or anything else; the environment can be any type of environment; the changes can form and be accumulated based on any set of rules.
In essence, it says things change, and certain things change "better" than others.
This is why there is social darwinism, memetics, Popper's Knowledge evolution, etc. I realize that rather than falsifying evolution it provides some circumstantial evidence, but that's not the point. The point is: it doesn't say anything. What we now call social darwinism actually predates the theory of evolution-- in fact, Darwin made an argument that can roughly be summarized as "We all know this sort of change happens in the economic realm, why not here? It's really the same sort of evolution."
You can apply the idea to literally anything that changes. Philosophy, science, marketing, pop culture, art, etc, etc-- Figure out the (intellectual/cultural/economic/political) environment, and you can see why the "victors" are the things that grew.
Is it easier to believe that the universe precisely follows simple, elegant and human discoverable laws, or that we impose simple and elegant structures on an unstructured (or weakly structured... or, dare I say, inelegantly structured) universe? What happens if the universe is simply, but inelegantly structured? If the different pieces sort of "klunk" together, rather than flowing smoothly? Can we really, actually, imagine our universe working that way? Or working according to rules that humans can't discover?
We do it because we need to impose structure on our universe in order to understand it. We cannot cope with-- cannot advance in-- a world without order. So we impose structures on our universe. Obviously, the elegant ones are the easiest to deal with. And this is where memetics/knowledge evolution come from: we learn to use more advanced structures to model the phenomena we are seeing, thus allowing us to account for more of the intricacies.
It's the smart ones who can impose structures on anything that go crazy and search for codes in the newspaper. The idea that there are a few simple laws in the background of all of the workings of the universe isn't that different from the idea that there are a few people in the background of all the workings of man, is it? What's the difference between Templar conspiracies and the search for the TOE? Humans are fickle, while universal laws can't decide-- they can only act?
***
On another note, I'm finally creating a "science" tag. I'll retroactively place posts in it at some point.
In essence, it says things change, and certain things change "better" than others.
This is why there is social darwinism, memetics, Popper's Knowledge evolution, etc. I realize that rather than falsifying evolution it provides some circumstantial evidence, but that's not the point. The point is: it doesn't say anything. What we now call social darwinism actually predates the theory of evolution-- in fact, Darwin made an argument that can roughly be summarized as "We all know this sort of change happens in the economic realm, why not here? It's really the same sort of evolution."
You can apply the idea to literally anything that changes. Philosophy, science, marketing, pop culture, art, etc, etc-- Figure out the (intellectual/cultural/economic/political) environment, and you can see why the "victors" are the things that grew.
Is it easier to believe that the universe precisely follows simple, elegant and human discoverable laws, or that we impose simple and elegant structures on an unstructured (or weakly structured... or, dare I say, inelegantly structured) universe? What happens if the universe is simply, but inelegantly structured? If the different pieces sort of "klunk" together, rather than flowing smoothly? Can we really, actually, imagine our universe working that way? Or working according to rules that humans can't discover?
We do it because we need to impose structure on our universe in order to understand it. We cannot cope with-- cannot advance in-- a world without order. So we impose structures on our universe. Obviously, the elegant ones are the easiest to deal with. And this is where memetics/knowledge evolution come from: we learn to use more advanced structures to model the phenomena we are seeing, thus allowing us to account for more of the intricacies.
It's the smart ones who can impose structures on anything that go crazy and search for codes in the newspaper. The idea that there are a few simple laws in the background of all of the workings of the universe isn't that different from the idea that there are a few people in the background of all the workings of man, is it? What's the difference between Templar conspiracies and the search for the TOE? Humans are fickle, while universal laws can't decide-- they can only act?
***
On another note, I'm finally creating a "science" tag. I'll retroactively place posts in it at some point.
Tuesday, August 5, 2008
My Servant Whom I love...
Isaiah 42:1-4:
Here is my servant, whom I uphold,
my chose one in whom I delight;
I will put my spirit on him and he will bring justice to the nations.
He will not shout or cry out, or raise his voice in the streets.
A bruised reed he will not break and a smoldering wick he will not snuff out.
In faithfulness he will bring forth justice;
he will not falter or be discouraged
till he establishes justice on earth.
In his law the islands will put their hope.
***
The note in my Bible says:
There is a twofold account of the coming Servant:
He is represented (1) as weak, despised, rejected, slain; and also (2) as a mighty conqueror, taking vengeance on the nations and restoring Israel. The former class of passages relate to the first advent and are fulfilled; the latter, to the second advent and are unfulfilled.
The chapter goes on with words similar to Isaiah 61: "to open eyes that are blind, to free captives from prison and to release from the dungeon those who sit in darkness."
I'm not sure if I agree with the note in Bible, about the second description being unfulfilled. I think that the idea that the "Coming Kingdom" is outside of our hands allows a lot of laziness within Christians. While that interpretation is not wrong, and is certainly not unBiblical, it does miss something. As with anything, especially the Scripture, there are multiple levels to anything-- multiple interpretations-- all of which must be taken into account. I am a firm believer that the nature of God has not and will not change; So Our Lord must have been a conqueror from the start. We must, as his followers be diligent, and conquer. But not, as those loyal to earthly kings, in a physical way. The battles we fight are not for land or lives or power. Nor are they intellectual battles, whose purpose is to spread knowledge of truth. They are battles of the Spirit, whose purpose is to spread the experience of Love.
Perhaps the note is correct, and that those things are unffulfilled, but if that is the case, it is only because we, as His Kingdom, have not fulfilled the prophesy.
Here is my servant, whom I uphold,
my chose one in whom I delight;
I will put my spirit on him and he will bring justice to the nations.
He will not shout or cry out, or raise his voice in the streets.
A bruised reed he will not break and a smoldering wick he will not snuff out.
In faithfulness he will bring forth justice;
he will not falter or be discouraged
till he establishes justice on earth.
In his law the islands will put their hope.
***
The note in my Bible says:
There is a twofold account of the coming Servant:
He is represented (1) as weak, despised, rejected, slain; and also (2) as a mighty conqueror, taking vengeance on the nations and restoring Israel. The former class of passages relate to the first advent and are fulfilled; the latter, to the second advent and are unfulfilled.
The chapter goes on with words similar to Isaiah 61: "to open eyes that are blind, to free captives from prison and to release from the dungeon those who sit in darkness."
I'm not sure if I agree with the note in Bible, about the second description being unfulfilled. I think that the idea that the "Coming Kingdom" is outside of our hands allows a lot of laziness within Christians. While that interpretation is not wrong, and is certainly not unBiblical, it does miss something. As with anything, especially the Scripture, there are multiple levels to anything-- multiple interpretations-- all of which must be taken into account. I am a firm believer that the nature of God has not and will not change; So Our Lord must have been a conqueror from the start. We must, as his followers be diligent, and conquer. But not, as those loyal to earthly kings, in a physical way. The battles we fight are not for land or lives or power. Nor are they intellectual battles, whose purpose is to spread knowledge of truth. They are battles of the Spirit, whose purpose is to spread the experience of Love.
Perhaps the note is correct, and that those things are unffulfilled, but if that is the case, it is only because we, as His Kingdom, have not fulfilled the prophesy.
Tuesday, July 29, 2008
On Lambda Calculus and Ordinal Numbers (Part 2)
So we're going to take a little detour and forget about functions for a moment. Instead, we're going to talk about ordinals. First, we need to talk about ordered sets really quick.
An ordered set is a set with order. I know...
So, a set is any collection of objects within the universe of discourse (i.e. if we're talking about numbers, we can't have a set with plates; but if we're talking about dishes we can't have a set with numbers). Typically, order doesn't matter, so the set {1,2,3} is the same as the set {2,3,1}. Also, the number of times an element shows up doesn't matter, so {1,1,1,2,3} is the same as {1,2,3} (is the same as {2,3,1}).
In an ordered set, the order does matter. Some relation applies to each set of two differentelements. I don't use the word pair, because in a mathematical pair, (a,b) is not the same as (b,a). Either a < b or b< a.
To be more for formal. Define a relation < on a set S. For any distinct members a,b of S, either (a,b) is in < , or (b,a) is in < .
An ordered set is well-ordered if the set, and every subset (ordered by the same relation) of it has a first element. For example the integers (including negative integers) ordered according to the normal definition of "< " are not well-ordered, because it does not have a first element. Neither are the natural numbers ordered backwards (i.e. {...,3,2,1,0}).
But the natural numbers ordered normally are, since any subset will have a first element.
An ordinal number is a "number" which designates a well-ordered set. (Well, any well-ordered set that satisfies the same properties).
Anyway, avoiding all the definitions and proofs, when you add two ordinals, attach the second set to the end of the first set. and "rename" the elements to avoid repetition. Notice that this is not commutative-- a+b is not necessarily the same as b+a.
Also, before you ask, numbers are defined as the set of all numbers that are before it in the usual ordering of the naturals. 0={} (empty set), 1={0} = {{}}, 2={0,1}, etc.
For example 1+ω (ω meaning infinity) is {0,0,1,2,...}. Renaming all of the elements after the first one gives us {0,1,2,...} = ω
However, +1 gives is {0,1,2,3,...,1}. The last element can't be relabeled to the "next element of ω" because ω has no last element. So we make 1 into 1' (or something) so we have {0,1,...,1'}, which is still infinite but has a first and a last element, so is not ω.
[note: from now on the relabeling of identical elements will be assumed, so ω+1={0,1,2,...,1} will be acceptable)
Now, multiplication works similarly. In a*b, a is appended to itself b times. E.g.
2*4 = {0,1,0,1,0,1,0,1} (with proper relabeling).
Again, 2*ω = {0,1,0,1,.....} Again, we get 2*ω = ω
But, ω*2 = {0,1,2,...,0,1,2,}. This is ω+ω which is not the same as ω, because we have two disjoint maximal subsets with no last element.
That's really all we need to know about ordinals: addition and multiplication are non-commutative, and for any finite a, a*ω = a+ω = ω, and ω*a = ω+ω+...+ω, (a times).
Now that that is out of the way, we can look back at lambda calculus.
The first thing to notice is that everything has a successor and a predecessor (There's a function for that, but it's messy and we don't need to see it.) And if a comes before b, and b comes before c, then a comes before c. Also, the predecessor of 0 is o-- which is to say, 0 is the first element.
What I'm trying to say is that each of the natural numbers in LC represents an ordinal number.
I said earlier that Y might act like an ordinal number. Namely, ω:
ω=Y=λf.(λx.f(xx)) λx.f(xx)
Since I haven't actually described how this works.Yf takes a function and returns f(Yf). That is, it returns itself with an extra f out front; where f is the function it was given. This also means that ff(Yf) -> f...ff(Yf). Hurray! An infinite number of f's.
We need to see if this is actually a valid encoding of ω. So, we need to make sure all of the following hold:
1+ω=ω, 2*ω=ω, ω+1 is distinct from ω, and ω*2=ω+ω.
Let's look back at our definition.
Addition:
plus m n := λm.λn.λf.λa. m f (n f a)
Multiplication:
times m n := λm.λn.λf. m (n f)
1+Y:
-> λf.λa. 1 f (Y f a) -> λf.λa. f (Y f a). There are a few rules of LC that allow us to throw out the a. I'm not sure how to describe them easily. It has to do with free and bound variables... I'll let you figure that out.
Anyway,
-> λf.f(Yf) which is what Yf "reduces" to. So 1+Y holds as expected
Y+1:
λf.λa. Y f (1 f a) -> λf. λa. Y f fa -> λf.λa.f (Yf) fa
The fa at the end don't go in the (Yf) parenthesis, because Y doesn't use them.
So what we have is equivalent to
λf. (Yf) f.
The a can be thrown out, but not the f, since it's the same as the argument to Y. This means, we have an extra f at the end. I.e. Y is not Y+1.
2*Y:
λf. Y (2 f) -> λf. (Y (ff))
This is our own major problem. I'm not entirely sure what to do with this, and it isn't going to reduce to (Yf) unless I'm very much mistaken.
Y*2:
λf. 2 (Yf). -> λf.(Yf)(Yf)
Y+Y = λf.λa. Y f (Y f a). Again, this becomes:
λf. Yf (Yf), rearranging our notation:
λf. (Yf)(Yf)
So we have a tricky 2*Y. Does that mean Y isn't ω? Probably. Does that mean Y isn't infinite? No. Especially since 2*Y is still going to end up infinite.
I doesn't matter anyway, since Delaney was talking about addition, and this clearly holds for Y=ω.
Future things:
Figure out how to do 2*Y. Figure out how to encode sets and cardinals in LC.
An ordered set is a set with order. I know...
So, a set is any collection of objects within the universe of discourse (i.e. if we're talking about numbers, we can't have a set with plates; but if we're talking about dishes we can't have a set with numbers). Typically, order doesn't matter, so the set {1,2,3} is the same as the set {2,3,1}. Also, the number of times an element shows up doesn't matter, so {1,1,1,2,3} is the same as {1,2,3} (is the same as {2,3,1}).
In an ordered set, the order does matter. Some relation applies to each set of two differentelements. I don't use the word pair, because in a mathematical pair, (a,b) is not the same as (b,a). Either a < b or b< a.
To be more for formal. Define a relation < on a set S. For any distinct members a,b of S, either (a,b) is in < , or (b,a) is in < .
An ordered set is well-ordered if the set, and every subset (ordered by the same relation) of it has a first element. For example the integers (including negative integers) ordered according to the normal definition of "< " are not well-ordered, because it does not have a first element. Neither are the natural numbers ordered backwards (i.e. {...,3,2,1,0}).
But the natural numbers ordered normally are, since any subset will have a first element.
An ordinal number is a "number" which designates a well-ordered set. (Well, any well-ordered set that satisfies the same properties).
Anyway, avoiding all the definitions and proofs, when you add two ordinals, attach the second set to the end of the first set. and "rename" the elements to avoid repetition. Notice that this is not commutative-- a+b is not necessarily the same as b+a.
Also, before you ask, numbers are defined as the set of all numbers that are before it in the usual ordering of the naturals. 0={} (empty set), 1={0} = {{}}, 2={0,1}, etc.
For example 1+ω (ω meaning infinity) is {0,0,1,2,...}. Renaming all of the elements after the first one gives us {0,1,2,...} = ω
However, +1 gives is {0,1,2,3,...,1}. The last element can't be relabeled to the "next element of ω" because ω has no last element. So we make 1 into 1' (or something) so we have {0,1,...,1'}, which is still infinite but has a first and a last element, so is not ω.
[note: from now on the relabeling of identical elements will be assumed, so ω+1={0,1,2,...,1} will be acceptable)
Now, multiplication works similarly. In a*b, a is appended to itself b times. E.g.
2*4 = {0,1,0,1,0,1,0,1} (with proper relabeling).
Again, 2*ω = {0,1,0,1,.....} Again, we get 2*ω = ω
But, ω*2 = {0,1,2,...,0,1,2,}. This is ω+ω which is not the same as ω, because we have two disjoint maximal subsets with no last element.
That's really all we need to know about ordinals: addition and multiplication are non-commutative, and for any finite a, a*ω = a+ω = ω, and ω*a = ω+ω+...+ω, (a times).
Now that that is out of the way, we can look back at lambda calculus.
The first thing to notice is that everything has a successor and a predecessor (There's a function for that, but it's messy and we don't need to see it.) And if a comes before b, and b comes before c, then a comes before c. Also, the predecessor of 0 is o-- which is to say, 0 is the first element.
What I'm trying to say is that each of the natural numbers in LC represents an ordinal number.
I said earlier that Y might act like an ordinal number. Namely, ω:
ω=Y=λf.(λx.f(xx)) λx.f(xx)
Since I haven't actually described how this works.Yf takes a function and returns f(Yf). That is, it returns itself with an extra f out front; where f is the function it was given. This also means that ff(Yf) -> f...ff(Yf). Hurray! An infinite number of f's.
We need to see if this is actually a valid encoding of ω. So, we need to make sure all of the following hold:
1+ω=ω, 2*ω=ω, ω+1 is distinct from ω, and ω*2=ω+ω.
Let's look back at our definition.
Addition:
plus m n := λm.λn.λf.λa. m f (n f a)
Multiplication:
times m n := λm.λn.λf. m (n f)
1+Y:
-> λf.λa. 1 f (Y f a) -> λf.λa. f (Y f a). There are a few rules of LC that allow us to throw out the a. I'm not sure how to describe them easily. It has to do with free and bound variables... I'll let you figure that out.
Anyway,
-> λf.f(Yf) which is what Yf "reduces" to. So 1+Y holds as expected
Y+1:
λf.λa. Y f (1 f a) -> λf. λa. Y f fa -> λf.λa.f (Yf) fa
The fa at the end don't go in the (Yf) parenthesis, because Y doesn't use them.
So what we have is equivalent to
λf. (Yf) f.
The a can be thrown out, but not the f, since it's the same as the argument to Y. This means, we have an extra f at the end. I.e. Y is not Y+1.
2*Y:
λf. Y (2 f) -> λf. (Y (ff))
This is our own major problem. I'm not entirely sure what to do with this, and it isn't going to reduce to (Yf) unless I'm very much mistaken.
Y*2:
λf. 2 (Yf). -> λf.(Yf)(Yf)
Y+Y = λf.λa. Y f (Y f a). Again, this becomes:
λf. Yf (Yf), rearranging our notation:
λf. (Yf)(Yf)
So we have a tricky 2*Y. Does that mean Y isn't ω? Probably. Does that mean Y isn't infinite? No. Especially since 2*Y is still going to end up infinite.
I doesn't matter anyway, since Delaney was talking about addition, and this clearly holds for Y=ω.
Future things:
Figure out how to do 2*Y. Figure out how to encode sets and cardinals in LC.
Sunday, July 27, 2008
On Lambda Calculus and Ordinal Numbers (part 1)
I'm still annoyed at the guy I posted about on Friday. Mostly, I'm annoyed about how he accuses mathematicians of being contrary to critical thought, and of being authoritarian. Those two things don't work in math-- you can preach something as dogma, but if you don't have proof (based on certain clearly stated axioms, and valid lines of reasoning), your words ring hollow.
The other thing I'm annoyed about is his callous rejection of transfinite theory. He seems to think Hilbert's Hotel was supposed to be an intuitive "proof" that infinities work the way they do, rather than a clarification of how they work. Also, he rejects infinities... absolutely and completely. As a formalist, I disagree with the rejection of anything interesting, but something else is crying out in pain at that idea-- perhaps I'm more of a Platonist than I think.
Anyway, I'd like to clarify my comment on the fixed point combinator (Also known as the Y-combinator), because... mostly because I'm a nerd. But also because I think it acts like an infinite ordinal.
I'm going to write this in two parts. The first will be on Lambda Calculus, the second on Ordinals and how to apply LC to ordinals. Hopefully, I can also discuss cardinals, but that may be a bit of a stretch.
First, a bit on (untyped) Lambda Calculus and Church Encoding: Lambda Calculus is a formalism for dealing with functions. For those who have heard of set theory, think of it as an analog, only everything is a function (or an argument of a function), rather than a set (or element of a set). The basics are as follows: You have some function f, and some argument a you apply the function to the argument and get a result ((f)a) -> b. So b is the result of evaluating f. It is defined based on how f is defined, so if we define ((f)a) -> a, then for any a we use we get the same value back (this function is called the identity, often just I.) When it isn't ambiguous, we can drop the parenthesis:
Ia is the same as ((I)a) which will evaluate a.
You can have any number of arguments:
(f)ab) -> c, or (..(f)a... z) -> A.
But, you can treat f as a function which returns a new function. Looking at the first example above [(f)ab -> c], we can say (f)a -> fa, where fa is a new function, which will take some new argument. In this way, a function taking any number of arguments can be turned into a series of "chained" functions, all returning new functions.
What's useful about this, is we can forget that non-functions exist (until it's convenient to do otherwise), and just work with functions that return other functions. Also, we can abuse our notation and use ((f)abcd...) to be ((((((f)a)b)c)d)...); the first is obviously cleaner.
Now we can also define what Church calls "abstractions". Abstraction is just a way of defining a function. It looks like:
f := λa.b
This is the same as what I showed earlier with ((f)a) -> b, but it's more precise notation (technically, my earlier notation means something different). What it says is f is a function which takes 1 argument and returns b. So the function I (the identity) is I := λa.a
We can do the same thing with multiple arguments:
f := λabc.d OR f := λa.λb.λc.d.
Notice how the second version can be interpreted to mean f takes one argument a, and returns a function which takes another argument, which returns a function....
I won't get into the gritty details, but there are only a few things we're allowed to do in lambda calculus:
*Define a function, using abstraction (and maybe a special notation... we'll cross that bridge when we come to it.),
*Rename variables (this is called alpha-conversion,). There are certain rules for this (which we don't need to pay attention to), but the only real purpose is to avoid confusion, and to make
proofs easier to follow.
*Apply abstract functions. (Called beta-reduction) This only comes into play with abstractions, but since we can only use functions which have been defined via abstraction (or in terms of already defined functions), this can be applied to any function. If we have some abstraction λx.fx (where f is some function we already know), applying the abstraction looks like this:
((λx.fx) a) -> fa.
All that really happens is we return what's after the "." with everything before the dot replaced with everything outside of the parenthesis.
*Replace equivalent expressions with each other (called eta-conversion). So, for example, I and λx.x can replace each other in any expression.
This comes into play when we have, for example ((I)a). To evaluate this:
((I)a) -> ((λx.x)a) , which is an application, that beta-reduces to a. This is, of course, mind-numbingly tedious, but in actual calculation, most of these steps are done implicitly
***
Now, it's a little more complicated than that, but not much. However, we can construct all of our numbers with it. I'll only focus on the natural numbers (0,1,2, etc), but it can be extended to integers, rationals and the reals using messy definitions in much the same way as in set theory.
Now, a "Church Numeral" is an encoding of a number using lambda calculus. Basically, the goal is to use LC to make definitions for all of our numbers (well, counting numbers) as well as multiplications, addition and exponentiation (exponentiation, just because it's easy). We'll ignore subtraction and division because they are actually rather messy. What we need are functions that interact the same way our numbers and operations interact.
Let's start with numbers. An easy way to define a number is to say that a number n takes a function, and applies it n times. This makes sense, but it isn't quite lambda calculus. So, we start with 0:
0 := λf.λa.a The astute reader will notice that what it returns is the identity:
0 := λf.I
1 is defined in the same way:
1 := λf.λa.fa Ignoring the a, we notice that this is the identity. The one difference is that we restrict the valid arguments to function values. (In general, f will always be a function. a can be, but whether it is necessarily a function is dependent on context, and will be clarified)
Now, we can define a successor. That is, the number after the number we're given. It takes (awkwardly) 3 arguments (see above about arguments, and see below about bracket notation):
SUCC := λn.λf.λa.f (nfa) (i.e. ((f) (((n)f)a))
What it says is given any number n, the successor is defined by applying the function argument of n (that is, f) to the other argument of n (that is, a) one extra time.
This is how we get to each number from the previous one. As an example:
[Don't worry if you can't follow my examples, they shouldn't actually be necessary.]
SUCC 2 -> (λn.λf.λa.f (nfa)) λf.λa.ffa -> λf.λa. f ((λf.λa.ffa) f a)
applying the "f a" at the end we get:
λf.λa. fffa; when we count the f's this means 3.
Now, using the same logic as the successor function, we can defined addition:
[M+N] := plus m n := λm.λn.λf.λa. m f (n f a) {remember that m and n are numbers}
The thing to notice about this is (n a) is the "a" in m. I.e. ((λf.λa.f...f a) f) (n a) -> (λa.f...fa) (n a) -> f....f...fa , where the first "f..." is m f's, and the "f...f" is n f's. So we have m+n f's, as we want.
Another example:
[2+3] -> (λm.λn.λf.λa. m f (n a)) 2 3 -> λf.λa. 3 f (2 f a) -> {dropping the leading λf.λa for cleanliness}
3 f ((λf.λa.ffa) f a) -> 3 f (ffa) -> ((λf.λa. fffa) f) ffa -> (λa.fffa) ffa ->{adding the λs back in} λf.λa.fffffa -> 5
Next,
[M*N] := times m n := λm.λn.λf. n (m f) {i.e. ((n) (m f))}
Notice that the function being applied n times is (m f). So, we have (remembering back to 2nd grade) n sets of m f's, or n*m f's. Again, just as we want.
Finally, exponentiation. I don't know if I'll work this into the second part... but we'll see.:
[M^N] := exp m n := λm.λn.λf n m f
This is very much like multiplication, only n is applied straight through (without the little "m f" detour), so What ends up happening is we get n copies of M all multiplied together.
Anyway, tomorrow I'll have a post about ordinals, and how to apply ordinal arithmetic to LC
The other thing I'm annoyed about is his callous rejection of transfinite theory. He seems to think Hilbert's Hotel was supposed to be an intuitive "proof" that infinities work the way they do, rather than a clarification of how they work. Also, he rejects infinities... absolutely and completely. As a formalist, I disagree with the rejection of anything interesting, but something else is crying out in pain at that idea-- perhaps I'm more of a Platonist than I think.
Anyway, I'd like to clarify my comment on the fixed point combinator (Also known as the Y-combinator), because... mostly because I'm a nerd. But also because I think it acts like an infinite ordinal.
I'm going to write this in two parts. The first will be on Lambda Calculus, the second on Ordinals and how to apply LC to ordinals. Hopefully, I can also discuss cardinals, but that may be a bit of a stretch.
First, a bit on (untyped) Lambda Calculus and Church Encoding: Lambda Calculus is a formalism for dealing with functions. For those who have heard of set theory, think of it as an analog, only everything is a function (or an argument of a function), rather than a set (or element of a set). The basics are as follows: You have some function f, and some argument a you apply the function to the argument and get a result ((f)a) -> b. So b is the result of evaluating f. It is defined based on how f is defined, so if we define ((f)a) -> a, then for any a we use we get the same value back (this function is called the identity, often just I.) When it isn't ambiguous, we can drop the parenthesis:
Ia is the same as ((I)a) which will evaluate a.
You can have any number of arguments:
(f)ab) -> c, or (..(f)a... z) -> A.
But, you can treat f as a function which returns a new function. Looking at the first example above [(f)ab -> c], we can say (f)a -> fa, where fa is a new function, which will take some new argument. In this way, a function taking any number of arguments can be turned into a series of "chained" functions, all returning new functions.
What's useful about this, is we can forget that non-functions exist (until it's convenient to do otherwise), and just work with functions that return other functions. Also, we can abuse our notation and use ((f)abcd...) to be ((((((f)a)b)c)d)...); the first is obviously cleaner.
Now we can also define what Church calls "abstractions". Abstraction is just a way of defining a function. It looks like:
f := λa.b
This is the same as what I showed earlier with ((f)a) -> b, but it's more precise notation (technically, my earlier notation means something different). What it says is f is a function which takes 1 argument and returns b. So the function I (the identity) is I := λa.a
We can do the same thing with multiple arguments:
f := λabc.d OR f := λa.λb.λc.d.
Notice how the second version can be interpreted to mean f takes one argument a, and returns a function which takes another argument, which returns a function....
I won't get into the gritty details, but there are only a few things we're allowed to do in lambda calculus:
*Define a function, using abstraction (and maybe a special notation... we'll cross that bridge when we come to it.),
*Rename variables (this is called alpha-conversion,). There are certain rules for this (which we don't need to pay attention to), but the only real purpose is to avoid confusion, and to make
proofs easier to follow.
*Apply abstract functions. (Called beta-reduction) This only comes into play with abstractions, but since we can only use functions which have been defined via abstraction (or in terms of already defined functions), this can be applied to any function. If we have some abstraction λx.fx (where f is some function we already know), applying the abstraction looks like this:
((λx.fx) a) -> fa.
All that really happens is we return what's after the "." with everything before the dot replaced with everything outside of the parenthesis.
*Replace equivalent expressions with each other (called eta-conversion). So, for example, I and λx.x can replace each other in any expression.
This comes into play when we have, for example ((I)a). To evaluate this:
((I)a) -> ((λx.x)a) , which is an application, that beta-reduces to a. This is, of course, mind-numbingly tedious, but in actual calculation, most of these steps are done implicitly
***
Now, it's a little more complicated than that, but not much. However, we can construct all of our numbers with it. I'll only focus on the natural numbers (0,1,2, etc), but it can be extended to integers, rationals and the reals using messy definitions in much the same way as in set theory.
Now, a "Church Numeral" is an encoding of a number using lambda calculus. Basically, the goal is to use LC to make definitions for all of our numbers (well, counting numbers) as well as multiplications, addition and exponentiation (exponentiation, just because it's easy). We'll ignore subtraction and division because they are actually rather messy. What we need are functions that interact the same way our numbers and operations interact.
Let's start with numbers. An easy way to define a number is to say that a number n takes a function, and applies it n times. This makes sense, but it isn't quite lambda calculus. So, we start with 0:
0 := λf.λa.a The astute reader will notice that what it returns is the identity:
0 := λf.I
1 is defined in the same way:
1 := λf.λa.fa Ignoring the a, we notice that this is the identity. The one difference is that we restrict the valid arguments to function values. (In general, f will always be a function. a can be, but whether it is necessarily a function is dependent on context, and will be clarified)
Now, we can define a successor. That is, the number after the number we're given. It takes (awkwardly) 3 arguments (see above about arguments, and see below about bracket notation):
SUCC := λn.λf.λa.f (nfa) (i.e. ((f) (((n)f)a))
What it says is given any number n, the successor is defined by applying the function argument of n (that is, f) to the other argument of n (that is, a) one extra time.
This is how we get to each number from the previous one. As an example:
[Don't worry if you can't follow my examples, they shouldn't actually be necessary.]
SUCC 2 -> (λn.λf.λa.f (nfa)) λf.λa.ffa -> λf.λa. f ((λf.λa.ffa) f a)
applying the "f a" at the end we get:
λf.λa. fffa; when we count the f's this means 3.
Now, using the same logic as the successor function, we can defined addition:
[M+N] := plus m n := λm.λn.λf.λa. m f (n f a) {remember that m and n are numbers}
The thing to notice about this is (n a) is the "a" in m. I.e. ((λf.λa.f...f a) f) (n a) -> (λa.f...fa) (n a) -> f....f...fa , where the first "f..." is m f's, and the "f...f" is n f's. So we have m+n f's, as we want.
Another example:
[2+3] -> (λm.λn.λf.λa. m f (n a)) 2 3 -> λf.λa. 3 f (2 f a) -> {dropping the leading λf.λa for cleanliness}
3 f ((λf.λa.ffa) f a) -> 3 f (ffa) -> ((λf.λa. fffa) f) ffa -> (λa.fffa) ffa ->{adding the λs back in} λf.λa.fffffa -> 5
Next,
[M*N] := times m n := λm.λn.λf. n (m f) {i.e. ((n) (m f))}
Notice that the function being applied n times is (m f). So, we have (remembering back to 2nd grade) n sets of m f's, or n*m f's. Again, just as we want.
Finally, exponentiation. I don't know if I'll work this into the second part... but we'll see.:
[M^N] := exp m n := λm.λn.λf n m f
This is very much like multiplication, only n is applied straight through (without the little "m f" detour), so What ends up happening is we get n copies of M all multiplied together.
Anyway, tomorrow I'll have a post about ordinals, and how to apply ordinal arithmetic to LC
Saturday, July 26, 2008
On One (and Infinity)
So, I once again find myself puzzled by people who cannot accept that .9... = 1. I'm not sure exactly what it is that draws mathematicians to this puzzle; it very well could be the average person's confusion-- similar to how many theologians felt the need to waste their time refuting The da Vinci Code.
Anyway, I stumbled across this, a mostly well written essay discussing the topic, with the author ultimately coming to the conclusion that he isn't entirely convinced of their equality.
Avoiding the painful first paragraph (which had me expecting an entertaining piece of non-sense), his argument comes down to a rejection of attainable infinities. viz, the philosophical ideal that infinity is outside of the realm of mathematics.
This is, I guess, a valid axiom, assuming of course that you're a platonist (or, as it turns out in this case, an intuitionist). A formalist can choose to reject the infinity axiom (or any axiom dealing with infinity-- thus, the idea of infinity), but he cannot claim it is wrong. Other philosophies of math run into similar ideological problems. Platonists can claim that infinity "doesn't exist", and I guess, looking at the average complaint with the .9... = 1 proofs comes down to "infinity doesn't exist!"-- an argument which only makes sense in a mathematical platonic context. Of course, Delaney's rejection is largely from the intuitionist persepcetive... I'll deal with that specifically in a bit.
Overall, I find it an annoyingly awkward position to take, and cannot believe that someone with the math training that Delaney has would take it. Allow me to explain:
First and foremost, we lose the concept of the decimal expansion (or maybe not...). This is a small loss for Delaney, as he rejects them anyway "With great temerity, I still hold that any decimal expansion is never exactly equal to pi. The decimal expansion is simply an approximation of pi."
Besides the implicit rejection of any non-finite decimal expansion, this seems to be a a confusion between number and numeral. A numeral is a symbolic encoding of a number-- that is a symbol (or group of symbols) which is interpreted to mean a number. It is, for lack of a better description, the "name" of a number. Just as the word "one" is symbolic of the mathematical object 1, any written decimal expansion is a symbolic description of the number under question. So, yes, any written numeral describing pi (either "pi" or "3.14159..." or anything else) will not actually equal the number itself. However, pi is a number with a value in the real (well-defined) number line. And a decimal expansion (as a mathematical object, rather than a symbolic representation) will have the exact value of pi. Whether or not a human can "read" this value on a page is immaterial-- the mathematical object which is a decimal expansion is identical to the mathematical object which is left as a greek letter. Again, they represent the exact same mathematical object. When doing algbera (or calculus, or anything else), you are working with mathematical objects, not any representation of the mathematical object. It is only when the representation is ambiguous (e.g. when a number is rounded) that problems arise, and that is not because of the objects, but because the representation is imprecise. [note: In rare instances, you do work with the representation, but in such pursuits (aptly named "metamathematics"), the representation is treated as a mathematical object, and is subject to it's own rules (like how working with the reals is different than working with vectors).]
Unless I'm mistaken, we are then forced to allow "infinite decimal expansions" as long as we allow the mathematical object they are identical to. If, on the other hand, we reject these objects, we are rejecting the irrationals (a hefty price to pay), as well as many of the rationals: any number which can be represented as p/q, where p is relatively prime to q, and q is relatively prime to both 2 and 5 will be disallowed (which is to say, 4 out of every 10 non-integer rationals)
We are left with a set of numbers which is not closed under any operation-- that is, we do not even have a group, unless we restrict ourselves to the integers... a boring mathematical universe indeed. Unless of course, we "diagonalize", which Delaney also rejects.
The point of that whole paragraph (which may have been lost somewhere) is that we either accept decimal expansions as valid mathematical objects which are [i]equal[/i] to their fractional counterpart, or we reject almost the whole number line.
In the same way as he questions the nature of decimal representations, he questions "epsilon" the elusive little infinitesimal that he (correctly) calls a "logical entity". Now, he suggests that the separation of epsilon between two numbers (e.g. .99... and 1) should be taken into account. Contrary to his likely expectation, I'm not inclined to disagree. The problem, of course, is that saying two numbers are unequal in a continuum (e.g. the real number line) means there is a number between them. There is no number between them. The "difference"-- epsilon-- is a representation of exactly that idea: they are at the same spot on the continuum, thus there is no mathematical difference between them.
If epsilon had any actual properties in the real number system, we as the mathematical community would be glad to take these into consideration, but adding it has the same effect as 0-- that is to say, no effect at all.
His argument (in the Repeating Nines essay) really hinges on these two concepts: Epsilon, and unattainable infinities. Beyond that, he seems to be bitter because he's on "the losing side" of the debate between "logicists and intuitionists". Of course there is no "losing side" in the first place, as there are plenty of intuitionists doing math right now. Delaney apparently just has trouble working in systems with rules different from those he thinks are "right"-- even most Platonists I know can work in a system they think is untrue. Why was Delaney unable to finish?
Beyond that, he tries to claim that "logicists" (I assume this is a blanket term for formalists and platonists) focus on paradoxes as a way to construct a system... How is this possible? As far as I know, set theory is built to avoid paradoxes. Russell's paradox is a bit of blow to Fregean set theory, but Quine's set theory deals with it quite well; it reduces the statement to nothing. Not a null statement-- but something entirely outside of the universe of discourse. Logicians don't focus on paradoxes, however they are, as you are certainly aware, something that must be dealt with when they arise.
I offer, as "evidence for infinity" the fixed point combinator (because a set-theoretical notation will click with him about the same as the rest of the set theoretical notions have). Y Defined (in an untyped lambda calculus) as Y->λf.(λx.f(xx))(λx.f(xx)). I won't go into the details of the operation, but if we are given Yf, we get f(Yf) which becomes f(f(Yf) -> f(f(...(f(Yf))..)) [I know, my parenthesis are backwards from Church's.] So, Yf= f(Yf) = f....(Yf)... No matter how many f's are started with, an infinite number are attained; that is calling y infinity: infinity +1 = infinity + 1000 = infinity. So, would a functional representation, rather than a set theoretic notation soothe his aching soul, and allow him to accept transfinite theory? Probably not; I'm sure he'll describe some other absurdity which "proves" infinite is unattainable.
He can work in a system whose maximal cardinality is Aleph-0 all he wants, and I won't restrict that, but it doesn't mean infinity is any less real than one-- less applicable, yes, but just as real (which is to say, entirely a mental construct, with certain agreed-upon properties.)
I know, it's rambly...
Anyway, I stumbled across this, a mostly well written essay discussing the topic, with the author ultimately coming to the conclusion that he isn't entirely convinced of their equality.
Avoiding the painful first paragraph (which had me expecting an entertaining piece of non-sense), his argument comes down to a rejection of attainable infinities. viz, the philosophical ideal that infinity is outside of the realm of mathematics.
This is, I guess, a valid axiom, assuming of course that you're a platonist (or, as it turns out in this case, an intuitionist). A formalist can choose to reject the infinity axiom (or any axiom dealing with infinity-- thus, the idea of infinity), but he cannot claim it is wrong. Other philosophies of math run into similar ideological problems. Platonists can claim that infinity "doesn't exist", and I guess, looking at the average complaint with the .9... = 1 proofs comes down to "infinity doesn't exist!"-- an argument which only makes sense in a mathematical platonic context. Of course, Delaney's rejection is largely from the intuitionist persepcetive... I'll deal with that specifically in a bit.
Overall, I find it an annoyingly awkward position to take, and cannot believe that someone with the math training that Delaney has would take it. Allow me to explain:
First and foremost, we lose the concept of the decimal expansion (or maybe not...). This is a small loss for Delaney, as he rejects them anyway "With great temerity, I still hold that any decimal expansion is never exactly equal to pi. The decimal expansion is simply an approximation of pi."
Besides the implicit rejection of any non-finite decimal expansion, this seems to be a a confusion between number and numeral. A numeral is a symbolic encoding of a number-- that is a symbol (or group of symbols) which is interpreted to mean a number. It is, for lack of a better description, the "name" of a number. Just as the word "one" is symbolic of the mathematical object 1, any written decimal expansion is a symbolic description of the number under question. So, yes, any written numeral describing pi (either "pi" or "3.14159..." or anything else) will not actually equal the number itself. However, pi is a number with a value in the real (well-defined) number line. And a decimal expansion (as a mathematical object, rather than a symbolic representation) will have the exact value of pi. Whether or not a human can "read" this value on a page is immaterial-- the mathematical object which is a decimal expansion is identical to the mathematical object which is left as a greek letter. Again, they represent the exact same mathematical object. When doing algbera (or calculus, or anything else), you are working with mathematical objects, not any representation of the mathematical object. It is only when the representation is ambiguous (e.g. when a number is rounded) that problems arise, and that is not because of the objects, but because the representation is imprecise. [note: In rare instances, you do work with the representation, but in such pursuits (aptly named "metamathematics"), the representation is treated as a mathematical object, and is subject to it's own rules (like how working with the reals is different than working with vectors).]
Unless I'm mistaken, we are then forced to allow "infinite decimal expansions" as long as we allow the mathematical object they are identical to. If, on the other hand, we reject these objects, we are rejecting the irrationals (a hefty price to pay), as well as many of the rationals: any number which can be represented as p/q, where p is relatively prime to q, and q is relatively prime to both 2 and 5 will be disallowed (which is to say, 4 out of every 10 non-integer rationals)
We are left with a set of numbers which is not closed under any operation-- that is, we do not even have a group, unless we restrict ourselves to the integers... a boring mathematical universe indeed. Unless of course, we "diagonalize", which Delaney also rejects.
The point of that whole paragraph (which may have been lost somewhere) is that we either accept decimal expansions as valid mathematical objects which are [i]equal[/i] to their fractional counterpart, or we reject almost the whole number line.
In the same way as he questions the nature of decimal representations, he questions "epsilon" the elusive little infinitesimal that he (correctly) calls a "logical entity". Now, he suggests that the separation of epsilon between two numbers (e.g. .99... and 1) should be taken into account. Contrary to his likely expectation, I'm not inclined to disagree. The problem, of course, is that saying two numbers are unequal in a continuum (e.g. the real number line) means there is a number between them. There is no number between them. The "difference"-- epsilon-- is a representation of exactly that idea: they are at the same spot on the continuum, thus there is no mathematical difference between them.
If epsilon had any actual properties in the real number system, we as the mathematical community would be glad to take these into consideration, but adding it has the same effect as 0-- that is to say, no effect at all.
His argument (in the Repeating Nines essay) really hinges on these two concepts: Epsilon, and unattainable infinities. Beyond that, he seems to be bitter because he's on "the losing side" of the debate between "logicists and intuitionists". Of course there is no "losing side" in the first place, as there are plenty of intuitionists doing math right now. Delaney apparently just has trouble working in systems with rules different from those he thinks are "right"-- even most Platonists I know can work in a system they think is untrue. Why was Delaney unable to finish?
Beyond that, he tries to claim that "logicists" (I assume this is a blanket term for formalists and platonists) focus on paradoxes as a way to construct a system... How is this possible? As far as I know, set theory is built to avoid paradoxes. Russell's paradox is a bit of blow to Fregean set theory, but Quine's set theory deals with it quite well; it reduces the statement to nothing. Not a null statement-- but something entirely outside of the universe of discourse. Logicians don't focus on paradoxes, however they are, as you are certainly aware, something that must be dealt with when they arise.
I offer, as "evidence for infinity" the fixed point combinator (because a set-theoretical notation will click with him about the same as the rest of the set theoretical notions have). Y Defined (in an untyped lambda calculus) as Y->λf.(λx.f(xx))(λx.f(xx)). I won't go into the details of the operation, but if we are given Yf, we get f(Yf) which becomes f(f(Yf) -> f(f(...(f(Yf))..)) [I know, my parenthesis are backwards from Church's.] So, Yf= f(Yf) = f....(Yf)... No matter how many f's are started with, an infinite number are attained; that is calling y infinity: infinity +1 = infinity + 1000 = infinity. So, would a functional representation, rather than a set theoretic notation soothe his aching soul, and allow him to accept transfinite theory? Probably not; I'm sure he'll describe some other absurdity which "proves" infinite is unattainable.
He can work in a system whose maximal cardinality is Aleph-0 all he wants, and I won't restrict that, but it doesn't mean infinity is any less real than one-- less applicable, yes, but just as real (which is to say, entirely a mental construct, with certain agreed-upon properties.)
I know, it's rambly...
Tuesday, July 15, 2008
Faithfulness...
God just pointed out to me a (sort of) new interpretation of Romans 8:35-39:
Who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or the sword? As it is written:
Who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or the sword? As it is written:
"for your sake we face death all day long;
we are considered as sheep to be slaughtered."
No, in all these things we are more than conquerors through him who loved us. For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, neither height nor depth, nor anything else in all creation, will be able to separate us from the love of God that is in Christ Jesus our Lord.
**
I guess it's not so much a new interpretation as it is a new revelation. Once we have tasted of that love. Once we feel that deep bittersweet longing so acutely-- so much more acutely than the bitter sehnsucht we as humans are cursed with-- it is impossible to forget it. Try as our heart might, and despite any laziness, or any apathy, or hardship or distractions, something will always bring that love back into our heart, and like the eager bride before her wedding, we shake with excitement.
God's faithfulness is amazing. It's so incredible how he seeks us despite everything. If only I could consistently nurture that relationship, instead of getting apathetic, I would see more of the amazing things I've seen...
we are considered as sheep to be slaughtered."
No, in all these things we are more than conquerors through him who loved us. For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, neither height nor depth, nor anything else in all creation, will be able to separate us from the love of God that is in Christ Jesus our Lord.
**
I guess it's not so much a new interpretation as it is a new revelation. Once we have tasted of that love. Once we feel that deep bittersweet longing so acutely-- so much more acutely than the bitter sehnsucht we as humans are cursed with-- it is impossible to forget it. Try as our heart might, and despite any laziness, or any apathy, or hardship or distractions, something will always bring that love back into our heart, and like the eager bride before her wedding, we shake with excitement.
God's faithfulness is amazing. It's so incredible how he seeks us despite everything. If only I could consistently nurture that relationship, instead of getting apathetic, I would see more of the amazing things I've seen...
Tuesday, June 17, 2008
Hurray, Postings!
Sorry, it's been a while since I've actually posted anything here. And this post is insubstantial...
link: Landscapes. It's a project I'll be slowly working on with time... So far, only one piece; another will be up tonight. After that, who knows.
link: Landscapes. It's a project I'll be slowly working on with time... So far, only one piece; another will be up tonight. After that, who knows.
Wednesday, May 14, 2008
Like Clockwork...
This is something I recently came up with; I wrote it down while I was at breakfast with Ellie (actually on the train back from breakfast). I'll be reading it Friday.
I've been inside that machine-- the one they say revolves the heavens about the earth-- I've watched its gears turn, traced the outlines of their teeth with my fingers; I've even stood on some. I watched a gear crack and fall, a spring loosed to follow its own path.
I looked up through the labyrinth of clockwork-- the endless cacophony of ticking and grinding-- and I thought to myself "This is not the machine which revolves the universe."
I've been inside that machine-- the one they say revolves the heavens about the earth-- I've watched its gears turn, traced the outlines of their teeth with my fingers; I've even stood on some. I watched a gear crack and fall, a spring loosed to follow its own path.
I looked up through the labyrinth of clockwork-- the endless cacophony of ticking and grinding-- and I thought to myself "This is not the machine which revolves the universe."
Tuesday, May 6, 2008
Imagination
I was thinking in the shower about a comment my physics professor made regarding funding for scientific research. "If the government doesn't fund it, who will?" I'll leave an argument for how stupid this question is for someone else to make, because I just don't care, but it got me thinking. We humans are so (read; sooooo) incapable of believing that things can work any way besides the way they do. We have really no imagination.
This set me thinking (since it was a physics professor) about quantum theory, and 20th century physics in general. I have no trouble accepting wave-particle duality, quantization, super-positioning, entanglement, the Uncertainty Principle, or all of the other fun names we have for physics concepts. I can accept them and understand them without the slightest twisting of my brain, because I can say with absolute conviction that a donut makes the same shape as a coffee cup; I can say with absolute conviction that 2*3=1; that 4*4 = 6; And I can say with a straight face that 4 or 5 dimensional space is "easy", and is just a special case of d-dimensional space. Imaginary numbers are as real as negative numbers are as real counting numbers (real in the intuitive sense, not the well-defined). I won't bet that a coin which has landed heads 1000 will land heads or tails on the next flip, unless I'm betting less money than you are.
I can turn a hollow sphere inside out. I can split a solid sphere into two spheres the same size and density of the first. And I have no super powers.
All of these different facts and systems that I've mentioned have different rules, follow different patterns, have different truths. Also, I'm expected to understand all of this before I leave college. So why would it not be the same with physics? Why could it not be the same with funding? Or art? Or the future? Why could it not be true that God follows rules which don't make sense to us?
We humans have such poor, poor imaginations. "Capacity for abstract thought," no! Where is the abstract thought?
There are times when I still think I should have been a physicist. It's so mind-numbingly intuitive; the math is easy, even at the quantum level. Oo! Group theory! I need to know that to get into graduate school; as well as ring theory and field theory, analysis, topology, and anything else you physicists have tried to play with. One of these days physicist will start using category theory to start kludging all these half-baked ideas of theirs together into a "coherent" whole, and that's when the rest of the world will know they're just making it up as they go along.
At lest mathematicians know their making it up; and are expected to.
Maybe I should still be a physicist; Maybe physics need someone who thinks they're all a bunch of idiots.
This set me thinking (since it was a physics professor) about quantum theory, and 20th century physics in general. I have no trouble accepting wave-particle duality, quantization, super-positioning, entanglement, the Uncertainty Principle, or all of the other fun names we have for physics concepts. I can accept them and understand them without the slightest twisting of my brain, because I can say with absolute conviction that a donut makes the same shape as a coffee cup; I can say with absolute conviction that 2*3=1; that 4*4 = 6; And I can say with a straight face that 4 or 5 dimensional space is "easy", and is just a special case of d-dimensional space. Imaginary numbers are as real as negative numbers are as real counting numbers (real in the intuitive sense, not the well-defined). I won't bet that a coin which has landed heads 1000 will land heads or tails on the next flip, unless I'm betting less money than you are.
I can turn a hollow sphere inside out. I can split a solid sphere into two spheres the same size and density of the first. And I have no super powers.
All of these different facts and systems that I've mentioned have different rules, follow different patterns, have different truths. Also, I'm expected to understand all of this before I leave college. So why would it not be the same with physics? Why could it not be the same with funding? Or art? Or the future? Why could it not be true that God follows rules which don't make sense to us?
We humans have such poor, poor imaginations. "Capacity for abstract thought," no! Where is the abstract thought?
There are times when I still think I should have been a physicist. It's so mind-numbingly intuitive; the math is easy, even at the quantum level. Oo! Group theory! I need to know that to get into graduate school; as well as ring theory and field theory, analysis, topology, and anything else you physicists have tried to play with. One of these days physicist will start using category theory to start kludging all these half-baked ideas of theirs together into a "coherent" whole, and that's when the rest of the world will know they're just making it up as they go along.
At lest mathematicians know their making it up; and are expected to.
Maybe I should still be a physicist; Maybe physics need someone who thinks they're all a bunch of idiots.
Friday, April 25, 2008
Combinatorics
Counting is fun. It's just really enjoyable to sit there and figure out the most "elegant" way to solve a problem. It's really, really amazing that there are so many ways to approach the same problem, and as long as they count the same thing, they're the same.
It also is finally starting to give me a solid grasp of isomorphism. Not that the idea of isomorphism is difficult, just that seeing isomorphisms sitting there in front of you is difficult.
I think graph theory will fully solidify that idea for me.
It also is finally starting to give me a solid grasp of isomorphism. Not that the idea of isomorphism is difficult, just that seeing isomorphisms sitting there in front of you is difficult.
I think graph theory will fully solidify that idea for me.
Saturday, April 19, 2008
Functional programming in Perl!
Perl has Lisp-like lexical closures and it supports currying.
Both of these happen through perlref, a new feature in perl 5. Essentially, how it works is you can make anything a scalar ($) variable... anything. Sort of.
You can make a scalar variable that points to anything, which is, for the programmer, almost the same thing.
This way you can make a variable that is the curry of a function... I'm not going to go into the details, because my understanding of Perl is, well, yeah... It was fun to find out as I was searching for something else, however.
***
Perl, it seems, is a really fun language, for mostly opposite reasons from scheme. Perl's beauty comes from the amount of pre-imposed structure, but a very large language to work with. Pretty Perl is sort of like really well-written blank verse: it is going to have some obtuse words and awkward grammar, but it still flows well.
Scheme, on the other hand, comes from a small rule set, but a language which can be molded and formed as you please. Scheme would be comparable to some of Cummings better works: It follows (almost) no rules of language, but it drags you around the program nonetheless.
...
Or something like that. The poetry = programming analogy really falls flat under any rigorous scrutiny, but I hope I got the point across:
Perl- a lot of rules, but a lot of ways to exploit them
Scheme- few rules, meaning absolute freedom to exploit anything and everything.
Both of these happen through perlref, a new feature in perl 5. Essentially, how it works is you can make anything a scalar ($) variable... anything. Sort of.
You can make a scalar variable that points to anything, which is, for the programmer, almost the same thing.
This way you can make a variable that is the curry of a function... I'm not going to go into the details, because my understanding of Perl is, well, yeah... It was fun to find out as I was searching for something else, however.
***
Perl, it seems, is a really fun language, for mostly opposite reasons from scheme. Perl's beauty comes from the amount of pre-imposed structure, but a very large language to work with. Pretty Perl is sort of like really well-written blank verse: it is going to have some obtuse words and awkward grammar, but it still flows well.
Scheme, on the other hand, comes from a small rule set, but a language which can be molded and formed as you please. Scheme would be comparable to some of Cummings better works: It follows (almost) no rules of language, but it drags you around the program nonetheless.
...
Or something like that. The poetry = programming analogy really falls flat under any rigorous scrutiny, but I hope I got the point across:
Perl- a lot of rules, but a lot of ways to exploit them
Scheme- few rules, meaning absolute freedom to exploit anything and everything.
Monday, April 14, 2008
Functions are the Numbers of the 21st Century
I went to two of the talks at Mengerfest today, (at the urging of a professor). They were unenchanting. Ellis's talk was a research talk, and so came with the dryness to be expected, while the material was interesting. Trefethen's talk would have been exceptionally interesting, were I a numerical analyst. Alas, I am no such creature.
After the talk someone commented about Trefethen's talk, and quoted Menger (apparently from a conversation): "Functions are the numbers of the 21st century."
That statement says a lot about the process of abstraction, and mathematical thought in general. Mathematics, (and it seems, with it, all of human thought) is tending toward increased abstraction. Originally integers were the epitome of mathematical abstraction; then rational and real numbers; then the Cartesian plane, complex numbers, and then sets.... With the 20th century developments of general abstract nonsense, topos theory and type theory, mathematics has climbed higher up this mountain than ever before.
There's a certain similarity in rules underlying all of these developments and abstractions-- they all work more or less the same. This implies (at least) one of three things:
1)The way we do math requires structures to exhibit similarities.
2)There is an inherent structure behind math... That is, the universe is design in such a way that there is a tremendously simple, yet elegantly complex structure.
3)The structures we impose are mental constructs. The fact that we are doing math requires these structures-- our mind cannot process abstract data in any other way.
Being me, I tend towards 1 or 3. 2 is too deterministic. 1, I think, presents more interesting applications: Is there another way to approach math, so that other structures develop?
And when I step back and examine the broader implications of 1,2, and 3, I realize, once again that it just doesn't matter. Math will still be conducted the same way; humans will continue to act roughly the same and life will continue as normal-- no interrupt to the daily scheduled programming.
If the different systems (that is, math, or the universe, or anything else) end up acting the same way, we have an isomorphism, right? So the systems can be considered the same, no matter how different the rules.
And I realized that more and more, this is why it's hard for me to give a shit when someone spouts self-righteous or otherwise ignorant nonsense: they'll continue acting the same. The system in which I said nothing at all is more or less the same as the system in which I correct their error, in the name of "Truth."
And at this point I've lost my train of thought... something on everything being the same, and so on and such... and I need to count some orbits.
Cheers.
After the talk someone commented about Trefethen's talk, and quoted Menger (apparently from a conversation): "Functions are the numbers of the 21st century."
That statement says a lot about the process of abstraction, and mathematical thought in general. Mathematics, (and it seems, with it, all of human thought) is tending toward increased abstraction. Originally integers were the epitome of mathematical abstraction; then rational and real numbers; then the Cartesian plane, complex numbers, and then sets.... With the 20th century developments of general abstract nonsense, topos theory and type theory, mathematics has climbed higher up this mountain than ever before.
There's a certain similarity in rules underlying all of these developments and abstractions-- they all work more or less the same. This implies (at least) one of three things:
1)The way we do math requires structures to exhibit similarities.
2)There is an inherent structure behind math... That is, the universe is design in such a way that there is a tremendously simple, yet elegantly complex structure.
3)The structures we impose are mental constructs. The fact that we are doing math requires these structures-- our mind cannot process abstract data in any other way.
Being me, I tend towards 1 or 3. 2 is too deterministic. 1, I think, presents more interesting applications: Is there another way to approach math, so that other structures develop?
And when I step back and examine the broader implications of 1,2, and 3, I realize, once again that it just doesn't matter. Math will still be conducted the same way; humans will continue to act roughly the same and life will continue as normal-- no interrupt to the daily scheduled programming.
If the different systems (that is, math, or the universe, or anything else) end up acting the same way, we have an isomorphism, right? So the systems can be considered the same, no matter how different the rules.
And I realized that more and more, this is why it's hard for me to give a shit when someone spouts self-righteous or otherwise ignorant nonsense: they'll continue acting the same. The system in which I said nothing at all is more or less the same as the system in which I correct their error, in the name of "Truth."
And at this point I've lost my train of thought... something on everything being the same, and so on and such... and I need to count some orbits.
Cheers.
Saturday, April 12, 2008
Philosophy of Math
I was thinking today about the various philosophies of math. Ideally, I'm a formalist, but more and more, I don't think that is quite right. Various other semi-formal positions seem to be a little closer to what I believe than formalism proper. Anyway, as I was considering this, I came to the conclusion that math would still be done by the same people, in the same way no matter which philosophy turned out to be "correct" (if there is one). With this realization I came full-circle, because it is such a formalist thing to decide: "it doesn't matter what it means outside of the system, because the system works the same way no matter what."
I'm also still bemused with Gödel's second theorem: For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. That is, any mathematical system that can express addition can not be proven consistent within the system. If it contains a statement proving its consistency, it means it is not consistent. Gödel's first theorem is likewise entertaining, but the basic idea of the second is just hilarious. It says that a mathematical system saying it is consistent is like someone saying "I have never lied." Everyone has lied at some point. Thus, we know that statement to be a lie... Only it's stronger in math.
I'm also still bemused with Gödel's second theorem: For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. That is, any mathematical system that can express addition can not be proven consistent within the system. If it contains a statement proving its consistency, it means it is not consistent. Gödel's first theorem is likewise entertaining, but the basic idea of the second is just hilarious. It says that a mathematical system saying it is consistent is like someone saying "I have never lied." Everyone has lied at some point. Thus, we know that statement to be a lie... Only it's stronger in math.
Friday, April 11, 2008
Perl... and the pearl of great worth
Schlomo just sent me the scripts he's using to pick apart Shakespeare plays. They're really not that bad, but since (presumably) he's the only one who has had to deal with them, his documentation consists of #{file does this}. It makes it severely difficult to follow, when there are 10 scripts which assume things about each other. They are fair assumptions, but not helpful when you don't have the context to understand...
****
On another note, God has been active lately. We're all growing closer together, and it seems more people are slowly joining our community. It's really exciting
****
On another note, God has been active lately. We're all growing closer together, and it seems more people are slowly joining our community. It's really exciting
Saturday, April 5, 2008
Romans 8 (NIV)
Therefore, there is now no condemnation for those who are in Christ Jesus, because through Christ Jesus the law of the Spirit of life set me free from the law of sin and death. For what the law was powerless to do in that it was weakened by the sinful nature, God did by sending his own Son in the likeness of sinful man to be a sin offering. And so he condemned sin in sinful man, in order that the righteous requirements of the law might be fully met in us, who do not live according to the sinful nature but according to the Spirit.
Those who life according to the sinful nature have their minds set on what that nature desires; but those who live in accordance with the Spirit have their minds set on what the Spirit desires. The mind of sinful man is death, but the mind controlled by the Spirit is life and peace;
the sinful mind is hostile to God. It does not submit to God's law, or can it do so. Those controlled by the sinful nature cannot please God.
You, however, are controlled not by the sinful nature, but by the Spirit, if the Spirit of God lives in you. And if anyone does not have the Spirit of Christ, he does not belong to Christ. But if Christ is in you, your body is dead because of sin, yet your spirit is alive because of righteousness. And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you.
Therefore, brothers, we have an obligation-- but it is not to the sinful nature, to live according to it. For if you live according to the sinful nature, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live, because those who are led by the Spirit of God are sons of God. For you did not receive a spirit that makes you a slave again to fear, but you received the Spirit of sonship. And by him we cry, "Abba, Father." The Spirit himself testifies with our spirit that we are God's children. Now if we are children, then we are heirs-- heirs of God and coheirs with Christ, if indeed we share in his sufferings in order that we may also share in his glory
I consider that our present sufferings are not worth comparing with the glory that will be revealed in us. The creation waits in eager expectation for the sons of God to be revealed. For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope that the creation itself will be liberated from its bondage to decay and brought into the glorious freedom of the children of God.
We know that the whole creation has been groaning as in the pains of childbirth right up to the present time. Not only so, but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for adoption as sons, the redemption of our bodies. For in this hope we are saved. But hope that is seen is no hope at all. Who hopes for what he already has? But if we hope for what we do not yet have, we wait for it patiently.
In the same way, the Spirit helps us in our weakness. We do not know what we ought to pray for, but the Spirit himself intercedes for us with groans that words cannot express. And he who searches our hearts knows the mind of the Spirit, because the Spirit intercedes for the saints in accordance with God's will.
And we know that in all things, God works for the good of those who love him, who have been called according to his purpose. For those God foreknew he also predestined to be conformed to the likeness of his Son, that he might be the firstborn among many brothers. And those he predestined, he also called; those he called, he also justified; those he justified, he also glorified.
What, then, shall we say in response to this? If God is for us, who can be against us? He who did not spare his own Son, but gave him up for us all-- how will he not also, along with him, graciously give us all things? Who will bring any charge against those whom God has chosen? It is God who justifies. Who is he that condemns? Christ Jesus, who died-- more than that, who was raised to life-- is at the right hand of God and is also interceding for us.
Who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or danger or sword? As it is written:
Those who life according to the sinful nature have their minds set on what that nature desires; but those who live in accordance with the Spirit have their minds set on what the Spirit desires. The mind of sinful man is death, but the mind controlled by the Spirit is life and peace;
the sinful mind is hostile to God. It does not submit to God's law, or can it do so. Those controlled by the sinful nature cannot please God.
You, however, are controlled not by the sinful nature, but by the Spirit, if the Spirit of God lives in you. And if anyone does not have the Spirit of Christ, he does not belong to Christ. But if Christ is in you, your body is dead because of sin, yet your spirit is alive because of righteousness. And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you.
Therefore, brothers, we have an obligation-- but it is not to the sinful nature, to live according to it. For if you live according to the sinful nature, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live, because those who are led by the Spirit of God are sons of God. For you did not receive a spirit that makes you a slave again to fear, but you received the Spirit of sonship. And by him we cry, "Abba, Father." The Spirit himself testifies with our spirit that we are God's children. Now if we are children, then we are heirs-- heirs of God and coheirs with Christ, if indeed we share in his sufferings in order that we may also share in his glory
I consider that our present sufferings are not worth comparing with the glory that will be revealed in us. The creation waits in eager expectation for the sons of God to be revealed. For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope that the creation itself will be liberated from its bondage to decay and brought into the glorious freedom of the children of God.
We know that the whole creation has been groaning as in the pains of childbirth right up to the present time. Not only so, but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for adoption as sons, the redemption of our bodies. For in this hope we are saved. But hope that is seen is no hope at all. Who hopes for what he already has? But if we hope for what we do not yet have, we wait for it patiently.
In the same way, the Spirit helps us in our weakness. We do not know what we ought to pray for, but the Spirit himself intercedes for us with groans that words cannot express. And he who searches our hearts knows the mind of the Spirit, because the Spirit intercedes for the saints in accordance with God's will.
And we know that in all things, God works for the good of those who love him, who have been called according to his purpose. For those God foreknew he also predestined to be conformed to the likeness of his Son, that he might be the firstborn among many brothers. And those he predestined, he also called; those he called, he also justified; those he justified, he also glorified.
What, then, shall we say in response to this? If God is for us, who can be against us? He who did not spare his own Son, but gave him up for us all-- how will he not also, along with him, graciously give us all things? Who will bring any charge against those whom God has chosen? It is God who justifies. Who is he that condemns? Christ Jesus, who died-- more than that, who was raised to life-- is at the right hand of God and is also interceding for us.
Who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or danger or sword? As it is written:
For your sake we face death all day long;
we are considered as sheep to be slaughtered.
we are considered as sheep to be slaughtered.
No, in all these things we are more than conquerors through him who loved us. For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, neither height nor depth, nor anything else in all creation will be able to separate us from the love of God that is in Christ Jesus our Lord.
Sylvia
I
The event of a lifetime is almost underway
The children laugh and sing, let them play.
The eyes of innocence look on and make another wish:
For wishes can come true when they are pure.
A candle burns, wax trickles down to rest on weathered wood
Between to chairs set facing through the glow.
The winter winter wind comes wandering in
To freeze the silent pose.
No matter how it tries, the candle glows.
No matter how it tries...
In one chair sits Sylvia
with sullen eyes and a weak chin,
next to Father Time with his shoulders black and grim.
They've been sitting there with steady glares
unblinking and unchanged,
contemplating something dark and lonely.
They've been sitting there for hours,
days, and, weeks, and months, and years,
waiting for their moment to arrive
The anthem plays a melody that's in and out of time.
The light that flickers only has one point of view
surrounding darkness makes itself a home.
The girl was of fine color and was shaped of gold;
the man was pallid white and set in stone.
The rose with frozen petals never withers in the sun.
The mind that never learns never forgets!
The winter wind comes wondering in
the freeze the heart and skin.
No matter how it tries the candle glows.
Twinkling lights and merry men are dancing in the street
as shining figures bustle to and fro.
The snow is softly falling like a blanket in the cold,
with fakes that warm the hearts of those who let them.
There is one place along the way that's dimly lit and grey,
but no one seems to notice in their joy.
The wintering wind comes wandering in to hear it's own self blow.
No matter how it tries, the candle glows.
In one chair sits Sylvia
with sullen eyes and a weak chin,
next to Father Time with his shoulders black and grim.
They've been sitting there with steady glares
unblinking and unchanged,
contemplating something dark and lonely.
They've been sitting there for hours,
days, and, weeks, and months, and years,
waiting for their moment to arrive.
Finally a slow grin played across her daring face,
and the old bag grew a look of sudden horror.
With that the old man died and smashed his wrinkled, balding head,
and the girl got up and left the room and went ot into the winter wind and...
II
She walked out through the snow.
She left the body there.
The flakes fell to her face,
and her face fell to the ground.
She listened as the chaos swelled around.
While the church ahead began to ring the bells
That thundering sound
lifted her to her feet and
she walked on.
She continued down the road.
The monotony had made restless beyond words.
She found a shovel and dug a big pothole.
Then a carriage crashed and made a big and deafening sound
Police gathered round.
The flakes fell to her face,
while her eyes
turned them to water.
The owner yelled and called her terrible names,
"Whore! Wench! What have you done?"
They cuffed her up and took her shovel away
but then the sun came out
and dried the stuff from her face.
She smiled sweet, apologized, and walked her self away.
Now you're hear in my heart
and you know who you are.
No one's going to clean up this mess:
the wheels have fallen off,
the current's gonna take you where you want,
but you keep on running,
The current's gonna make you who you are.
You know your smile's growing stronger,
while your stride is getting longer:
You're gonna make it out,
on your own.
~By the Butterfly Assassins
The event of a lifetime is almost underway
The children laugh and sing, let them play.
The eyes of innocence look on and make another wish:
For wishes can come true when they are pure.
A candle burns, wax trickles down to rest on weathered wood
Between to chairs set facing through the glow.
The winter winter wind comes wandering in
To freeze the silent pose.
No matter how it tries, the candle glows.
No matter how it tries...
In one chair sits Sylvia
with sullen eyes and a weak chin,
next to Father Time with his shoulders black and grim.
They've been sitting there with steady glares
unblinking and unchanged,
contemplating something dark and lonely.
They've been sitting there for hours,
days, and, weeks, and months, and years,
waiting for their moment to arrive
The anthem plays a melody that's in and out of time.
The light that flickers only has one point of view
surrounding darkness makes itself a home.
The girl was of fine color and was shaped of gold;
the man was pallid white and set in stone.
The rose with frozen petals never withers in the sun.
The mind that never learns never forgets!
The winter wind comes wondering in
the freeze the heart and skin.
No matter how it tries the candle glows.
Twinkling lights and merry men are dancing in the street
as shining figures bustle to and fro.
The snow is softly falling like a blanket in the cold,
with fakes that warm the hearts of those who let them.
There is one place along the way that's dimly lit and grey,
but no one seems to notice in their joy.
The wintering wind comes wandering in to hear it's own self blow.
No matter how it tries, the candle glows.
In one chair sits Sylvia
with sullen eyes and a weak chin,
next to Father Time with his shoulders black and grim.
They've been sitting there with steady glares
unblinking and unchanged,
contemplating something dark and lonely.
They've been sitting there for hours,
days, and, weeks, and months, and years,
waiting for their moment to arrive.
Finally a slow grin played across her daring face,
and the old bag grew a look of sudden horror.
With that the old man died and smashed his wrinkled, balding head,
and the girl got up and left the room and went ot into the winter wind and...
II
She walked out through the snow.
She left the body there.
The flakes fell to her face,
and her face fell to the ground.
She listened as the chaos swelled around.
While the church ahead began to ring the bells
That thundering sound
lifted her to her feet and
she walked on.
She continued down the road.
The monotony had made restless beyond words.
She found a shovel and dug a big pothole.
Then a carriage crashed and made a big and deafening sound
Police gathered round.
The flakes fell to her face,
while her eyes
turned them to water.
The owner yelled and called her terrible names,
"Whore! Wench! What have you done?"
They cuffed her up and took her shovel away
but then the sun came out
and dried the stuff from her face.
She smiled sweet, apologized, and walked her self away.
Now you're hear in my heart
and you know who you are.
No one's going to clean up this mess:
the wheels have fallen off,
the current's gonna take you where you want,
but you keep on running,
The current's gonna make you who you are.
You know your smile's growing stronger,
while your stride is getting longer:
You're gonna make it out,
on your own.
~By the Butterfly Assassins
Sunday, March 30, 2008
Why I'm Not a Scientist
Science, I've come to realize is not necessarily about "discovering Truth" as most people see it. Rather, it is a way to further the human study of the universe. By this I mean that a good scientific idea asks more questions than it answers-- allowing human questioning to continue forever. Correctness from the scientific perspective does not mean "It is absolutely true." Instead, it means "It fits the data (roughly), and it leads to a new area of study."
As an example (and I realize the absurdity in it), suppose it were proven, absolutely, that God exists, and everything happens only because "God says so"-- there is no discernable logic, no actual pattern to predict in God's plan. It allows for only one question "Why this way?" which is to question to ineffable will of God.
Science, cannot and would not accept this conclusion; our pursuit of knowledge about the universe would be pointless. Doubtless, people would still continue to practice science, creating mathematical models of the phenomenon around them.
What's interesting here is that they would likely continue to find ever more "accurate" models of the working universe, even though they would know that these models do not, in truth, contribute to any understanding of the universe-- it is already understood. This implies a couple of things:
1)We find patterns in everything, whether they exist or not.
2)Any sufficiently complex language can model anything; even if there is no pattern to model.
3)The link between science and math is artificial.
Ramsey theory suggests that in any large random system (which the universe described about would be, for all practical purposes) there will be some sort of pattern, even if it is not inherent in the design or mechanism of the system. The real problem arises when we ask "Which pattern is it?" This question cannot be answered without enumerating the entire system, an obviously impossible task when confronted with the universe.
Science then, would be conducted in absolutely the same way if it were both useless and contrary to truth, yet scientists trumpet Logical Positivism as something that will ultimately unlock the key to Life, the Universe, and Everything.
As an example (and I realize the absurdity in it), suppose it were proven, absolutely, that God exists, and everything happens only because "God says so"-- there is no discernable logic, no actual pattern to predict in God's plan. It allows for only one question "Why this way?" which is to question to ineffable will of God.
Science, cannot and would not accept this conclusion; our pursuit of knowledge about the universe would be pointless. Doubtless, people would still continue to practice science, creating mathematical models of the phenomenon around them.
What's interesting here is that they would likely continue to find ever more "accurate" models of the working universe, even though they would know that these models do not, in truth, contribute to any understanding of the universe-- it is already understood. This implies a couple of things:
1)We find patterns in everything, whether they exist or not.
2)Any sufficiently complex language can model anything; even if there is no pattern to model.
3)The link between science and math is artificial.
Ramsey theory suggests that in any large random system (which the universe described about would be, for all practical purposes) there will be some sort of pattern, even if it is not inherent in the design or mechanism of the system. The real problem arises when we ask "Which pattern is it?" This question cannot be answered without enumerating the entire system, an obviously impossible task when confronted with the universe.
Science then, would be conducted in absolutely the same way if it were both useless and contrary to truth, yet scientists trumpet Logical Positivism as something that will ultimately unlock the key to Life, the Universe, and Everything.
Wednesday, March 26, 2008
Correspondences. I
I have opened a dialog with one of the few people who asks truly probing questions of people, and makes truly profound statements about the same. She has agreed to let our emails become public. I will post them in the form: {email from me; email from Jamie}. This way, Jamie is the first to see my responses. I find it only fair as they're directed at her. Also, any and all editing from the original is simply to present the same words in a more readable format. All typos, miswordings, etc., remain intact. Also, i've gotten rid of any quotation i've deemed superfluous. Jamie, if i've made an error, please correct me. This first one contains the first 4 emails.
Anyway:
Jamie!
This is Cory, How are you?
You're someone I want to keep in contact with, because it's hard to find
someone who so actively digs into the human brain/soul/heart/whatever...
Anyway, 2 things:
Would you be able to compile a list of my quotes that you wrote down
while you were here? It's always interesting to see how you have
impressed others...
I have a blog of sorts; ramblings and rants on various topics. It's at
onag.blogspot.com
It may present to you probing questions to ask.
Cheers,
Cory
*****
"How do you know whether you are hitting your own personal boundaries, or you're hitting a limit where you can't go farther"
"Everyone has some sort of void in their soul...."
"...that the things that are hilarious are taken far too seriously by far too many people..."
"We have no objective way of determining if our reality is the same as someone else's"
I haven't thought about the first quote yet because I'm terrified that there might actually be a limit where ''you'' can't go farther. That would fuck with my current philosophy of the world. Yet the very act of being terrified implies there is such a limit....I guess I'm just reluctant to reconstruct my notions of the world this month. I think at this point you were referring to some kind of inner personal limit however and not such concepts as the universe and different dimensions and whatnot. That makes is slightly easier to answer, yet is it even necessary to ask questions about the expanse of space seeing as how whatever knowledge we get doesn't affect its very nature...(or does it?) or ours?
-side tangent- why do we even look for answers in general? Is there ever some kind of permanent answer? If we do not ever actually know everything-which we cannot- or even real answers, ones that are really the Truth and not some kind of answers we have decided on for our own life construction and stability, it is very improbably that we could ever actually know them...so what is the point of questions (ah ha) of searching? What makes someone decide on a certain set of values on which to base their lives?
Back on semi-track. Is this limit you were referring to some kind of inner limit? Some 'soul' limit?
Your other quotes suffered a tragic (they were introduced to my car before they made it to my journal) death.
*****
Yeah, my language was kind of vague... it wouldn't do in a proof. Let's try again:
"How does one know whether they are hitting the boundaries of their own comfort, or an insurmountable-- universal-- mental limit." (i use the singular, ambiguous "they", because it is less awkward than the alternatives.)
It's a scary thing to think about; we have some desire-- some need-- to know everything; what happens if we can't?
I don't know whether this limit is some sort of "soul limit" some sort of paradox of self-reference-- i.e. a boundary on our capacity for metacognition.
So. On questions. This is one that has bugged me for quite some time. I've managed to sidestep the question (more, push it out of the way so i don't have to figure it out) by saying i like to learn because there's beauty in it. This is, in part, a cop out. There's truth to it, but that's only a part of the drive.
The real reason is that it seems (at least temporarily) to fill the void in my soul. Hereafter, i'll refer to this void as nostalgia or despair; i'll explain this idea later. Anyway, whenever i think of the quest for knowledge, i think of Unamuno's The Tragic Sense of Life, which i recommend you read if you have questions about our quest for knowledge. Essentially, it's about the very existential idea of despair, the source of that despair and how we use knowledge to combat that despair. I either completely agree or completely disagree with everything he has to say, but it's an interesting read either way.
Anyway, despair:
A lot of philosophers have discussed the concept itself directly, and i find it difficult to think that any philosopher has hoped for anything greater than to understand it. We have, as i've stated, a void in our soul. Despair (the state of being) itself is the recognition of that void, and the conflict created by trying to understand the nature of that void, as well as the struggle to fill or remove that void.
The Christian answer to "what causes despair?" is that we have a longing for an intimacy with God-- an intimacy, which, being fallen creatures, eludes us in except in brief flashes and moments of clarity. This is, i think, an adequate assessment. The general idea can be generalized to be compatible with more humanist philosophies by saying "we have a longing for inclusion in something greater than ourselves." All of human action can be reduced to a (perceived) fulfillment of this desire. The Christian has the benefit of having found a somewhat satisfactory explanation of the cause, but this still does not help us with the solution to the question "How do we fill the void?" The answer, of course, would be "intimacy with God," but what does that /really/ mean?
One my list of problems with science is the attempt to explain away this desire for depth as some sort of result of our biology and mechanical construction. I find this to be a dangerous and morbid simplification of the human condition. It contributes nothing to the discussion of despair, nor does it provide some sort of solution to the "problem". It only gives an excuse for ignoring it, to push it aside. Ignoring something as fundamental to the human condition as despair is a good way to find yourself lost and empty. I think this is where science as a discipline and way of knowing is right now. (There is, of course, some confirmation bias in this characterization of science, but it's something to ponder.)
The more general approach (inclusion in something greater) is more difficult to find any answers to. Why do we need this inclusion? What do we mean, "something greater than ourselves"? How do we gain this inclusion? Alternatively, how do we satisfy this desire? Can this desire be, permanently, or at least satisfactorily fulfilled?
I guess these questions are the next problem in our discussion.
On a related, but perhaps more uplifting note:
It has been suggested (by i forget whom) that hope is impossible without despair. There is a lot of truth in that idea. Hope is the desire (perhaps even need) for fulfillment of something which is outside of your control-- the fulfillment of something greater than yourself. This desire brings you, in some way, into this fulfillment. In other words, hope is the desire for the work of, and thus inclusion in, something greater than yourself. It is a (partial?) fulfillment of despair. Thus, without despair, there can be no hope.
The question, then, is: can we rise above this despair->hope->despair... cycle, or is it the case that "the cure for pain is in the pain"? And, if we can rise above, is it really something we should desire?
(the quote is from "The Cure for Pain" by mewithoutYou; i'd recommend looking up the lyrics to many of their songs...)
I may have had something else to say, but I forgot it.
Cheers,
Cory
*****
Yeah, my language was kind of vague... it wouldn't do in a proof. Let's try again
Seriously. I don't know how you could have made such an error.
It's a scary thing to think about; we have some desire-- some need-- to know everything; what happens if we can't?
EXCATLY!!!!! Yet, could it not be argued-well is obviously can-that life as the potentially beautiful thing that it can be is made possible, maybe even defined by the fact that there never seems to be any concrete answers, except the ones that are semi-hypcritical such as "change is the only constant"? And the search for answers that don't really exist, or maybe do for a limited period of time, defines us......I'm going to stop on the incoherent train of though.
Anyway:
Jamie!
This is Cory, How are you?
You're someone I want to keep in contact with, because it's hard to find
someone who so actively digs into the human brain/soul/heart/whatever...
Anyway, 2 things:
Would you be able to compile a list of my quotes that you wrote down
while you were here? It's always interesting to see how you have
impressed others...
I have a blog of sorts; ramblings and rants on various topics. It's at
onag.blogspot.com
It may present to you probing questions to ask.
Cheers,
Cory
*****
"How do you know whether you are hitting your own personal boundaries, or you're hitting a limit where you can't go farther"
"Everyone has some sort of void in their soul...."
"...that the things that are hilarious are taken far too seriously by far too many people..."
"We have no objective way of determining if our reality is the same as someone else's"
I haven't thought about the first quote yet because I'm terrified that there might actually be a limit where ''you'' can't go farther. That would fuck with my current philosophy of the world. Yet the very act of being terrified implies there is such a limit....I guess I'm just reluctant to reconstruct my notions of the world this month. I think at this point you were referring to some kind of inner personal limit however and not such concepts as the universe and different dimensions and whatnot. That makes is slightly easier to answer, yet is it even necessary to ask questions about the expanse of space seeing as how whatever knowledge we get doesn't affect its very nature...(or does it?) or ours?
-side tangent- why do we even look for answers in general? Is there ever some kind of permanent answer? If we do not ever actually know everything-which we cannot- or even real answers, ones that are really the Truth and not some kind of answers we have decided on for our own life construction and stability, it is very improbably that we could ever actually know them...so what is the point of questions (ah ha) of searching? What makes someone decide on a certain set of values on which to base their lives?
Back on semi-track. Is this limit you were referring to some kind of inner limit? Some 'soul' limit?
Your other quotes suffered a tragic (they were introduced to my car before they made it to my journal) death.
*****
Yeah, my language was kind of vague... it wouldn't do in a proof. Let's try again:
"How does one know whether they are hitting the boundaries of their own comfort, or an insurmountable-- universal-- mental limit." (i use the singular, ambiguous "they", because it is less awkward than the alternatives.)
It's a scary thing to think about; we have some desire-- some need-- to know everything; what happens if we can't?
I don't know whether this limit is some sort of "soul limit" some sort of paradox of self-reference-- i.e. a boundary on our capacity for metacognition.
So. On questions. This is one that has bugged me for quite some time. I've managed to sidestep the question (more, push it out of the way so i don't have to figure it out) by saying i like to learn because there's beauty in it. This is, in part, a cop out. There's truth to it, but that's only a part of the drive.
The real reason is that it seems (at least temporarily) to fill the void in my soul. Hereafter, i'll refer to this void as nostalgia or despair; i'll explain this idea later. Anyway, whenever i think of the quest for knowledge, i think of Unamuno's The Tragic Sense of Life, which i recommend you read if you have questions about our quest for knowledge. Essentially, it's about the very existential idea of despair, the source of that despair and how we use knowledge to combat that despair. I either completely agree or completely disagree with everything he has to say, but it's an interesting read either way.
Anyway, despair:
A lot of philosophers have discussed the concept itself directly, and i find it difficult to think that any philosopher has hoped for anything greater than to understand it. We have, as i've stated, a void in our soul. Despair (the state of being) itself is the recognition of that void, and the conflict created by trying to understand the nature of that void, as well as the struggle to fill or remove that void.
The Christian answer to "what causes despair?" is that we have a longing for an intimacy with God-- an intimacy, which, being fallen creatures, eludes us in except in brief flashes and moments of clarity. This is, i think, an adequate assessment. The general idea can be generalized to be compatible with more humanist philosophies by saying "we have a longing for inclusion in something greater than ourselves." All of human action can be reduced to a (perceived) fulfillment of this desire. The Christian has the benefit of having found a somewhat satisfactory explanation of the cause, but this still does not help us with the solution to the question "How do we fill the void?" The answer, of course, would be "intimacy with God," but what does that /really/ mean?
One my list of problems with science is the attempt to explain away this desire for depth as some sort of result of our biology and mechanical construction. I find this to be a dangerous and morbid simplification of the human condition. It contributes nothing to the discussion of despair, nor does it provide some sort of solution to the "problem". It only gives an excuse for ignoring it, to push it aside. Ignoring something as fundamental to the human condition as despair is a good way to find yourself lost and empty. I think this is where science as a discipline and way of knowing is right now. (There is, of course, some confirmation bias in this characterization of science, but it's something to ponder.)
The more general approach (inclusion in something greater) is more difficult to find any answers to. Why do we need this inclusion? What do we mean, "something greater than ourselves"? How do we gain this inclusion? Alternatively, how do we satisfy this desire? Can this desire be, permanently, or at least satisfactorily fulfilled?
I guess these questions are the next problem in our discussion.
On a related, but perhaps more uplifting note:
It has been suggested (by i forget whom) that hope is impossible without despair. There is a lot of truth in that idea. Hope is the desire (perhaps even need) for fulfillment of something which is outside of your control-- the fulfillment of something greater than yourself. This desire brings you, in some way, into this fulfillment. In other words, hope is the desire for the work of, and thus inclusion in, something greater than yourself. It is a (partial?) fulfillment of despair. Thus, without despair, there can be no hope.
The question, then, is: can we rise above this despair->hope->despair... cycle, or is it the case that "the cure for pain is in the pain"? And, if we can rise above, is it really something we should desire?
(the quote is from "The Cure for Pain" by mewithoutYou; i'd recommend looking up the lyrics to many of their songs...)
I may have had something else to say, but I forgot it.
Cheers,
Cory
*****
Yeah, my language was kind of vague... it wouldn't do in a proof. Let's try again
Seriously. I don't know how you could have made such an error.
It's a scary thing to think about; we have some desire-- some need-- to know everything; what happens if we can't?
EXCATLY!!!!! Yet, could it not be argued-well is obviously can-that life as the potentially beautiful thing that it can be is made possible, maybe even defined by the fact that there never seems to be any concrete answers, except the ones that are semi-hypcritical such as "change is the only constant"? And the search for answers that don't really exist, or maybe do for a limited period of time, defines us......I'm going to stop on the incoherent train of though.
I don't know whether this limit is some sort of "soul limit" some sort
of paradox of self-reference-- i.e. a boundary on our capacity for
metacognition.
hmmm.....well, I think in order to even start going about an answer to that question we need to define what a 'soul' is.
So. On questions. This is one that has bugged me for quite some time.
I've managed to sidestep the question (more, push it out of the way so i
don't have to figure it out) by saying i like to learn because there's
beauty in it. This is, in part, a cop out. There's truth to it, but
that's only a part of the drive.
Agreed.
Anyway, despair:
A lot of philosophers have discussed the concept itself directly, and i
find it difficult to think that any philosopher has hoped for anything
greater than to understand it.
Thats very eloquent of you.
We have, as i've stated, a void in our
soul. Despair (the state of being) itself is the recognition of that
void, and the conflict created by trying to understand the nature of
that void, as well as the struggle to fill or remove that void.
The Christian answer to "what causes despair?" is that we have a longing
for an intimacy with God-- an intimacy, which, being fallen creatures,
eludes us in except in brief flashes and moments of clarity. This is, i
think, an adequate assessment. The general idea can be generalized to be
compatible with more humanist philosophies by saying "we have a longing
for inclusion in something greater than ourselves."
I felt like making a counter-point at this point in time, not with the idea of what was said, but merely because I felt the need to disagree....which brings me to a more interesting question. If two people agree on an answer, but from two completly different points of view, is it really the same answer?
can be reduced to a (perceived) fulfillment of this desire. The
Christian has the benefit of having found a somewhat satisfactory
explanation of the cause, but this still does not help us with the
solution to the question "How do we fill the void?" The answer, of
course, would be "intimacy with God," but what does that really mean?
One my list of problems with science is the attempt to explain away this
desire for depth as some sort of result of our biology and mechanical
construction. I find this to be a dangerous and morbid simplification of
the human condition. Itcontributes nothing to the discussion of
despair, nor does it provide some sort of solution to the "problem". It
only gives an excuse for ignoring it, to push it aside. Ignoring
something as fundamental to the human condition as despair is a good way
to find yourself lost and empty.
Nice quote. Yet, by filling this void, or trying to make it 'go away' are we not still trying to ignore it or push it aside?
The more general approach (inclusion in something greater) is more
difficult to find any answers to. Why do we need this inclusion? What do
we mean, "something greater than ourselves"? How do we gain this
inclusion? Alternatively, how do we satisfy this desire? Can this desire
be, permanently, or at least satisfactorily fulfilled?
I guess these questions are the next problem in our discussion.
Or does it need to be filled? What if, op, just kidding, way to address that in the next paragraph. Or, somewhat. What if it is this void that not only gives us hope, an hard quality to try to define,? My current theory is that it is this void, this despair that makes life so incredibly, for lack of a better word and approaching class, amazing and difficult and different. I'm just going to ramble now. For awhile, basically my whole life, I've been searching for a way to fill this void, comfort the despair. But more recently I decided to try to embrace it, live whole-heartedly in it. Do you think such a thing is possible? Its rather strange. I don't know what to think about it. In which case there is no need to rise above it but rather incorporate it into one's being.
On a related, but perhaps more uplifting note:
It has been suggested (by i forget whom) that hope is impossible without
despair. There is a lot of truth in that idea. Hope is the desire
(perhaps even need) for fulfillment of something which is outside of
your control-- the fulfillment of something greater than yourself.
[...]
Also, assuming this conversation continues in the direction it is going,
could I post it online?
Feel free. ..... (novel concept :)
I can tell you have absolutly nothing to say about the world. Its interesting. I've missed alot of your ideas this time around because you include so many in each sentence and I'm late to class, but I wonder if you need a reply to the questions you ask, or merely the oportunity to fully do so. Because there is such a ...life? in the moment of asking questions.
Ellie has probably already asked you a question similar to this one-
but if you could have an answer, figure out any one thing, what would it be?
Tuesday, March 25, 2008
Lambda Calculus...
Is weird. Really weird.
Why is it that we can think of sets of sets intuitively, but operating on operators is nearly impossible to conceptualize?
On a related note: It's ridiculous how exactly LISP notation follows Church's function notation. Even things that aren't related (necessarily) to lambda calculus are borrowed from him.
Why is it that we can think of sets of sets intuitively, but operating on operators is nearly impossible to conceptualize?
On a related note: It's ridiculous how exactly LISP notation follows Church's function notation. Even things that aren't related (necessarily) to lambda calculus are borrowed from him.
Saturday, March 22, 2008
Grace, Childishness and Service
Sully wrote this a few months ago; it's one of many truly inspired writings that have flown through him. I hope he doesn't mind me posting it here...
On thought, before I post it. At one point, he mentions knowing something versus knowing about something. I bring up this point often, but only I believe it is a crucial concept to understand. German has different words for each idea. "I know something" would be translated "Ich weiss etwas". The infinitive of the word (to know) is wissen. It implies knowledge of a concept or idea; one would also say "Ich weiss Tom" to mean that he knows about Tom-- has knowledge regarding who Tom is, etc.
"I know Tom" as we say in English would be translated "Ich kenne Tom." The infinitive is kennen. It implies a familiarity, an understand of and closeness with. "Ich kenne etwas" (I am familiar with something) has a similar-- although weaker-- meaning to "I grok something."
Sully brings up this point (briefly) in the essay. God calls us to know Him-- Gott Kennen; not to know about Him-- Gott Wissen.
Anyway, Cheers (I know it's long...):
** The definition and presence of Grace **
The foundation of a Christian life is Christ. This means, more than anything, that we are founded on Grace, for that is Christ’s purpose. He came for us who were unworthy of the favor of God’s presence. We were unworthy not by judgment, but by action because we chose to step from that Presence into the darkness that is absent Him. Thereby we drew ourselves away from God, not He from us.
Having stepped into darkness, we were lost and unable to ever find Him there, for though we may or may not be able to persevere the breadth and width of darkness by religion and other machination, God does not exist in that darkness and thereby can not be found by us who do exist in darkness. Christ is God reaching through the darkness to find us and offer to us His hand that we might again walk with Him. For that was certainly his purpose, that God and all God’s love and blessings should come to us who were otherwise eternally incapable of finding that Presence – this is Grace.
By Grace we are able to enjoy a Fellowship and a Blessing that we could never attain by our own strength. By Grace we who are only capable of at best the most righteous unrighteousness, should be considered (and by being considered, made) truly righteous and able to be near Him who can not coincide with unrighteousness. By Grace, through mysteries indefinite and unresolvable, the presence of God comes without the need of our own understanding or thought.
All Grace requires is choice and by choice action: The choice of accepting the hand outstretched through darkness to us. The choice to recognize the presence of that hand and recognize the necessity of that hand for ever finding God. The choice to give up our comfortable familiarity with uncomfortable darkness and step into an uncomfortable unfamiliarity with all that is truly secure and comfortable. The choice to continue laying every step beside God’s after being brought beside him. The choice to not again step back into darkness. And the choice, when again stepped back into darkness, to accept the loving hand instantly reaching down through darkness to pick us back up from our stumble. Grace is attained only by this choice. Grace is attained only by this action. If we do not choose to accept the hand and choose to walk beside God as he leads and choose to let him pick us back up and choose to simply recognize his hands and our darkness, we have no part with Grace. We then have no part with God and no part with ever being brought out of the darkness. We then choose and by action have sentenced ourselves to living eternally absent God and living eternally absent the Heaven where he does exist.
So it is that only by a persistent walk with God and acceptance of Grace and fellowship with Christ we can receive life. As Christians, we say these things and accept these ideas with lips that swear our hearts know that truth. However, on this point we are liars, for the majority of us are not graceful people. It is this absence of Grace that makes us besmear the label of “Christians”, depend yet on our own strength, judge with envy or malice those around us, and ultimately lose salvation by desecrating our faith down to the same worthlessness of all other religions.
And yet we say, “Yes, I am imperfect, I know that. But that’s just the point. I am allowed to be an imperfect sinner and yet have God and Salvation.” And yet, in saying this statement, otherwise full of truth, we are completely and utterly wrong. For this statement only applies to those who attempt to overcome their imperfection. It only applies to those who begrudge and hate their sin and wish it washed away. Essentially, it only applies to those who hate the darkness and long for the light. As we often say it, and as it was said above, what we really say and really believe is that we are free to yet live in darkness and still have the gift of God’s presence. And this is this same as saying that darkness can exist within full and absolute light - a contradiction that a child can easily see and make fun of, and yet we make ourselves as dumb as adults to believe it.
The true nature of Grace and the true and only power of “Being saved by Grace” are caught up entirely in the calling of Levi. Jesus walks through the city, sees Levi sitting at his tax booth, and walking up to him, says, “Follow me.” Levi could never have brought Jesus there. He could never have found the offering of fellowship with him. And that is why it is an offering of Grace that Jesus walks up and asks him to follow. However, the bestowing and acceptance of this Grace only occurs, that is the fellowship of a life spent in the presence of God only occurs, when Levi responds and “leaving everything, he rose and followed him.” If he had not followed he could not have possibly walked with and talked with and known who Jesus was - he would have refused Grace. Certainly he would have ultimately heard an awful lot about Jesus and possibly even developed an awful lot to say in favor of the things he did, but he would not have know him. This is how we have become as Christians today. We have heard a lot of about God and have developed a lot to say about God, but we know him not. And why? Simply because we choose not to accept Grace and walk with him. We choose to keep our place in darkness rather than be lifted up into Light.
All of us who are offended by these words testify by our own offense that Grace is not in us. For Grace can not be offended, for it is the absence of judgment and prejudice and pride. Grace is the administration of God’s love and as such bears all things that love has and carries nothing that love does not. As Christ says through Paul, “Love is patient and kind; love does not envy or boast; it is not arrogant or rude. It does not insist on its own way; it is not irritable or resentful; it does not rejoice at wrongdoing, but rejoices with the truth. Love bears all things, believes all things, hopes all things, endures all things.” And yet, how many of us love like this? If we do not appear to love like this, then it must certainly also appear that Grace is not in us.
** The presence of self-righteousness in the absence of Grace **
Self-righteousness is the opposite of Grace, just as the darkness of sin is the opposite of God’s light. Self-righteousness is the belief that ones’ own efforts can bring them into the presence of God. It is the belief that Light can be found in darkness. It is the perpetual selfish searching for things that do not exist at the cost of what little does exist. And yet, is it so surprising that so many are self-righteous? Should they be judged and dismissed for being self-righteous? No! Rather, those who are neither in Grace nor self-righteous are all the more foolish. For the self-righteous recognizes the need to find the Divine above all else and merely disregards the Only Way. The other is foolish enough to also disregard and lie about their own hunger. If one does not accept Grace then what other strength can you possibly believe will ever bring you life but your own? And, as for those in Grace, they will never look in judgment upon the self-righteous but rather look with eyes of patient and empathetic love, just as they will look upon all who are outside Grace.
The pattern of self-righteousness is this, all who are deemed beneath one’s righteousness are generally ‘loved’ and all those who are deemed unfairly above one’s righteousness are always enviously hated. All who are beneath the righteousness of the self-righteous are loved in the most perverse sense of benign indifference. They serve the self-righteous person in that by being beneath him or her, they are believed to lift him or her up closer to the Divine. As such, the self-righteous are willing to be benevolent to these under-souls and also match their benevolence with a thousand times the evil by stepping on the backs of these under-souls and press-ganging them to abide by whatever boundaries of self-righteousness they suggest will lead to the Divine. Those who will not be press-ganged into the self-righteous person’s particular religion, or otherwise suggest the true unrighteousness of the self-righteous, become competition and are hated.
Those who follow the self-righteous person’s views but appear more righteous can either be hated, or accepted in the same way as the under-souls. By being above they can seem closer to the Divine and thereby offer both an affirmation of the self-righteous person’s methods as well as present a human ladder for the self-righteous person to pull themselves higher. In that case those above are ‘loved.’ Alternatively, those above may be seen as obstacles of separation to the divine and by their presence confirm how unrighteous the self-righteous person truly is. In such a case, it is but the logical choice to hate and envy.
Those people who are both outside the self-righteous person’s beliefs of righteousness, appear to be about equivalently as righteous, and who suggest that the self-righteous person is unrighteous, are hated with passion. This is particularly true if such a person otherwise abides by nearly the exact same patterns and beliefs of righteousness. For, in this case, not only is the self-righteous person in struggle for closeness to the Divine, but they are being told that their struggling is incorrect and thereby become even less righteous than they had thought they were.
Of course, all of these words can be confusing, and on this particular previous point it is perhaps of especial benefit to provide an example: Sam believes that it is ok to drink alcohol with friends and be a Christian because Sam otherwise follows all the proper rules of conduct and goes to church and participates in a small group. Hunter, from Sam’s small group, tells Sam that drinking alcohol is ungodly and sinful and that Sam is a poor Christian unless Sam repents of drinking alcohol. They have an argument on this point and begrudge each other about it. Later, Sam sees their pastor give recognition to Hunter for being an exemplary Christian. Sam now enviously hates Hunter. In this case, both Sam and Hunter are self-righteous and thereby both are absent of Grace and thereby also absent of Christ. They have forsaken Christ’s hand for belief that following the straightest lines drawn in darkness will give them a path to Him. They merely argue about whose line is straighter. And yet, both are in fact in darkness. And yet, the hand of God and Christ’s Grace is right there reaching down to them, begging them to take it again or for the first time, depending on if they ever took it before.
Another example that perhaps will touch the reader’s soul better is this: Sam and Hunter have never officially met but are aware of each other’s presence as members of the same church. Hunter has been working hard helping her church. She has been making bulletins and organizing meetings and orchestrating prayer gatherings and leading a small group. Hunter has never seen Sam do anything for the Church. Hunter hears their pastor say that Sam is an incredible Christian and that Sam means a lot to the body of Christ. Hunter now envies Sam and while she will not admit it, actually hates her simply by the fact that she refuses to love her. In this case it appears that Hunter is the only one that is self-righteous and the only one whose monstrous efforts do more to help her reject Grace than bring her close to Christ.
For a final look to expound upon this point of self-righteousness and the absence of Grace, I will return to the example of Levi: Jesus looks across the market and sees Levi, a tax collector, and walking past Rabbi’s and the most ‘righteous’ and worthy of men, goes up and invites the rather ‘unrighteous’ Levi, not the Rabbi’s and Scribes, to walk with him and absorb his love. This simple fact that God is ignorant of all personal effort is what makes grace into Grace. And does this seem wrong? Well, from the framework of self-righteousness it is wrong. If you believe that your personal effort is necessary then you will begrudge those who have God’s love despite lack of effort. That exchange of love is proof that you and all your pride are wrong and that your ideas and efforts are worthless.
The trouble with all this is that it leaves us at the point of the contemporary church, which has accepted that it is not by our efforts but by God’s effort that we know Him. And again, this is true. What is wrong is that we have extrapolated this to mean that Faith only requires God’s effort and not our own. Which really means that we have rejected the idea that we need to take God’s hand to be lifted up by Grace. This leaves us at the point of being able to stay in sin and conduct our lives of darkness, living in richness and sloth saying, “If God intends differently, then He will make it so.” And the truth is that God will make it so. He will overwhelm Earth by the presence of Heaven and the old will go. The new will come – and we will not be a part of it because we did not step from the old when he asked us to walk with Him. We will be in Hell for having rejected Grace.
Now, if we have rejected Grace, then it would seem, as mentioned before, that most of us would fall back into self-righteousness. And that has been outlined somewhat above and many will be able to prove or lie well enough that they are not part of that self-righteousness. However, for us Christians that have rejected Grace by the false belief that we need make no action and that it is all about God, we are left with an entirely different sort of self-righteousness. We come to the point of saying that we have Grace simply by believing in it. That is, we think that it is what we think that grants us God’s love - so we have descended to a self-righteousness not of physical actions but of thoughts.
We argue theology day by day. We pick and choose our spiritual heroes. We form lines in the sand over how we understand particular points. We become upset and horrified at people thinking differently than us - And we are self-righteous and absent Grace and know not Christ.
Now these points are particularly subtle, extraordinarily so. This is why this is the latest and greatest of Satan’s defeats to our Faith. For is it necessary that we believe in Christ and the necessity of Grace and in the love of God? Yes, most definitely – And yet we can believe such words and not have any of those! In fact, we can believe many many more words than all that and have even less!
The trouble comes down to the fact that all these thoughts still have not accepted Grace. You can believe all you want that Christ’s hand is reaching down to you and that it is necessary for you to take it, but if you do not take it you simply do not have it. The act of taking that hand does requires the thought to take that hand and we have failed by equating simply thinking about taking that hand with actually taking it. So in this way we are left in a world of thoughts and absent Grace.
Our self-righteousness becomes one of imagining that if we think hard enough and strong enough and with enough conviction about the means and method of taking God’s hand and the shape of that hand and about where that hand is going to lift us, we will have attained what that hand offers. In simpler terms, if someone invites me and my friend to a walk in the park and we say no and instead imagine walking in the park, have we actually walked in the park? I might make a better guess than my friend at what the weather is like in the park and my friend might make a better guess at how long the walk will be and we both might well enough predict where the walk would traverse, but have either of us actually walked in the park? And yet, we delude ourselves into believing that if we imagine well enough and accurately enough about the walk in the park it is the same as walking in the park! We as Christians believe that if we imagine and believe and know enough things about God and Christ and Grace we will have those things!!!
And so it is that we argue and envy and despise anyone who contradicts us or our thoughts on Christ. We hate and judge in proportion to how overtly contradictory their assertion is to whatever ideas we have accepted. And in all of this, we know the least part of Christ and Grace. To be fair, we may even know accurately about these things, but we do not know these things! Do you meet someone’s strange assertions about God with curiosity and love, or at the very least sympathy? If not, then you are intellectually self-righteous and know not grace. For if you knew Grace you would not need your self-righteousness and would not need to violently argue or judge your fellow human brother or sister. You would not feel threatened. You would not feel frustrated. You would not be absent of love. You would not be absent of Christ. – And yet, even here, in this error, Christ waits with loving arms, not just hands, waiting to pick us up and let us truly know him and love!
To see how incredible this Grace is, let us return again to Levi. When Jesus walked up to Levi, was it necessary for Levi to know this was Jesus? If Levi did not know that this was Jesus. If in contradiction to all other evidence he thought it was some random bum off the street, and yet followed, would he have received any less of the blessings of Christ? Of course not! You see that the Grace is in the following, not in the knowing or thinking. Now, if Levi were to have made believe that Jesus did not exist and that no one was calling him to follow, would he have missed Grace? Yes. However, as long as we recognize Christ’s presence and follow, whether we believe him teacher, beggar, liar, or even Jesus, nothing else matters – there is the same benefit for all perspectives! Levi has shown us that all that matters is following, and that as long as we follow, false beliefs and thoughts can do nothing to us! Such is the wonder of Grace!
** A note on the usefulness of correction **
All of the discussion presented so far has relied greatly on what Grace does not look like to provide proof of its absence. Henceforth, it is of interest to talk, in more hopeful and brilliant life, about what Grace is and how it benefits. In both the former and proceeding cases of negative and positive, the contradictions presented are for the purpose of showing the need to seek out and accept Christ and Grace. On this point, it is the nature of Grace and Christ to point out errors and fallacies, which we Christians certainly do well enough. However, the entire urgency of Graceful pointing is absolutely for the purpose of directing hearts to Christ. By pointing out lack of love to the Pharisees, Christ was pointing out the need for love and thereby the need for himself. It was out of love and the desire to lead others into His Grace that Christ corrected others. This is the only useful means of contradiction. To contradict as we do today, simply for the sake of ‘asserting truth’, is to waste breath and at its best, is merely a means of sustaining self-righteousness. To contradict for the sake of opening another’s eyes that they might see Christ and then long after Him better, not just for ‘opening another’s eyes’, is the only benefit of contradiction. Contradiction as a means purely of judging another and finding them incorrect is a motivation of pride and sin. In one is self-service, in the other love and selflessness.
Further, the nature of Christ does not actually care about the correction of things pointed out. That is, errors are irrelevant in contrast to the need of Christ they point out. In this way Grace does not seek to instruct others in a manner of correcting appearances, but in the manner of seeking Christ. If anything is amiss, Grace instructs ignorance of that error and focus on Christ and from the focus on Christ correction of that error (and correction of many other errors unmentioned and unperceivable) will come. In this way one does not say, “You are not caring for you neighbor” with the intent of the offender actually then going and caring for their neighbor. Rather, they say it with intent that they go and accept the Grace of Christ. And then following this acceptance, the offender will be filled with such love that can not but help caring for their neighbor, (and their brother, and their parents, and those living in distant cities, and all of creation.) All of this is the manner of Christ’s correction, and it cares least of all about focus on correcting any error pointed out. It abides that error quietly in love for the sake of love that the one in wrong may find God’s love. This is Grace, and if it were not this way, none of us could ever stand under the burden of all our faithlessness.
** The necessity and birth of Childishness by Grace **
Grace is God giving us his love and blessings without any other requirement upon us than that we acknowledge and follow Christ. The blessings of God are the assurance of his ultimate rescue through all things, the assurance of his ultimate eternal presence, and the assurance of his ultimate unending sustenance in the realm of Heaven. In a word, Security. The blessings of God are Security. They are a security unlike any security the world offers. This Security is what people long after when they seek the Divine through the plans of all their self-righteousness. And, quite simply, it is for this reason that Grace frees us from self-righteousness and any effort of our own to attain Security. We have Security as a perpetual guarantee written upon our hearts and are thereby free to give up all the struggles of the world.
Now, if we have Security without the need of self-righteousness, what things does this means we can give up and cast off? Mostly, it means that we can give up all the rules imposed by the self-righteous and we can give up the need for strength. For truly, the only purpose of gathering strength in this world is for self-righteousness. In the world of Christ’s Light, He desires to be our strength and wash our feet and give us ease. This does not mean that we will not gain strength, for actually we gain far more strength by Grace than we ever would have by our most violent efforts – again, such is Grace. What this simply means is that we need not be focused on gathering and accounting for strength.
The greatest antonym known today for strength and pedantry of rules, is childishness. And what is childishness? Childishness is the naïve belief that one is protected, secure, can play whatever, can run free without a single care and all this for merely trusting completely and again naively in the absolute power and love of a parent. Effectually, Childishness is then Christianity, for God is our Perfect Parent.
Now, some will say that there are still rules even here, and that will be gotten to, but for the mean time consider that all of this is exactly what Christ intended when he called us to meet him like children. We were called to fear not the world’s judgment, to play each day of our life, and to depend completely and ultimately on him with the same clinginess of any child. For that is exactly what it means to be a child. And if we do truly believe in Christ and accept Grace and have Security, then why should we not be this way? As everything we are to have comes through Grace by God’s power and not the world’s, then we really do not need fear anything the world can do to us. If we are freed by Grace from the struggles and efforts of attaining Security, then what is left to do but play each and every moment of our lives? And if all of this does come by Grace from Christ, then how can we not ravenously and fearfully check every moment to see that Christ is always there beside us within grasp? This is all the pattern of a true Christian life. These are the trappings of Grace.
Childishness can not exist within the world of strength and self-righteousness. Pure childishness requires the complete and perfect service of another. If self-righteousness is all about what we can do for ourselves, and a child is meant to trust to have to do nothing for themselves, then self-righteousness seeks to kill the child. And how does it do this, rather, how do we in all our self-righteousness do this? Day by day we demand and require children to learn how to support themselves. We tell them day by day to play less, to do more for themselves, and to behave in a manner of fearing what the world will do to them. We do not teach them that they can be children and have Security, but rather that Security is only attained by killing the child in them. We teach them this because we were taught this because the world taught us this. And that is exactly the point; the world taught us this, not Christ. If we truly had Grace, we would not teach this to our children and to each other. We would simply have no need to teach and propagate this living death of self-righteousness. We would see how much better it is to become children. More than that, we would be instantly transformed by Grace into children as that is all that would be left to us after being lifted out of darkness.
In the world of the self-righteous, the weak serve the strong. For the strong force the weak to serve, doling out strength and the promise of Security as a reward for service. Strength is a necessity to force one’s way to the top and hopefully somehow attain Security. And no weak or strong person can do anything to change this system because there is always someone stronger enforcing the system.
In the world of Grace, the strong serve the weak. For the strong have no need of serving themselves since they (and the weak) have Security. And as the strong are strong, it makes little sense that it should be any other way than that they should serve the weak. Quite simply, the strong are much better at serving the weak than the weak are at serving the strong. By the presence of Love through Grace, it is the desire of everyone to serve everyone else. There is no frustration in this service and no conflict over serving - as that is not the loving desire to serve another but the hateful and selfish desire to serve one’s own plans and pride. Graceful service bears total patience, total empathy, and total joy. No one bemoans their weakness. No one bemoans the modesty of their service - for cares about the gloriousness of one’s abilities and the grandness of service are cares only of the self-righteous who still struggle for Security. Those in Grace do not only think they are freed, but are actually freed from this care. And, in the world of Grace, since each person has Security, each person rarely has any need, and play predominates and play becomes service. In truth, play is nothing more than the desire to serve expressing itself through each player pretending to have a need to be served. Each pretends to be less than they are that the other might be more. This is true playfulness and true Childishness. All of this is true Grace.
Now, as far as rules are concerned within the freedom of Childishness in Christ, rules are the commands of God. However, they are no different than the rules of earthly parents: “Look both ways before you cross the street”, “Brush your teeth before you go to bed”, “Don’t hit your sister”. Each of these rules is intended for our protection and uplifting and each of these rules requires no more strength or effort than we already have. And this is the difference between the world’s rules and the rules of the true God. They are not burdensome, are not filled with malice, and do not demand our ‘growth’ and the death of our childish nature. As such, these rules are nothing more than God in all his strength serving us who are weaker, and require in no way self-righteousness or strength in trade for love. If they did, they would be rules of the world and not gifts of God.
Grace creates the Childishness commanded by Christ. Without this Childishness, there cannot be Grace. And if there is not Grace, there is not Christ. And yet, with Grace, the constantly offered Grace, look what life there is! Look, to be a child, to be free, to have Security, and to have Christ, all for ever! And what does Christ ask!? Merely that you follow him and accept Grace!
On thought, before I post it. At one point, he mentions knowing something versus knowing about something. I bring up this point often, but only I believe it is a crucial concept to understand. German has different words for each idea. "I know something" would be translated "Ich weiss etwas". The infinitive of the word (to know) is wissen. It implies knowledge of a concept or idea; one would also say "Ich weiss Tom" to mean that he knows about Tom-- has knowledge regarding who Tom is, etc.
"I know Tom" as we say in English would be translated "Ich kenne Tom." The infinitive is kennen. It implies a familiarity, an understand of and closeness with. "Ich kenne etwas" (I am familiar with something) has a similar-- although weaker-- meaning to "I grok something."
Sully brings up this point (briefly) in the essay. God calls us to know Him-- Gott Kennen; not to know about Him-- Gott Wissen.
Anyway, Cheers (I know it's long...):
** The definition and presence of Grace **
The foundation of a Christian life is Christ. This means, more than anything, that we are founded on Grace, for that is Christ’s purpose. He came for us who were unworthy of the favor of God’s presence. We were unworthy not by judgment, but by action because we chose to step from that Presence into the darkness that is absent Him. Thereby we drew ourselves away from God, not He from us.
Having stepped into darkness, we were lost and unable to ever find Him there, for though we may or may not be able to persevere the breadth and width of darkness by religion and other machination, God does not exist in that darkness and thereby can not be found by us who do exist in darkness. Christ is God reaching through the darkness to find us and offer to us His hand that we might again walk with Him. For that was certainly his purpose, that God and all God’s love and blessings should come to us who were otherwise eternally incapable of finding that Presence – this is Grace.
By Grace we are able to enjoy a Fellowship and a Blessing that we could never attain by our own strength. By Grace we who are only capable of at best the most righteous unrighteousness, should be considered (and by being considered, made) truly righteous and able to be near Him who can not coincide with unrighteousness. By Grace, through mysteries indefinite and unresolvable, the presence of God comes without the need of our own understanding or thought.
All Grace requires is choice and by choice action: The choice of accepting the hand outstretched through darkness to us. The choice to recognize the presence of that hand and recognize the necessity of that hand for ever finding God. The choice to give up our comfortable familiarity with uncomfortable darkness and step into an uncomfortable unfamiliarity with all that is truly secure and comfortable. The choice to continue laying every step beside God’s after being brought beside him. The choice to not again step back into darkness. And the choice, when again stepped back into darkness, to accept the loving hand instantly reaching down through darkness to pick us back up from our stumble. Grace is attained only by this choice. Grace is attained only by this action. If we do not choose to accept the hand and choose to walk beside God as he leads and choose to let him pick us back up and choose to simply recognize his hands and our darkness, we have no part with Grace. We then have no part with God and no part with ever being brought out of the darkness. We then choose and by action have sentenced ourselves to living eternally absent God and living eternally absent the Heaven where he does exist.
So it is that only by a persistent walk with God and acceptance of Grace and fellowship with Christ we can receive life. As Christians, we say these things and accept these ideas with lips that swear our hearts know that truth. However, on this point we are liars, for the majority of us are not graceful people. It is this absence of Grace that makes us besmear the label of “Christians”, depend yet on our own strength, judge with envy or malice those around us, and ultimately lose salvation by desecrating our faith down to the same worthlessness of all other religions.
And yet we say, “Yes, I am imperfect, I know that. But that’s just the point. I am allowed to be an imperfect sinner and yet have God and Salvation.” And yet, in saying this statement, otherwise full of truth, we are completely and utterly wrong. For this statement only applies to those who attempt to overcome their imperfection. It only applies to those who begrudge and hate their sin and wish it washed away. Essentially, it only applies to those who hate the darkness and long for the light. As we often say it, and as it was said above, what we really say and really believe is that we are free to yet live in darkness and still have the gift of God’s presence. And this is this same as saying that darkness can exist within full and absolute light - a contradiction that a child can easily see and make fun of, and yet we make ourselves as dumb as adults to believe it.
The true nature of Grace and the true and only power of “Being saved by Grace” are caught up entirely in the calling of Levi. Jesus walks through the city, sees Levi sitting at his tax booth, and walking up to him, says, “Follow me.” Levi could never have brought Jesus there. He could never have found the offering of fellowship with him. And that is why it is an offering of Grace that Jesus walks up and asks him to follow. However, the bestowing and acceptance of this Grace only occurs, that is the fellowship of a life spent in the presence of God only occurs, when Levi responds and “leaving everything, he rose and followed him.” If he had not followed he could not have possibly walked with and talked with and known who Jesus was - he would have refused Grace. Certainly he would have ultimately heard an awful lot about Jesus and possibly even developed an awful lot to say in favor of the things he did, but he would not have know him. This is how we have become as Christians today. We have heard a lot of about God and have developed a lot to say about God, but we know him not. And why? Simply because we choose not to accept Grace and walk with him. We choose to keep our place in darkness rather than be lifted up into Light.
All of us who are offended by these words testify by our own offense that Grace is not in us. For Grace can not be offended, for it is the absence of judgment and prejudice and pride. Grace is the administration of God’s love and as such bears all things that love has and carries nothing that love does not. As Christ says through Paul, “Love is patient and kind; love does not envy or boast; it is not arrogant or rude. It does not insist on its own way; it is not irritable or resentful; it does not rejoice at wrongdoing, but rejoices with the truth. Love bears all things, believes all things, hopes all things, endures all things.” And yet, how many of us love like this? If we do not appear to love like this, then it must certainly also appear that Grace is not in us.
** The presence of self-righteousness in the absence of Grace **
Self-righteousness is the opposite of Grace, just as the darkness of sin is the opposite of God’s light. Self-righteousness is the belief that ones’ own efforts can bring them into the presence of God. It is the belief that Light can be found in darkness. It is the perpetual selfish searching for things that do not exist at the cost of what little does exist. And yet, is it so surprising that so many are self-righteous? Should they be judged and dismissed for being self-righteous? No! Rather, those who are neither in Grace nor self-righteous are all the more foolish. For the self-righteous recognizes the need to find the Divine above all else and merely disregards the Only Way. The other is foolish enough to also disregard and lie about their own hunger. If one does not accept Grace then what other strength can you possibly believe will ever bring you life but your own? And, as for those in Grace, they will never look in judgment upon the self-righteous but rather look with eyes of patient and empathetic love, just as they will look upon all who are outside Grace.
The pattern of self-righteousness is this, all who are deemed beneath one’s righteousness are generally ‘loved’ and all those who are deemed unfairly above one’s righteousness are always enviously hated. All who are beneath the righteousness of the self-righteous are loved in the most perverse sense of benign indifference. They serve the self-righteous person in that by being beneath him or her, they are believed to lift him or her up closer to the Divine. As such, the self-righteous are willing to be benevolent to these under-souls and also match their benevolence with a thousand times the evil by stepping on the backs of these under-souls and press-ganging them to abide by whatever boundaries of self-righteousness they suggest will lead to the Divine. Those who will not be press-ganged into the self-righteous person’s particular religion, or otherwise suggest the true unrighteousness of the self-righteous, become competition and are hated.
Those who follow the self-righteous person’s views but appear more righteous can either be hated, or accepted in the same way as the under-souls. By being above they can seem closer to the Divine and thereby offer both an affirmation of the self-righteous person’s methods as well as present a human ladder for the self-righteous person to pull themselves higher. In that case those above are ‘loved.’ Alternatively, those above may be seen as obstacles of separation to the divine and by their presence confirm how unrighteous the self-righteous person truly is. In such a case, it is but the logical choice to hate and envy.
Those people who are both outside the self-righteous person’s beliefs of righteousness, appear to be about equivalently as righteous, and who suggest that the self-righteous person is unrighteous, are hated with passion. This is particularly true if such a person otherwise abides by nearly the exact same patterns and beliefs of righteousness. For, in this case, not only is the self-righteous person in struggle for closeness to the Divine, but they are being told that their struggling is incorrect and thereby become even less righteous than they had thought they were.
Of course, all of these words can be confusing, and on this particular previous point it is perhaps of especial benefit to provide an example: Sam believes that it is ok to drink alcohol with friends and be a Christian because Sam otherwise follows all the proper rules of conduct and goes to church and participates in a small group. Hunter, from Sam’s small group, tells Sam that drinking alcohol is ungodly and sinful and that Sam is a poor Christian unless Sam repents of drinking alcohol. They have an argument on this point and begrudge each other about it. Later, Sam sees their pastor give recognition to Hunter for being an exemplary Christian. Sam now enviously hates Hunter. In this case, both Sam and Hunter are self-righteous and thereby both are absent of Grace and thereby also absent of Christ. They have forsaken Christ’s hand for belief that following the straightest lines drawn in darkness will give them a path to Him. They merely argue about whose line is straighter. And yet, both are in fact in darkness. And yet, the hand of God and Christ’s Grace is right there reaching down to them, begging them to take it again or for the first time, depending on if they ever took it before.
Another example that perhaps will touch the reader’s soul better is this: Sam and Hunter have never officially met but are aware of each other’s presence as members of the same church. Hunter has been working hard helping her church. She has been making bulletins and organizing meetings and orchestrating prayer gatherings and leading a small group. Hunter has never seen Sam do anything for the Church. Hunter hears their pastor say that Sam is an incredible Christian and that Sam means a lot to the body of Christ. Hunter now envies Sam and while she will not admit it, actually hates her simply by the fact that she refuses to love her. In this case it appears that Hunter is the only one that is self-righteous and the only one whose monstrous efforts do more to help her reject Grace than bring her close to Christ.
For a final look to expound upon this point of self-righteousness and the absence of Grace, I will return to the example of Levi: Jesus looks across the market and sees Levi, a tax collector, and walking past Rabbi’s and the most ‘righteous’ and worthy of men, goes up and invites the rather ‘unrighteous’ Levi, not the Rabbi’s and Scribes, to walk with him and absorb his love. This simple fact that God is ignorant of all personal effort is what makes grace into Grace. And does this seem wrong? Well, from the framework of self-righteousness it is wrong. If you believe that your personal effort is necessary then you will begrudge those who have God’s love despite lack of effort. That exchange of love is proof that you and all your pride are wrong and that your ideas and efforts are worthless.
The trouble with all this is that it leaves us at the point of the contemporary church, which has accepted that it is not by our efforts but by God’s effort that we know Him. And again, this is true. What is wrong is that we have extrapolated this to mean that Faith only requires God’s effort and not our own. Which really means that we have rejected the idea that we need to take God’s hand to be lifted up by Grace. This leaves us at the point of being able to stay in sin and conduct our lives of darkness, living in richness and sloth saying, “If God intends differently, then He will make it so.” And the truth is that God will make it so. He will overwhelm Earth by the presence of Heaven and the old will go. The new will come – and we will not be a part of it because we did not step from the old when he asked us to walk with Him. We will be in Hell for having rejected Grace.
Now, if we have rejected Grace, then it would seem, as mentioned before, that most of us would fall back into self-righteousness. And that has been outlined somewhat above and many will be able to prove or lie well enough that they are not part of that self-righteousness. However, for us Christians that have rejected Grace by the false belief that we need make no action and that it is all about God, we are left with an entirely different sort of self-righteousness. We come to the point of saying that we have Grace simply by believing in it. That is, we think that it is what we think that grants us God’s love - so we have descended to a self-righteousness not of physical actions but of thoughts.
We argue theology day by day. We pick and choose our spiritual heroes. We form lines in the sand over how we understand particular points. We become upset and horrified at people thinking differently than us - And we are self-righteous and absent Grace and know not Christ.
Now these points are particularly subtle, extraordinarily so. This is why this is the latest and greatest of Satan’s defeats to our Faith. For is it necessary that we believe in Christ and the necessity of Grace and in the love of God? Yes, most definitely – And yet we can believe such words and not have any of those! In fact, we can believe many many more words than all that and have even less!
The trouble comes down to the fact that all these thoughts still have not accepted Grace. You can believe all you want that Christ’s hand is reaching down to you and that it is necessary for you to take it, but if you do not take it you simply do not have it. The act of taking that hand does requires the thought to take that hand and we have failed by equating simply thinking about taking that hand with actually taking it. So in this way we are left in a world of thoughts and absent Grace.
Our self-righteousness becomes one of imagining that if we think hard enough and strong enough and with enough conviction about the means and method of taking God’s hand and the shape of that hand and about where that hand is going to lift us, we will have attained what that hand offers. In simpler terms, if someone invites me and my friend to a walk in the park and we say no and instead imagine walking in the park, have we actually walked in the park? I might make a better guess than my friend at what the weather is like in the park and my friend might make a better guess at how long the walk will be and we both might well enough predict where the walk would traverse, but have either of us actually walked in the park? And yet, we delude ourselves into believing that if we imagine well enough and accurately enough about the walk in the park it is the same as walking in the park! We as Christians believe that if we imagine and believe and know enough things about God and Christ and Grace we will have those things!!!
And so it is that we argue and envy and despise anyone who contradicts us or our thoughts on Christ. We hate and judge in proportion to how overtly contradictory their assertion is to whatever ideas we have accepted. And in all of this, we know the least part of Christ and Grace. To be fair, we may even know accurately about these things, but we do not know these things! Do you meet someone’s strange assertions about God with curiosity and love, or at the very least sympathy? If not, then you are intellectually self-righteous and know not grace. For if you knew Grace you would not need your self-righteousness and would not need to violently argue or judge your fellow human brother or sister. You would not feel threatened. You would not feel frustrated. You would not be absent of love. You would not be absent of Christ. – And yet, even here, in this error, Christ waits with loving arms, not just hands, waiting to pick us up and let us truly know him and love!
To see how incredible this Grace is, let us return again to Levi. When Jesus walked up to Levi, was it necessary for Levi to know this was Jesus? If Levi did not know that this was Jesus. If in contradiction to all other evidence he thought it was some random bum off the street, and yet followed, would he have received any less of the blessings of Christ? Of course not! You see that the Grace is in the following, not in the knowing or thinking. Now, if Levi were to have made believe that Jesus did not exist and that no one was calling him to follow, would he have missed Grace? Yes. However, as long as we recognize Christ’s presence and follow, whether we believe him teacher, beggar, liar, or even Jesus, nothing else matters – there is the same benefit for all perspectives! Levi has shown us that all that matters is following, and that as long as we follow, false beliefs and thoughts can do nothing to us! Such is the wonder of Grace!
** A note on the usefulness of correction **
All of the discussion presented so far has relied greatly on what Grace does not look like to provide proof of its absence. Henceforth, it is of interest to talk, in more hopeful and brilliant life, about what Grace is and how it benefits. In both the former and proceeding cases of negative and positive, the contradictions presented are for the purpose of showing the need to seek out and accept Christ and Grace. On this point, it is the nature of Grace and Christ to point out errors and fallacies, which we Christians certainly do well enough. However, the entire urgency of Graceful pointing is absolutely for the purpose of directing hearts to Christ. By pointing out lack of love to the Pharisees, Christ was pointing out the need for love and thereby the need for himself. It was out of love and the desire to lead others into His Grace that Christ corrected others. This is the only useful means of contradiction. To contradict as we do today, simply for the sake of ‘asserting truth’, is to waste breath and at its best, is merely a means of sustaining self-righteousness. To contradict for the sake of opening another’s eyes that they might see Christ and then long after Him better, not just for ‘opening another’s eyes’, is the only benefit of contradiction. Contradiction as a means purely of judging another and finding them incorrect is a motivation of pride and sin. In one is self-service, in the other love and selflessness.
Further, the nature of Christ does not actually care about the correction of things pointed out. That is, errors are irrelevant in contrast to the need of Christ they point out. In this way Grace does not seek to instruct others in a manner of correcting appearances, but in the manner of seeking Christ. If anything is amiss, Grace instructs ignorance of that error and focus on Christ and from the focus on Christ correction of that error (and correction of many other errors unmentioned and unperceivable) will come. In this way one does not say, “You are not caring for you neighbor” with the intent of the offender actually then going and caring for their neighbor. Rather, they say it with intent that they go and accept the Grace of Christ. And then following this acceptance, the offender will be filled with such love that can not but help caring for their neighbor, (and their brother, and their parents, and those living in distant cities, and all of creation.) All of this is the manner of Christ’s correction, and it cares least of all about focus on correcting any error pointed out. It abides that error quietly in love for the sake of love that the one in wrong may find God’s love. This is Grace, and if it were not this way, none of us could ever stand under the burden of all our faithlessness.
** The necessity and birth of Childishness by Grace **
Grace is God giving us his love and blessings without any other requirement upon us than that we acknowledge and follow Christ. The blessings of God are the assurance of his ultimate rescue through all things, the assurance of his ultimate eternal presence, and the assurance of his ultimate unending sustenance in the realm of Heaven. In a word, Security. The blessings of God are Security. They are a security unlike any security the world offers. This Security is what people long after when they seek the Divine through the plans of all their self-righteousness. And, quite simply, it is for this reason that Grace frees us from self-righteousness and any effort of our own to attain Security. We have Security as a perpetual guarantee written upon our hearts and are thereby free to give up all the struggles of the world.
Now, if we have Security without the need of self-righteousness, what things does this means we can give up and cast off? Mostly, it means that we can give up all the rules imposed by the self-righteous and we can give up the need for strength. For truly, the only purpose of gathering strength in this world is for self-righteousness. In the world of Christ’s Light, He desires to be our strength and wash our feet and give us ease. This does not mean that we will not gain strength, for actually we gain far more strength by Grace than we ever would have by our most violent efforts – again, such is Grace. What this simply means is that we need not be focused on gathering and accounting for strength.
The greatest antonym known today for strength and pedantry of rules, is childishness. And what is childishness? Childishness is the naïve belief that one is protected, secure, can play whatever, can run free without a single care and all this for merely trusting completely and again naively in the absolute power and love of a parent. Effectually, Childishness is then Christianity, for God is our Perfect Parent.
Now, some will say that there are still rules even here, and that will be gotten to, but for the mean time consider that all of this is exactly what Christ intended when he called us to meet him like children. We were called to fear not the world’s judgment, to play each day of our life, and to depend completely and ultimately on him with the same clinginess of any child. For that is exactly what it means to be a child. And if we do truly believe in Christ and accept Grace and have Security, then why should we not be this way? As everything we are to have comes through Grace by God’s power and not the world’s, then we really do not need fear anything the world can do to us. If we are freed by Grace from the struggles and efforts of attaining Security, then what is left to do but play each and every moment of our lives? And if all of this does come by Grace from Christ, then how can we not ravenously and fearfully check every moment to see that Christ is always there beside us within grasp? This is all the pattern of a true Christian life. These are the trappings of Grace.
Childishness can not exist within the world of strength and self-righteousness. Pure childishness requires the complete and perfect service of another. If self-righteousness is all about what we can do for ourselves, and a child is meant to trust to have to do nothing for themselves, then self-righteousness seeks to kill the child. And how does it do this, rather, how do we in all our self-righteousness do this? Day by day we demand and require children to learn how to support themselves. We tell them day by day to play less, to do more for themselves, and to behave in a manner of fearing what the world will do to them. We do not teach them that they can be children and have Security, but rather that Security is only attained by killing the child in them. We teach them this because we were taught this because the world taught us this. And that is exactly the point; the world taught us this, not Christ. If we truly had Grace, we would not teach this to our children and to each other. We would simply have no need to teach and propagate this living death of self-righteousness. We would see how much better it is to become children. More than that, we would be instantly transformed by Grace into children as that is all that would be left to us after being lifted out of darkness.
In the world of the self-righteous, the weak serve the strong. For the strong force the weak to serve, doling out strength and the promise of Security as a reward for service. Strength is a necessity to force one’s way to the top and hopefully somehow attain Security. And no weak or strong person can do anything to change this system because there is always someone stronger enforcing the system.
In the world of Grace, the strong serve the weak. For the strong have no need of serving themselves since they (and the weak) have Security. And as the strong are strong, it makes little sense that it should be any other way than that they should serve the weak. Quite simply, the strong are much better at serving the weak than the weak are at serving the strong. By the presence of Love through Grace, it is the desire of everyone to serve everyone else. There is no frustration in this service and no conflict over serving - as that is not the loving desire to serve another but the hateful and selfish desire to serve one’s own plans and pride. Graceful service bears total patience, total empathy, and total joy. No one bemoans their weakness. No one bemoans the modesty of their service - for cares about the gloriousness of one’s abilities and the grandness of service are cares only of the self-righteous who still struggle for Security. Those in Grace do not only think they are freed, but are actually freed from this care. And, in the world of Grace, since each person has Security, each person rarely has any need, and play predominates and play becomes service. In truth, play is nothing more than the desire to serve expressing itself through each player pretending to have a need to be served. Each pretends to be less than they are that the other might be more. This is true playfulness and true Childishness. All of this is true Grace.
Now, as far as rules are concerned within the freedom of Childishness in Christ, rules are the commands of God. However, they are no different than the rules of earthly parents: “Look both ways before you cross the street”, “Brush your teeth before you go to bed”, “Don’t hit your sister”. Each of these rules is intended for our protection and uplifting and each of these rules requires no more strength or effort than we already have. And this is the difference between the world’s rules and the rules of the true God. They are not burdensome, are not filled with malice, and do not demand our ‘growth’ and the death of our childish nature. As such, these rules are nothing more than God in all his strength serving us who are weaker, and require in no way self-righteousness or strength in trade for love. If they did, they would be rules of the world and not gifts of God.
Grace creates the Childishness commanded by Christ. Without this Childishness, there cannot be Grace. And if there is not Grace, there is not Christ. And yet, with Grace, the constantly offered Grace, look what life there is! Look, to be a child, to be free, to have Security, and to have Christ, all for ever! And what does Christ ask!? Merely that you follow him and accept Grace!
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Cory Knapp.