## Quantum Mechanics I: Interference

A bunch of people have been asking me about the interpretation of QM.  Now, every interpretation of QM predicts (or claims to predict) the same experimental results in any experiment (or at least, any realistically feasible experiment).  Otherwise they wouldn't be rival interpretations, they would be rival theories, and we would just do an experiment to see who is right.

So before discussing what QM actually means, it's good to get the ground rules down—the ones that all physicists agree are the right ones to use to predict the results of actual experiments.

Let's suppose we're doing a physics experiment, which I am going to describe in an extremely abstract way, because that's the kind of person I am.  A (somewhat idealized) way to describe a certain class of experiments is as follows:  We start out by preparing the initial configuration of the apparatus to be in some particular configuration (or "state"), let's call it A.  For example, we start with a radioactive atom.  We can let the experimental apparatus evolve on its own, isolated from the rest of the world, until it reaches some final configuration.  Then we look inside, e.g. 10 minutes later, and check what the current state of the experiment is.  Perhaps the atom has now decayed into something else.  Let's call this final configuration B.

Several aspects of this description are clearly idealizations.  There are always some limitations in our ability to control and/or know the initial condition A, the system is never going to be completely isolated from the rest of the world no matter how hard I try.  And in some experimental setups this may be a good thing—that is, we may want to deliberately reach in to measure and/or adjust the system, part way through its "time evolution".  (Unlike biologists, we physicists use the word evolution any time anything changes!)   And, a the end of the process, we're never going to be able to measure the final outcome with perfect precision either.

But I'm a theorist so I can ignore the messiness of real life, whenever it pleases me to do so.

Now if the laws of physics were deterministic (and if we know what they are, and we know the initial state completely precisely...) then in principle we could simply solve all of the relevant equations and find out what exactly the final state would be.   So after 10 minutes, A will become some particular B with probability 1.

(In practice, this calculation is often impossible because of phenomena like chaos where (in some systems, not others) the final outcome depends very, very sensitively on the initial conditions.  For chaotic systems, you need to know the initial conditions to exponential precision in order to predict the future.   This is why we can't predict the weather accurately for more than about a few days out, because the number of digits accuracy you'd need to measure things at is proportional to the number of days!)

But the actual laws of physics are stranger than that.  They are not deterministic.  I think I've read that some Philosophers of Science claimed that Determinism was an important assumption underlying the possibility of doing Science at all.  Well, Determinism is false, and yet we scientists still have jobs.  I know, 20-20 hindsight, but it was still a dumb thing to say (if anyone in fact ever said it, which I haven't done the research to confirm...).

So let's try again.  Once again, we'll set up our experiment in a particular initial state A.  But now, there are several possible final states B, B', B'' etc.  Let's suppose we want to calculate the probability for some specific one: B.  So the sane, sensible way of doing this, would be to think of all the different ways that A could evolve in time to become B.  To actually do calculations, you apply the rules of probability theory:

NORMAL PROBABILITY THEORY:

• For any particular process by which A can evolve to B (a history), we find all the things which happened in that history, and calculate the probability of each individual event (using our knowledge of the laws of physics, as worked out from experiment or theory)  Then we multiply those probabilities to calculate the probability of that particular history.
.
• If there is more than one distinct history going from A to B, then we add up the probabilities of each history (since each of them are separate possible ways to get B), to get the total probability of observing B.
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• At the end of the day, the probabilities for all possible final outcomes should add up to 1.

Note that, since probabilities are between 0 and 1, multiplying them makes them smaller, as befits situations where multiple unlikely things need to happen to get from A to B.  On the other hand, adding them makes them bigger, as makes sense if there's more than one way for something to happen.

This is the sort of probability theory which would make sense a priori to our rational minds.  The kind from which one can prove theorems like Bayes' theorem.  But the universe doesn't really work that way either!

One way to think about QM is that it's like a Behind the Looking Glass version of probability theory, where things almost work like how you expect them to,  but not quite.  The basic weird idea of quantum mechanics that instead of assigning each path from A to B a probability (which is a real number between 0 and 1) you assign to each path an amplitude (which is a complex number whose absolute value is less than or equal to 1).

A complex number can be thought of as just a vector lying in a two-dimensional plane.  In order to specify it, you need to know how long it is (the "absolute value" of the complex number) and what direction it points in (the "phase" of the complex number).  Of course, if the absolute value is zero, then the phase is meaningless, since the complex number is just 0.

In QM, the absolute value squared of an amplitude represents the probability for an event to happen.  This is called the Born rule, and it is the necessary interface for getting actual predictions about the world out of the theory.

So let's suppose you have two different possible ways to go from A to B.  (A classic example is the double slit experiment, where a particle passes through a screen which has two holes in it, and then reaches one of several possible locations on the detector.)

If the two possible histories have the same phase, then they constructively interfere and the probability of B happening is more than you would expect, from adding up the probabilities of the two histories.  On the other hand, if the two possible histories have opposite phases, then they destructively interfere, and the final probability is less than you would expect.  In fact, if the amplitudes are equal and opposite, then the total probability of getting to B is exactly 0!

(More generally, amplitudes constructively interfere if they are at an acute angle in the complex plane, and destructively interfere if they are at an obtuse angle.  For right angles, the Pythagorean Theorem + the Born Rule tells you that you get the naive expected answer from just adding up the probabilities.)

So, to summarize, instead of doing the thing that makes sense, you do this instead:

QUANTUM PROBABILITY THEORY:

• For any particular process by which A can evolve to B (a history), we find all the things which happened in that history, and calculate the amplitude for each individual event (using our knowledge of the laws of physics, as worked out from experiment or theory.)  Then we multiply those amplitudes to calculate the probability of that particular history.
.
• If there is more than one history going from A to B, then we add up the amplitudes of each history (since each of them are separate possible ways to get B), to get the total probability of ending up at B.
.
• The probability of observing B is given by taking the absolute value squared of the total amplitude.  Unlike amplitudes, this is always a real number between 0 and 1.  Also, the laws of physics are chosen so that, at the end of the day, the probabilities of all possible final outcomes still add up to 1.  (This requirement is called unitarity).  QM may be weird, but it's not that weird.

(You may wish to go back and compare this, point by point, with the Not-Batshit-Crazy-Probability-Theory earlier in the post.)

So, if you have a system with N different initial states (and therefore N possible final states), you can specify the time evolution over any given time $t$ by writing all of the possible transition amplitudes from each possible initial state A, A', A''... to B, B', B''... in an N x N matrix $U(t)$, with complex numbers in each slot.  If you know about the math of matrices, this matrix is required to be unitary: $UU^\dagger = U^\dagger U = I$.  That's what enforces unitarity, the rule that probabilities add to 1 no matter which state you start with.

On the other hand, if you want to know which specific states are allowed, or which specific unitary matrix to use, then you need to specify a particular quantum mechanical theory, e.g. a harmonic oscillator, or Quantum Electrodynamics., or the Standard Model.  QM is a framework for constructing theories, not a specific theory.  Just like Newton's Law $F = ma$, or the rules of classical physics, are a general framework; only experiments can tell you which particular forces actually exist in Nature.

In the next post of the series, I'll spell out some of the implications of this framework, and then maybe I'll be in a position to talk about interpretation.

Posted in Physics | 13 Comments

## Descent of the Word

The Stoic philosophers are known for insisting that the Passions be ruled by Reason, and that one should avoid fretting over anything one can't control.  What is less well known is that they believed in Ethical Monotheism, or something close to it.

Greek religion, not being based on revelation, did not have clear and precise doctrines about the divine.  When a Greek person said "Zeus" they might mean the limited mythological god that appears in Homer (the one who was born at a particular time, squabbled with other deities and had scandalous love affairs) or they might mean the Highest God, who is above all and created everything—what we would call God.  Sometimes one and the same document is inconsistent enough to have it both ways.  e.g. the Phaenomena of Aratus, which begins by invoking the Zeus who fills all of Nature and provides all blessings, later (with a somewhat embarrassed "if,  indeed, the story be true") recounts myths about Zeus being hid in a cave during his childhood!

When St. Paul preached his Sermon on Mars Hill to the Athenians, he could have told them that their religion was completely false and wrong, that Zeus was completely different from the God of Christianity.  Instead he chose to begin with those aspects of Greek culture which pointed to the true God.

There have been pious men in all cultures who have realized that there is one highest God, worthy of all worship.  And whether they prayed to El or Yahweh or Zeus or Brahman or Allah, he heard them.  (If it were forbidden to use the name of pagan deities to refer to Yahweh, then we'd better stop using the term Deity, since Deus is just another form of Zeus.  For that matter we'd better stop saying "God", since that term also originally came from pagan worship ceremonies.  And we would have to throw out the New Testament as well, since it uses θεος (theos), which was also used for polytheistic gods!)  Please note, I am not saying that all religions are the equal or the same; there are many important differences between religions and it matters which one we believe.  Yes, the Jews are the Chosen People.  But God is not only the god of the Jews, but of the pagans also.

With that excessively long introduction, I now present the "Hymn to Zeus", written by Cleanthes (c. 330 - c. 230 BC), the second leader of Stoicism:

Most glorious of the immortals, invoked by many names, ever all-powerful,
Zeus, the First Cause of Nature, who rules all things with Law,
Hail! It is right for mortals to call upon you,
since from you we have our being, we whose lot it is to be God's image,
we alone of all mortal creatures that live and move upon the earth.
Accordingly, I will praise you with my hymn and ever sing of your might.
The whole universe, spinning around the earth,
goes wherever you lead it and is willingly guided by you.
So great is the servant which you hold in your invincible hands,
your eternal, two-edged, lightning-forked thunderbolt.
By its strokes all the works of nature came to be established,
and with it you guide the universal Word of Reason which moves through all creation,
mingling with the great sun and the small stars.
O God, without you nothing comes to be on earth,
neither in the region of the heavenly poles, nor in the sea,
except what evil men do in their folly.
But you know how to make extraordinary things suitable,
and how to bring order forth from chaos; and even that which is unlovely is lovely to you.
For thus you have joined all things, the good with the bad, into one,
so that the eternal Word of all came to be one.
This Word, however, evil mortals flee, poor wretches;
though they are desirous of good things for their possession,
they neither see nor listen to God's universal Law;
and yet, if they obey it intelligently, they would have the good life.
But they are senselessly driven to one evil after another:
some are eager for fame, no matter how godlessly it is acquired;
others are set on making money without any orderly principles in their lives;
and others are bent on ease and on the pleasures and delights of the body.
They do these foolish things, time and again,
and are swept along, eagerly defeating all they really wish for.
O Zeus, giver of all, shrouded in dark clouds and holding the vivid bright lightning,
rescue men from painful ignorance.
Scatter that ignorance far from their hearts.
and deign to rule all things in justice.
so that, honored in this way, we may render honor to you in return,
and sing your deeds unceasingly, as befits mortals;
for there is no greater glory for men
or for gods than to justly praise the universal Word of Reason.

Cleanthes recognizes that God created the world through his Word, that we are created in his image, that we ought to behave rationally but are foolishly drawn to evil, and that we need for God to intervene to show us the true way, and to save us.

About 250 years later, the Deity heard this prayer and answered it by sending his Word to save us from our sins.  The Word had always existed and was the light of the world.  But now it entered the world.  St. John writes:

In the beginning was the Word,
and the Word was with God,
and the Word was God.
He was with God in the beginning.
All things were created through Him,
and apart from Him not one thing was created
that has been created.
Life was in Him,
and that life was the light of men.
That light shines in the darkness,
yet the darkness did not overcome it.

There was a man named John
who was sent from God.
He came as a witness
to testify about the light,
so that all might believe through him.
He was not the light,
but he came to testify about the light.
The true light, who gives light to everyone,
was coming into the world.

He was in the world,
and the world was created through Him,
yet the world did not recognize Him.
He came to His own,
and His own people did not receive Him.
But to all who did receive Him,
He gave them the right to be children of God,
to those who believe in His name,
who were born,
not of blood,
or of the will of the flesh,
or of the will of man,
but of God.

The Word became flesh
and took up residence among us.
We observed His glory,
the glory as the One and Only Son from the Father,
full of grace and truth.
(John testified concerning Him and exclaimed,
“This was the One of whom I said,
‘The One coming after me has surpassed me,
because He existed before me.’”)
Indeed, we have all received grace after grace
from His fullness,
for the law was given through Moses,
grace and truth came through Jesus Christ.
No one has ever seen God.
The One and Only Son—
the One who is at the Father’s side—
He has revealed Him.

No one has ever came to God, except through his Word; the Word which existed beforehand and gave light to Enoch, Melchizedek, Jethro, Job, Epimenides, Socrates, Plato, Cleanthes, and any of the other pagans who sought after God but who lived before Christ's birth.  But now that Jesus has come into the world, we are rescued from ignorance, and we can now place our faith explicitly in a tangible salvation which was revealed to us from Heaven.

"But blessed are your eyes, because they see; and your ears, because they hear.  For truly I say to you that many prophets and righteous men desired to see what you see, and did not see it, and to hear what you hear, and did not hear it."  (Matt. 13:16-17)

Thank you, Jesus for coming into the world.  And Merry Christmas to everyone!

Posted in Theology | 9 Comments

## Some More Random Stuff

I guess a suitably random place to start is here:

♦  Speaking of children and languages, here's an article about St. Tolkein's history of inventing languages, found on a website devoted to describing all of the languages of Middle Earth.

♦  On the topic of Inklings, St. Lewis wrote a propaganda essay, “The Norse Spirit in English Literature”, with the goal of reconciling Iceland to having been invaded by the British during WWII.  Although, probably the essay reflected his real beliefs, since he was a huge affectionado of Norse literature, as discussed in his autobiography Surpised by Joy, as well as the essay "First and Second Things" (which can be found in God in the Dock, or better yet in the more complete collection C.S. Lewis, Essay Collection and Stories, if you find a cheap enough copy.)

♦  Speaking of which, if you ever time travel back to the WWII era, and need to know who is likely to be a Nazi sympathizer (assuming you can't easily hop back to the future to check their wikipedia articles), here is your definitive guide.  Somewhat revealing concerning its assumptions about social class stratifications which no longer exist in the same form in contemporary America... yet I feel there is still something universal to be learned about totalitarian impulses, which can be extracted from this bundle of prejudices.

♦  Speaking of propagandists, a professional metaphor maker talks about tools of the trade.

♦  And a warning about the use of metaphors to explain science.  Of course, people often think they are getting rid of metaphors and talking literally, when really they are merely changing which metaphor they are using...

♦  A chemist blogs humorous descriptions of substances which no sane chemist should ever work with.  Some samples:

Sand Won't Save You This Time (about Chlorine Trifloride; here's a video.)
Dioxygen Difluoride

And if you liked being terrified by those, here are some more...

♦  If you prefer metaphorical explosions, here's a form of therapy where you insult and challenge the other person, so that they argue against you and thus become more positive and self-confident?  Pretty sure this is not for everyone, but sometimes reverse psychology can do wonders.  Not too surprisingly, it doesn't work properly unless you do it with love and humor.

♦  Sometimes a sense of conventional responsibility (avoiding risks) can make a person do terrible things (such as killing their own offspring through the sin of abortion).

In a similar vein, I'm reminded of a certain woman I knew in college, who was taught by her mother that it was "irresponsible" to marry someone and have kids, before you are in your 30s and have built up a successful career.  (Never mind that biology makes it easier to start a family when you're younger!)  Of course, she still fell in love with people and dated them in the meantime, breaking the heart of one of my friends along the way.

Perhaps we modern people could use to refocus our sense of duty a bit, away from guilt about lack of our own self-advancement, and more towards an old-fashioned sense of "doing the right thing" by other people?

♦  Another of my friends from college has a new blog about the intersection of ecology and theology.

♦  Speaking of theologians, did you know that St. Thomas Aquinas wrote a short book entirely on the question of whether the world could have been eternal?

♦  Speaking of ecology, an interview with Hayao Miyazaki.  (If you haven't seen any of his movies, you should drop whatever it is you are doing now, and watch one.)

♦  Speaking of St. John's College, I was recently besmazzled when I learned that a fellow alumnus (St. Ben Sasse) has managed to get himself elected to the U.S. Senate!  (He has also studied at some lesser institutions such as Harvard, Oxford, and Yale.)

In accordance with tradition, he remained silent for a year after his election, observing the institution.  Then he got up and delivered an insightful, nonpartisan speech describing some of the issues with the Senate as an institution.  (I was able to figure out his partisan affiliation from reading the speech, but it was reasonably subtle.)

I first encountered the speech as it was linked from Sun and Shield, and then when he started talking about Socrates, I said to myself "Could it possibly be???  A Johnnie in the Senate?  But we're so tiny and insignificant in the world's eyes!"  And then I checked his wikipedia page and sure enough, he had an M.A. from St. John's in Annapolis.  (The Masters is basically a condensed version of the undergraduate program).

♦  Arrow's Theorem says that there are no perfect voting systems involving at least 2 voters and at least 3 choices.  They always sometimes lead to paradoxical results.  An example of such a voting paradox arose recently in the 3rd circuit court of appeals.  Be sure to read this comment.  Be sure to scroll down to the comment by "L Pseudonymous" about hypothetical future judges Alpha, Beta, and Gamma...

Regarding the resolution of the paradox, I think for a court of appeals, issue voting makes a lot more sense than outcome voting.  In a legal system based on precedent, we want judges to be focussed on making the rules that make the most sense, not focussed on which parties should win in any given case.  It also makes it easier to determine what precedent is set in future cases.

It especially makes sense to separate votes on standing (i.e. whether the party is sufficiently affected by the situation to be allowed to sue) from the merits of the case (i.e. who is right about the law).  If there's no standing, the Judges have no jurisdiction and are required to dismiss the suit without considering the merits.  (That's because Article III of the US constitution only empowers Judges to decide "Cases" and "Controversies" between actual affected parties, not to issue advisory opinions on abstract questions of law.)

But what if a majority thinks there is standing, and a minority doesn't?  It doesn't seem reasonable that the minority shouldn't be allowed to have an opinion about the merits of the case, once the court has definitively (and precendentially) decided by majority vote that standing exists.  (The other rule would lead to perverse incentives: Judges would be tempted to find standing so that their opinion about the merits could be considered.)

One potential problem with issue voting in general, is that the power to decide which way the "issues" are listed, may determine the outcome of the case.  In fact I seem to recall it's a theorem, that any time there's a voting paradox, the person who decides which order the yes/no  questions are presented in (assuming people vote honestly) can always control the final outcome.   But the distinction between standing and the merits is so fundamental to US judicial proceedings (and the order to consider them in is also clear), that at least these two stages can be separated, without such ambiguity.

♦  An article about the eccentricities of J.H. Conway, one of the greatest living mathematicians.  Most famous among outsiders for his cellular automaton "Life", but he also made important contributions to Group Theory, invented Surreal Numbers (useful for the theory of games), and a bunch of other things.

♦  And on the topic of games, here's a free game you can download, invented by a group of radical Bayesians, to see if your probability estimates are properly calibrated.  It's like a trivia game, but you have to decide how sure you are that your guess is right, and the scoring system is designed so that honest play is the best strategy (but you don't need to understand why, in order to enjoy the game).

## Baths

Dear Aron,

I hope and pray you are doing fine.

I will try to keep my comments short (perhaps more will come later, for what they are worth).

In 2 Chronicles 4:5 of the KJV (King James Version), one will see "received and held three thousand baths."

Can you comment on it - such as if you find anything significant in it?

Thank you.

i7sharp

Dear i7sharp,
This verse refers to the basin in Solomon's temple (sometimes called the "Sea"), which the priests were to use for ceremonial washing, before beginning their work on the daily sacrifices and offerings, as commanded in the Torah.

A "bath" was an ancient Hebrew measure of liquids. Unfortunately, we don't know exactly how big it was, but the early biblical commenters put it at around four or eight gallons.  So what I find most significant here is that this is a LOT of water; around ten thousand gallons!  It would have been a very impressive sight.

(Some of the measurements in the Temple may have numerological significance, but I don't see any particularly obvious meaning associated to the the number 3,000.  Also, the parallel passage in 1 Kings 7:26 has 2,000 baths instead; one of those minor discrepencies which maybe indicates that the Hebrew historians weren't quite as concerned with precision of detail as a modern historian might be.)

Spiritually speaking, the items in the Temple all prefigure the work of Christ. Water is used to wash away filth, so the giant basin of water represents the vast mercy of God, big enough to wash away the worst sins.  The fact that the priests had to wash before beginning their duties, shows the necessity of repentance before we can draw near to God.

In the New Covenant, we are reminded of the same symbolic truth by the ritual of Baptism.  However, unlike the priests (who had to wash many times), Christians are baptized only once, in order to show that Christ's sacrifice is more effective than animal sacrifice.  It is capable of causing a permanent cleansing of the human heart, even though of course we do need to continually seek forgiveness regarding day-to-day issues.  As Jesus said, "A person who has bathed all over does not need to wash, except for the feet, to be entirely clean" (John 13:10).  In the same way, Christians need to repent of the sins that arise from time to time, but we should do so in a way which does not deny the work which God has already done in us.

In the Book of Revelation, items from the Temple reappear in the visions to show that the true temple of God is in Heaven.  In particular, there is a Glassy Sea before the throne (4:6), which is associated with the victory of God's saints (15:2).

So that's what I see in this passage. Y ou can find more commentaries on BibleHub.

Posted in Theology | 1 Comment

## Bible Reading Plans

Dear Aron,

Can I ask, how do you know so much about the Bible? I've seen your comments before and I get the impression that you know a lot about theological history too. I've always been telling myself I should learn more but I don't where to start?

Best regards,
Andrew

I'll save the Church history stuff for another post, and focus on advice on how to read the Bible here.

As for why I know so much about the Bible, well, I've been interested in theology since my childhood, so that helps.  My Mom was a Sunday school teacher, who talks about the bible all the time, and I also did bible quizzing as a teenager.  I'm a fast reader.  And ah like to think ah'm purty smart two!

But there are many saints who had none of these advantages, and still know the Bible like the back of their own hand.  God promises wisdom to all those who ask him for it, without doubting.  So don't give up!

Reading theological commentaries on the Bible is also helpful, but it is not as important as actually reading the Bible yourself.  If you want to become very knowledgeable about the Scriptures, the first priority is to actually read the entire Bible.  Then do it again.  That'll put you in a better position to judge whether the theologians are bullshitting you or not.

Oh, how I love your law! I meditate on it all day long.
Your commands are always with me and make me wiser than my enemies.
I have more insight than all my teachers, for I meditate on your statutes.
I have more understanding than the elders, for I obey your precepts.
I have kept my feet from every evil path so that I might obey your word.
I have not departed from your laws, for you yourself have taught me.
How sweet are your words to my taste, sweeter than honey to my mouth!
I gain understanding from your precepts; therefore I hate every wrong path.
(Psalm 119 מ (mem))

When you're reading, first make sure you understand the basic meaning of the text.  A Bible with footnotes or study notes may be helpful here, as long as you don't let it disrupt your flow when you don't need it.  (And as long as the person writing the notes doesn't have a theological agenda of their own.  The NIV Study Bible notes are evangelical but otherwise pretty neutral.  Avoid the Scofield Reference Bible, which has a complicated "end times" agenda)  Study bibles also tend to have maps, charts, and introductions to the individual books, which may (or may not) be helpful.  If you get stuck, you can also start looking at alternate translations or commentaries.

If you get confused, feel free to slow down a bit and process more carefully.  You don't necessarily need to understand everything, but if you're reading St. Paul's letter to the Romans, or something else complicated, you might need to work through the ideas verse by verse just to make sure you understand the basic ideas being expressed.  If you do feel you understand the basic literal meaning, then start asking yourself why questions instead.  Or ask yourself how it fits into the big context of the whole story of the Bible.  In a good Bible study or Sunday school class, people do this together as a group, with an experienced guide.

I never did all that much verse memorization, but that doesn't mean you shouldn't!

But if I partly remember a passage that seems relevant for what I'm thinking about, I always try to look it up immediately (if I don't remember where it is, using cross-references, Google search, or a concordance).  Same thing if I'm reading part of the Bible and it reminds me of another part.  Some bibles have "cross-references" to specifically tell you which verses are related to which other verses.  Here's a visualization of a collection of 63,779 cross references, by Sts. Chris Harrison and Christoph Römhild:

You can see from this that the Bible functions more like a neural network than like an ordinary book.  When you know it well enough, it becomes like the third hemisphere of your brain, a bundle of connected ideas which you can use to think, not only about itself, but about other things.  It's your job to figure out what are the cross-references with the things in your own life.

Start by picking a translation which you're comfortable with.  This is a matter of taste, since some translations are more literal, and some are easier to read, and you need to pick the compromise which is right for you.  Before you settle on one, check to make sure you like both the way it translates prose (ordinary narration) and poetry (the Psalms and most of the Old Testament prophets).

No translation is perfect, but here are some I can personally vouch for, at a given level of the accuracy/readability tradeoff.  In order from most to least literal:

more literal
Shocken Bible (Jewish, preserves a lot of Hebraic style, currently Genesis–Kings only)
New King James Version (or the original KJV if you're okay with archaic language)

compromise
Holman Christian Standard Version
New International Version (I grew up with this; I prefer 1984 to 2011)
(New) Jerusalem Bible (a Catholic version, I've only read the old version)

paraphrase (very readable but not as accurate for serious study)
JB Philips (New Testament only)
New Living Version (this or the Philips are particularly good for Paul's letters)
Contemporary English Version (very easy to read, particularly good for OT history)

(But I don't recommend St. Eugene Peterson's "The Message" unless you are incapable of reading anything not "written up" in bestseller cliches, or have read the bible a million times before and need an electrical shock.  This is in a category of its own, way more nonliteral than any of the other paraphrases.  His introductions to the books are pretty decent though.)

I've already discussed some of the issues that come up with translation choices before, in my post on why "word-for-word" translations are impossible.  If you don't know ancient Greek or Hebrew, the next best way to "check" a translation you aren't sure about, is to consult what the same verse says in other translations.  If there's a difference, you know people don't all agree.

Below are some possible methods for reading through the Bible.  I'm basing some of these recommendations off of these suggestions of St. Tim Isbell, the pastor of the church I grew up in.

1) Straight Through

You could just read through all the books in the order they appear in the Table of Contents.  But I don't actually recommend it.  First of all, within each Testament, the books are sorted by genre, not always chronologically.   There's nothing theologically special about that order, in fact the Old Testament books appear in a different order in Jewish bibles.  In the usual Christian order, the Old Testament is sorted into Torah, History, Wisdom, and Prophecy, while the New Testament has a similar order: Gospels, Acts, Letters, and Revelation.

There's nothing wrong with reading cover-to-cover if you want to, but there are several disadvantages:

• One is that you won't get to the New Testament until you're 3/4 of the way throughwhich means if you give up early, you won't get to it at all!
• Another potential problem is that it tends to group similar books together, so you may get bogged down, and it won't be as interesting as if you mix things up from different parts of the Bible.  It's kind of like eating only meat on Monday, vegetables on Tuesday, fruit on Wednesday, carbs on Thursday, and desert on Friday.  It's better to mix things up a bit.
• Also, some books of the bible have identical or nearly identical passages.  For example, the book of Chronicles includes a bunch of summaries of earlier books of the bible, and has chapters which are identical to chapters in Samuel and Kings.  Similarly, the three synoptic gospels (Matthew, Mark, and Luke) are all based on a common outline of Jesus' life.  Unless you are specifically interested in comparing-and-contrasting the similar passages, it makes more sense to space these books out, rather than reading them right next to each other.
• Although Genesis and Exodus are mostly pretty interesting, a lot of people get bogged down in Leviticus because of all the weird laws about sacrifices and what to do about leprosy.  Personally I find that stuff fascinating, but you might think it's tedious.  This leads to a general rule: if you get bored with anything, go ahead and skim it on your first pass.  You can always come back to it again later.  Yes, reading the whole Bible is good, but if you aren't looking forward to reading the next chapter, you might end up putting it off 1 Chronicles 1-9 (nine chapters of genealogy) for months.  If you can't do it, better to keep engaged and moving forward.

2) Arbitrary Order

As an alternative to reading cover to cover, you may wish to simply read the books in a random order, according to your whim and/or the guidance of the Holy Spirit.  Once you've finished a book, simply mark it with a bullet point in the Table of Contents, so you know which ones are left to read.  Then you can try to use up each category of book at about the same time.  And you'll still get the same feeling of accomplishment when you've read it all!

3) Storyline Plan

For a first pass through the Bible, you may wish to just focus on the Storyline, the books which contain the main narrative of the Bible.

St. Tim writes that:

If you've never read the whole Bible story, or if your grasp of Bible stories is all jumbled, then start with this plan. It is valuable to grasp an overview of the whole Bible – and you can do this reading only the most action-packed 30% of the Bible. The other 70% contains alternative views of the same history, side commentaries written by prophets, and poetry. Reading this 30% of the Bible takes about 30 hours. If you read it like you’d read a novel, in 20 minutes a day you’ll grasp the whole Bible story in just 3 months!

For this you just need to read the following list of books in order:

New Testament:
Any of Matthew, Mark or Luke (your choice).
Acts

Old Testament:
Genesis
Exodus
Leviticus chapter 10
Numbers 9-27
Deuteronomy 27-34
Joshua
Judges
Ruth
1-2 Samuel
1-2 Kings
Jonah
Daniel 1-6
Ezra
Nehemiah
Esther

When you've finished, put a bullet point by each of the books you've completed.  Congratulations, you're now about 1/3 of the way done with the Bible!  When you start reading the other books, you can start by looking at a timeline to remind yourself when they were written, to put them into context.

4) Augmented Lectionary Plan

In this section I will describe my mother's plan for reading through the entire Bible while simultaneously following the "lectionary readings" associated with each Sunday.

First let me explain what the lectionary is.  It's a rule for deciding which Scriptures to read on a given Sunday.  A bunch of liturgical churches in North America, including the Episcopalians, Lutherans, United Methodists, Catholics, etc. have all agreed to use the same cycle of readings, the Revised Common Lectionary (RCL), in their public scripture readings every Sunday.

The readings follow the six seasons of the traditional Christian calender, each of which reflects on a particular part of Christ's life:

Advent—anticipating the coming of Christ (including the second coming)
Christmas—the Incarnation
Epiphany—Christ revealed publicly
Lent—his life of discipline and self-sacrifice
(culminating in Holy Week, his last week of life)
Easter—his resurrection and appearances to the disciples
Pentecost (or Ordinary Time)his reign in heaven, and continued work through the Church

So if you think it would be cool to have your Scripture readings match what the time of year, and what a bunch of other Christians are reading, this plan may be for you,

In order to cover as much of the Bible as possible, the Lectionary cycles through 3 years, one focussing on each of the 3 synoptic Gospels (Matthew, Mark, or Luke).  (The Gospel of John is distributed through all 3 years.)  So the cycle approximately repeats every three years, although not exactly due to things like the date of Easter.

Each Sunday has a Gospel and Epistle reading, and usually 2 choices of Old Testament readings, one which goes through the history consecutively, and the other chosen to match thematically with the New Testament reading.  When I use this plan, I like reading them both, and also reading any verses they skipped over.

Unfortunately, even in 3 years the RCL doesn't actually cover the whole Bible, since they tend to focus only on parts suitable for public reading.  Worse still, there seems to be a theological agenda to shield congregations from difficult, violent, or upsetting parts of scripture.  For example, they often excerpt psalms (unless one is reading Psalm 119, where I understand why people might lose patience, the proper unit of a Psalm is the whole Psalm!) and when they do, it is almost always the violent or cursing parts which they remove.   (The most ironic example I know is the reading for Revelation 22:12-14, 16-17, 20-21, which deliberately skips the verse threatening people with Hell.  So of course, they also had to skip the verse saying that if anyone removes anything from the book, they'll be punished with all the plagues in the book.  Who knew that liturgists had the same kind of God-defying kahunas that the Pharaoh of the Exodus had?)

On the other hand, churches that read the Lectionary are still exposed to a significantly more diverse set of Scriptures than the average church that doesn't follow the lectionary.  Unless a church takes special effort to include the whole Bible in their worship, they will normally tend to focus on only a very small subset of the Bible; mostly the nicer parts of the New Testament plus a few very selected and stereotypical pieces of the Old Testament.  (And your pastor is definitely not going to impress me with his extensive knowledge of the minor prophets by preaching on Haggai 1 when he thinks the congregation isn't tithing enough.  I've already heard that sermon several times already.  If you have to preach, try picking a passage you've seldom or never heard preached before.  In some ways it's actually easier to extract the obvious message from a new passage, than to try to say something about the Woman at the Well or the Good Samaritan that nobody's ever said before.)

Sometimes people say, but doesn't having a specific scripture for each Sunday stifle the guidance of the Holy Spirit?  (In that case, why have a private Bible reading plan either?)  My answer: if the Holy Spirit leads you to a particular text, you should definitely listen to that and not do what you were planning to do.  But we need a plan to cover the more normal situation, where there isn't an obvious revelation from God.

Anyway, for those who want to read the entire Bible AND follow the Lectionary, my Mom has created a plan for augmenting the lectionary readings so that you end up read the whole Bible every 3 years.  She does it by mixing in readings during the week which go through various books of the Bible.  Make sure to read the FAQ as well.  The rate is slow enough that if you get off track, you can catch up.  This plan is good for people who are already familiar with the main storyline of the Bible, and want to think about the connections between different parts of the Bible.

The next church year starts, not on Jan 1, but THIS UPCOMING SUNDAY (Nov 29, 2015).  So if you want to start on this plan, that would be an excellent time to start!  But you could jump on board at any time.

5) Chronological Order

You can buy Bibles which purport to put the Scriptures in chronological order.  I've never read through one of these, but it sounds fun.

It raises an interesting question: do you put the books in the order they were written in, or the order the events described in them happened?  (Sometimes these very different, e.g. the Epistles were probably written before the Gospels, but the Gospels describe the life of Jesus which was before when the Epistles were written.  Also, sometimes nobody really knows when a text was written, and people are just guessing.  So caveat emptor.

6) Anything else

Anything else you want to do?  Great!  Whichever plan actually gets you reading more of the Bible, that's the best plan for you!  People are welcome to share their own ideas in the comments.

Another idea: if you're going through the Bible a second time, one way to mix things up is to pick a different translation from your usual choice.  Or, you could try to be on the lookout for a particular broad theme (e.g. Messianic prophecies and other foreshadowings of the New Covenant).  If your chosen theme is narrower (e.g. biblical feasts and fasts) you might focus on the particular parts of the Bible which are relevant for that theme.

Appendix: A Brief Synopsis of the Bible
A while back I wrote this synopsis of the Bible for a Muslim friend, which I reproduce in a revised form here:

The Old Testament books are sorted by genre (The Jews have the same books but sort them into a different order). The first 5 books are called the Torah, the Pentateuch, or the Law of Moses, and are traditionally attributed to Moses. They are a mixture of narrative and laws:

GENESIS — narrative of Creation, the Fall of Adam & Eve, the flood, God choosing the patriarchs Abraham, Isaac, and Jacob, and Jacob's 12 sons who become the 12 tribes of Israel. One of the 12 is Joseph; his brothers sell him into slavery, but Joseph ends up in charge of all Egypt, and rescues Egypt and his family from famine.
EXODUS — The Israelites are oppressed and enslaved, God chooses Moses (and his brother Aaron) to lead them out of slavery, with many dramatic miracles. Introduces the Jewish Passover, the 10 commandments and other laws, construction of the "tabernacle" or tent in which God met them. The Israelites make an idol of the golden calf; God tells Moses he will kill all of them and start over with him, but Moses intercedes by praying to God, and God forgives them.
LEVITICUS laws concerning priests, sacrifices, clean and unclean animals, and other rituals
NUMBERS The Israelites wander around in the desert, sinning many times. God tells them to invade the Promised Land (Canaan), but they don't believe they can do it, and try to stone Moses. Again Moses has to intercede. God makes the Israelites wander around in the desert for 40 years, so that only the children under 20 can enter the land. The people complain of thirst: God tells Moses to speak to a rock and cause water to come out. Moses is so angry with them, he strikes the rock with his staff instead. For this sin, Moses is not allowed to enter the Promised Land.
DEUTERONOMY Moses delivers more laws as the people are about to enter the Promised Land. Most importantly, to "Hear O Israel, the Lord our God, the Lord is One", not to make idols, and to "Love the Lord your God with all your heart, soul, and strength". The death of Moses.

The next set of books are the Historical Books which deal with the history of Israel after the death of Moses.  They seem to be partly based on the writings of prophets like Samuel, Nathan, Gad, Ahijah, Iddo, Shemaiah, Isaiah, and Jeremiah, as well as the official court records and other sources.  Within this section, the books are in rough chronological order.

JOSHUA Moses's assistant Joshua leads the conquest of the Promised Land.
JUDGES a period of time before there were any kings. The Israelites repeatedly became idoloters, were invaded by foreigners who oppressed them as a punishment, and then were rescued by heroes (called "judges") chosen by God.
RUTH a romance story between a Moabite woman named Ruth and an Israelite man Boaz, who marry and become ancestors of David.
1 SAMUEL the life of Samuel, a prophet who was the last judge of Israel. The people demand a king "like the other nations". Samuel says that the king will oppress them, and is displeased because God is supposed to be their King. God says to do it anyway, and Samuel anoints Saul as king. However Saul disobeys God & is replaced by David, the shepherd and musician. David serves in Saul's court, and becomes close friends with Saul's son Jonathan, but eventually Saul tries to kill David, who runs away and refuses to harm Saul. Saul sins by consulting a medium, and is rebuked by the ghost of Samuel. Saul is then killed in battle alongside Jonathan, and David becomes King.
2 SAMUEL the reign of David. David serves God with all his heart, and is blessed by God. David decides to build a permanent building for God. The prophet Nathan tells him that instead God will build David's house, that he will never take away his love from David, that David will have a descendent who will reign forever, and that his son will build a Temple for him. Later, David sins by committing adultery with Bathsheba and killing her husband. Nathan rebukes him, David repents, and God forgives him so that he will not die, but as a punishment tells him that "the sword will never depart from your house".
1&2 KINGS The reign of Solomon, a son of David by Bathsheba. God tells Solomon he can ask for anything he wants, and Solomon asks for wisdom. He becomes the wisest person who had ever lived, and builds God's Temple. He becomes rich and famous and has hundreds of wives. However, in his old age, his wives lead him into idolatry and worshiping other gods, and as a result God divides his kingdom so that his descendents have 2 tribes (called "Judah") while the other 10 tribes become a different country (called "Israel"). The book goes on to describe the kings in Judah (some of which followed God) and the kings in Israel (who nearly all didn't). The prophets Elijah and Elisha protest against the wicked king Ahab in Israel. After several more generations, Assyria conquers Israel. Later Babylon captures Judah, and leads the Jews into captivity for 70 years, as prophesied by Jeremiah.
1&2 CHRONICLES another perspective on the same history.
EZRA & NEHEMIAH After the Persians conquered the Babylonians, they allowed the Jews to come back and rebuilt their Temple and city wall. Under the influence of the righteous priest Ezra, the Jews commit to follow only God and to obey the Law of Moses.
ESTHER The Persian King Xerxes takes a Jewish girl as his Queen. She courageously intercedes to prevent a genocide of the Jews plotted by Xerxes wicked advisor Haman.

The next set of books are called "Wisdom Literature" because they include practical perspectives on what life is like:

JOB — a dialogue about a righteous man Job, who is very rich and powerful. God allows Satan to take away everything he owns, to kill his children, and to afflict him with a horrible disease, to see if he will still serve God. Job's 3 friends come and tell him he should repent because he must have sinned. Job argues with them, saying he was righteous and complains bitterly against God, and asks God to vindicate him. At the end, God comes down and, instead of explaining himself, asks Job questions about Nature which he can't answer, and Job cannot reply. Finally, God says that he is angry with Job's friends "because they did not speak rightly about me, as my servant Job has", and requires that Job offer sacrifices for them so that they can be forgiven. God restores Job's wealth to twice what it was before.
PSALMS a book of 150 hymns (songs) for the Temple worship, about half by King David and the rest by other people. Mostly prayers of human beings to or about God.
PROVERBS advice about living a good life, mostly short sayings by Solomon and others.
ECCLESIASTES philosophy attributed to Solomon about how earthly life is meaningless, so you should find contentment wherever you can, while still obeying God.
SONG OF SONGS erotic poetry celebrating love, also attributed to Solomon.

The final set of Old Testament books are called the Prophets (even though obviously prophets were involved in the other books too, these books are usually involve the message of God given to a single, specific prophet):

The "major prophets" (called that only because their books are longer than the others, not because they are necessarily more important) are ISAIAH, JEREMIAH, EZEKIEL, and DANIEL. Of the 12 "minor prophets", the first 9 are HOSEA, JOEL, AMOS, OBADIAH, JONAH, MICAH, NAHUM, HABAKKUK,   These prophesied before or during the Babylonian exile, warning the Israelites that they would be punished for their idolatry. The nations would then be punished for their sins, and finally God would restore Israel under the reign of the Messiah, David's descendent, who will cause all nations to worship the one true God, and will reign forever.  The last 3 minor prophets, HAGGAI, ZECHARIAH, and MALACHI, were sent to encourage the people after they returned from exile, in the Ezra-Nehemiah period. Malachi was the last prophet of the Old Testament, after than there was a silence of no prophets for 400 years, before the New Testament.

[Catholics accept a few additional books in the Old Testament beyond those listed here.  These were written in the Intertestamental Period: after the death of Malachi, but before the birth of John the Baptist.  They are not included in most Protestant or Jewish Bibles.  Among them, the books I've found most interesting are 1-2 Maccabees which provide some useful historical context for this period, and Wisdom and Sirach (a.k.a. Ecclesiasticus) which are additional books of wisdom/proverbs.  The others are Tobit and Judith (fictional historical romances with obvious anachronisms), Baruch (supposedly written by Jeremiah's secretary) and various Additions to the books of Esther and Daniel.  The Orthodox accept a few more.  But I wouldn't worry about any of these until you've read the books that all Christians accept!]

The New Testament:

This begins with the 4 Gospels of MATTHEW, MARK, LUKE, and JOHN which are biographies of Jesus, the descendent of King David. Beginning with the ministry of John the Baptist, and Jesus' Baptism, they go on to describe Jesus' teachings and miracles. Then comes the Passion, in which Jesus entered Jerusalem, was betrayed to the Jewish and Roman leaders, condemned for our sins and crucified. Then he came back to life again, and appeared to his disciples after the Resurrection, commissioning them to preach the gospel to all nations.

St. Luke also wrote a book called ACTS which describes the early Church after Jesus' ascension into heaven, and how the Holy Spirit came to live inside of every person who believes in Jesus as the Messiah. It also tells about the ministries of the apostles St. Peter (Jesus' disciple) and St. Paul (who persecuted Christians until he had a vision of the resurrected Jesus appearing to him). It describes a vision in which Peter saw a sheet come down from heaven, and to kill and eat unclean animals. Peter protested, but the vision was repeated 3 times. The point of the vision was to explain how God was now going to accept non-Jewish people into the church. It is also why Christians do not have dietary restrictions about clean and unclean animals, like Jews and Muslims do.

Then there the letters (called "Epistles") which give practical instructions for living the Christian life, as part of the Church, in light of the salvation that has come to us through Jesus.  There are 13 letters by Paul to different churches or individual Christians (ROMANS, 1-2 CORINTHIANS, GALATIANS, EPHESIANS, PHILIPPIANS, COLOSSIANS, 1-2 THESSALONIANS, 1-2 TIMOTHY, TITUS, PHILEMON), plus HEBREWS, an anonymous letter to Jewish Christians which was traditionally attributed to Paul but most scholars think it was probably written by someone else in his circle.  There's also 1 letter by JAMES (Jesus' brother), 2 by PETER, 3 by JOHN, 1 by JUDE (another brother of Jesus). These letters describe the theology and practice of the apostles, and we regard them as inspired because Jesus promised that the Holy Spirit would guide the apostles.

Finally, there is a book called REVELATION which involves prophecies about the end of the world (among other things), attributed to St. John in his old age.

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