## My take on Loop Quantum Gravity

A friend of mine from St. John's College, who was recently accepted to a physics doctoral program at Penn State, asked me what my opinion of Loop Quantum Gravity is.  I replied be email, and then I decided, why not tell the world!

Now, Loop Quantum Gravity is the main rival to String Theory as an attempt to quantize gravity, although it only commands about a tenth of the resources that String Theory does.  The people who work on it tend to have more of a General Relativity background than a Particle Physics background, and this tends to influence what types of problems they are trying to solve.

Warning: Unlike my other physics posts, I have made no attempt to make my commentary here accessible to non-physics people.  (Yes, that means every other time I wrote a physics post and nobody understood it, I was the one to blame for not making it accessible enough!)

Einstein's theory of general relativity is background free, meaning that it does not start with any absolute background space or time, but instead allows the spacetime geometry to be dynamically constructed from the evolution of the metric.  A theory of quantum gravity ought to be similar---it ought to be expressed in a way which doesn't depend on the prior specification of any spacetime metric.  I think this is really important, but no one really knows how to do this.  There are many ideas, but they all have various difficulties.

In principle, I think the idea of LQG---to build spacetime out of a discrete, quantum structure---is a very elegant and moving idea.  (I first got interested in quantum gravity by reading the online writings of John Baez, who used to work on LQG.)  Also, the LQG people have a very beautiful quantization of space at one time, in terms of spin networks.  Essentially, by doing a step-by-step quantization of GR at one time (minus the dynamics), making only a few arbitrary choices, they were able to obtain spin networks.  I'm sure you [i.e. the friend I was writing to---AW] know what these are, but let me assure you that they are beautiful and have some deep connections to geometrical ideas.

The next step in the construction of LQG is to decide what the dynamics are.  Technically, this is done either (A) by choosing a "Hamiltonian constraint" in parallel with the Hamiltonian formulation of GR, or (B) in the spin-foam formalism, by postulating some sort of sum over histories assigning an action to each spin foam.  It is here which we encounter the major problem: There is no agreement over how to implement the dynamics!  There are many ideas, but no consensus on what to do.  Implementing dynamics seems to involve some arbitrary choices.  Some of the proposed solutions seem to me obviously wrong (e.g. see Smolin's criticism of Thiemann's Hamiltonian constraint: arXiv:gr-qc/9609034).  There is also a serious danger that by choosing the wrong dynamics, one breaks the diffeomorphism invariance of the theory.  In the Hamiltonian approach this manifests itself in so-called "anomalies in the constraint algebra", while in the spin foam approach it is unclear whether the inner product obtained from the sum over histories really has the necessary gauge invariance.  I summarized these problems in passing, with citations, in the

Thus---even leaving aside the critical hard problem of whether and how a continuum spacetime can emerge from a discrete description (a problem aggravated by the fact that it is difficult to see how any discrete model of spacetime besides causal sets could possibly preserve Lorentz invariance, see arXiv:gr-qc/0605006)---I would say that LQG really doesn't exist yet as a well-defined theory.  Unless you consider dynamics to be an unimportant part of a theory.  And finding sensible dynamics is a really hard problem, perhaps impossible.

Yet, despite the lack of dynamics, there's no end of papers where people do specific applications, like count black hole entropy, or even attempt to do quantum cosmology (basically by truncating the theory to a finite number of degrees of freedom, and then quantizing those degrees of freedom in a way which is "loopy" in spirit).  But all of these things are totally provisional until one can embed them in an actual theory with dynamics.   People used to be really interested in solving these hard problems, but I feel like a lot of them have now given up and are seeking more limited goals.  This is a shame, since I think progress can only come by facing the hard issues head on.  And maybe by showing some flexibility in how the theory is formulated.

Once one has the dynamics, again one can say nothing about the real world until one has identified the correct vacuum state.  An arbitrarily constructed "weave" state that happens to look like some Riemannian geometry doesn't cut it.  You have to figure out how to identify the right vacuum state---the one with lowest energy (once you figure out how to define that!).  Many deep questions here!  I think most people in LQG are asking all the wrong questions.

One can put too much emphasis on quantizing gravity---really that's backwards, we need the classical theory to emerge from the quantum theory, not vice versa.  When people calculate discrete area and volume spectra for spin network edges and vertices, they've got things backwards.  These are just some operators at the Planck scale.  The really interesting question is not, how much "area" is associated with each spin, but how many of each type of spin crosses a given area of the vacuum state (if such a thing even exists).

I despise the ignorant bigotry which most string theorists show towards LQG, even though LQG barely exists as a theory.  Their contempt is undeserved.  The LQG people are trying to do something genuinely harder---to reconstruct spacetime from first principles.  We don't know how to formulate string theory except by means of strings propagating in some background spacetime, or via dualities like AdS/CFT.  Since the theory has gravitons, with a diffeomorphism gauge symmetry, it's clear to me there has to be some background free formulation of string theory, but no one has any idea what this would look like.  And most string theorists don't even understand why it is important.

Personally (and unexpectedly for me) I've found that as someone who studies black hole theormodynamics, I can interface better with string theorists than with LQG people---the ones who are really interested in fundamental concepts, like Don Marolf and others at UCSB, for example---even though I don't really consider myself a string theorist.  This may be a bit of a conceit at this point, since I've now written multiple papers on AdS/CFT.  My heart is more strongly devoted to the types of ideas LQG people explore, but my mind
recognizes that they really haven't made all that much progress.

Posted in Physics | 3 Comments

## Yet More Random Stuff

I've been staying home sick with some horrible cough for about 3 weeks now.  One would think that this would be quite conducive to blogging, but when I'm running a fever I find it hard to concentrate enough to produce mental output.  (Mental input, like books and movies, is fine).

Fortuantely—either because of taking antibiotics, or for some other reason—I'm beginning to feel much better, so here's a post, consisting of links which I've found interesting since the last time I did links:

• Of This and Other Worlds blogs on the Problem of Susan in the Narnia books.  The Superversive adds some interesting personal testimony.
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• A New York Times article on computer software that supposedly grades essays.  Anyone who thinks that computer programs can substitute for human graders is completely misinformed about the point of essays.  Which is always to communicate some sort of meaning through organized thought.  This is something that no computer can do, prior to the development of some actual AI overlords.  The best it could possibly do is check for pretentious vocabulary, correct bad grammar (badly) and enforce meaningless and stupid rules about how many paragraphs there must be.   No machine could possibly check for the presence of an interesting thesis supported by coherent argument based on plausible evidence.  There are probably some things you could measure which are corollated with being a good writer, but even this will cease once students learn how to flatter the machine.
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The sad thing is that there are probably human teachers who grade this superficially.  Although, even they could probably tell if the sentences didn't actually fit together in any way (besides beginning with words like "Moreover").
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Out of curiosity, I just went and checked the webpage of the discern program to see what their alogorithm was.  It's machine learning based on sample essays which are already graded.  Oh my.  That means neither the student nor the classroom instructor will even know what criterion the machine is using.
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• What St. Lewis (in his capacity as a literary scholar) thought of the Puritans.
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• You've probably heard how the first man in space, Soviet cosmonaut Yuri Gagarin, said that didn't enocunter God there.  As if God were literally located in the sky.  Well, it turns out, the whole story was a Soviet lie; St. Yuri was an Orthodox Christian.  More details here.
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• A haunting article, by and about a woman who acts the part of a sick patient for medical students.  This is one of the best written narratives I've read in quite some time.
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• An interesting (and to me inspiring) letter from missionary St. Anthony Norris Groves (1795-1853) to crankish schismatic (St?) John Nelson Darby (1800-1882) on the topic of Christian unity.
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Darby was one of the first people to teach that Christians would be raptured into heaven 7 years prior to the Second Coming of Christ, a belief almost completely unheard of prior to Darby.  This is part of a detailed scheme called Dispensationalism, popular in American Fundamentalist circles, which is based on that idea that apparent contradictions in the Bible should be resolved by assigning different texts to one of seven different covenants or "dispensations" in which God treats people differently.  This way of thinking leads them to construct an elaborate timeline of End Times events (a suprise Rapture, followed by 7 years of Tribulation, followed by the Second Coming, followed by 1000 years of The Milennium [this one at least has a  foundation in a literal reading of the Book of Revelation], and then finally the Final Judgement).  Oddly enough, people think that this elaborate scheme comes from reading the Bible literally as a fundamentalist should, even though no one who read the Bible without influence from Darby would ever come to this elaborate scheme on their own.
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More relevantly to this letter, Darby went on to found a small denomination of his own which excommunicated nearly everybody else.
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• St. Maxime is a Stylite monk with a much better way to isolate himself from the World.  Make sure to click through the slide show.
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• An article on the simplicity of God (hint: it doesn't mean that he is easy to understand).  Consider me firmly in the "classical theism" camp.  I consider the idea that God is just a person like us, but pure spirit and infinitely powerful etc., to be idolatrous.  True, we humans are the image of God.  The converse is not true: God is not to be conceived as being in our image.
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• The New York magazine interviews St. Antonin Scalia.  There was an interesting moment where Scalia brings up that he believes in the Devil.  The interviewer acts a bit incredulous, and asks:

Isn’t it terribly frightening to believe in the Devil?
You’re looking at me as though I’m weird. My God! Are you so out of touch with most of America, most of which believes in the Devil? I mean, Jesus Christ believed in the Devil! It’s in the Gospels! You travel in circles that are so, so removed from mainstream America that you are appalled that anybody would believe in the Devil! Most of mankind has believed in the Devil, for all of history. Many more intelligent people than you or me have believed in the Devil.

## Must Science be Falsifiable?

There's a common notion floating around, due to Karl Popper, that scientific theories are characterized by the fact that they are falsifiable.  The idea is that it is never possible to verify a scientific theory (i.e. the sun always comes up) because one day it might not happen.  But it is possible that the sun might not come up some day, and then the theory is falsified.  It must then be rejected, and replaced with something more complicated.

Now, let me confess right away that I have not gotten this idea by reading any of Popper's writings.  It is an idea which has been popularized in the scientific community.  You see, everyone knows what Popper said without having read any of it ourselves.  It could be that if I actually read Popper's books, my idea of what he said would be falsified.  So let me confine myself in this post to discussing Popperism as commonly understood.

If a theory is unfalsifiable (that is, if no experiment you could possibly perform would rule it out, then according to Popperism it is not a scientific theory.  Among those who subscribe to Scientism, this is usually assumed to be A BAD THING™.  (The way some people talk, if a theory is unfalsifiable, that means it is false!)

People often characterize bogus pseudoscientific ideas as unfalsifiable, because of the tendency of people who believe in them not to subject them to rigorous scrutiny.  But this is clearly an oversimplification.  True, there is such a thing as mystical Woo-Woo from which no definable predictions can be made, either because the ideas are not precise enough or because they don't relate to any actually observable phenomena.  But many psuedoscientific ideas, such as homeopathy, reflexology, or astrology, can be tested experimentally, it's just that the people who believe in them don't like the results when people do!)  I've heard people refer to Young Earth Creationism (YEC) as unfalsifiable.  I think their reasoning must be the following:

1. YEC is unscientific and wrong.

2. I've been taught that when ideas are unscientific, the reason is because they are unfalsifiable.

3. Therefore, YEC is unfalsifiable.

In fact, though, the real problem with YEC is that it IS falsifiable, and in fact has been falsified many time over. If the universe were created about 6,000 years ago and we have to get all of the layers of fossils and rock from a single planet-wide Flood about 4,500 years ago, then there are a gazillion problems with observation.  It contradicts the results of almost every branch of science which tells us anything about the past.  (Adding bizarre extra ideas, like God created the earth with fossils in it in order to trick us into believing in evolution, may make YEC unfalsifiable, but it might be better to characterize this as pigheaded refusal to accept reasonable falsification.)

[Fun fact: if you interpret all of the genealogies in Genesis as being literal, with no gaps—which of course I don't—then it follows that when Abraham was born, all of his patrilineal ancestors were still alive, back to the tenth generation (Noah)!  (This is using the Masoretic Hebrew text that omits Cainan, who is included in the Septuagint Genesis and Luke.)]

All right, digression over.

Clearly there is something right about the idea that theories ought to be falsifiable, yet not confirmable with certainty.  Major scientific theories usually deal with generalities: they make predictions for a large (perhaps infinite) number of different situations.  Normally, it is not possible to verify them in all respects, because even if it works well in many cases, it could always be an approximation to something else.

On the other hand, I think there are some scientific ideas which are verifiable but not falsifiable.  Here's an example:

Ring Hypothesis: Somewhere in this universe or another, there exists a planet with a ring around it.

I submit to you that: 1) our observation of Saturn verifies the Ring Hypothesis, 2) when scientists verify a proposition by looking through a scientific instrument, that counts as Science, and 3) no possible observation could have falsified the Ring Hypothesis.  (Even restricting to the Milky Way, eliminating planets with rings would be a tall order, impossible with current technology.)  Therefore, there are scientific propositions which are verifiable but not falsifiable.

On the other hand, even if an experiment "falsifies" a theory, it could be that the experiment rather than the theory is wrong. As Einstein once said "Never accept an experiment until it is confirmed by theory".  This witticism may seem to turn science on its head, but nevertheless it has a bit of truth to it.  A while back, there was an experimental observation which seemed to suggest that neutrinos travel faster than light.  Soon there were many papers on the arxiv trying to explain the anomaly.  But it turned out, not surprisingly, that there was an error in the measuring devices.  Usually, when a well-tested theory is in conflict with an experiment, and the anomaly has no particularly good theoretical explanation, it is the experiment which is wrong.  Not always, but usually.

What this means is that we need a more flexible set of ideas in order to discuss falsification and verification.  In particular, we ought to accept that falsification and verification can come in degrees—observations can make an idea more or less probable, without reducing the probability to exactly 0 or 1.  The accumulation of enough experimental data against a theory should make you reject it, but it may be able to withstand one or two anomalous measurements.

The quick answer is that one ought to use Bayes' Theorem instead.  This is a general rule for updating beliefs, taking into account both our prior expectations and observation.  This goes not just for Science, but also for everything else.  The only thing that makes Science special is that, due to a number of special circumstances, the process of testing through observation is particularly easy to do.

Even though falsification is not the best way to think about Science, it still works pretty well in many cases.  In a later post, I hope to explain the connection between Bayes' Theorem and falsification.  Usually we should expect good theories of the universe to be falsifiable, but in certain situations they don't have to be.  Bayes' Theorem can be used to understand both the general rule, and why there are exceptions.

Posted in Scientific Method | 12 Comments

## New Job, excuses for not posting

This is just a quick note to say I've been travelling a lot recently (to Japan to visit the IPMU and to Seattle for Christmas) and have also been very busy with job application stuff.

I've accepted an exciting new postdoc at the Institute for Advanced Study near Princeton, starting in the fall.  Also, this Wed I am scheduled to interview for a faculty position at U Chicago.  We'll see if any of the planes are actually going there on Tuesday.   "Winter Storm Ion" is apparently going to provide the Windy City with the coldest temperatures in 20 years.  I went to the Sports Authority to get some cold-weather gear so I don't die.

I'm hoping to get back to blogging later this month.

Posted in Blog | 5 Comments

## The "nuclear option" was illegal

Last week, 52 Senate Democrats voted to get rid of the filibuster for Presidential nominations to certain positions—in particular for Lower and Appelate Court Nominees, but not for Supreme Court nominees.  This move was branded as the "nuclear option" back when Republicans threatened to do it (but did not) during the Bush presidency.  It was completely and unabashedly illegal, and those Senators who voted for it (most of whom denounced it vigorously when Republicans proposed it) should be ashamed of themselves.  This post will explain why their decision was contrary to the law.

The most important law in the United States is the Constitution.  It takes precedence over all other laws, and describes under what conditions laws can be made.  It says among other things that

Each House [of Congress] may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a Member.  (Article I. Sec. 5)

In accordance with this proviso, each of the two houses of Congress has adopted a set of rules, which they use regulate debate, votes, and other matters (kind of like Robert's Rules of Order, but the details are different).  The Sentate Rules can be found here (Rule 22 being the most important for issues surrounding the filibuster).

Since the Constitution authorizes the Senate to make Rules for itself, these Rules are just as much binding law as ordinary federal legislation is.  The only possible exception would be if a Rule contradicted the Constitution.  In that case, the Rule would be invalid.  For example, if the Senate passed a Rule saying that they could expel Senators with a majority vote, then this rule would be invalid, since the section of the Constitution which I quoted requires a 2/3 vote.  But on most procedural issues, the Constitution is silent so the Senate gets to decide.

The important Rules to know about are the following:

• Technically, it only takes a majority of the Senate (if all are present 51, 50 with the VP) to pass Bills, to approve a Presidential Nominee, or to change the Rules, but this is only once debate on the Bill or Rule ends.

The hard way to end debate (which almost never happens) is to give two chances to each Senator to speak as long as they can on the topic (without taking breaks to go to the bathroom!).  This was used to pass the Civil Rights act of 1957, after Senator Strom Thurmand spoke for 24 hours and 18 minutes.  (This was a real filibuster, the kind where you read biscuit recipies, which almost never happens these days.)

The easy way is to invoke cloture, which limits the time left for debate:

• It takes a 3/5 majority (60 votes) to end debate on most topics,
• except on a motion to change the Senate Rules, which takes a 2/3 majority (67 votes).

So, practically speaking it takes 60 votes to do anything in the Senate.  This Rule forces the Majority Party to have to reach out at least a little bit to the Minority Party when they pass legislation.  Otherwise the Minority might refuse to vote for end debate (and this is what is usually called a "filibuster" in these boring times).

• At any time, the Senate may agree to temporarily waive a Rule, but this requires a unanimous vote.
• If there is a question about what a Rule means, the Presiding Officer gets to interpret the rule.  However, the matter can then be appealed (without debate) to the entire Senate, and by a majority vote they can sustain or reject the decision.

Now notice this.  It takes a 2/3 vote to change the Rules (really to end debate on a Rule change).  But it only takes a majority vote to interpret the Rules.  This makes sense: when the Senators vote to change a rule, they are exercising a legislative function, deciding what the rule ought to be.  When the Senators vote on interpeting the rules, they are excercising a quasi-judicial function.  Essentially, they are the "Supreme Court" which decides what the Rules mean.  When making this vote, surely they are morally bound to judge honestly, and decide, not what they think the rule ought to be, but what it actually is.  Otherwise, there would be no need for a higher threshold in order to actually change the rules.

But the power to interpret necessarily includes the power to misinterpret the Rules.  This can be used to abolish any Rule by majority vote—not by amending it, but simply by interpreting it not to apply, even when it clearly does apply.

This is the "nuclear option".  The way it plays out was as follows.  Majority Leader Reid raises a Question of Order asking whether the Rules permit him to end debate on a Judicial Nominee with only a majority vote.  Patrick Leahy, the Presiding Officer, rules that according to Senate Rules and precedents, the answer is No—the Rules clearly state that a 3/5 vote is required.  So Reid appeals the decision to the main body of the Senate.  The Senate voted 52-48 to overrule the decision of the Presiding Officer (among the 52 being Leahy himself!).  Bye bye filibuster for Judicial Nominees.  (3 Democrats had the integrity to vote against, and of course so did the Republicans.)

Note that no actual change to the text of the Rule occured.  It was only "reinterpreted", in a Humpty Dumpty-esque act of linguistic power:

'When I use a word,' Humpty Dumpty said in rather a scornful tone, 'it means just what I choose it to mean—neither more nor less.'

'The question is,' said Alice, 'whether you CAN make words mean so many different things.'

'The question is,' said Humpty Dumpty, 'which is to be master—that's all.'     (St. Lewis Carroll, Through the Looking Glass)

I have asserted that there can be no actual justification for the Senate's interpretation of Rule 22 .  There are only 2 possible ways the decision could be correct.  Either: A) Rule 22 has a special exception for certain Judicial Nominees or else B) Rule 22's 3/5 vote requirement is unconstitutional when used to filibuster Judical Nominees.  (But apparently  not Supreme Court and Executive Branch Nominees?!?)

Option (A) is clearly absurd.  Rule 22 gives the threshold to "bring to a close the debate upon any measure, motion, other matter pending before the Senate".  Clearly the approval of a Judicial Nominee  "a measure, motion, or other matter".

Option (B) is only slightly less absurd.  The Constitution says that the President:

shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law.  (Article II. Sec. 2)

The argument here is that "Advice and Consent" implicitly includes the idea of a majority vote.  This is a rather weak argument, since "majority vote" is nowhere included in the text.  Whereas the statement about the Senate making its own Rules is quite explicit.  So this interpretation of the Constitution seems quite dubious.

Even if the Constitution did require a majority vote for Nominees, there is absolutely no good reason why this should apply to some types of Nominees but not others.  Nor is it obvious why the filibuster would be constitutional for legislation, since the majority vote requirement could just as easily be read into the power of the Senate to pass Bills.  But if the filibuster is unconstitutional in general, this would be a rather surprising thing to find out now, after 170 years of precendent to the contrary.

The fact is that those who voted for the "nuclear option" knew perfectly well that it was of extremely dubious legality.  They didn't do it because they genuinely believed in it.  They did it for political reasons, as a naked act of political power.

Can any justification can be made for this act?  Let me make some points about what is and is not relevant:

1. Whether or not the "nuclear option" was legal cannot possibly depend on how frequently the Republicans were filibustering Judicial nominees.  It is a question of legal interpretation, not a question of history.  The unfair tactics and hypocrisy of the other side is irrelevant.
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2. Besides, the Opposition Party in a democracy is allowed to use any legal tactic in order to delay or obstruct legislation.  If their obstruction is unwise, unprecedented, immoral, or hypocritical, voters may take note and respond.  But excessive use of a legal tactic on one side cannot justify use of an illegal tactic on the other side.
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3. The liberty of a free people depends on the fact that government officials do not consider themselves above the law, but instead obey it.  Without this social norm, restrictions on the government (such as the Bill of Rights) would be meaningless.  This social norm is therefore far more important than nearly all of the minor partisan squabbles which could tempt one political party to abadon it.
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4. There may be extreme circumstances which may justify illegal actions, but "There are hypocritical obstructionists in Congress" doesn't qualify.  That's way too common of an occurence to justify anything!
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5. When I call the nuclear option illegal, I don't mean that the Senate doesn't have the power to interpret its own rules, or that this decision doesn't stand as a precedent from now on.  If the Supreme Court were to rule 5-4 that the First Amendment allows the government to ban books, this act would be legal in the sense that they are charged with interpreting the Constitution, yet still wrongly decided in the sense that it directly violates the text they are charged with interpeting.
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If we further suppose that the Supreme Court knew perfectly well that the decision was erroneous, but did it anyway in order to spite their political opponents, then that would be a pretty close analogue to what the Senate just did.
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6. Strictly speaking, it is the act of banning books which is unconstitutional, not the decision itself.  Similarly, the Senate decision is tantamount to an illegal violation of the Senate rules, but since it is the highest court for interpreting its own Rules, there is another sense in which what it did is now de facto legal.
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This doesn't make much practical difference, though.  A completely lawless use of the power to interpret is exactly the same as if there were no law at all.
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7. My views are not based on which party is in charge.  I was vehemently opposed to the "nuclear option" when Republicans proposed it, and I am still opposed now.

All told, it is a dark day for the Republic.  The trouble is, these days both Parties hate each other so much that they spend all their time thinking about how the other side is hypocritical, without noticing that they also chang their position whenever it is convenient.  (See Kerr's Law).  Political expediency trumps truth.  I'll spare you all the juicy quotes from the Senators who flip flopped on this issue when the Party roles were reversed.

Instead I will remind us of the words of the Master whom most of those in Congress claim to serve:

How can you say to your brother, ‘Let me take the speck out of your eye,’ when all the time there is a plank in your own eye?  You hypocrite, first take the plank out of your own eye, and then you will see clearly to remove the speck from your brother’s eye.  (Matthew 7:4-5)

Posted in Politics | 2 Comments

## How to Construct Laws of Physics

Suppose you want to write down the laws of physics.  How would you go about it?

What?  You want to do some experiments first?  Forget about that.  This is theoretical physics.  Let's not worry about pedantic things like what the actually correct laws of physics are.  Instead, let's try to ask what they should look like more generally.  What are the ground rules for trying to construct laws of physics?

(Of course, in reality we do get these ground rules from experiment.  The way it works is, we make up rules to describe lots of specific systems which we actually measure, and then eventually we get some idea of what the meta-rules are, i.e. the rules for constructing the rules.  But let's just try to make something up here, and see how close we get to reality.)

Let's try to do this step by step.  Let's take for granted the existence of a spacetime.  In the first step, we need to decide what kind of entites there are moving around in this spacetime.  Since we're on the hook for giving an exact description, we'd better start with something which is mathematically simple.  For example, we could postulate that there are a bunch of point particles flying around.  If there are $N$ particles, and space is 3 dimensional, then we can describe all of their positions with $3N$ parameters.  (We can then think of the universe as a point moving around in a $3N$ dimensional space, called configuration space.)

Or maybe there's a bunch of strings wiggling around.  Or perhaps there are fields, whose values are defined at each point of space.  (In these cases, we will need an infinite number of parameters to describe what is going on at each moment of time!   But don't worry—since we won't be doing any actual calculations, this won't necessarily make things any harder.)

All right.  Now that we've decided what kind of stuff we have, we need to know how it changes with time.  For this we need to write down equations of motion.

We could write down an equation involving one derivative of positions with respect to time.  This would determines the velocity of each piece of particle/string/field/whatever in terms of its position.  But that won't be like real physics since real physical objects have inertia.  Stuff keeps on trucking until a force acts on it.  This means that the future motion of an object doesn't just depend on where it is right now, but also on how fast it is going.

So instead we need to write down an equation involving two derivatives of the position with respect to time.  This will determine the acceleration of each entity, as a function of its position and/or velocity.  That's a bit more like real life.   (In other words, to work out what happens we need to know about both the positions and velocities.  If we have $N$ particles, this is a $6N$ dimensional space called phase space.)

So you could just sit down and write down some second-order differential equation equation involving acceleration, and call that the laws of physics.  But most of these would still be qualitatively different from the fundamental laws of actual physics.  For example, nothing would stop you from including friction terms which would cause the motion of objects to slow down as time passes.  For example, if we have a particle moving along the x-axis, we could write down an equation like this:

This would cause the particle to slow down as time passes.  But in reality, friction only ever happens when some object rubs up against another object.  The motion doesn't disappear, it just goes into the other object.  This is related to Newton's Third Law, a.k.a conservation of momentum.

So physics has more rules then one might think are really necessary.  You can't just write down any old equations of motion.  They have to be special, magical equations, which satisfy certain properties.

We could just make some giant list of desired laws.  But that would be rather ad hoc.  Instead, physicists try to derive all of the magic from some simple framework.  We've just seen that just writing down equations of motion is not the best framework since it doesn't guarantee basic physics principles like conservation laws.

There are two particularly simple frameworks which can be used.  For most systems these are equivalent, and you can derive one framework from the other.  I'm just going to summarize these at lightening speed here:

• Lagrangian mechanics:  Here the fundamental concept is the action, a number

obtained by integrating some function $L(x,\,\dot{x})$ of the positions and velocities over all moments of time.  ($L$ is called the "Lagrangian", and is normally equal to the kinetic energy minus the potential energy).  The basic rule is that a small change $\delta x(t)$ in the paths of particles/strings/fields/whatever in any finite time interval $t_\mathrm{initial} < t < t_\mathrm{final}$ should leave the action unchanged, to first order (i.e. up to terms linear in $\delta x(t)$).  In other words:

Here $x$ can be any of the position parameters in the theory.  Once you write down a single equation specifying $S$, all of the equations of motion for all entities are determined by this rule.
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As a simple example, consider a point particle moving along a 1-dimensional coordinate $x$, with a potential $V(x)$ which depends on your position.  This might describe a train sliding frictionlessly along a roller coaster track, where $x$ is the length measured along the track and $V(x)$ is proportional to its height measured from the ground. The Lagrangian is kinetic energy minus potential energy:

The rule here is that given the initial and final locations of the train in some short time interval, the train moves in a way that minimizes the total action of its trajectory—which implies by basic principles of calculus that small variations of the path have to leave the action unchanged.
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Imagine if you are walking from your house to a shop.  You leave your house and 9 am, and you need to be at the shop at exactly 10 am.  You don't like walking too quickly, because it expends too  energy.  On the other hand, if it's a bit chilly you might also prefer to spend more time in sunny areas, and less time in shady areas.  What would you do?  If you want to maximize your happiness (or minimize your unhappiness), you would compromise by walking more quickly in the shade than in the sun.  Similarly, if we fancifully suppose that the train had a soul and that it preferred to spend more time up high (so long as it gets to its destination on time), we would then have an explanation for why the train lingers at the higher parts of the track.  More generally, when the potential energy is higher the kinetic energy is less—one can prove that the total energy is conserved.
• Hamiltonian mechanics: The fundamental concept here is that all parameters in physics come in conjugate'' pairs.  For example, for a regular particle, the conjugate variable to the position $x$ is the momentum $p = m\dot{x}$, while the conjugate variable to momentum is minus the position, $-x$.  (That minus sign is important: without it conservation laws wouldn't work properly.)  The variable which is conjugate to time is known as the "Hamiltonian" $H$—this turns out to be nothing other than the total energy of the system (kinetic plus potential).   It turns out that if you know the Hamiltonian $H(x,\,p)$ as a function of the positions and their conjugate momenta, you can work out everything that happens.  You work out the equations of motion with the rule (called "Hamilton's equations" that the change of a parameter with respect to time, equals the change of the energy with respect to the conjugate variable.  In other words:

The minus sign in the second equation means that position is to momentum as momentum is to minus momentum, just like I told you.
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A consequence of "Hamilton's equations" is that, assuming $H$ does not depend on some particular position coordinate $x$, $\partial H / \partial x = 0$ and so $p$ is conserved.  More generally, Hamilton's second equation says that the "force" $\dot{p}$ is zero when the gradient (i.e. derivative) of $H$ with respect to position is zero.  Similarly, if the gradient of $H$ with respect to the $p$ coordinate is zero, then $\partial H / \partial p = 0$, and Hamilton's first equation says that the velocity $\dot{x}$ is zero.  If the fomula for kinetic energy is the usual nonrelativistic formula $p^2 / 2m$ (written as a function of the momentum $p$ instead of $\dot{x}$ since this formulation of physics is all about $p$'s), this tells us that the "velocity" is zero when the momentum is zero.
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More generally, Hamilton's equations tell you that if you graph out the 2 dimensional phase space of a particular pair of x-p coordinates, the trajectory of the system in the x-p plane is at right angles to the direction of the gradient of $H$, and equal in size to the gradient.  This means that the system always moves along a direction where $H$ isn't changing, and so $H$ is conserved (unless we make it an explicit function of time, in which case we would have to write it as $H(x,\,p\,t)$).

From either of these two equivalent formulations of physics, there is a famous theorem first proved by Emmy Noether.  She showed that any time $H$ or $L$ has a symmetry which shifts some parameter, its conjugate parameter is conserved (it doesn't change with time).  I've already shown you some specific examples (symmetry with respect to $x$ shifts makes $p$ be conserved, symmetry with respect to $t$ shifts makes $H$ be conserved).  This is the most important theorem in all of theoretical physics.

If you just start by trying to write down equations of motion for your laws of physics, you can't prove Noether's theorem.  It just doesn't work.  Since you don't have a notion of conjugate quantities, you can't even get started.  Many important physical concepts such as energy, momentum, mass, force, and so on won't even be defined.  So there's a lot more to life than the equations of motion.

## Some Mythical Conflicts between Science and Religion

A couple posts elsewhere refuting a common Medieval-bashing trope, that the Medieval Church tried to suppress scientific ideas, in a series of mythical conflicts between Science and Religion, by historian of Science St. James Hannam.

On the same site, Tim O' Neill writes some further commentary along the same lines, in the course of reviewing Hannam's book God's Philosophers.

Of course, even if stupid religious people had been persecuting scientists for the last fourteen millennia, it wouldn't make the least bit of difference to the question of whether the two sets of ideas are compatible.  That is a philosophical, not a historical question.

## The Achievement Gap

Sometimes educators talk about "closing the achievement gap" which separates high and low performing students.  There are documented gaps in educational outcomes on the basis of e.g. economic classes, race, etc.  Some of these gaps lead to serious social problems down the road.  But even if we somehow produced a society which had equal outcomes for every factor in the current Politically Correct List of Superficial Ways to Classify People™, there would still be high-performing students and low-performing students.  Educators don't like this sort of situation, because they don't want to feel like they are failing some of their students.

Now, there are two ways to close a gap.  One is to take the students who are doing badly, and teach them better.  The other way is to take the students who are doing better, and teach them worse.  Or at least, don't pay any special attention to them, since the goal is to produce equality.  This shows the danger of adopting equality as a goal.  Inequality is defined as a difference between two people.  Adopting equality as a goal means you are trying to benefit one person as compared to another.  If all better-off students were worse off, there would be less to feel bad about.

Instead, we ought to adopt the goal of benefiting all students.  But especially the ones who are most capable of benefiting from education.  This is, primarily, the more intelligent and motivated students.  People with (small "d") democratic sensibilities don't want to hear this.  But as St. Lewis writes in an essay on "Democratic Education":

Equality (outside mathematics) is a purely social conception. It applies to man as a political and economic animal.  It has no place in the life of the mind.  Beauty is not democratic; she reveals herself more to the few than to the many, more to the persistent and disciplined seekers than to the careless.  Virtue is not democratic; she is achieved by those who pursue her more hotly than most men.  Truth is not democratic; she demands special talents and special industry in those to whom she gives her favors.  Political democracy is doomed if it tries to extend its demand for equality into these higher spheres.  Ethical, intellectual, or aesthetic democracy is death.

A truly democratic education—one which will preserve democracy—must be, in its own field, ruthlessly aristocratic, shamelessly `high-brow'.  In drawing up its curriculum it should always have chiefly in view the interests of the boy who wants to know and can know.  (With very few exceptions, they are the same boy.  The stupid boy, nearly always, is the boy who does not want to know.)  It must, in a certain sense, subordinate the needs of the many to the needs of the few, and it must subordinate the school to the university.  Only thus can it be a nursery of those first-class intellects without which neither a democracy nor any other State can thrive.

The goal of leaving No Child Left Behind sounds enlightened, but leaving some children behind is in fact a necessary logical corollary of teaching children difficult subjects.  If your only goal is not to abandon the children who are behind, then you will abandon those who are ahead: the ones who are actually interested in learning.  Most people, believe it or not, forget most of what they were made to learn in school.  The future philosophers, scientists, authors, judges, and so on will actually remember (part of) their education and apply it.

This is not to say that education is unimportant for the masses.  A certain quantum of literacy and comprehension is necessary to survive in the world.   By definition, a democracy has votes, and a certain degree of education is necessary to vote wisely.  When broad sections of society are deprived of a good education, and become a permanent underclass, society suffers.  The heroic teachers who try to remedy this, by volunteering to teach failing students, are worthy of our respect.  It is a valuable project, but it ought not to have such an exclusive monopoly on our thinking that we forget the need to teach those most capable of learning.

But aren't those students going to be learning anyway, in pretty much whatever environment you put them in?  To some extent, yes.  But it makes a difference what you think is the point of education.  The current goal is to produce a system in which any student can succeed if they really try.  This means lots of busywork, and a hefty amount of grade inflation (rewarding the consistently effortful, and punishing those who take chances on difficult subjects).  It does not necessarily mean teaching critical thinking.  Teaching people to be "good at school" can mean be a sort of slave-mentality, while the goal of a liberal arts education is to produce people who can think for themselves.  This involves a sort of paradox: you have to teach people to teach themselves.  Ignoring a student's needs is one way to try to encourage this, but it is not the best way.

Let me be autobiographical for a moment, just to give a concrete example. I hope that I am now old enough and fulfilled enough to be beyond any resentment, but I feel that a specific example will be helpful, and I am the example I know best.

In the area of mathematics, I was "left ahead" as a child for pretty much my entire school career before I started taking college classes.  The teachers recognized my knowledge, but none of them did what was required to give me sufficiently advanced material.  I suppose they probably had their hands full with the students who needed more help with the assigned curriculum.

Eventually, in the 7th grade, someone put me into the 8th grade Algebra class.  It was too late—I was already starting to do Calculus by then.  I was too bored by the subject to do any homework, so the teacher failed me, even though she knew I knew all the material.  She thought I was lazy and needed study skills, which was true, although this was hardly the correct motivator to produce them.  I had to repeat the class again in 8th grade.   I was mortified, but fortunately none of my classmates knew about the situation.  I still didn't do any homework (through guilt-ridden procrastination and deception, not through a firmly decided upon rebellion), but this time she recommended me into the Honors Geometry class in the 9th grade.

This time homework was only 10% of the grade, but the extremely formulaic and tedious standards for proofs docked me another 10% or so on the exams.  (See A Mathematicians Lament for an important critique of the way we teach Geometry and other mathematical subjects.)  That got me to a C+.  As a result I was looking at having to take the non-honors version of the next course in the sequence Algebra II.  (Los Altos High School had a policy against skipping classes).  Bear in mind that, on my own, I was learning Maxwell's equations,  General Relativity, and Quantum Mechanics at this point.

There was a standardized test to overcome the C, but in a school full of overachievers it was deliberately designed to be impossible.  Too many questions in too short of a time.  I knew immediately, before getting the results, that it wasn't going to fly.  I was going to be steamrollered under the wheels of an formalistic bureaucracy which was unable to make a plain human diagnosis of the sort of student I was: lazy but brilliant.  I was terrified that I would never receive the help I needed to succeed at what I already knew I wanted to do in life: theoretical physics.

Some wandered in desert wastelands,
finding no way to a city where they could settle.
They were hungry and thirsty,
and their lives ebbed away.
Then they cried out to the Lord in their trouble,
and he delivered them from their distress.
He led them by a straight way
to a city where they could settle.
Let them give thanks to the Lord for his unfailing love
and his wonderful deeds for mankind,
for he satisfies the thirsty
and fills the hungry with good things.
(Psalm 107:4-9.  Read the whole thing!)

So I cried out to the Lord to save me, and he rescued me from my afflictions.  The instruments of his salvation were as follows: Although I complained of the inhuman bureaucracy, in fact there was an excellent academic counselor at the school who knew my situation and advised me to apply to the Foothill Middle College Program, basically a way to flunk out of high school into the local community college. They only take Juniors and Seniors, so I had to skip my Sophomore year.  No regrets!

When I went there, Foothill finally gave me an actual placement test, and I got the highest result and so placed into Calculus 1A (I made an arrangement with the prof to skip the classes and take the final: with Calculus 1B I finally got to new material).

My weird education story doesn't end there, but this was a critical turning point.  It happened because at some point certain educators cared enough design and implement a program for people like me.  For this and many other gifts I give thanks to the Head Teacher:

I love the Lord, for he heard my voice;
he heard my cry for mercy.
Because he turned his ear to me,
I will call on him as long as I live.
(Psalm 116:1-2)