Must Science be Falsifiable?

by Aron Wall Posted on January 29, 2014

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 | 36 Comments

New Job, excuses for not posting

by Aron Wall Posted on January 5, 2014

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 unlawful

by Aron Wall Posted on November 27, 2013

[Edit 12/28/17---In the title and throughout, I have replaced "illegal" with "unlawful", in order to avoid the possible implication that the travesty below would be subject to criminal punishment, something I did not intend to imply.  Since I wrote this article, the Republicans have followed suit by nuking the filibuster for Supreme Court nominees as well.---AW.]

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 unlawful, 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 (but, obviously, they only bind the Senate itself, not the rest of us).  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).

Some additional important Rules:

  • 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, it wouldn't make any sense for there to be 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 the Republicans also voted against.)

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 is "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 unlawful 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 unlawful actions, but "There are hypocritical obstructionists in Congress" doesn't qualify.  That's way too common of an occurrence to justify anything!
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  5. When I call the nuclear option unlawful (unruleful), 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.
    .
    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 unlawful 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

by Aron Wall Posted on November 19, 2013

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 determine 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:

\ddot{x} = -\dot{x}.

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

    S = \int L(x,\,\dot{x})\,dt

    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:

    \frac{\delta S}{\delta x(t)} = 0.

    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.
    .
    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:

    \frac{m\dot{x}^2}{2} - V(x).

    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 at 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 much 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:

    \frac{dx}{dt} = \frac{\partial H}{\partial p},\qquad\frac{dp}{dt} = -\frac{\partial H}{\partial x}.

    The minus sign in the second equation means that position is to momentum as momentum is to minus position, just like I told you.
    .
    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.

Posted in Physics | 2 Comments

Some Mythical Conflicts between Science and Religion

by Aron Wall Posted on November 14, 2013

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.

Posted in History, Links, Reviews | Leave a comment