Pillar of Science II: Elegant Hypotheses

by Aron Wall Posted on November 3, 2012

Scientific Theories must be Elegant.

Since there are always infinitely many different hypotheses which fit any set of data, there must be some prior beliefs which we use to decide between them.  Any hypothesis which has an excessive number of entities or postulates is unappealing, and gives rise to the suspicion that it works because of special pleading or force-fitting the data rather than because it has any deep connection with Nature.  So all else being equal, scientists prefer hypotheses which are simple, uniform, common-sensical and aesthetically pleasing.

At least part of this requirement is captured in the principle known as Occam's razor, which in the original form proposed by Occam translates to “Entities are not to be multiplied without necessity”.  Of course, one may be forced to postulate complexities if the data rules out any simpler hypothesis, but even here one must pick among the simplest of an infinite number of possible explanations for the same data.

This criterion of elegance is informed by previous scientific work as well as by a priori considerations.  It also varies from field to field: a particle physicist should be much more reluctant to postulate a new force of nature than a cellular biologist is to postulate a new kind of organelle.

Because many important scientific theories have greatly defied prior expectations, it is best not to turn these a priori expectations into hard and fast rules which would prevent too many hypotheses from being considered altogether.  Instead, scientists mainly use intuition and rules-of-thumb to judge which theories are worth considering.

There are many famous cases where the elegance of a new theory was used to predict confidently the results of an experiment.  Einstein once quipped about Planck that

...he did not really understand physics, during the eclipse of 1919 he stayed up all night to see if it would confirm the bending of light by the gravitational field [as predicted by Einstein].  If he had really understood the general theory of relativity, he would have gone to bed the way I did.

Nevertheless, ultimately the criterion of elegance is subordinate to observations.  It doesn't matter how beautiful or simple your theory is, if it gets the facts wrong.  To be sure, sometimes experiments turn out to be wrong too, especially when they go against fundamental principles of theory (like the recent supposedly faster-than-light neutrinos thing).  But if, in the long run, experimental observation can't correct our prejudices, then there's no point in doing science.   Nature may be beautiful but that doesn't mean that she (or her Creator) cares about our personal aesthetic of how things should be run.  In the greatest popularized physics lectures of all time, Feynman advises that:

Finally, there is this possibility: after I tell you something, you just can't believe it.  You can't accept it.  You don't like it.  A little screen comes down and you don't listen anymore.  I'm going to describe to you how Nature is—and if you don't like it, that's going to get in the way of your understanding it.  It's a problem that physicists have learned to deal with: They've learned to realize that whether they like a theory or they don't like a theory is not the essential question.  Rather, it is whether or not the theory gives predictions that agree with experiment.  It is not a question of whether a theory is philosophically delightful, or easy to understand, or perfectly reasonable from the perspective of common sense.  The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense.  And it agrees fully with experiment.  So I hope you can accept Nature as She is—absurd.

I'm going to have fun telling you about this absurdity, because I find it delightful.  Please don't turn yourself off because you can't believe Nature is so strange.  Just hear me all out, and I hope you will be as delighted as I am when we're through.

Excellent advice for anyone who wants to see the world scientifically.  Perhaps you can already see some implications for religious views, but we'll go into that some other time.

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All Saints Day Roundup

by Aron Wall Posted on November 1, 2012

In honor of All Saints Day, here are some links to the saints on my blogroll:

This blog has its own canonization policy: every serious Christian, whom I refer to by name in the 3rd person, is a "Saint" (e.g. St. Faraday).  This policy is inspired by how the word "saints" was used in the early church to refer to ordinary Christians, e.g. St. Paul addresses one of his letters to "the saints in Ephesus", meaning every person in the congregation.  It emphasizes the fact that the Holy Spirit dwells inside every person who gives their life over to Jesus in order to become one of his Fathers's children.

The Hebrew word qadosh means something sacred which is set apart and dedicated to God's service, while the English word holy is related to whole or wholesome.  In its most proper sense, holiness is a property of God alone, and expresses that he is Good, not just in some conscientious ethical sense, but in the sense of a numinous, awe-inspiring Otherness which, for those fortunate enough to experience it, overpowers us with its majestic glory and weightiness.   The bodies of the "saints" are living Temples in which the Holy One dwells, and we become holy in a derivative sense, sanctified because of his presence inside of us.

Imagine a pond, which has some sort of flowers growing on its surface (a little like water lilies).  Most of these flowers float aimlessly on the surface, but some of them grow stems downwards in to the water.  This makes them rather awkwardly shaped at first, but when the stems reach the ground, they attach to the solid earth underneath.  From then on, the flowers share in the Solidity of the ground beneath.  They no longer drift with the surface currents, and they receive nutrients from below as well as above.  This is only an analogy, but perhaps it gives an idea of the kind of difference that holiness makes to a life.

When I call all Christians saints, this is to bring home the awareness of this astonishing fact.  It is not intended to deny that we all struggle in many ways with sin and bad habits, grieving his Spirit, and that we are therefore in constant need of forgiveness, from God and from one another.

Nor is it intended to deny that some people, because of their fellowship with Jesus, through suffering and joy, become especially holy in a way that serves as a special example of holiness to the rest of us.  I think of St. "Father John", the priest of Holy Trinity Orthodox church of Santa Fe, who cannot be in the same room with anyone without expressing deep love for them.

Nor do I mean to imply that only religious people can be ethical—if by ethics, one means a conscientious effort to be courageous, kind, honest, generous and self-controlled.  However, nonreligious people cannot be, and are not even trying to be, holy in the sense described above—unless indeed they have a relationship with God without knowing it.  (For we must never forget, that even before a person has a relationship with God, God is still having a relationship with them.  Like a host at a party, he provides them with food, drink, and entertainment, and if they happen to be ungrateful or mistreat the other guests, he takes it personally.)  For Christians, ethics comes out of holiness, because of God's love for us; it does not come out of conscientiousness.  That is the most important distinction between religious and nonreligious ethics.

Posted in Links, Theology | 4 Comments

Time as the Fourth Dimension?

by Aron Wall Posted on October 28, 2012

You've probably heard that time is the fourth dimension.  What does it mean?  It should seem rather fishy that time should be the same sort of thing as a spatial dimension.  We all know that you can only go in one direction in time—towards the future!  Time is measured in seconds, space in meters, etc.  It turns out though, that time can be thought of as the same sort of thing as space, but not exactly: there's just one tiny change that makes everything turn out different.

Let's start with two dimensional boring old Euclidean geometry of the sort one learns about in high school. It's called two dimensional because you can specify the location of a  point using two numbers (x,\,y).  Now the most important equation—from which everything else about geometry follows—is the Pythagorean theorem.  This allows you to compute the distance r between any two points (x_1,\,y_1) and (x_2,\,y_2).  If we define \Delta x = x_2 - x_1 and \Delta y = y_2 - y_1, then we can think of \Delta x and \Delta y as the sides of a right-angled triangle.  Then the Pythagorean theorem says that then the distance is

r = \sqrt{(\Delta x)^2 + (\Delta y)^2}.

Now it turns out that the exact same formula works in three dimensional Euclidean space as well; you just have to add in the z coordinate:

r = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}.

Now our world has only three dimensions of space, so far as we know (though that hasn't stopped physicists from exploring ideas such as string theory where there are more dimensions of space).  Nevertheless, nothing stops us from imagining that there are more spatial directions, say four.  (If you haven't read the classic Flatland, go do it now.  Geometry as social satire!)  In that case, one would simply give the extra dimension a new name (say w since we're out of letters at the end of the alphabet) and then write down

r = \sqrt{(\Delta w)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}.

Voilá!  Four dimensional space has arrived!

What has not yet arrived is spacetime.  Each of these dimensions are all exactly the same (it doesn't matter which one we call "first", "second", "third", or "fourth" because of rotational symmetry).  There is no notion of past or future.  To get a time coordinate, we have to do something to make it special.  And what we do is very simple, we just put in a minus sign:

s = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - (\Delta t)^2}.

In deference to custom, I've renamed r as the separation s, but that's just a change of notation.  What you should be noticing instead is how similar this equation looks to the last one.  In fact, just as all of Euclidean geometry essentially follows from the Pythagorean theorem, so all of what's called Special Relativity, or the geometry of Minkowski space (x,\,y,\,z,\,t) essentially follows from this one equation.  (Minkowski was the name of Einstein's math teacher, who first pointed out this way of understanding relativity.)

Now, what is the effect of this minus sign?  It turns out that it changes things rather a lot.  In Euclidean geometry, the thing inside of the square-root is always positive.  But because of the minus sign, the s^2 of two different points can be either positive, negative, or zero:

  • If s^2 > 0, then its square root represents the distance between the two points, just like in Euclidean geometry.  In this case we say that the two points are "spacelike" separated.
  • If s^2 < 0, then the thing under the square-root is negative, so when we take the square-root we get an imaginary answer.  Fortunately, that's okay!  We just flip the sign of the thing under the square root, and intepret it as a duration instead of a distance.  In this case the two points are "timelike" separated.
  • If s^2 = 0, then there is neither distance nor time between the two points.  In this case, we say that the two points are "lightlike" separated.  That's because rays of light (in a vacuum) travel along paths whose points are lightlike separated—in a sense, because they are travelling equally in space and time, but they experience no time or distance as they travel.

I should say that I'm using units here where the speed of light (normally called c) is equal to one.  That means that e.g. if we measure time in seconds, we have to measure space in light-seconds (the distance light travels in one second).  Otherwise, the beautiful equations would get all cluttered up with c's flying all over the place.

We're used to dividing up time into three parts relative to ourselves: past, present, and future. The present is just an infinitesimal sliver, so in a sense this division is into two parts: points to the past have \Delta t < 0 compared to you, while points to the future have \Delta t > 0 compared to you.

However, special relativity tells us you have to chop up spacetime in a more complicated way.  Bearing in mind that you each live in a particular place as well as a particular time, you can chop up spacetime into three different regions.  The future is points that are timelike separated to you and have \Delta t > 0; these are the points of spacetime that you can affect.  The past is points that are timelike but have \Delta t < 0; these are the points that can affect you.  Then there is elsewhere, the points that are spacelike separated.  These points can neither affect, nor be affected, by each other.  The three regions are separated by the "light cone", which consists of the points that you could send a lightray to (or from).  I'm too lazy to draw a picture right now, but you can see a pretty good explanation here.

Next up, we'll talk about rotations.  As always, readers are free to ask me questions in the comments box.

Posted in Physics | 10 Comments

Pillar of Science I: Repeatable Observations

by Aron Wall Posted on October 27, 2012

Scientific Hypotheses are Tested by means of Repeatable Observations.

Every scientific hypothesis ultimately stands or falls based on whether it fits with observed data.  The best kind of data comes from experiment, in which the details of the situation are specified by the scientist and may be freely repeated or varied in order to see how the outcome changes.  Some people might say that experiments are a necessary part of science, but I don't agree.   That's because certain fields of Science don't seem to have them.

Unfortunately controlled experiments are not always feasible for historical sciences such as Geology or Evolutionary Biology, or for sciences which study objects that cannot be easily affected by humans such as Astronomy or Seismology.  These sciences make up for the lack of experimentation by the fact that there is a large and ever increasing amount of observational data to test different hypotheses.  So you can keep getting new data in a way that's reminiscent of the experimental sciences.

In the worst case all the observable effects of a hypothesis are mediated through a unique event. Such hypotheses are usually not subject to repeatable scrutiny, but there can be exceptions.  A notable example relates to the highly regarded, but not completely established, theory of inflation in the early universe (a tiny fraction of a second after the so-called "Big Bang singularity", which as far as we know was the beginning of the universe.).  In this theory there is a phenomenon known as Cosmic Variance, which implies that certain aspects of the theory can only be tested once, because the universe only happened once.  Nothing stops you from looking again, but you'll see the same thing so it doesn't tell you anything new.  (More precisely, there are two kinds of error, experimental error coming from randomness or bias in the experimental measurement, and cosmic variance coming from the way the early universe happened to evolve.  You can beat down the former by doing more or better experiments, but you can't get rid of the the latter no matter how many experiments you do.)

Inflation says that, shortly after the Big Bang, the universe went through a period of extremely rapid exponential growth, caused by fields whose particle-excitations are too massive to be detectable in any forseeable particle accelerator.  (This field is creatively named the "Inflaton". )  The inflationary hypothesis predicts that at the end of the period of inflation, the matter fields should be in a particular state, subject to statistical fluctuations originating from quantum mechanical uncertainty.  As the universe expanded, these fluctuations were stretched out to enormous distance scales, whose size is now comparable to the entire observable universe.   They were originally measured by the COBE (Cosmic Background Explorer) and WMAP (Wilkinson Microwave Anisotropy Probe) satellites, which measured the cosmic background radiation coming from different parts of the sky.

By looking in different directions in the sky, COBE or WMAP could detect tiny variations in the cosmic background radiation (one part in 10^{-5}), which seems to mostly confirm the inflationary model.  However, because these fluctuations come from a one-time-only event, if they should happen to have formed in a statistically unusual way, there is no way of compensating for this by repeating the experiment and averaging out all the results.  Although repeating the experiment can reduce the uncertainty due to measurement error, there remains a residual uncertainty, the Cosmic Variance, which can never be reduced because it is always the same every time the experiment is performed.

Someone who was a stickler could say that the theory of inflation "doesn't count as Science" because you can't get an arbitrarily large amount of data on it.  But that would be silly, since it's so similar to other kinds of Science, and you can still repeat in the sense of looking again.

So the requirement that Science be repeatable is flexible, depending on the nature of the particular thing being studied.  Sometimes we have experiments; sometimes we have a lot of data; sometimes we have a finite amount of data that can never be increased, and we just do the best we can.  But if it doesn't ultimately rely on observations, it isn't Science.

Posted in Scientific Method | 3 Comments

The Pillars of Science

by Aron Wall Posted on October 26, 2012

In the old days, the word "Science" used to just mean "knowledge", no matter how you got the knowledge, or what it was about.   But these days, we use the word to mean a particular kind of academic subject, distinguished from e.g. knowledge of history, literature, philosophy, or law.

Although grandiose philosophical ideas about how to process all forms of evidence often invoke the word “scientific” for its selling-power, it is less confusing to restrict the term “Science” to mean what used to be called “Natural Philosophy”, i.e. the study of Nature through research disciplines whose methodology is sufficiently similar to e.g. Physics, Chemistry, Geology, and Biology etc.  To avoid confusion, I will always use the term "Science" in this modern sense.

Science is an uncannily accurate method for separating truth from error.  Its accuracy does not come from luck, but from a scientific culture which demands carefully checking hypotheses to see which ones fit the world the best. This culture operates by certain rules.  These rules were not imposed for philosophical reasons by some enlightened lawgiver (shut up, Bacon!).  Instead, they grew up gradually over several millennia.  Eventually people figured out, by a process of trial and error, which methods work best for finding and testing new ideas.

In the series of posts that follows, I will describe six "Pillars of Science": 1) Repeatable Observations, 2) Elegant Hypotheses, 3) Approximate Models, 4) Precise Descriptions, 5) Ethical Integrity, and 6) Community Examination.  Any one of these ideals can be culturally revolutionary standing alone, but together they promote a truth-seeking culture capable of the astonishing discoveries we've all heard about.

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