Monthly Archives: November 2012

Does your vote make a difference?

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Tomorrow is election day here in the US, so I’m going to have a post about elections.  It’s strictly nonpartisan—people seeking bile can look elsewhere.

People often say things like “I’m not voting because it’s unlikely that one vote could make a difference”.  Sometimes they sound rather cynical about it, which I find a little odd, seeing as the whole point of democracy is that no individual gets to dictatorially make all the decisions for everyone.  But let’s suppose they are just trying to be practical.  Is this a rational attitude to take?

There are many possible reasons to vote, even when your vote won’t make a difference.   For example, it could be urged that voting fulfils a civic duty, that casting an informed vote forces you to become educated about the issues of the day, and to come to a decision about them.  It could also be urged that there is some political responsibility being exercised by the members of the majority even when the majority is by greater than one, or a value of protest in voting for the minority position.  Presumably people who “waste” their vote on third parties, or other hopeless causes, are primarily thinking about the communicative effects of voting.

But let’s leave all this aside, and just think about the sort of power that comes from exercising a vote that changes the outcome.  This only happens when 1) the votes for two options are tied, so that you cast the deciding vote, or else 2) the votes for two options differ by one, so that you can accept that result or else cause a tie (the resolution of ties depends on the election system.  If ties are resolved randomly, the final result will change half the time).  For simplicity, let’s assume the election has just two options: Yes and No.

How unlikely this is depends on the number of voters V, and the odds that the election will end up tied (within one vote).  Voters don’t vote randomly, instead the more popular candidate will usually receive more votes. A certain fraction \(x\) of the voters will choose to vote Yes, while \(1-x\) will vote No.  Since we aren’t certain what \(x\) will be in advance, we have to model it by a probability distribution over possible values of \(x\).  This is a function \(p(x)\) describing the probability density of getting any particular outcome \(x\).  (Here I’m assuming that \(V\) is large so that \(x\) can be approximated as a continous variable.  Note that because \(p(.5)\) is a probability density rather than a probability, it can be greater than 1).

The odds of your vote making a difference can now be calculated to be:$$\frac{p(.5)}{V}$$This is a small number.  However, how important would it be if you got lucky and your vote did make a difference?  The results of the election presumably affect a large number of people, roughly comparable to the number \(V\) of voters.  So the factors cancel out, and the expected effect of your vote does not depend on the size of V!  Instead it is proportional to \(p(.5)\), which is equal to 100 times the odds that the vote lies within the critical percentile between 49.5% and 50.5%.

Another way of putting this, is if it would be worthwhile for everyone to miss their lunch to vote, so that everyone can get (what you believe is) a political benefit, then it is worthwhile for you to vote so that maybe everyone can get the political benefit.  Here I’m assuming you are voting for altruisic reasons, to benefit everyone.  The expected selfish benefits to you personally are indeed tiny.

It might still be rational to abstain from voting if 1) the odds of the election being close are small, or 2) the expected difference in benefit from the options is small.  These factors obviously depend on the specific election, but in general one expects that important issues are at stake.  So unless the outcome is a foregone conclusion, it is indeed rational to vote because of its deciding effects.

Does that mean that, if you’re a resident of a non-Swing-State like California, you shouldn’t vote, because (regardless of whether you support or oppose him) we all know that the state will go to Barack Obama?  If you said that, shame on you!  Go look at the local races instead of just thinking about the Presidency.  Although the number \(V\) of voters affected by local politics is much less, the odds of you making a difference are proportionally greater.  The size cancels out, so you should regard local, state, and national races as being approximately equally important.  (Although I suppose one could argue that the President is twice as important, because he determines foreign policy as well as domestic policy.)

The Ten Symmetries of Spacetime

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Previously, I described the main formula of Special Relativity: $$s^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 – (\Delta t)^2.$$This formula tells us the amount of distance squared between two points (if \(s^2 > 0\)) or the amount of duration squared (if \(s^2 < 0\)).  (By using some trigonometry we can also use this formula to figure out the size of angles, so this encodes everything about the geometry).  All the crazy time dilation and distance contraction effects you’ve probably heard about are encoded in this formula.

Today I want to talk about the symmetries of spacetime.  What I mean by a symmetry is this: a way to change the coordinates \((t,\,x,\,y,\,z)\) of spacetime in a way that leaves the laws of physics the same.  Now I haven’t told you what the laws of physics are, but the important thing is that they depend on the geometry of spacetime.  So that means that we need to check in what ways we can change the coordinates of spacetime without changing the formula for \(s^2\).

The first kind of symmetry is called a translation.  This consists of simply shifting the coordinate system e.g. one meter to the right, or one second to the future.  This doesn’t affect the formula for \(s^2\) since it only depends on the coordinate differences \(\Delta t\), \(\Delta x\) etc.  We can write a time translation like this: $$ t^\prime = t + a,$$ i.e. the new time parameter \(t^\prime\) equals the old one plus some number \(a\).  Similarly, the three possible kinds of spatial translations are:$$ x^\prime = x + b; \\ y^\prime = y + c; \\ z^\prime = z + d.$$By choosing the numbers a, b, c, d, arbitrarily, one obtains a four dimensional space of possible translation symmetries.

The second kind of symmetry is more complicated, but you’ve certainly heard of it before—it’s called a rotation.  If we have two spatial coordinates, then we can rotate them by some angle \(\theta\) (measured in radians), which leaves all the distances the same.  The algebraic formula for a rotation looks like this:$$x^\prime = \phantom{-}\cos (\theta) \, x + \sin (\theta) \,y; \\ y^\prime = -\sin (\theta) \,x + \cos (\theta) \,y.$$That involves some trigonometry, but things look a bit simpler if we take the angle \(\theta\) to be a really tiny parameter \(\epsilon\), and just consider the resulting infinitesimal coordinate changes \(\delta x \approx (x^\prime – x)\): $$\delta x = \phantom{-}\epsilon\,y; \\ \delta y = -\epsilon\,x.$$ Translated into English, that says that if you rotate the y-axis of your coordinate chart a little bit towards the x-axis, you have to rotate the x-axis a little bit away from the y-axis (or vice versa if \(\epsilon\) is negative).  I’m too lazy to draw this, but if for some reason you can’t visualize it, a little bit of fidgeting with any rigid flat object should convince you.

Now actually we have three different spatial coordinates: x, y, and z.  That means that you can actually rotate in 3 different ways: along the x-y plane, the y-z plane, and the z-x plane.  Of course there are other angles you can rotate at as well, but they are all just combinations of those three; in other words the space of possible rotations is 3-dimensional.

But now, what about the time direction?  It would feel terribly lonely if it were left out, and in fact it is also possible to rotate spacetime about the t-x plane, the t-y plane, and the t-z plane.  However, remember how time is not quite the same as space?  Instead, it’s just like space except for a funny minus sign.  So not surprisingly, the formula for a rotation also has a funny minus sign—or rather, a funny absence of a minus sign: $$\delta t = \epsilon\,x; \\ \delta x = \epsilon\,t.$$So if you rotate the t-axis towards the x-axis (which corresponds to changing your coordinate system so that you are travelling at a constant speed), then the x-axis has to rotate towards the t-axis (which means that your notion of simultaneity has to change as well).  If you know how to integrate this with calculus, you can get the effects of a finite “rotation” in space (called a Lorentz boost) through an “angle” \(\chi\):$$t^\prime = \cosh (\chi) \, t + \sinh (\chi) \,x; \\ x^\prime = \sinh (\chi) \, t + \cosh (\chi) \,x.$$In the above, cosh and sinh are functions similar to cosine and sine but defined using hyperbolas instead of circles.

So this rotation has some wierd properties: It describes a crazy world (ours!) in which things rotate in hyperbolas instead of circles.  That’s because of the minus sign in the formula for \(s^2\) above, which makes it so the points of equal distance (or duration) correspond to hyperbolas instead of circles.  This has some additional consequences: 1) Because hyperbolas are infinitely long, the “hyperbolic angle” \(\chi\) ranges from \(-\infty\) to \(+\infty\), unlike circular angles which come back to where you started after you rotate through \(2\pi\) radians.  2) Because the two axes both move towards (or both move away) from each other, when you do a really big rotation it scrunches everything up towards \(t = x\) or \(t = -x\).  What this means is that when you accelerate objects more and more, they don’t go arbitrarily fast.  Instead they just get closer and closer to the speed of light.

In conclusion, spacetime has 10 kinds of symmetry: 4 kinds of translations and 6 kinds of rotations.  The space of possible symmetries is 10 dimensional.  It is called the Poincaré group.

P.S. In this whole discussion I have ignored the possibility of reflection symmetries such as \(t \to -t\) or \(x \to -x\).  These are also symmetries of the formula for \(s^2\), but they are discrete rather than continuous—there’s no such thing as a “small” reflection the way you can have a small rotation.  Adding these in doesn’t change the fact that the Poincare group is 10 dimensional.  However, these transformations are actually NOT symmetries of Nature.  They are violated by our theory of the weak force.  The only discrete symmetry like this which is preserved by the weak force is CPT: the combination of time reflection, space reflection, and switching matter and antimatter.

Pillar of Science II: Elegant Hypotheses

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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.

All Saints Day Roundup

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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.