I’m not sure I understand matter

Posted on 2012-01-24 by Gilbert

Catholic thought has been deeply entwined with Thomism for centuries and while the association has loosened it is still very present.  So as a reactionary Catholic I’m of course sympathetic to that philosophy.  But the truth is, my understanding of it is quite deficient.  I have read parts of Thomas’s and Aristotle’s  works and some popular explanations but not enough for a full understanding.

Typically I read up on some questions I’m interested in until I can make sense of it. But that’s basically using famous works as a quarry, gathering material to patch up my thinking.  I’m not really interested in what Aquinas said because he said it, but because it is often true.  At least for a layman like me, that seems like a reasonable way to do it.  But sometimes I’m not sure if I understood the original answer or just reinterpreted some proof-texts to mean an answer that makes sense to me.

One example where I’m unsure if my interpretation fits the original system is matter. It probably doesn’t mean what we call by the same name in modern physics, but then what does it mean? I tend to look at this by looking at the function the concept serves in the original context and then trying to figure out if I know of anything serving that function in reality.

So one reason to talk about matter and the related concept of form is explaining creation and destruction. Consider, for example, a paper clip. If I bend it out of form I have a piece of wire that is no longer a paper clip. It may also be melted down and remolded into some other object. In both cases the paper clip as such ceases to exist. But I still have a new object made of steel and in fact made of the same steel the paper clip was previously made of.  Now we just stick a label on it: What perished gets called the form and what survived gets called the matter of the paper clip. The paper clip was made up of both and that kind of combination gets called a substance[1]. So in our example the matter of the paper clip is steel and the form might be something like “a thin, kinked wire forming two loops elastically connected”.

One consequence of this is that what matter is in any given context depends on the substance we’re analyzing. For example, a chunk of steel is the matter of the paper clip, but that chunk of steel is itself a substance composed of form and matter. The matter would be atoms of iron and carbon and possibly some others and the form would concern, for example, the crystal structure and such things. This relativity is, I think, what the distinction between proximate and remote matter is getting at.

The extreme end of this continuum would be matter devoid of any form, which is called prime matter. Prime matter as such can’t exist or even be imagined, because it is, by definition, the pure potential to exist in some form. Basically neither forms nor prime matter directly exist, they are rather aspects of the substances that do directly exist.  Fitting either to objects known in, say, modern physics would be missing the point.

So far I’ve just been parroting the standard definitions, now I’ll try to apply it and possibly prove I don’t really get it: How far down the chain can we actually go? What is, so to say, the maximally remote matter actually existing? The knee-jerk answer would be elementary particles, but I think that is wrong. The problem is that elementary particles can sometimes be created and destroyed. The canonical example would be photons, that is the particles of which light is made of. The screen I’m presently staring at doesn’t have a reservoir of photons, it (or some part in it) is creating them as it sends them to me. And then they hit my retina and cease to exist. But they don’t simply appear and disappear without trace. They do, for example, carry energy from the screen to my retina. And I think in this framework that would make the energy part of their matter[2]. The photons are made of energy, that energy exists before and after them, so it is pretty much by definition matter in relation to them.

But how can I say the energy survives the end of the photon? Whatever absorbed the photon doesn’t keep the energy around separately, it just gets added to the energy it already has.  I think the answer is that energy is conserved, i.e. the total amount of energy is the same before during and after the photons existence. But then I could make the same argument for other conserved quantities. So, for example, charge and spin would also be maximally remote kinds of matter in the metaphysical sense. I’m not so sure I can extend the same treatment to vectorial quantities. I thought, for example, about momentum  being a kind of matter, so that a substance being in motion would be equivalent to it being conjoined to the motion kind of matter. The problem with this interpretation is that two frontally colliding objects with momentum \(p\) and \(-p\) stop. Total momentum is conserved, because it was \(p-p=0\) before the collision and is \(0+0=0\) after it.  But the momenta of the two objects cease to be and that is not something matter should do[3]. So my best guess is that the maximally remote kinds of matter actually existing correspond to scalar conserved quantities. Nice slogan, eh?

Now the question is, could a Thomist who really knows his stuff agree with that?  And the answer is I don’t know. My understanding is compatible with the usage of the word matter in all arguments I know of, but then I obviously don’t know all arguments. So maybe the real Thomist would tell me this is bunk because of some extremely well known text I missed.  On the other hand maybe he would dismiss it as trivial stuff every serious metaphysicist knows about. So I have an understanding of matter and it serves me, but I’m still not sure I actually understand what other people meant by matter.

Footnotes    (↵ returns to text)

  1. Note that I am limiting myself to physical substances. It gets more complicated with, for example, the matter of sacraments.↵
  2. You might have a spontaneous association of \(E=mc^2\), but that is entirely superficial. I’m talking metaphysics here. It’s not about about energy being equivalent to mass, it’s about them falling in a common philosophical category whether they are equivalent or not. Also, in a minute I’ll be arguing for further members of that category not equivalent to either mass or energy.↵
  3. If you just thought of antimatter your associations are as loose as mine. I’m talking of a philosophical concept that just happens to share a name with a physical concept descended from it.↵
Posted in Armchair philosophy | Tagged | 4 Comments

Slippery slopes, statistics and the telephone game

Posted on 2012-01-12 by Gilbert

Leah Libresco of Unequally Yoked recently told her readers:

In statistical news, social scientists have not found any evidence of doctor assisted suicide launching us down a slippery slope.  The rates of people electing medically-assisted suicide has stayed flat in Oregon and the Netherlands (and in the Netherlands, there’s about 15 years worth of data).

These data aren’t a rebuttal to anyone who opposes doctor-assisted suicide for all people, but, if I’ve got any readers who were concerned the procedure would pressure the elderly to choose death rather than be a burden, how long would this trend need to last for you to consider it persuasive?

Now I’m one of the people she refers to in the second paragraph and would be opposed to Euthanasia even if there was no slippery slope. But I still think the slippery slope argument is very important. Not only does it add a practical problem, I also think that leading to further evil is a good heuristic for dubious act being evil in the first place.  So while not the only reason, the slippery slope is  one reason we know euthanasia is evil in the first place.

But does that slippery slope actually exist? Leah says no and she has a source to back her up.  Her reference is a post at TYWKIWDBI which in turn quotes an article by Peter Singer[1], which in turn relies on the report[PDF] of a Royal Society of Canada expert panel on end-of-life decision making. That report in turn relies on various official data sourced of which we will talk in due course. My headline promised you a telephone game and now you know the players, but is the data transmission actually faulty? And if yes, where did the transmission errors actually happen?

Well, the relevant part of what Leah read at TYWKIWDBI is this:

In The Netherlands, voluntary euthanasia accounted for 1.7% of all deaths in 2005 – exactly the same level as in 1990…

In Oregon, where the Death with Dignity Act has been in effect for 13 years, the annual number of physician-assisted deaths has yet to reach 100 per year, and the annual total in Washington is even lower…

[I have deleted their emphasis because I quote only the emphasized part.] “The rates are flat” certainly seems like a good summary of this. But note, that the snippet doesn’t say that explicitly. The Netherlands rate in 2005 was the same as in 1990 but it doesn’t tell us anything about what happened in between. And on Oregon and Washington we get an absolute number not yet surpassed but nothing about what the actual numbers were. As we’ll see in a moment both these silences are telling.  But Leah’s interpretation clearly is the one the text naturally suggests, so I’ll say we have no reason to blame her.

TYWKIWDBI, too, is innocent. The quote is taken directly and faithfully from Peter Singer’s article. The shortening is correctly marked and doesn’t change the meaning. There is no omitted context that would change the message. In short they didn’t add anything and therefore didn’t add anything wrong.

Peter Singer is where it gets interesting. Let’s start with the Netherlands.

If we could rely on the numbers reported through official channels (prosecutors for part of the period and later commissions set up for that purpose) there would be no discussion. These numbers go up, period. Problem is, reporting practices used to be abysmal and have gradually improved. So how much of the increase is in actual euthanasias[2] and how much is just improved reporting of what always happened? Answering this question is part of why the Dutch government commissioned five anonymous polls of doctors in 1990, 1995, 2001, 2005, and 2010. The results of these polls are extrapolated to the national rates most people rely on. [3] The results of the 2010 poll don’t seem to be out yet [Update: They are now, look at my follow-up post], so the Canadian panel had four data points, of which Peter Singer quotes the first and the last, noting they give the same rate.  To end the suspense, the report says

In 1990, 1.7% of all deaths were preceded by voluntary euthanasia, as compared to 2.4% in 1995, 2.6% in 2001 […]. This trend reversed in 2005, when again 1.7% of all deaths were the result of voluntary euthanasia[…]

This is interesting. Yes, the 2005 rate is back to its 1990 value. but that’s a decrease after it had been going up for 10 years.  And yes, that reversal follows the 2001 change from a non-prosecuting regime to official legalization. So slippery slope believers like me need to explain why the rate went down again after the law got worse, and I’ll be doing so infra. But the data  doesn’t quite fit with the other story either. After all, even the most ardent euthanasia supporter wouldn’t argue that legalization propels us up the slippery slope.[4]  Basically the actual data should send us looking for other factors explaining the change.

So is Peter Singer the message-corrupting player of the telephone game? Well yes and no. No, because the Canadian panel actually argued against fear of the slippery slope and used this data in partial support of its argument. But their argument is more complex then “rate flat, therefore no slippery slope”. At least right now I won’t engage the real argument beyond saying it leaves me unconvinced. I couldn’t do it justice without derailing my train of thought.  But here we have the yes part of my last question. With a standard length column, of which he used one third on a touching lead-in story and another third to trash Christianity, Peter Singer didn’t have space do the real argument justice either. So he simplified it to a news bite of two data points and the conclusion.  Like most news bites it isn’t technically false. And like most news bites it stems from a desire to communicate something similar to a real argument. And as with most news bites the similarity turns out to be pretty remote. So his summary did set readers up for a serious misunderstanding.

But back to the main question: Why did the rate go back down? Here it helps to look at the last step of the telephone game. The Canadian panel takes its data from the report[PDF] of the dutch panel that collected it. The report is, unsurprisingly, in Dutch, but it has an English summary.  And that summary actually has a plausible explanation. I’ll number the reasons for easier reference:

One of the most remarkable findings in the practical investigation is the decrease in euthanasia and assistance in suicide in 2005 compared to 2001 and 1995. […] This decline is linked to a number of other developments. First of all,[1.] the absolute number of deaths in 2005 was less than in 2001, whereas the proportion of persons aged 80 and over (euthanasia and assistance in suicide occur relatively infrequently in this age group) was actually greater. Secondly, [2.] an increase was found in other methods of controlling the symptoms of patients in the terminal phase of their lives, such as continuous deep sedation. In addition, the majority of the physicians thought [3.] there is a clear connection between improvements in palliative care and the decrease in life-terminating action taken by physicians. And finally, [4.] the decrease in the number of cases of euthanasia can probably partly be explained in part by changes in knowledge and opinions on the effects of morphine, which means that physicians are probably less inclined to attribute a life-shortening result to morphine. This has resulted in a decrease in the number of cases of life termination using morphine. We note here that in such cases, it is rather a question of a different appreciation of physicians of their own actions than an actual change in behaviour.

Let’s look at these reasons. #1 is a change in the eligible demographic. If anything it gives cause for the frightening suspicion that euthanasia declined because most of the eligible candidates have already been killed. But it could also be simple demographic change and I haven’t checked the data to see which story is more likely. Either way it isn’t an argument against a slippery slope. #2 has a dark side I’ll talk about later, but for now note that #2&#3 both talk of better alternatives having become available.  Great thing, but not an argument against slippery slopes. On a somewhat bitter note those aren’t newly invented options, just options now more widely available. Perhaps they should have, like,  been made available before, you know, people needing them got killed instead. And finally #4 is, as they say, better knowledge and not a real change.  So all in all none of these reasons contradict the slippery slope story.

Now one might hope these advances have reversed the trend permanently, but a look at the official statistics shows it isn’t so.  They have been rising again, and this time better reporting can’t be blamed alone. 2910 Euthanasias were officially reported in the year 2010, compared to the estimated 2325 for 2005 of which ca. 80% had been reported. Even in the best case, where reporting is now perfect, the euthanasia rate is up by a roughly a quarter  since the 2005 result on which the idea of it having stabilized rested.[5]

This isn’t all. As I already mentioned, the second reason quoted for the 2005 decrease has a dark side.  Continuous deep sedation, one of the alternatives that has increased, means keeping the patient unconsciously sedated until death. In itself, this is not a moral problem. But at least in the Netherlands it is more often than not combined with the decision to forgo hydrating and feeding the patient as unnecessary medical treatment. And if we know they will die within a day anyway, as about half of them do, that might even be legitimate. But otherwise it is simply a more distanced way of killing. Looking at it from this angle it is very disturbing that 7.1% of Dutchmen dieing in 2005 were under continuous deep sedation, up from 6.0% in 2001.  It’s hard to figure out how much of this is killing and how much legitimate palliative care.  But there is at least a very real possibility that the killings just moved to another name and protocol.

So all in all we have seen that the rise of euthanasia in the Netherlands has at best been temporary interrupted by factors unrelated to legalization.

And in Oregon and Washington? Well both Oregon[pdf] and Washington[pdf] publish official reports of their acts’s implementation. In Oregon both dispensations and deaths were higher than ever before. The Washington regime only took effect in 2009 so the rate increases in the second year don’t prove any trend yet. If the point is supposed to be that the numbers are smaller than in the Netherlands, that would be an apples and oranges comparison. For one, the American regimes are much stricter. But more importantly, assisted suicide (patient takes prescribed poison) isn’t as common as euthanasia (doctor directly kills patient) in the Netherlands either.

So on Oregon and Washington too, Peter Singer cited statistics that are technically true but give a very wrong impression.

To sum up my message, euthanasia rates have a clear upwards trend, we shouldn’t rely on Peter Singer to summarize complex circumstances and it is easy to fall for “facts” produced by telephone games.

Footnotes    (↵ returns to text)

  1. Yes, that Peter Singer, the guy who thinks infanticide is OK but eating meat isn’t.↵
  2. Or what the Dutch call euthanasias. They define the term narrowly to exclude people who get euthanized involuntarily.  Since we are talking about growth rates rather than absolute numbers, that isn’t relevant to our present discussion.↵
  3. I have one nit-pick here, that doesn’t relate to the rest of the post: Those polls are voluntary and had response rates of about 3/4. This is not a bad response rate, but enough to give us the usual problem of voluntary polls: If the propensity to answer is correlated with the question asked, the calculated rates are wrong.  I do actually suspect that kill-happy doctors are less answer-happy, thus giving understated rates. But I can’t prove that, it might be just the other way around.  For our purposes this is less relevant, because we are interested in changes rather than absolute numbers.↵
  4. Or maybe I’m too optimistic here. I have heard the analogue argument for legalized abortion and the right-wing equivalent is, of course,  supply-side economics.  So make that the most ardent rational supporter.↵
  5. Or maybe a little less, because total deaths might have gone up. I’m too lazy to  look that number up, but it can’t make that large a difference.↵
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The BBC thinks I’m morally lenient

Posted on 2012-01-04 by Gilbert

The BBC offers a quiz to Test Your Morality (HT: Brian Green at The Moral Mindfield).

Basically they present you with some scenarios of what someone has done and ask you to give 1-10 ratings of your sense of wrongness, of anger, and of disgust, as well as your desire to avoid and to punish such a person. Then it tells you what you told it.  They are also selling it as testing a scientific hypothesis, but I don’t take that angle all that serious.

So according to the BBC I’m below average in all measures of condemnation, but especially in anger.  It would be great if that was true, but I happen to know myself better than the BBC and it isn’t.

Nobody takes this kind of Internet test seriously anyway, but I’ll spoil the fun and give a reason for dismissing it: They ask you to rate various emotions on a 1 to 10 scale that isn’t calibrated to anything. This is not unusual in the social sciences, but at least for evaluating individuals it’s bunk. I think the normal reaction is to put some imagined act at 10 and then to compare to that.  And I suspect my imagination runs a little wilder than mosts. I think I can see some evidence of this on the punishing scale.  I interpreted it as if I was the competent authority, because otherwise punishing would simply be non of my business and I would have to check 1 everywhere.  I suppose most people would do the same instinctively and that gives an objective calibration, because we know what kind of punishments governments dole out.  And I was indeed closer to average on this scale than on the ones calibrated only by imagination.

Still it’s fun. Go try it.

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Presidential status check

Posted on 2012-01-04 by Gilbert

Christian Wulf, the president of Germany (a figurehead in our constitutional order) is presently caught in a bit of a scandal. Everyone is wondering if and when he will resign.  If he actually resigns I might use the selection of his successor as an excuse to explain the German constitution on this blog. But for now, what if you aren’t obsessive about German news?

Well, then here’s the website for you: istchristianwulffnochimamt.de efficiently conveys the one bit of information everybody wants to know. IstChristianWulfNochImAmt is German for IsChristianWulfStillInOffice.  At present the indicator stands at “Ja” (yes) if he should resign it would presumably change to “Nein”.

 

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Laughing all the way to church

Posted on 2012-01-03 by Gilbert

Behold, my reader, this scene from Monty Python’s The Meaning of Life:

I was reminded of this of this scene on reading Between Heaven and Mirth: Why Joy, Humor, and Laughter Are at the Heart of the Spiritual Life by James Martin, SJ.  Just a few days ago I learned about that books existence from a review at Leah Libresco’s Unequally Yoked.

Wishful thinker that I am, I ignored both the book’s official description and the review, jumped to the conclusion that this must be mainly a collection of Catholic jokes, and immediately bought the Kindle edition.

Now imagine, if you can, me, in my pajamas, Kindle in one hand, nice cup of tea in the other, lying down for some light bedtime entertainment. And the first thing I read is an extensive explanation of how humor has a legitimate place in Christian spirituality.

I couldn’t agree more! We could indulge in Catholic jokes anytime we wanted! That’s what being a Catholic is all about! That’s why it’s the  Church for me! That’s why it’s  the Church for anyone who respects the individual and the individual right to decide for him- or herself! And Catholicism doesn’t stop at a simple joke, I can read irreverent and biting jokes if I want. I can go down the road, anytime I want, and walk into the next book store, and hold my head up high, and say in a loud, steady voice: “Sir, I want you to sell me a joke collection! In fact today I’ll have it all sarcastic. For I am a Catholic!”

Now to be clear, it’s all my fault that I ended up playing both roles in that video. Only my own lewd fantasy tricked me into expecting a book different from the one both promised and delivered. Also, the book does contain some good jokes, it’s just that they are not the main focus. So it’s probably about time to stop the self-pity.

Anyway, having read a book about it and now being all informed on the relation of joy, humor and spirituality, I suddenly feel qualified to offer my own profound thoughts both on the book and the question. You get my opinion on the book in this here post and my additional thoughts on humor and religion in two upcoming ones.

A bit more than one third of the  book is spent on an argument that humor has a rightful place in spirituality. This is done in the way we Catholics argue for basically anything we argue for: An intro chapter catches our attention and explores both everyday and spiritual connections of humor. Then follows a chapter explaining away the humor-hostile parts of our tradition. Next is some scriptural support for humor (actually the first of three scriptural “studies in joy” interspersed between the chapters), followed by a chapter on humor in tradition as seen through the saints and a chapter listing some practical advantages of humor, which is not too far from an argument from natural law and general revelation. Don’t let my jestful description of our standard argumentative pattern mislead you, I actually find this kind of argument pretty darn convincing. And while I didn’t need any convincing in the first place, Father Martin’s argument surely could have done the job.

There is however one detail I’d like to grumble about. He never really says so, but to me it read like humor was an obligatory part of spiritual life. I don’t believe that. Yes, God is funny, but God is also musical and I’m not. That lack of talent doesn’t separate me from god and a lack of humor wouldn’t either.  Near the book’s beginning we hear the horror story of an old pre-councilar Jesuit reprehending one of Father Martin’s friends that “all levity is excessive”. Now to be honest, if you feel like that grump I probably don’t want to hang out with you. But I’m pretty sure Jesus does. And not just because he will fix you, but because that kind of sober and earnest obligation-discharging, too, is is a path to him.

Having established that humor and religion ought to be integrated Fr. Martin goes on to give us advice on how to do it. The theme broadens here.  While the theoretical argument for humor was about humor specifically, the practical parts are about joy in a broader sense with humor as an important instance of it.

This advice part is where the book turns out to be  – and I mean that the good way – profoundly Jesuitical. See, the question of how to be joyful is not all that different from the question of how to get closer to the source and perfection of all joy, which is God. And advice on humor and something is always to a large part advice on the something. So the discussion of joy, humor and various parts of life is quite naturally suffused with more general spiritual guidance.

For example chapter five is on “how vocation, service and love can lead to joy”. In answering that question it naturally also explains the Christian idea of vocation, service and love.  Likewise chapter nine gives advice on how to pray joyfully, but in doing so it first of all gives advice on how to pray.

Sandwiched between these  chapters on the “big” spiritual questions are three more on the plainer everyday business of how to live a joyful spiritual life individually (chapter eighth) and and as a church (chapter six) as well as on overcoming difficulties one might face in doing so (chapter seven). The subtext of spiritual guidance is weaker here, but still present.

So altogether there is a good Catholic argument for a joyful spiritual life, a how-to guide on doing it and some  unobtrusive general spiritual guidance. I haven’t tried all the advice offered, but the parts I can judge are mainly good advice. And it’s an entertaining read, with some jokes, though not enough for a joke collection. Add it all up and I am quite satisfied with having cheated myself into buying this book.

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Quantum sophistry

Posted on 2011-12-30 by Gilbert

A few days ago, we got to know the members of the St. Hypotheticus drinking and nerdery club. Today we listen in on their inaugural meeting. They have just finished the administrative part and are now about to enter on the drinking and nerdery.

Carl: (Knocking at a beer keg) All right, I already prepared the drinking …

[applause]

Carl: …now for the nerdery. Any ideas?

Bob: I have a question about quantum mechanics and philosophy.

Albert: (groan) You know, most I heard about this is bunk. It has far less philosophical consequences than you might think.

Bob: I know. The thing is, shortly I was accosted by some young folks at the college who where looking for an analytical philosopher…

Matthew: …Uhu…

Bob: An analytical Thomist does totally count, and besides if it was so important they could have asked, couldn’t they? Anyway, they asked if one could empirically prove two particles indistinguishable even in principle and I said no. Then they walked away laughing and talking about configurations and flows and distinguishable wave functions. Most of it went over my head, but I googled it and it turns out they were working from a script. I brought some copies, so I wondered what y’all have to say about that.

Matthew: Argh, it’s the…

Albert: … many-worlders …

Clara: … Bayesians …

Matthew: …Gnostics.

Bob: Um yes, there is a lot of strange stuff on that site, but I don’t even know how much of that the nice folks who followed the script actually believe. So can we look at the point? That script claims:

If you have a particle P1 and a particle P2, and it’s possible in the experiment for both P1 and P2 to end up in either of two possible locations L1 or L2, then the observed distribution of results will depend on whether “P1 at L1, P2 at L2” and “P1 at L2, P2 at L1” is the same configuration, or two distinct configurations. If they’re the same configuration, we add up the amplitudes flowing in, then take the squared modulus. If they’re different configurations, we keep the amplitudes separate, take the squared moduli separately, then add the resulting probabilities. As (1 + 1)2 != (12 + 12), it’s not hard to distinguish the experimental results after a few trials.

Clara: Well first of all someone should tell them Bayesian geniuses that neither \(\left(1+1\right)^2=4\) nor \(\left(1^2+1^2\right)=2\) are acceptable values for a probability. You know, those are supposed to be between 0 and 1.

Carl: Yeah, but the point seems to be that addition and modulus squaring don’t commute, and that much seems true.

Albert: Is there some spin \(\frac{1}{2}\) stuff in there?

Bob: It explicitly says half-integer spins don’t matter.

Albert: That sure is a relief, otherwise our author might have wounded his fingers typing up all the math. Anyway, other than in the detail Clara just noted, that quote is quite true. But the claim it is supposed to support just doesn’t follow from it, except with an implicit wrong assumption that it couldn’t be just one configuration even if the particles are distinguishable in principle.

Bob: I’m not quite sure I get that. Can we look at this in more detail?

Albert: OK, the most famous mathematical objects we have in quantum mechanics are wave functions.  In the simplest case they describe  single spinless particles and then they are just  functions that gives you a complex number for every point in  space.

Jenny:  Why complex?

Albert: So that they can interfere. But don’t worry about that now, it isn’t that important for what we’re talking about today.

Jenny: OK.

Kate: Fine, so what do we do with these complex numbers?

Albert: Take their modulus (don’t worry about that step for now) and square it. And that gives us a probability density.

Matthew: Um, density?

Albert: Yeah, there are infinitely many points, so we can’t give probability to every one, or else we would have trouble with them adding up to more than one, and we can’t be more than sure the particle is somewhere. But the math trick isn’t that important, if there where only finitely many points this would simply be a probability. Now there are some neat things we can do with wave functions. One thing we can do with them is multiply them with (complex) numbers. So for example if \(\psi\) is a valid wave function, then so is \(5\psi\).

Clara: Whoa! That can’t work, you’re bringing in those probabilities larger than one again!

Albert: OK, we do some normalization. We said for a wave function \(\psi\left(\vec x\right)\) the probability density for finding the particle at \(\vec x\) is \(\left|\psi\left(\vec x\right)\right|^2\).  Now if those probabilities would add up to more than one, whatever they add up to is called \(\left\langle\psi\middle|\psi\right\rangle\). There’s a meaning to that notation, but today we’ll just treat it as a name. Then instead of \(\left|\psi\left(\vec x\right)\right|^2\) our probability density is actually \(\frac{\left|\psi\left(\vec x\right)\right|^2}{\left\langle\psi\middle|\psi\right\rangle}\).

Clara: OK, now you just canceled out the factor you had allowed in. What is this good for again?

Albert: Quite true, alone it would be useless. But we can also add wave functions. So if \(\psi_1\) and \(\psi_2\) are valid wave functions, then so is \(\psi\) with \(\psi\left(\vec x\right):=\psi_1\left(\vec x\right)+\psi_2\left(\vec x\right)\).

Clara: Aha! So you’re telling me they form a vector space.

Albert: Yes, though I didn’t want to confuse everyone else with that terminology. So now we can write some wave functions in terms of others. What we do next is select some set of wave functions as a base[1] and then write all the others by multiplying and adding those. To put the same in more pompous words, every wave function can be written as a linear combination of base functions.

Now, for many-particle systems it works in basically the same way.

Bob: Care to elaborate on that?

Albert: I think discussing it for two particles will do.  The principles generalize to an arbitrary number of particles, but then the math gets a bit more complicated.

Bob: Fine by me.

Albert: The way it works for two particles is is basically the same as for one. We just get a wave function depending on two locations and then the probability density for particle 1 being at \({\vec x}_1\) and particle 2 being at \({\vec x}_2\) is \(\left|\psi\left({\vec x}_1,{\vec x}_2\right)\right|^2\).

Bob: Well mathematically it sounds very similar, but only because we haven’t talked about any particular wave functions and operators in either case. If we don’t have any connection between the particular objects for different numbers of particles, then the one-particle case just isn’t a simplification of the two-particle case.  And then by extension it would never make sense to talk of a part of a whole because the whole wouldn’t just be more than but actually unrelated to the sum of its parts. Since we can’t conceive of the whole universe at once that would make all physics impossible.

Albert: Right, but luckily[2] there is such a connection. It turns out, we can express the base functions of the two-particle system by those of two one-particle systems. Now how we do that depends on whether the particles are distinguishable and that is what those folks you met where getting at.

Bob: Now it gets interesting.

Albert: Yup. If the particles are distinguishable it’s really easy. In that case if we have a base function \(\psi_1\left({\vec x}_1\right)\) for particle one and a base function \(\psi_2\left({\vec x}_2\right)\) for particle two, then \(\psi\left({\vec x}_1,{\vec x}_2\right)=\psi_1\left({\vec x}_1\right)\psi_2\left({\vec x}_2\right)\) is a base function for the two-particle system and all two-particle base functions can be expressed in that way. Since we don’t always want to mention the arguments I’ll introduce some notation here: Instead of \(\psi\left({\vec x}_1,{\vec x}_2\right)=\psi_1\left({\vec x}_1\right)\psi_2\left({\vec x}_2\right)\) we’ll write \(\psi=\psi_1\otimes\psi_2\).

Bob: I psi[3]. But of course this can’t work if they are indistinguishable, because if \(\psi_1\neq\psi_2\) we would have the two particles distinguished right there.

Albert: Right. In the two-particle case it turns out that if \(\psi_1\) and \(\psi_2\) are one-particle base functions, then  \(\psi=\frac{1}{\sqrt{2}}\left(\psi_1\otimes\psi_2\pm\psi_2\otimes\psi_1\right)\) is a two-particle base function. It’s only one base function, it’s either \(+\) or \(-\) depending on what kind of particle we’re talking about.

Bob: Slow now, I don’t quite understand this form. First, what is \(\frac{1}{\sqrt{2}}\) doing in there?

Albert: You remember how a few minutes ago we talked about dividing out \(\langle\psi|\psi\rangle\) to make sure all the probabilities added up to 1? Well, most of the time it is easier to multiply the whole function  by some constant that makes  \(\langle\psi|\psi\rangle=1\), so that we don’t have to bother with that step. In this case \(\frac{1}{\sqrt{2}}\) is that constant.

Bob: Fine. And the part in the parentheses?

Albert: That part takes care of the indistinguishability. See, if even in principle nobody knows which particle is where, then the probabilities really shouldn’t change if we swap them. So we want \(\left|\psi\left({\vec x}_1,{\vec x}_2\right)\right|=\left|\psi\left({\vec x}_2,{\vec x}_1\right)\right|\) and this form guarantees precisely that.

Bob: Great.

Albert: There is one more feature I would like to draw your attention to. Please note \(\psi_1\otimes\psi_2\) and \(\psi_2\otimes\psi_1\) are base functions for the system of two distinguishable particles. So the base functions for the system of indistinguishable particles are linear combinations of those for the system of distinguishable particles.  That means all wave functions for indistinguishable particles are also valid wave functions for distinguishable particles.

Bob: But not the other way around?

Albert: Nope, not the other way around. In fact if \(\psi_1\neq\psi_2\) then \(\psi_1\otimes\psi_2\) and \(\psi_2\otimes\psi_1\) themselves are wave functions for the system of distinguishable particles that don’t work for indistinguishable particles.

Bob: You know, the wave functions for indistinguishable particles being a subset of those for distinguishable ones makes sense.  After all we can always not distinguish distinguishable particles, but distinguishing indistinguishable ones is, by definition, impossible.

Albert: Exactly. But now look at this: Even if they would be distinguishable in principle, we can prepare two particles we can’t practically distinguish into an indistinguishable configuration. But if we could prepare them into any of the additional states available to distinguishable particles that would amount to distinguishing them. Since we have assumed we know of no method of distinguishing them, we also know of no method of preparing them to any of the additional states they might have available if they are distinguishable in principle. So whether our inability to distinguish them is an intrinsic feature or our ignorance, the set of configurations we can prepare is identical. Both cases correspond to only one alternative of the experiment they were proposing and therefore it can’t be used to tell the difference between those two cases.

Bob: So you can’t be sure the particles we now hold indistinguishable will never turn out to be distinguishable after all?

Albert: I can think that very unlikely chiefly because of Friar Ockham’s razor, but no, I can’t be absolutely sure of it.

Bob: It’s nice to be right.

Albert: Ahem. The argument you gave for your position was actually invalid. It’s only by luck that you turned out to be technically correct.

Bob: That is, of course, always the best kind of correct.

Footnotes    (↵ returns to text)

  1. Well, complete orthonormal system actually, since some of the vectors might be infinite linear combinations of the base vectors. But if you know that difference you don’t need me to explain it and if you don’t you won’t care.↵
  2. Actually the connection is not by luck or even by providence but by mathematical necessity. But we won’t go into that now.↵
  3. ha, ha↵
Posted in Socratic dialogues | Tagged , , | 4 Comments

Meet the cast

Posted on 2011-12-27 by Gilbert

One genre I’d like to try on my new blog is Socratic dialogue. The first one will go up on Friday.  This post gives an overview of the characters and background I created for that purpose.

The parishioners at St. Hypotheticus mostly like one another. But many of them are colorful characters and sometimes maintaining their sympathy requires them to keep a bit of a distance.  For example, at last summer’s parish picnic some of the nerdier parishioners got  started on math jokes. It began small, with Carl  quipping “Let \( \varepsilon<0 \) .” But soon the slightly intoxicated nerds where boring the crowd with math puns. After half an hour of hearing about Piano curves, Abelian soups, and Bananach spaces, Sally made them stop. Sally is the organizational backbone of the parish. She does most of the work and even Fr. Bill is afraid of her. The parish picnic, said Sally, was supposed to be family friendly fun and fellowship[1]. If they wanted drinking and nerdery they could separately meet for that purpose. That was the beginning of the St. Hypotheticus drinking and nerdery club.

The inaugural members are:

Carl, Sally’s Husband. Since Sally isn’t a member he got stuck with the presidency, i.e. with buying the beer. Carl is an electrical engineer and model railroad enthusiast. When they married  Carl reluctantly agreed to live by Sally’s very conservative interpretation of  1 Cor. 11:3. But after 20 years of headship he sometimes wonders if there might be some nerve damage.

Jenny, the social justice Catholic. She is always carrying a petition for everyone else to sign. People who agree with her pretend they don’t, just to see her get agitated. Occasionally she tries to bribe potential signatories with books. The deal is worth taking, because she’s read everything and knows who will like what.  Still in high school, Jenny plans to study psychology in college.

Bob, a starving Thomist philosopher barely surviving by adjuncting at a nearby college. Sally regularly pesters him to get rid of his ratty clothes, arguing that Carl’s cast-offs would still be a massive improvement. Bob doesn’t see the need. The remote final cause of better clothing, he argues, is marriage and look where that led Carl.

Matthew, the grumpy news-junkie.  Matthew likes to joke he doesn’t attend mass in the extraordinary form because he won’t be satisfied before they go back to Aramaic. A Student of economics at St. Hieronymus university Matthew subscribes to several magazines and hundreds of blogs. While reading he fumes about people never learning. After all, he maintains, every political philosophy now extant is a replay of an early Church heresy. Still underage, he’s dissatisfied about not being allowed to  participate in the club’s drinking.

Clara, who’s dual goals in life are practical research into creating the perfect pizza and the eradication of the Bayesian heresy. Clara works as an actuary and is alway busy preparing for exams.

Albert, who found his dream job as a teacher of math and sciences at the local Catholic high school.  Albert loves explaining things. He’s not too particular about his victims actually wanting to know what he explains to them. Most of his free time is spent on hobby electronics.

Kate, Albert’s newly-wed wife. A lawyer and Harry Potter fan, Kate is working on her opus magnum on the constitution of magical Britain. She also maintains the drinking and nerdery club needs a proper charter and is trying to recruit a second member of the drafting committee.

Somewhat unrealistically the club will not be seen praying on screen.  That is because I don’t want to give the impression of invoking divine sanction for the viewpoints I’ll be arguing for.

On friday we will listen in on the club’s inaugural meeting, where they will discuss quantum mechanics and philosophy.

 

Footnotes    (↵ returns to text)

  1. Sally is also auspicious about all alliterations absolutely anywhere.↵
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The stable probably stank

Posted on 2011-12-25 by Gilbert

The midnight mass at Christmas Eve attracts a lot of people who don’t otherwise go to mass.  Most priests use the opportunity for a missionary homily, designed to explain as much as possible to the cultural Christians while still preserving the expected sentimentality and high culture earnestness they have come for.  In my experience these homilies are rather boring. I also suspect they mostly go over the heads of their intended audience.

Except this year, where the priest at the midnight mass I attended did it a bit more concrete than usual. I’d like to share the gist of his homily:

Christ was born in a stable.  Stables normally stink. So probably God decided to become man in a literally stinky place. This doesn’t make stinkiness any less stinky, but it also didn’t stop Christ from being born there.

As Christ came to man in general he also wants to come to everyone of us. Doing so, he will find our hearts too are a pretty stinky place. We are often selfish, resentful, cruel, ungrateful &c. This doesn’t stop Christ from wanting to be born in us. This doesn’t mean our stinkyness isn’t real.  But it does mean it is not a hindrance for him wanting to be born in us.  But if he is born in us he won’t remain a child forever. He will grow to instruct and lead us.

In doing this, Christ is also setting an example for us. Often we find other people stinking in the same figurative sense as ourselves. The example of Christ teaches us not to thumb our noses in disgust but to be accepting of the person, stinkiness and all, and to help them, even if the stinkiness itself is a bad thing.

In addition to being an image of each of us, the stable is also an image of all of us, i.e. of the Church.  It stinks in the Church too. (This is particularly true in Germany where we are still in the kind of scandal the American Church has mostly worked through.) The stink is real and shouldn’t be denied.  But it is also no reason for despair,  because this is precisely the kind of situation to which Christ has come.

This simple image brings the message through a lot better than all the more abstract Christmas sermons I ever heard.  Of course it involves telling the congregation they stink, but we all seem to have taken it as slightly funny rather than offensive.

I think going into that kind of detail is beautifull. The abstract theological explanations sure have their place, but it is awfully easy to drown the message in sentimental intellectuality while missing it at the gut level.  Really, Christmas is that concrete and messy, that is what the incarnation is all about.

Posted in Meditations | Tagged , | 1 Comment

Let’s go

Posted on 2011-12-25 by Gilbert

I’ve been thinking about starting a blog for a while and now I’m finally doing it. For now I have an about page and the first real post will be going up shortly. Let’s see where it leads.

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