# The Universe of Discourse

Mon, 20 Aug 2018

I hope and expect this will be my last post on this loathsome subject.

### Diocesan vs. religious priests

In the previous article, I said:

let us say that there are currently around 1620 priests total in the other six dioceses. (This seems on the high side, since my hand-count of Pittsburgh priests contains only about 200. I don't know what to make of this.)

I wonder if it is because my hand counts, taken from diocesan directory pages, include only diocesan priests, where the total of 2500 also includes the religious priests. I should look into this.

I think this is probably correct. The list of Pittsburgh priests has a section the the bottom headed “Religious Priests Serving in the Diocese”, but it is empty.

### Kennedy's Catholic Directory

While attempting to get better estimates for the total number of active priests, I located part of The Official Catholic Directory of P.J. Kennedy & Sons for the year 1980, available on the Internet Archive. It appears that this is only one volume of many, and unfortunately IA does not seem to have the others. Luckily, though, this happens to be the volume that includes information for Philadelphia, Pittsburgh, and Scranton. It reports:

(p. 695)(p. 715)(p. 892)
Priests: diocesian active in diocese 831 547 374
Priests: active outside diocese 33 29 13
Priests: retired, sick or absent: 121 60 52
Diocesan Priests in foreign missions 2 2 26
Religious Priests resident in diocese 553 243 107
Religious Priests from diocese in foreign missions: 93 78 26
Total Priests in Diocese 1631 957 572

I have no idea how authoritative this is, or what is the precise meaning of “official” in the title. The front matter would probably explain, but it does not appear in the one volume I have. The cover also advises “Important: see explanatory notes, pp. vi–viii”, which I have not seen.

The Directory also includes information for the Ukrainian Catholic Archepathy of Philadelphia, which, being part of a separate (but still Catholic) church is separate from the Philadelphia Roman Catholic dioceses. (It reports 127 total priests.) It's not clear whether to absorb this into my estimate because I'm not sure if it was part of the total number I got from the Pennsylvania Catholic Conference.

But I am not going back to check because I feel there is no point in trying to push on in this way. An authoritative and accurate answer is available from the official census, and my next step, should I care to take one, should be to go to the library and look at it, rather than continuing to pile up inaccurate guesses based on incomplete information.

### Sipe's earlier estimate

Jonathan Segal points out that A.W. Richard Sipe, a famous expert on clergy sexual abuse, had estimated in 1990 that about 6% of U.S. priests has sexually abused children. This is close to my own estimate of 6.1% for the six Pennsylvania dioceses. Most of this agreement should be ascribed to luck.

### Final remark

In 1992, Magda Davitt, the Irish singer formerly known as Sinéad O'Connor, became infamous for protesting systematic Catholic sex abuse by tearing up a photograph of John Paul II on live American television. She was almost universally condemned. (The Wikipedia article has a few of the details.)

This week, America: The Jesuit Review, which claims to be “the leading Catholic journal of opinion in the United States”, reported:

Pope Francis issued a letter to Catholics around the world Monday condemning the "crime" of priestly sexual abuse and its cover-up and demanding accountability…

The Vatican issued the three-page letter ahead of Francis' trip this weekend to Ireland, a once staunchly Roman Catholic country where the church's credibility has been damaged by years of revelations that priests raped and molested children with impunity and their superiors covered up for them.

A lot of people owe Sinéad O'Connor a humble apology.

Fri, 17 Aug 2018

The grand jury report on Catholic clergy sexual abuse has been released and I have been poring over it. The great majority of it is details about the church's handling of the 301 “predator priests” that the grand jury identified.

I have seen several places the suggestion that this is one-third of a total of 900. This is certainly not the case. There may be 900 priests there now, but the report covers all the abuse that the grand jury found in examining official records from the past seventy years or so. Taking a random example, pages 367–368 of the report concern the Reverend J. Pascal Sabas, who abused a 14-year-old boy starting in 1964. Sabas was ordained in 1954 and died in 1996.

I tried to get a good estimate of the total number of priests over the period covered by the report. Information was rather sketchy. The Vatican does do an annual census of priests, the Annuarium Statisticum Ecclesiae, but I could not find it online and hardcopies sell for around €48. Summary information by continent is reported elsewhere [2], but the census unfortunately aggregates North and South America as a single continent. I did not think it reasonable to try to extrapolate from the aggregate to the number of priests in the U.S. alone, much less to Pennsylvania.

Still we might get a very rough estimate as follows. The Pennsylvania Catholic Conference says that there were a total of just about 2500 priests in Pennsylvania in 2017 or 2016. The Philadelphia Archdiocese and the Altoona-Johnstown Diocese are not discussed in the grand jury report, having been the subject of previous investigations. The official websites of these two dioceses contain lists of priests and I counted 792 in the Philadelphia directory and 80 in the Altoona-Johnstown directory, so let us say that there are currently around 1620 priests total in the other six dioceses. (This seems on the high side, since my hand-count of Pittsburgh priests contains only about 200. I don't know what to make of this.)

[ Addendum 20180820: I wonder if it is because my hand counts, taken from diocesan directory pages, include only diocesan priests, where the total of 2500 also includes the relgiious priests. I should look into this. ]

Suppose that in 1950 there were somewhat more, say 2160. The average age of ordination is around 35 years; say that a typical priestly career lasts around 40 years further. So say that each decade, around one-quarter of the priesthood retires. If around 84% of the retirees are replaced, the replacement brings the total number back up to 96% of its previous level, so that after 70 years about 75% remain. Then the annual populations might be approximately:

$$\begin{array}{rrrrr} \text{year} & \text{total population} & \text{retirements} & \text{new arrivals} & \\ 1950 & 2160 & 540 & 453 \\ 1960 & 2073 & 518 & 434 \\ 1970 & 1989 & 497 & 417 \\ 1980 & 1909 & 477 & 400 \\ 1990 & 1832 & 458 & 384 \\ 2000 & 1758 & 439 & 368 \\ 2010 & 1687 & 421 & 354 \\ \hline \text{(total)} & & 3350 & 2810 \\ \end{array}$$

From the guesses above we might estimate a total number of individual priests serving between 1950 and 2018 as !!2160 + 2810 - 70 = 4900!!. (That's 2160 priests who were active in 1950, plus 2810 new arrivals since then, except minus !!354\cdot20\% \approx 70!! because it's only 2018 and only 20% of the new arrivals for 2010–2020 have happened so far.)

So the 301 predator priests don't represent one-third of the population, they probably represent “only” around !!\frac{301}{4900} \approx 6.1\%!!.

The church's offical response is availble.

[ Addenda: An earlier version of this article estimated around 900 current priests instead of 1620; I believe that this was substantially too low. Also, that earlier version incorrectly assumed that ordinations equalled new priests, which is certainly untrue, since ordained priests can and do arrive in Pennsylvania from elsewhere. ]

[ Addendum 20180820: Some followup notes. ]

Tue, 14 Aug 2018

A few years ago Katara was very puzzled by traffic jams and any time we were in a traffic slowdown she would ask questions about it. For example, why is traffic still moving, and why does your car eventually get through even though the traffic jam is still there? Why do they persist even after the original cause is repaired? But she seemed unsatisfied with my answers. Eventually in a flash of inspiration I improvised the following example, which seemed to satisfy her, and I still think it's a good explanation.

Suppose you have a four-lane highway where each lane can carry up to 25 cars per minute. Right now only 80 cars per minute are going by so the road is well under capacity.

But then there is a collision or some other problem that blocks one of the lanes. Now only three lanes are open and they have a capacity of 75 cars per minute. 80 cars per minute are arriving, but only 75 per minute can get past the blockage. What happens now? Five cars more are arriving every minute than can leave, and they will accumulate at the blocked point. After two hours, 600 cars have piled up.

But it's not always the same 600 cars! 75 cars per minute are still leaving from the front of the traffic jam. But as they do, 80 cars have arrived at the back to replace them. If you are at the back, there are 600 cars in front of you waiting to leave. After a minute, the 75 at the front have left and there are only 525 cars in front of you; meanwhile 80 more cars have joined the line. After 8 minutes all the cars in front of you have left and you are at the front and can move on. Meanwhile, the traffic jam has gotten bigger.

Suppose that after two hours the blockage is cleared. The road again has a capacity of 100 cars per minute. But cars are still arriving at 80 per minute, so each minute 20 more cars can leave than arrived. There are 600 waiting, so it takes another 30 minutes for the entire traffic jam to disperse.

This leaves out some important second-order (and third-order) effects. For example, traffic travels more slowly on congested roads; maximum safe speed decreases with following distance. But as a first explanation I think it really nails the phenomenon.

Wed, 08 Aug 2018

[ Previously: [1] [2] ]

In my original article, I said:

I was fairly confident I had seen something like this somewhere before, and that it was not original to me.

Jeremy Yallop brought up an example that I had definitely seen before.

In 2008 Conor McBride and Ross Paterson wrote an influential paper, “Idioms: applicative programming with effects” that introduced the idea of an applicative functor, a sort of intermediate point between functors and monads. It has since made its way into standard Haskell and was deemed sufficiently important to be worth breaking backward compatibility.

McBride and Paterson used several notations for operations in an applicative functor. Their primary notation was !!\iota!! for what is now known as pure and !!\circledast!! for what has since come to be written as <*>. But the construction

$$\iota f \circledast is_1 \circledast \ldots \circledast is_n$$

came up so often they wanted a less cluttered notation for it:

We therefore find it convenient, at least within this paper, to write this form using a special notation

$$[\![ f is_1 \ldots is_n ]\!]$$

The brackets indicate a shift into an idiom where a pure function is applied to a sequence of computations. Our intention is to provide a sufficient indication that effects are present without compromising the readability of the code.

On page 5, they suggested an exercise:

… show how to replace !![\![!! and !!]\!]!! by identifiers iI and Ii whose computational behaviour delivers the above expansion.

They give a hint, intended to lead the reader to the solution, which involves a function named iI that does some legerdemain on the front end and then a singleton type data Ii = Ii that terminates the legerdemain on the back end. The upshot is that one can write

iI f x y Ii


and have it mean

(pure f) <*> x <*> y


The haskell wiki has details, written by Don Stewart when the McBride-Paterson paper was still in preprint. The wiki goes somewhat further, also defining

 data J = J


so that

iI f x y J z Ii


now does a join on the result of f x y before applying the result to z.

I have certainly read this paper more than once, and I was groping for this example while I was writing the original article, but I couldn't quite put my finger on it. Thank you, M. Yallop!

[ By the way, I am a little bit disappointed that the haskell wiki is not called “Hicki”. ]

In the previous article I described a rather odd abuse of the Haskell type system to use a singleton type as a sort of pseudo-keyword, and asked if anyone had seen this done elsewhere.

Joachim Breitner reported having seen this before. Most recently in LiquidHaskell, which defines a QED singleton type:

 data QED = QED
infixl 2 ***

(***) :: a -> QED -> Proof
_ *** _ = ()


so that they can end every proof with *** QED:

singletonP x
=   reverse [x]
==. reverse [] ++ [x]
==. [] ++ [x]
==. [x]
*** QED


This example is from Vazou et al., Functional Pearl: Theorem Proving for All, p. 3. The authors explain: “The QED argument serves a purely aesthetic purpose, allowing us to conclude proofs with *** QED.”.

Or see the examples from the bottom of the LH splash page, proving the associative law for ++.

I looked in the rest of the LiquidHaskell distribution but did not find any other uses of the singleton-type trick. I would still be interested to see more examples.

A friend asked me the other day about techniques in Haskell to pretend to make up keywords. For example, suppose we want something like a (monadic) while loop, say like this:

      while cond act =
cond >>= \b -> if b then act >> while cond act
else return ()


This uses a condition cond (which might be stateful or exception-throwing or whatever, but which must yield a boolean value) and an action act (likewise, but its value is ignored) and it repeates the action over and over until the condition is false.

Now suppose for whatever reason we don't like writing it as while condition action and we want instead to write while condition do action or something of that sort. (This is a maximally simple example, but the point should be clear even though it is silly.) My first suggestion was somewhat gross:

      while c _ a = ...


Now we can write

      while condition "do" action


and the "do" will be ignored. Unfortunately we can also write while condition "wombat" action and you know how programmers are when you give them enough rope.

But then I had a surprising idea. We can define it this way:

      data Do = Do
while c Do a = ...


Now we write

      while condition
Do action


and if we omit or misspell the Do we get a compile-time type error that is not even too obscure.

For a less trivial (but perhaps sillier) example, consider:

    data Exception a = OK a | Exception String

data Catch = Catch
data OnSuccess = OnSuccess
data AndThen = AndThen

try computation Catch handler OnSuccess success AndThen continuation =
case computation of OK a        -> success >> (OK a) >>= continuation
Exception e ->            (handler e) >>= continuation


The idea here is that we want to try a computation, and do one thing if it succeeds and another if it throws an exception. The point is not the usefulness of this particular and somewhat contrived exception handling construct, it's the syntactic sugar of the Catch, OnSuccess, and AndThen:

    try (evaluate some_expression)
Catch (\error -> case error of "Divison by zero" -> ...
... )
OnSuccess ...
AndThen ...


I was fairly confident I had seen something like this somewhere before, and that it was not original to me. But I've asked several Haskell experts and nobody has said it was familar. I thought perhaps I had seen it somewhere in Brent Yorgey's code, but he vehemently denied it.

So my question is, did I make up this technique of using a one-element type as a pretend keyword?

[ Addendum: At least one example of this trick appears in LiquidHaskell. I would be interested to hear about other places it has been used. ]

[ Addendum: Jeremy Yallop points out that a similar trick was hinted at in McBride and Paterson “Idioms: applicative programming with effects” (2008), with which I am familiar, although their trick is both more useful and more complex. So this might have been what I was thinking of. ]

Thu, 02 Aug 2018

A professor of mine once said to me that all teaching was a process of lying, and then of replacing the lies with successively better approximations of the truth. “I say it's like this,” he said, “and then later I say, well, it's not actually like that, it's more like this, because the real story is too complicated to explain all at once.” I wouldn't have phrased it like this, but I agree with him in principle. One of the most important issues in pedagogical practice is deciding what to leave out, and for how long.

Kids inevitably want to ask about numerical infinity, and many adults will fumble the question, mumbling out some vague or mystical blather. Mathematics is prepared to offer a coherent and carefully-considered answer. Unfortunately, many mathematically-trained people also fumble the question, because mathematics is prepared to offer too many answers. So the mathematical adult will often say something like “well, it's a lot of things…” which for this purpose is exactly not what is wanted. When explaining the concept for the very first time, it is better to give a clear and accurate partial explanation than a vague and imprecise overview. This article suggests an answer that is short, to the point, and also technically correct.

In mathematics “infinity” names a whole collection of not always closely related concepts from analysis, geometry, and set theory. Some of the concepts that come under this heading are:

• The !!+\infty!! and !!-\infty!! one meets in real analysis, which can be seen either as a convenient fiction (with “as !!x!! goes to infinity” being only a conventional rephrasing of “as !!x!! increases without bound”) or as the endpoints in the two-point compactification of !!\Bbb R!!.

• The !!\infty!! one meets in complex analysis, which is a single point one adds to compactify the complex plane into the Riemann sphere.

• The real version of the preceding, the “point at infinity” on the real projective line.

• The entire “line at infinity” that bounds the real projective plane.

• The vast family of set-theoretic infinite cardinals !!\aleph_0, \aleph_1, \ldots!!.

• The vast family of set-theoretic infinite ordinals, !!\omega, \omega+1, \ldots, \epsilon_0, \ldots, \Omega, \ldots!!.

I made a decision ahead of time that when my kids first asked what infinity was, I would at first adopt the stance that “infinity” referred specifically to the set-theoretic ordinal !!\omega!!, and that the two terms were interchangeable. I could provide more details later on. But my first answer to “what is infinity” was:

It's the smallest number you can't count to.

As an explanation of !!\omega!! for kids, I think this is flawless. It's briefand it's understandable. It phrases the idea in familiar terms: counting. And it is technically unimpeachable. !!\omega!! is, in fact, precisely the unique smallest number you can't count to.

How can there be a number that you can't count to? Kids who ask about infinity are already quite familiar with the idea that the sequence of natural numbers is unending, and that they can count on and on without bound. “Imagine taking all the numbers that you could reach by counting,” I said. “Then add one more, after all of them. That is infinity.” This is a bit mind-boggling, but again it is technically unimpeachable, and the mind-bogglyness of it is nothing more than the intrinsic mind-bogglyness of the concept of infinity itself. None has been added by vagueness or imprecise metaphor. When you grapple with this notion, you are grappling with the essence of the problem of the completed infinity.

In my experience all kids make the same move at this point, which is to ask “what comes after infinity?” By taking “infinity” to mean !!\omega!!, we set ourselves up for an answer that is much better than the perplexing usual one “nothing comes after infinity”, which, if infinity is to be considered a number, is simply false. Instead we can decisively say that there is another number after infinity, which is called “infinity plus one”. This suggests further questions. “What comes after infinity plus one?” is obvious, but a bright kid will infer the existence of !!2\cdot\omega!!. A different bright kid might ask about !!\omega-1!!, which opens a different but fruitful line of discussion: !!\omega!! is not a successor ordinal, it is a limit ordinal.

Or the kid might ask if infinity plus one isn't equal to infinity, in which case you can discuss the non-commutativity of ordinal addition: if you add the “plus one” at the beginning, it is the same (except for !!\omega!! itself, the picture has just been shifted over on the page):

But if you add the new item at the other end, it is a different picture:

Before there was only one extra item on the right, and now there are two. The first picture exemplifies the Dedekind property, which is an essential feature of infinity. But the existence of an injection from !!\omega+1!! to !!\omega!! does not mean that every such map is injective.

Just use !!\omega!!. Later on the kid may ask questions that will need to be answered with “Earlier, I did not tell you the whole story.” That is all right. At that point you can reveal the next thing.