# The Universe of Discourse

Sun, 27 Nov 2022

One day I asked Lorrie if she thought that Schroeder actually grew up to be a famous concert pianist. We agreed that he probably did. Or at least Schroeder has as good a chance as anyone does. To become a famous concert pianist, you need to have talent and drive. Schroeder clearly has talent (he can play all that Beethoven and Mozart on a toy piano whose black keys are only painted on) and he clearly has drive. Not everyone with talent and drive does succeed, of course, but he might make it, whereas some rando like me has no chance at all.

That led to a longer discussion about what became of the other kids. Some are easier than others. Who knows what happens to Violet, Sally, (non-Peppermint) Patty, and Shermy? I imagine Violet going into realty for some reason.

As a small child I did not understand that Lucy's “psychiatric help 5¢” lemonade stand was hilarious, or that she would have been the literally worst psychiatrist in the world. (Schulz must have known many psychiatrists; was Lucy inspired by any in particular?) Surely Lucy does not become an actual psychiatrist. The world is cruel and random, but I refuse to believe it is that cruel. My first thought for Lucy was that she was a lawyer, perhaps a litigator. Now I like to picture her as a union negotiator, and the continual despair of the management lawyers who have to deal with her.

Her brother Linus clearly becomes a university professor of philosophy, comparative religion, Middle-Eastern medieval literature, or something like that. Or does he drop out and work in a bookstore? No, I think he's the kind of person who can tolerate the grind of getting a graduate degree and working his way into a tenured professorship, with a tan corduroy jacket with patches on the elbows, and maybe a pipe.

Peppermint Patty I can imagine as a high school gym teacher, or maybe a yoga instructor or massage therapist. I bet she'd be good at any of those. Or if we want to imagine her at the pinnacle of achievement, coach of the U.S. Olympic softball team. Marcie is calm and level-headed, but a follower. I imagine her as a highly competent project manager.

In the conversation with Lorrie, I said “But what happens to Charlie Brown?”

“No, why?”

“To everyone's great surprise, Charlie Brown grows up to be a syndicated cartoonist and a millionaire philanthropist.”

Of course she was right. Charlie Brown is good ol' Charlie Schulz, whose immense success suprised everyone, and nobody more than himself.

Charles M. Schulz was born 100 years ago last Saturday.

[ Addendum 20221204: I forgot Charlie Brown's sister Sally. Unfortunately, the vibe I get from Sally is someone who will be sucked into one of those self-actualization cults like Lifespring or est. ]

Sat, 26 Nov 2022

I was delighted to learn some time ago that there used to be giant wombats, six feet high at the shoulders, unfortunately long extinct.

It's also well known (and a minor mystery of Nature) that wombats have cubical poop.

Today I wondered, did the megafauna wombat produce cubical megaturds? And if so, would they fossilize (as turds often do) and leave ten-thousand-year-old mineral cubescat littering Australia? And if so, how big are these and where can I see them?

A look at Intestines of non-uniform stiffness mold the corners of wombat feces (Yang et al, Soft Matter, 2021, 17, 475–488) reveals a nice scatter plot of the dimensions of typical wombat scat, informing us that for (I think) the smooth-nosed (common) wombat:

• Length: 4.0 ± 0.6 cm
• Height: 2.3 ± 0.3 cm
• Width: 2.5 ± 0.3 cm

Notice though, not cubical! Clearly longer than they are thick. And I wonder how one distinguishes the width from the height of a wombat turd. Probably the paper explains, but the shitheads at Soft Matter want £42.50 plus tax to look at the paper. (I checked, and Alexandra was not able to give me a copy.)

Anyway the common wombat is about 40 cm long and 20 cm high, while the extinct giant wombats were nine or ten times as big: 400 cm long and 180 cm high, let's call it ten times. Then a propportional giant wombat scat would be a cuboid approximately 24 cm (9 in) wide and tall, and 40 cm (16 in) long. A giant wombat poop would be as long as… a wombat!

But not the imposing monoliths I had been hoping for.

Yang also wrote an article Duration of urination does not change with body size, something I have wondered about for a long time. I expected bladder size (and so urine quantity) to scale with the body volume, the cube of the body length. But the rate of urine flow should be proportional to the cross-sectional area of the urethra, only the square of the body length. So urination time should be roughly proportional to body size. Yang and her coauthors are decisive that this is not correct:

we discover that all mammals above 3 kg in weight empty their bladders over nearly constant duration of 21 ± 13 s.

What is wrong with my analysis above? It's complex and interesting:

This feat is possible, because larger animals have longer urethras and thus, higher gravitational force and higher flow speed. Smaller mammals are challenged during urination by high viscous and capillary forces that limit their urine to single drops. Our findings reveal that the urethra is a flow-enhancing device, enabling the urinary system to be scaled up by a factor of 3,600 in volume without compromising its function.

Wow. As Leslie Orgel said, evolution is cleverer than you are.

However, I disagree with the conclusion: 21±13 is not “nearly constant duration”. This is a range of 8–34s, with some mammals taking four times as long as others.

The appearance of the fibonacci numbers here is surely coincidental, but wouldn't it be awesome if it wasn't?

[ Addendum: I wondered if this was the only page on the web to contain the bigram “wombat coprolites”, but Google search produced this example from 2018:

Have wombats been around for enough eons that there might be wombat coprolites to make into jewelry? I have a small dinosaur coprolite that is kind of neat but I wouldn't make that turd into a necklace, it looks just like a piece of poop.

]

[ Addendum 20230209: I read the paper, but it does not explain what the difference is between the width of a wombat scat and the height. I wrote to Dr. Yang asking for an explantion, but she did not reply. ]

Tue, 08 Nov 2022

I haven't done one of these in a while. And there have been addenda. I thought hey, what if I ask Git to give me a list of commits from October that contain the word ‘Addendum’. And what do you know, that worked pretty well. So maybe addenda summaries will become a regular thing again, if I don't forget by next month.

Most of the addenda resulted in separate followup articles, which I assume you will already have seen. ([1] [2] [3]) I will not mention this sort of addendum in future summaries.

• In my discussion of lazy search in Haskell I had a few versions that used do-notation in the list monad, but eventually abandoned it n favor of explicit concatMap. For example:

      s nodes = nodes ++ (s $concatMap childrenOf nodes)  I went back to see what this would look like with do notation:  s nodes = (nodes ++) . s$ do
n <- nodes
childrenOf n


Meh.

• Regarding the origin of the family name ‘Hooker’, I rejected Wiktionary's suggestion that it was an occupational name for a maker of hooks, and speculated that it might be a fisherman. I am still trying to figure this out. I asked about it on English Language Stack Exchange but I have not seen anything really persuasive yet. One of the answers suggests that it is a maker of hooks, spelled hocere in earlier times.

(I had been picturing wrought-iron hooks for hanging things, and wondered why the occupational term for a maker of these wasn't “Smith”. But the hooks are supposedly clothes-fastening hooks, made of bone or some similar finely-workable material. )

The OED has no record of hocere, so I've asked for access to the Dictionary of Old English Corpus of the Bodleian library. This is supposedly available to anyone for noncommercial use, but it has been eight days and they have not yet answered my request.

I will post an update, if I have anything to update.

Fri, 04 Nov 2022

I keep getting lost in the maze of Haskell's numeric types. Here's the map I drew to help myself out. (I think there might have been something like this in the original Haskell 1998 report.)

(PNG version) (Original DOT file (The SVG above is hand-edited graphviz output))

Ovals are typeclasses. Rectangles are types. Black mostly-straight arrows show instance relationships. Most of the defined functions have straightforward types like !!\alpha\to\alpha!! or !!\alpha\to\alpha\to\alpha!! or !!\alpha\to\alpha\to\text{Bool}!!. The few exceptions are shown by wiggly colored arrows.

## Basic plan

After I had meditated for a while on this picture I began to understand the underlying organization. All numbers support !!=!! and !!\neq!!. And there are three important properties numbers might additionally have:

• Ord : ordered; supports !!\lt\leqslant\geqslant\gt!! etc.
• Fractional : supports division
• Enum: supports ‘pred’ and ‘succ’

Integral types are both Ord and Enum, but they are not Fractional because integers aren't closed under division.

Floating-point and rational types are Ord and Fractional but not Enum because there's no notion of the ‘next’ or ‘previous’ rational number.

Complex numbers are numbers but not Ord because they don't admit a total ordering. That's why Num plus Ord is called Real: it's ‘real’ as constrasted with ‘complex’.

## More stuff

That's the basic scheme. There are some less-important elaborations:

Real plus Fractional is called RealFrac.

Fractional numbers can be represented as exact rationals or as floating point. In the latter case they are instances of Floating. The Floating types are required to support a large family of functions like !!\log, \sin,!! and π.

You can construct a Ratio a type for any a; that's a fraction whose numerators and denominators are values of type a. If you do this, the Ratio a that you get is a Fractional, even if a wasn't one. In particular, Ratio Integer is called Rational and is (of course) Fractional.

## Shuff that don't work so good

Complex Int and Complex Rational look like they should exist, but they don't really. Complex a is only an instance of Num when a is floating-point. This means you can't even do 3 :: Complex Int — there's no definition of fromInteger. You can construct values of type Complex Int, but you can't do anything with them, not even addition and subtraction. I think the root of the problem is that Num requires an abs function, and for complex numbers you need the sqrt function to be able to compute abs.

Complex Int could in principle support most of the functions required by Integral (such as div and mod) but Haskell forecloses this too because its definition of Integral requires Real as a prerequisite.

You are only allowed to construct Ratio a if a is integral. Mathematically this is a bit odd. There is a generic construction, called the field of quotients, which takes a ring and turns it into a field, essentially by considering all the formal fractions !!\frac ab!! (where !!b\ne 0!!), and with !!\frac ab!! considered equivalent to !!\frac{a'}{b'}!! exactly when !!ab' = a'b!!. If you do this with the integers, you get the rational numbers; if you do it with a ring of polynomials, you get a field of rational functions, and so on. If you do it to a ring that's already a field, it still works, and the field you get is trivially isomorphic to the original one. But Haskell doesn't allow it.

I had another couple of pages written about yet more ways in which the numeric class hierarchy is a mess (the draft title of this article was "Haskell's numbers are a hot mess") but I'm going to cut the scroll here and leave the hot mess for another time.

[ Addendum: Updated SVG and PNG to version 1.1. ]