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Wed, 30 Dec 2020
Benjamin Franklin and the Exercises of Ignatius
Recently I learned of the Spiritual Exercises of St. Ignatius. Wikipedia says (or quotes, it's not clear):
This reminded me very strongly of Chapter 9 of Benjamin Franklin's Autobiography, in which he presents “A Plan for Attaining Moral Perfection”:
So I wondered: was Franklin influenced by the Exercises? I don't know, but it's possible. Wondering about this I consulted the Mighty Internet, and found two items in the Woodstock Letters, a 19th-century Jesuit periodical, wondering the same thing:
(“Woodstock Letters” Volume XXXIV #2 (Sep 1905) p.311–313) I can't guess at the main question, but I can correct one small detail: although this part of the Autobiography was written around 1784, the time of which Franklin was writing, when he actually made his little book, was around 1730, well before the suppression of the Society. The following issue takes up the matter again:
(“Woodstock Letters” Volume XXXIV #3 (Dec 1905) p.459–461) Franklin describes making a decision by listing, on a divided sheet of paper, the reasons for and against the proposed action. And then a variation I hadn't seen: balance arguments for and arguments against, and cross out equally-balanced sets of arguments. Franklin even suggests evaluations as fine as matching two arguments for with three slightly weaker arguments against and crossing out all five together. I don't know what this resembles in the Exercises but it certainly was striking. [Other articles in category /book] permanent link Sat, 26 Dec 2020
My big mistake about dense sets
I made a big mistake in a Math Stack Exchange answer this week. It turned out that I believed something that was completely wrong. Here's the question, are terminating decimals dense in the reals?. It asks if the terminating decimals (that is, the rational numbers of the form !!\frac m{10^n}!!) are dense in the reals. “Dense in the reals” means that if an adversary names a real number !!r!! and a small distance !!d!!, and challenges us to find a terminating decimal !!t!! that is closer than !!d!! to point !!r!!, we can always do it. For example, is there a terminating decimal !!t!! that is within !!0.0000001!! of !!\sqrt 2!!? There is: !!\frac{14142135}{10^7} = 1.4142135!! is closer than that; the difference is less than !!0.00000007!!. The answer to the question is ‘yes’ and the example shows why: every real number has a decimal expansion, and if you truncate that expansion far enough out, you get a terminating decimal that is as close as you like to the original number. This is the obvious and straightforward way to prove it, and it's just what the top-scoring answer did. I thought I'd go another way, though. I said that it's enough to show that for any two terminating decimals, !!a!! and !!b!!, there is another one that lies between them. I remember my grandfather telling me long ago that this was a sufficient condition for a set to be dense in the reals, and I believed him. But it isn't sufficient, as Noah Schweber kindly pointed out. (It is, of course, necessary, since if !!S!! is a subset of !!\Bbb R!!, and !!a,b\in S!! but no element of !!S!! between these, then there is no element of !!S!! that is less than distance !!\frac{b-a}2!! of !!\frac{a+b}a!!. Both !!a!! and !!b!! are at that distance, and no other point of !!S!! is closer.) The counterexample that M. Schweber pointed out can be explained quickly if you know what the Cantor middle-thirds set is: it's the set of midpoints of the deleted intervals. I was going to do a whole thing with diagrams for people who don't know the Cantor set, but I think what follows will be simpler. Consider the set of real numbers between 0 and 1. These can of course be represented as decimals, some terminating and some not. Our counterexample will consist of all the terminating decimals that end with !!5!!, and before that final !!5!! have nothing but zeroes and nines. So, for example, !!0.5!!. To the left and right of !!0.5!!, respectively, are !!0.05!! and !!0.95!!. In between (and around) these three are: $$\begin{array}{l} \color{darkblue}{ 0.005 }\\ 0.05 \\ \color{darkblue}{ 0.095 }\\ 0.5 \\ \color{darkblue}{ 0.905 }\\ 0.95 \\ \color{darkblue}{ 0.995 }\\ \end{array}$$ (Dark blue are the new ones we added.) And in between and around these are: $$\begin{array}{l} \color{darkblue}{ 0.0005 }\\ 0.005 \\ \color{darkblue}{ 0.0095 }\\ 0.05 \\ \color{darkblue}{ 0.0905 }\\ 0.095 \\ \color{darkblue}{ 0.0995 }\\ 0.5 \\ \color{darkblue}{ 0.9005 }\\ 0.905 \\ \color{darkblue}{ 0.9095 }\\ 0.95 \\ \color{darkblue}{ 0.9905 }\\ 0.995 \\ \color{darkblue}{ 0.9995 }\\ \end{array}$$ Clearly, between any two of these there is another one, because around $0.XXXX5$ we've added $0.XXXX05$ before and $0.XXXX95$ after, which will lie between $0.XXXX5$ and any decimal with fewer digits before it terminates. So this set does have the between-each-two-is-another property that I was depending on. But it should also be clear that this set is not dense in the reals, because, for example, there is obviously no number of this type that is near $0.7$. (This isn't the midpoints of the middle-thirds set, it's the midpoints of the middle-four-fifths set, but the idea is exactly the same.) Happy New Year, everyone! [Other articles in category /math] permanent link This tweet from Raffi Melkonian describes the appetizer plate at his house on Christmas. One item jumped out at me:
I wondered what that was like, and then I realized I do have some idea, because I recognized the word. Basterma is not an originally Armenian word, it's a Turkish loanword, I think canonically spelled pastırma. And from Turkish it made a long journey through Romanian and Yiddish to arrive in English as… pastrami… For which “spicy beef prosciutto” isn't a bad description at all. [Other articles in category /lang/etym] permanent link Tue, 15 Dec 2020The world is so complicated! It has so many things in it that I could not even have imagined. Yesterday I learned that since 1949 there has been a compact between New Mexico and Texas about how to divide up the water in the Pecos River, which flows from New Mexico to Texas, and then into the Rio Grande. New Mexico is not allowed to use all the water before it gets to Texas. Texas is entitled to receive a certain amount. There have been disputes about this in the past (the Supreme Court case has been active since 1974), so in 1988 the Supreme Court appointed Neil S. Grigg, a hydraulic engineer and water management expert from Colorado, to be “River Master of the Pecos River”, to mediate the disputes and account for the water. The River Master has a rulebook, which you can read online. I don't know how much Dr. Grigg is paid for this. In 2014, Tropical Storm Odile dumped a lot of rain on the U.S. Southwest. The Pecos River was flooding, so Texas asked NM to hold onto the Texas share of the water until later. (The rulebook says they can do this.) New Mexico arranged for the water that was owed to Texas to be stored in the Brantley Reservoir. A few months later Texas wanted their water. "OK," said New Mexico. “But while we were holding it for you in our reservoir, some of it evaporated. We will give you what is left.” “No,” said Texas, “we are entitled to a certain amount of water from you. We want it all.” But the rule book says that even though the water was in New Mexico's reservoir, it was Texas's water that evaporated. (Section C5, “Texas Water Stored in New Mexico Reservoirs”.) [Other articles in category /law] permanent link Thu, 10 Dec 2020
I had some free time yesterday, so that is what I did. The creek is pretty. I did not see anything that appeared to be white clay. Of course I did not investigate extensively, or even closely, because the weather was too cold for wading. But the park was beautiful. There is a walking trail in the park that reaches the tripoint itself. I didn't walk the whole trail. The park entrance is at the other end of the park from the tripoint. After wandering around in the park for a while, I went back to the car, drove to the Maryland end of the park, and left the car on the side of the Maryland Route 896 (or maybe Pennsylvania Route 896, it's hard to be sure). Then I cut across private property to the marker. The marker itself looks like this: As you see, the Pennsylvania sides of the monument are marked with ‘P’ and the Maryland side with ‘M’. The other ‘M’ is actually in Delaware. This Newark Post article explains why there is no ‘D’:
This does not explain the whole thing. The point was first marked in 1765 by Mason and Dixon and at that time Delaware was indeed part of Pennsylvania. But as you see the stone marker was placed in 1849, by which time Delaware had been there for some time. Perhaps the people who installed the new marker were trying to pretend that Delaware did not exist. [ Addendum 20201218: Daniel Wagner points out that even if the 1849 people were trying to depict things as they were in 1765, the marker is still wrong; it should have three ‘P’ and one ‘M’, not two of each. I did read that the surveyors who originally placed the 1849 marker put it in the wrong spot, and it had to be moved later, so perhaps they were just not careful people. ] Theron Stanford notes that this point is also the northwestern corner of the Wedge. This sliver of land was east of the Maryland border, but outside the Twelve-Mile Circle and so formed an odd prodtrusion from Pennsylvania. Pennsylvania only reliquinshed claims to it in 1921 and it is now agreed to be part of Delaware. Were the Wedge still part of Pennsylvania, the tripoint would have been at its southernmost point. Looking at the map now I see that to get to the marker, I must have driven within a hundred yards of the westmost point of the Twelve-Mile Circle itself, and there is a (somewhat more impressive) marker there. Had I realized at the time I probably would have tried to stop off. I have some other pictures of the marker if you are not tired of this yet. [ Addendum 20201211: Tim Heany asks “Is there no sign at the border on 896?” There probably is, and this observation is a strong argument that I parked the car in Maryland. ] [ Addendum 20201211: Yes, ‘prodtrusion’ was a typo, but it is a gift from the Gods of Dada and should be treasured, not thrown in the trash. ] [Other articles in category /misc] permanent link |