primeideal | Entries tagged with math

In one of Martin Gardner's math columns, he wrote about inductive reasoning and the difficulties trying to generalize from very abstract, closed systems to the real universe. Scientists observe many similar phenomena, and then they try to form rules or hypotheses that explain the overarching pattern. When you make a hypothesis like "All X are Y," then finding a single case of an X that is not Y would falsify the theory, but each individual discovery of a new X that is Y doesn't prove the theory conclusively, only adds more evidence. Gardner cites Paul Berent for examples of how this can go wrong:

A man 99 feet tall is discovered. He is a confirming instance of "All men are less than 100 feet tall," yet his discovery greatly weakens the hypothesis. Finding a normal-size man in an unlikely place (such as Saturn's moon Titan) is another example of a confirming instance that would weaken the same hypothesis.

I think I sometimes have this problem in social settings. I make a new acquaintance. Hypothesis: we have things in common and we can maybe become friends. Evidence: they like talking about baseball, I also like talking about baseball, good! That's a step in the right direction. Another piece of evidence: they complain about a politician I don't like and voted against. This should be another piece of evidence in favor of the hypothesis, but I feel like when I update my probabilities, the hypothesis is now less likely to be true. There's a higher chance that this person has exacting litmus tests for ideological purity, and even if I don't disappoint them on this axis, I will on some other issue, so I might as well get it over with now and back away before either of us are burned further. Obviously this creates lots of problems, like, I don't want to play devil's advocate and defend the politician who I didn't even vote for...

Multiply this by a higher scaling factor if there's a risk that the other person is being hyperbolic/sarcastic/deliberately doomerist for the purposes of ???/The Neurotypicals Are At It Again.

I don't know if there are any of you reading this, but this is the kind of thing I would post on normal social media if I still did that, but I don't, so.

There will be a lunar eclipse overnight (early hours of Thursday morning). Totality will peak around 1:59 Central time (but it'll also be total for about half an hour in either direction).

This means that if you're awake at 3/14 1:59, you will see the moon hit your eye like a big pizza pi.

...no nation ranks higher in its collective passion for statistics. In Japan, statistics are the subject of a holiday, local and national conventions, awards ceremonies and nationwide statistical collection and graph-drawing contests. "This year," said Yoshiharu Takahashi, a Government statistician, "we had almost 30,000 entries. Actually, we had 29,836."

Andrew H. Malcolm, New York Times, 1977, quoted in Edward Tufte, "The Visual Display of Quantitative Information."

I'm pretty sure I read a Tufte book (not sure whether it was this one or a different one) six years ago, but I wrote down the title wrong. (I also read a standalone chapter/reprint from another book more recently, which contrasted good use of data display [John Snow researching the cholera epidemic] to bad use of data display [the Challenger disaster]). Anyway, my review from 2018 is definitely the same review I'd give to this book, which is: "Examples of what not to do were good, the suggested “best practices” felt a little radical at times though."

The subtitle of the book suggests that it will be about pocket calculators. Actually "the world's first pocket calculator" appears on page 201 of 278 (excluding endnotes). The scope of the book is much broader, in terms of "tools people use to help them calculate," from the abacus through spreadsheet applications.
The writing style is kind of tangential, not in a pretentious "let me show you my learnings" way but in a rabbit-holey "let me show you my learnings" way.
The layout of the book has pretty chapter art, which is cute and a nice touch.
There's a cool diagram of Pascal's triangle (which, of course, was not originally discovered by Pascal) using Chinese counting rod numerals, published in Japan in 1712. (Similar to this, although this one is even earlier, from 1303.)
Johannes Kepler gets a shoutout, which may or may not be useful for fiction purposes.
The description of slide rules and their basis in logarithms was useful as well, as someone who is too young to have used a slide rule :)
The second half is a little dryer than the first, more corporate-speak. The focus-but-not-really on "pocket calculators" means he has an excuse for fast-forwarding through a lot of computer development, but there's less of a clear narrative.
Grace Hopper did not define the "mechanical error" sense of a "bug" (Thomas Edison used the word in a similar way in 1878); she did, of course, immortalize the irony of "the problem this time really was caused by a moth!"
Just in general, "pocket" size is really limiting. Especially for women who already don't have enough pockets :P
There are a couple cases where the author points out "yeah we don't even know anything of the woman who made this suggestion/this inventor's mother, society is really sexist sometimes :("
A quote from an engineer who worked on the Sumlock ANITA in 1961: "Experienced calculator operators were slow to appreciate the technological breakthrough because they could not get used to the lack of 'feel' and noise that was inherent in mechanical machines. The same phenomenon was experienced by typists changing from manual to electric typewriters." The same issue we have today with electric cars!

This book was extremely targeted at me. It's a riff on Raymond Queneau's "Exercises in Style," except it's math, different methods used to solve a cubic equation. Includes such methods as Medieval (European and Islamic), Arborescent (the "Gentzen Trees" which I learned about, briefly, in logic), Outsider (crank emails), Research Seminar (impenetrable), and Tea (what happens after the Research Seminar, and may be slightly easier to understand, or at least feature math puns). Highly recommended if you are literally me.

I was recently doing some Wikipedia research and learned that when Kurt Gödel left Vienna for the last time in late 1939, he travelled to Princeton the long way, because World War II was already making an Atlantic transit infeasible. He took the Trans-Siberian Railway to Japan, then sailed to San Francisco, and then crossed the US by train!

Fortuitously, I found a new Gödel biography shortly after, although this saga doesn't get much more elaboration. There are, however, too many funny asides about math/philosophy/history to quote. I'll go with this one:

During the Circle's discussions, Neurath would constantly try to enforce discipline by banging his fist furiously on the table and interrupting if anyone dared utter one of the words on his long list of proscribed terms, such as "idea," "ideal," even "reality." He ultimately tried the patience even of the ever-patient Schlick, who would plead, "Dear Neurath, please permit us!" Neurath responded by making a small card with the letter "M" on it which he would silently hold up whenever the discussions strayed into the forbidden area of metaphysics. After several weeks of this he announced, "I can abbreviate the procedure still further if I instead hold up a card with 'non-M' on it when you don't speak metaphysics."

And then I happened across "The Last Shadow," the culmination of the Ender's Game/Ender's Shadow series. I hadn't known that this was published this year, possibly because people in my circles aren't keeping up with new Orson Scott Card releases, possibly because I'm lazy about getting new physical books. Anyway, the last two Ender/Bean books, "Ender in Exile" and "Shadows in Flight," were released in 2008 and 2012 respectively (and "Exile" comes well before "Speaker of the Dead" in IC time), so this has been in the works for a while. (There was also "Children of the Fleet" in 2017 which was "what happened to Battle School after the war," and was...not that great.)

( Spoilers for the entire saga )

Wolfgang Paul was a German physicist who won the Nobel Prize in 1989.

Wolfgang Pauli was an Austrian physicist who won the Nobel Prize in 1945. No relationship. So of course, Paul called Pauli his "imaginary part!" :D

James Stewart was a Canadian mathematician who wrote some very popular calculus textbooks: if you've ever seen "the textbook with the violin on the cover and the hole in it that looks like a curvy-S integral sign," that's Stewart's book.

He was also a violinist himself, a gay rights activist who was involved in protests during the 70s, and built his own architecturally-significant house with walls shaped like the curvy-S integral sign!

Two ideas that have been swirling around in my head but I haven't found the right words to combine, until now:

Some of the mathy/data science things I'm learning about fall into the linear algebra category. Linear algebra is something that does give us an answer to the "when are we ever going to use this anyway." Let's say you're Netflix. You have lots of data about which people have watched which movies (maybe how they rated them, too), and you'd like to be able to recommend movies to people. You can build a matrix* where each row represents a person, and each column represents a movie, and there's a 1 in that row and column if the user has watched the movie. The problem with this is that it's very high-dimensional--there are thousands of movies and millions of users, so that's going to be difficult to manipulate. Additionally, the grid is "sparse"--most people haven't seen any given movie, so most of the entries will be zero. What you'd like to do is simplify the data into a more usable form.

One trick is called "singular value decomposition." You can "factor" the matrix, let's call it M, into the product of three other matrixes, U*G*F. U measures the relationship of users to genres--if someone liked Blade Runner, and also liked Star Wars, their relationship with the "science fiction" genre will be high. G measures the genres themselves, and is a "diagonal" matrix which is easy to work with--the largest numbers in the diagonal represent the most clearly defined genres. And F measures the films' relationship to genres; something like "The Princess Bride" might be a 70% match for the "romcom" genre but also a 50% match for the "fantasy" genre.

Now, you can make G smaller by only taking the top 10 genres (or some other number), and then cut off the extra rows and columns of U and F accordingly. Instead of M=U*G*F, you now have M (approximately) u*g*f, where little u, g, and f are much smaller and easier to work with. Success! The tradeoff is that this is only an approximation--some users' tastes will fall outside clear genres, and some movies might be difficult to categorize. But if you're a big-data company, that's the sacrifice you have to work with.

So why is this on my mind (other than that I should be learning some coding stuff for work)? Political science, of course.

Duverger's law says that, if you have a first-past-the-post system (most votes wins, like the way people vote for members of Congress in the US, but not like party-list parliamentary systems), you should expect a two-party system to emerge. Why? Because if you try to bring in a third party, you're likely to split the vote and see someone you really dislike win instead.

This means that the political divide in these countries seems to boil down to a one-dimensional spectrum. In the US, if you're more to the "left," overall, you'll probably vote for the Democratic candidate. If you're more to the "right," you'll probably vote for the Republican.

However, there are really lots of different issues people have opinions on. Political compass/Nolan Chart type diagrams try to view two dimensions, one correlating to economic freedom and one correlating to social freedom. A stereotypical Democrat might prefer fewer drug laws and higher taxes on the rich to fund schools. A stereotypical Republican probably prefers neither. A Libertarian might want both fewer drug laws and lower taxes, and so which party they vote for depends on which issues they feel more strongly about (or if one of the candidates is a complete lunatic, hypothetically speaking).

In NationStates (which is great, by the way), the non-economic axis is further subdivided into "personal freedom" (same-sex marriage, religious freedom) and "political freedom" (voting rights, censorship of anti-government views...), which means nations can inhabit one of 27** types of 3x3x3 nation categories with low/medium/high freedoms in personal/political/economic freedoms. And so on.

And it doesn't stop there--any issue has the potential to be partisan, and many people's opinions aren't correlated with their parties on every issue. So if I (hypothetically speaking) agree more with the Republican candidate 20% of the time, more with Democratic candidate 70% of the time, and have very moderate views 10% of the time, I'm going to vote for the Democrat, not 7 times out of 10, but 10 times out of 10.*** Because even if I were to help the Republican get elected sometimes, they wouldn't magically only carry out the policies I like.

In other words, politics is dimensionality reduction! We take our complex, multidimensional, not-always-agreeing beliefs in a multidimensional space, and project them down onto a one-dimensional line (or maybe more if you have a party list), and then go "okay, guess I'm voting for the Democrat, then."

The problem is when you try to project back. Given a single voter on the Democratic side of the spectrum, can you work backwards to say "well, she must agree with me about hate speech and abortion rights and police reform, because we voted for the same person"? No, you can't! Because she might have based her votes on the candidates' policies on climate change or nuclear disarmament or wealth taxes, and it's not obvious that any of these things have anything to do with each other. If you're politically engaged, it's easy to say "well someone who believes X also believes Y, because the Republican platform is in favor of both X and Y," but for many people--even many politically informed people--that just isn't how it works. And so trying to find which n-dimensional point turned someone into a "moderate Democrat" is a perilous exercise.

*not The Matrix, which I did watch on Netflix
**there are actually 28; if your nation is a "Queendom" and would otherwise be a "Father Knows Best State" you actually get to be a "Mother Knows Best State" :D
***I feel like there was a paper about how simpler is better ("mixed strategies" is relevant here) but I can't find it.

I created this journal on January 10, 2012. I'm not exactly sure what prompted me to do so--I wound up making an Ao3 account shortly after, to crosspost fic and comment on an amazing Yuletide fic for a tiny fandom I'd just read--but there I was.

On other sites, I was/am "Ember Nickel," but I felt like I needed a change, so I went with "prime ideal." I had just encountered that concept in my abstract algebra textbook, it refers to a certain type of set. (The classic example is "multiples of a certain prime number X, with respect to all the integers." So if X=5, then the set {5, 10, 15, 20... 0, -5, -10...} within the larger set {...-2, 1, 0, 1, 2, 3...} because 5 is prime.)

Plus it just sounded cool. Like...idealism. And "prime" as in "important," like that thing from Star Trek. (I'm not into Star Trek.)

Fast forward 7 years, and I'm proud to say that now I have a PhD in math. And yes, some of my research worked on prime ideals in algebra!

I'm not staying in academia, but I'm amused by the way that worked out. Subconscious bias directing me down this path since it's a part of my virtual identity? Maybe.

(I have a long essay/ramble coming soon, but I've been wanting to share this for a while, just got distracted by other stuff.)