Mathematics | Chris Aldrich

No doubt many have already seen that Springer has released about 500 books for free during the COVID-19 pandemic. According to Springer, these textbooks will be available free of charge until at least the end of July.

A bit of Googling will reveal people who’ve already written some code to quickly download them all in bulk as well. I’m happy with doing things manually as there’s only a handful of the 8GB of textbooks I’m interested in.

Browsing through, I’ll note a few that look interesting and which foodies like my friend Jeremy Cherfas may enjoy. (Though I suspect he’s likely read them already, but just in case…)

Food Analysis, ed. S. Suzanne Nielsen
Food Analysis Laboratory Manual by S. Suzanne Nielsen
Brewing Science: A Multidisciplinary Approach by Michael Mosher and Kenneth Trantham
Food Fraud Prevention: Introduction, Implementation, and Management by John W. Spink

Bookmarked At the Interface of Algebra and Statistics by Tai-Danae Bradley (arXiv.org)

This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals recover classical marginal probabilities. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. We decode this information, and show it is akin to conditional probability, and then investigate the extent to which the eigenvectors capture "concepts" inherent in the original joint distribution. The theory is then illustrated with an experiment that exploits these ideas. Turning to a more theoretical application, we also discuss a preliminary framework for modeling entailment and concept hierarchy in natural language, namely, by representing expressions in the language as densities. Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. The parallels are not coincidental, and a common blueprint is found in category theory. We close with an exposition on free (co)completions and how the free-forgetful adjunctions in which they arise strongly suggest that in certain categorical contexts, the "fixed points" of a morphism with its adjoint encode interesting information.

Listened to Two Schools in Marin County by Kai Wright and Marianne McCune from The United States of Anxiety | WNYC Studios

Last year, the California Attorney General held a tense press conference at a tiny elementary school in the one working class, black neighborhood of the mostly wealthy and white Marin County. His office had concluded that the local district "knowingly and intentionally" maintained a segregated school, violating the 14th amendment. He ordered them to fix it, but for local officials and families, the path forward remains unclear, as is the question: what does "equal protection" mean?

- Eric Foner is author of The Second Founding

Hosted by Kai Wright. Reported by Marianne McCune.

Thank you Kai and Marianne. Hearing stories like this really makes me furious that we haven’t figured out how to do these things better. Having some common stories and history to help bring out our commonness certainly helps in getting us past the uncomfortableness we all must feel. Perhaps once we’re past that we might all be able to come up with solutions?

I’m reminded of endothermic chemical reactions that take a reasonably high activation energy (an input cost), but one that is worth it in the end because it raises the level of all the participants to a better and higher level in the end. When are we going to realize that doing a little bit of hard work today will help us all out in the longer run? I’m hopeful that shows like this can act as a catalyst to lower the amount of energy that gets us all to a better place.

Example of an endothermic reaction. nigerianscholars.com / CC BY-SA

This Marin county example is interesting because it is so small and involves two schools. The real trouble comes in larger communities like Pasadena, where I live, which have much larger populations where the public schools are suffering while the dozens and dozens of private schools do far better. Most people probably don’t realize it, but we’re still suffering from the heavy effects of racism and busing from the early 1970’s.

All this makes me wonder if we could apply some math (topology and statistical mechanics perhaps) to these situations to calculate a measure of equity and equality for individual areas to find a maximum of some sort that would satisfy John Rawls’ veil of ignorance in better designing and planning our communities. Perhaps the difficulty may be in doing so for more broad and dense areas that have been financially gerrymandered for generations by redlining and other problems.

I can only think about how we’re killing ourselves as individuals and as a nation. The problem seems like individual choices for smoking and our long term health care outcomes or for individual consumption and its broader effects on global warming. We’re ignoring the global maximums we could be achieving (where everyone everywhere has improved lives) in the search for personal local maximums. Most of these things are not zero sum games, but sadly we feel like they must be and actively work against both our own and our collective best interests.

The Message Matters: A bone to pick with Jonah Goldberg about positively framing mathematics

In the opening of The Remannt episode “American Dreams, Populist Screams” (beginning at about 03:08) Jonah Goldberg and his guest go out of their way to talk about the moral and social bad that negative framing can have specifically on children, then expand it to adults, and then finally society at large.

They’re talking broadly about the negative messaging around the idea that the American dream is dead.

“People would understand that that kind of message can have a deleterious impact on someone’s life path. Right? The same principle applies even when you send that message to grownups.”

Then in the next breath, Jonah says:

“We promised our listeners there would be very little to no math on this podcast, but um, uh…”

Here he is essentially telegraphing to his audience, “we’re not going to expose you to the scary math”, “why do math?”, “math is hard”, “you can’t do math”. He is specifically providing a negative framing for mathematics. His audience subtly hears “Math is bad!”–a message which is regularly heard, not just here, but nearly everywhere in our society including in our schools–often while it’s being taught. He does it again at 12:38 into the show and even suggests fast forwarding his own show to skip over the math portion! (A portion which doesn’t really appear by the way.)

So which is it Mr. Goldberg? Positive framing or negative?

Can we be a little less anti-math in the future? Some might suggest that being bad at math can make it immensely harder to take risks, to do the hard work, to have the American Dream. Didn’t the American Dream and associated ideas of American exceptionalism mean we could do anything–including mathematics?!

Otherwise let’s go on telling our children as you say:

“the game is rigged, you should just grab what you can, and […] not worry about being a good person or not worry about being a hard worker, or any of these kinds of things. Take the easy path because you’ll never get ahead.” ❧

Going forward, let’s always frame math in a positive light.

I’d much rather hear regular messages that math is useful, math is productive, math is interesting, math is comprehensible, math is doable, math can be easy, math is fun! Or if you prefer a more nationalist, pro-capitalist positive framing: Math is American. Math will keep us on top. Math will get us there.

Math is good for our children, it’s good for adults, it’s good for society.

Read Why Don’t Polls Have More Information About Black Voters? by Kevin Drum (Mother Jones)

Rashawn Ray wants us to stop treating African Americans as a monolithic group:

Black Americans vote on par or higher than their state population. They represent a significant share of Democratic voters, especially in states like South Carolina (nearly 60%). Despite representing this large voting bloc, polls such as Quinnipiac continue to frame black Americans as a monolithic group, while disaggregating white people by age, political identification and education.

I argue it is important to see the heterogeneity of black Americans. Others agree. Professor Eddie Glaude Jr said: “We have to be more nuanced in how we talk about black voters. I would love to see the breakdown of the Q poll. Age. Class. Etc.” Rolling Stone writer Jamil Smith said, “I’ve examined the newest Quinnipiac poll very thoroughly … and unfortunately, it does not break down black voters by age, class, education, or even gender. Just ‘Black.’ White respondents receive more nuanced treatment in the poll.”

The problem here is not one of racism, but of statistics. The average poll reaches about a thousand people. Of those, about 13 percent are likely to be black. If you then break things down by, say, age, you’ll have only about 30-40 respondents in each group. Unfortunately, as the group size goes down, the margin of error for each group goes up. In this case, the margin of error for each of the age groups is upwards of 15-20 percent, which makes the results useless. It would be a dereliction of duty to even report them.

Some polls oversample blacks and Hispanics to avoid this problem, but that’s expensive. It’s usually done infrequently, and only for surveys specifically aimed at reporting the views of one ethnic group. So don’t blame Quinnipiac for this. It’s a problem of arithmetic and money, not bad faith.

📖 I’m 10% done reading Economy, Society, and Public Policy by CORE Team

Finished chapter one. I like that this text has so many linked resources, but some of the links to the sister texts make me think I’d be getting a deeper and more technical understanding by reading them instead of this more introductory text. Still, this has some tremendous value even as a refresher.

Annotations from Unit 1 Capitalism and democracy: Affluence, inequality, and the environment

Government bodies also tend to be more limited in their capacity to expand if successful, and are usually protected from failure if they perform poorly. ❧

They can expand in different ways however. Think about the expansion of empires of Egypt, Rome, and the Mongols in the 12th Century. What caused them to cease growing and decrease? What allowed them to keep increasing?
Annotated on February 10, 2020 at 04:50PM

Capitalism is an economic system that can combine centralization with decentralization. ❧

How can we analogize this with the decentralization of the web and its economy?
Annotated on February 10, 2020 at 04:50PM

Market competition provides a mechanism for weeding out those who underperform. ❧

Note how this has failed in the current guilded age of the United States where it is possible for things to be “too big to fail”.
Annotated on February 10, 2020 at 04:50PM

First, because capital goods do not fall from the sky: all countries that have successfully moved from poverty to affluence have done so, of necessity, by accumulating large amounts of capital. We will also see that a crucial feature of capitalism is who owns and controls the capital goods in an economy. ❧

Annotated on February 10, 2020 at 03:11PM

Yet some things that we value are not private property—for example, the air we breathe and most of the knowledge we use cannot be owned, bought, or sold. ❧

Annotated on February 10, 2020 at 04:49PM

We should be sceptical when anyone claims that something complex (capitalism) ‘causes’ something else (increased living standards, technological improvement, a networked world, or environmental challenges), just because we can see there is a correlation. ❧

Great and ridiculous examples of this can be found at https://www.tylervigen.com/spurious-correlations
Annotated on February 10, 2020 at 08:59PM

Figure 1.16 ❧

Note the dramatic inconsistency of the scale on the left hand side. What is going on here?
Annotated on February 10, 2020 at 09:23PM

Firms should not be owned and managed by people who survive because of their connections to government or their privileged birth: Capitalism is dynamic when owners or managers succeed because they are good at delivering high-quality goods and services at a competitive price. This is more likely to be a failure when the other two factors above are not working well. ❧

Here is where we’re likely to fail in the United States by following the example of Donald Trump, who ostensibly has survived solely off the wealth of his father’s dwindling empire. With that empire gone, he’s now turning to creating wealth by associating with the government. We should carefully follow where this potentially leads the country.
Annotated on February 10, 2020 at 09:31PM

In some, their spending on goods and services as well as on transfers like unemployment benefits and pensions, accounts for more than half of GDP. ❧

What is the Government’s proportion of the US GDP presently?
Annotated on February 10, 2020 at 09:34PM

James Bronterre O’Brien, told the people:‘Knaves will tell you that it is because you have no property, you are unrepresented. I tell you on the contrary, it is because you are unrepresented that you have no property …’ ❧

great quote
Annotated on February 10, 2020 at 09:53PM

Yet some things that we value are not private property—for example, the air we breathe and most of the knowledge we use cannot be owned, bought, or sold. ❧

Annotated on February 10, 2020 at 04:49PM