Recent reading
Jan. 29th, 2023 02:35 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
luzulaGirig-Sverige by Andreas Cervenka (2022)
A journalistic survey of the depressing changes in the Swedish economy and distribution of wealth over the last 30-40 years. Sweden still has fairly high taxes on income from work, but the taxes on capital and/or income from capital are very low and in some cases zero. Unsurprisingly, the rich are getting richer. Ugh.
Children of Time by Adrian Tchaikovsky (2015)
Read for book club. This is a reread, and I confess that I skipped or skimmed the parts about the humans, and only read the parts about the spiders. I love the spiders, they are great, and you can tell that the author is having so much fun with their biological and cultural evolution. By contrast, the humans on the generation spaceship are kind of boring. I have read and enjoyed the sequel before, and I see that there is now a third book out. *reserves it at the library*
Calculus Reordered by David Bressoud (2019)
Time for some math history! This was so great. A standard calculus course of today usually starts out with limits, then derivatives, then integrals. But this is not at all the order in which those concepts historically appeared. I knew some of what was in this book, but there was a lot of new detail-level stuff for me! So actually integrals were the first to appear, in the sense of adding up smaller and smaller pieces of easily calculable area or volume in order to find out the area/volume of some geometrical object. The ancient Greeks did this, and they were actually quite rigorous about it, in the sense that they showed that the area/volume of their object could be neither smaller nor larger than their formula showed. In that sense, they were quite close to modern math, even though in other ways their geometrical understanding was quite different. Derivatives were the next to appear, in the context of calculating velocities. In the 17th century, the connection between these two were realized by Newton and Leibniz (i e that you can calculate integrals by taking antiderivatives), which made it much easier since you don't have to find a tailored approach to every geometrical object.
There were tensions between the people who thought you had to keep to Archimedean rigor, and those who played fast and loose with infinitesimals. In the 18th century, Euler was one of the latter. Of him, the author says: ‘Euler carried the Bernoulli’s acceptance of infinitesimals and infinities to a dangerous extreme, yet he found his way through this minefield with a dexterity that is often breathtaking.’ There are some examples of his daring sleight of hand with infinite series that had my jaw dropping (and this is the guy that Frederick the Great thought was boring! But perhaps he didn't have much appreciation of math.)
In the beginning of the 19th century, people realized that the foundation was shaky, and that this could lead to results that were false, or that were not understood. For example, Fourier, with the series named after him, found an infinite sum of continuous functions where the partial sums converge to a discontinuous f, and where the sum of the derivatives of the terms is not equal to the derivative of f. Abel said, in 1825: ‘My eyes have been opened in the most surprising manner. If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without foundation. It is true that most of it is valid, but that is very surprising.’ The person who pioneered putting all this on a rigorous foundation was Cauchy, who first came up with our modern definition of limit. People had been working with limits before, but with more intuitive definitions. Abel said of Cauchy: ‘Cauchy is crazy, and there is no way of getting along with him, even though right now he is the only one who knows how mathematics should be done. What he is doing is excellent, but very confusing.’ (Cauchy was born in the middle of the French Revolution, and his father was high up in the city police, so they had to flee Paris…)
Derivatives and integrals were then defined using this rigorous definition of limit instead, and continuity, a concept that mathematicians had not been much interested in before, came into focus. During the rest of the 19th century, people were more and more concerned about finding what were the actual assumptions needed in order to draw conclusions in calculus—Cauchy had not got everything right the first time. This entailed finding ever more complicated counterexamples, and finding that sets of real numbers can be stranger than anyone could have imagined. Poincaré lamented in 1889: ‘in earlier times, when we invented a new function it was for the purpose of some practical goal. Today, we invent them expressly to show the flaws in our forefathers’ reasoning, and we draw from them nothing more than that.’ But at the end of it, calculus had a truly rigorous foundation.
As well as being interesting, this book is very well written! The author has a talent for presenting math in a clear and easily understandable way. You just need to have taken (er, and retained) a beginning level university calculus course to read it.
A journalistic survey of the depressing changes in the Swedish economy and distribution of wealth over the last 30-40 years. Sweden still has fairly high taxes on income from work, but the taxes on capital and/or income from capital are very low and in some cases zero. Unsurprisingly, the rich are getting richer. Ugh.
Children of Time by Adrian Tchaikovsky (2015)
Read for book club. This is a reread, and I confess that I skipped or skimmed the parts about the humans, and only read the parts about the spiders. I love the spiders, they are great, and you can tell that the author is having so much fun with their biological and cultural evolution. By contrast, the humans on the generation spaceship are kind of boring. I have read and enjoyed the sequel before, and I see that there is now a third book out. *reserves it at the library*
Calculus Reordered by David Bressoud (2019)
Time for some math history! This was so great. A standard calculus course of today usually starts out with limits, then derivatives, then integrals. But this is not at all the order in which those concepts historically appeared. I knew some of what was in this book, but there was a lot of new detail-level stuff for me! So actually integrals were the first to appear, in the sense of adding up smaller and smaller pieces of easily calculable area or volume in order to find out the area/volume of some geometrical object. The ancient Greeks did this, and they were actually quite rigorous about it, in the sense that they showed that the area/volume of their object could be neither smaller nor larger than their formula showed. In that sense, they were quite close to modern math, even though in other ways their geometrical understanding was quite different. Derivatives were the next to appear, in the context of calculating velocities. In the 17th century, the connection between these two were realized by Newton and Leibniz (i e that you can calculate integrals by taking antiderivatives), which made it much easier since you don't have to find a tailored approach to every geometrical object.
There were tensions between the people who thought you had to keep to Archimedean rigor, and those who played fast and loose with infinitesimals. In the 18th century, Euler was one of the latter. Of him, the author says: ‘Euler carried the Bernoulli’s acceptance of infinitesimals and infinities to a dangerous extreme, yet he found his way through this minefield with a dexterity that is often breathtaking.’ There are some examples of his daring sleight of hand with infinite series that had my jaw dropping (and this is the guy that Frederick the Great thought was boring! But perhaps he didn't have much appreciation of math.)
In the beginning of the 19th century, people realized that the foundation was shaky, and that this could lead to results that were false, or that were not understood. For example, Fourier, with the series named after him, found an infinite sum of continuous functions where the partial sums converge to a discontinuous f, and where the sum of the derivatives of the terms is not equal to the derivative of f. Abel said, in 1825: ‘My eyes have been opened in the most surprising manner. If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without foundation. It is true that most of it is valid, but that is very surprising.’ The person who pioneered putting all this on a rigorous foundation was Cauchy, who first came up with our modern definition of limit. People had been working with limits before, but with more intuitive definitions. Abel said of Cauchy: ‘Cauchy is crazy, and there is no way of getting along with him, even though right now he is the only one who knows how mathematics should be done. What he is doing is excellent, but very confusing.’ (Cauchy was born in the middle of the French Revolution, and his father was high up in the city police, so they had to flee Paris…)
Derivatives and integrals were then defined using this rigorous definition of limit instead, and continuity, a concept that mathematicians had not been much interested in before, came into focus. During the rest of the 19th century, people were more and more concerned about finding what were the actual assumptions needed in order to draw conclusions in calculus—Cauchy had not got everything right the first time. This entailed finding ever more complicated counterexamples, and finding that sets of real numbers can be stranger than anyone could have imagined. Poincaré lamented in 1889: ‘in earlier times, when we invented a new function it was for the purpose of some practical goal. Today, we invent them expressly to show the flaws in our forefathers’ reasoning, and we draw from them nothing more than that.’ But at the end of it, calculus had a truly rigorous foundation.
As well as being interesting, this book is very well written! The author has a talent for presenting math in a clear and easily understandable way. You just need to have taken (er, and retained) a beginning level university calculus course to read it.
(no subject)
Date: 2023-01-30 05:23 am (UTC)I am, shamefully, very fine about skipping! I'll keep an eye out. Maybe if I do read it at some point it will motivate me to go back to some of the math...