cahn

I'm trying to use my other account at least occasionally so I posted about my Yuletide gifts there, including the salon-relevant 12k fic that features Fritz, Heinrich, Voltaire, Fredersdorf, Saint Germain, Caroline Daum (Fredersdorf's wife), and Groundhog Day tropes! (Don't need to know canon.)

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From:

luzula

This is totally 18th century relevant, I promise. : ) And sorry for posting new things while not having read all your new things (though I continue to read and appreciate the thread about Eugene and the terrible frat boys of Versailles).

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-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 Fritz 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 partial sum of the derivatives of the terms does not converge 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.

Edited Date: 2023-01-29 01:48 pm (UTC)

From:

mildred_of_midgard

Quick note because I am way behind on salon and also ordering from archives:

(and this is the guy that Fritz thought was boring! But perhaps he didn't have much appreciation of math.)

Fritz was about as math-phobic as they come. He appreciated other people doing math, but it was something he struggled with personally. I would be *shocked* if he ever had any understanding of calculus.

Note: His father treated applied math as something that allowed you to predict, say, where your cannonballs would land, and thus very important for warfare, so Fritz did have some math forced down his throat in the period in which he was rebelling against his father. So that *might* have played a role in his dislike of doing math. But given his sponsoring of people like Maupertuis and Euler, I'm going to conclude that he was also just not very good at math. The same way he had German forced down his throat, but it also turned out that he wasn't linguistically gifted enough to pick up Italian or Latin or anything once he was on his own.

You just need to have taken (er, and retained) a beginning level university calculus course to read it.

Me: Well, I definitely took it... :P

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.

That reminds me, a book

cahn and I loved that you might also really like: Seduced by Logic: Emilie Du Châtelet, Mary Somerville, and the Newtonian Revolution. It's 18th century relevant! Emilie is a salon favorite! And the author has a talent for presenting math in a clear and easily understandable way.

From:

luzula

Ah, okay! So I guess Fritz recognized math as an important part of science which one might want to patronize, but was bad at it himself.

I'm sure you could read the book if you wanted, given that you have taken several years of math, but it seems you are busy with other things. : P

Thanks for the book rec, it does sound interesting!

From:

mildred_of_midgard

Yes, exactly. Want to know what Fritz did really appreciate, and why he thought Euler was boring? Sparkling conversation, witty repartee. It was something his society valued, and he felt pressured to live up to the expectation himself. Voltaire was briiillllliant at this. Euler...not so much.

I'm sure you could read the book if you wanted, given that you have taken several years of math,

I took the several years of math! What I retained is another question. But I was able to follow your summary!

but it seems you are busy with other things. :P

Yes. Yes, you could say that. :P But if I'm ever looking for a book on this subject, I'll check out your rec.

From:

cahn

I swear I will catch up to all the comments eventually, but I wanted to comment on this because YAY math history :D I feel like I knew some (though certainly not all!) of this stuff before, but had forgotten even the parts that I knew. (Apropos of which, have you read Journeys through Genius? I read it in high school and loved it -- it's a sort of survey of famous theorems/proofs, and does go a bit into some of the history, and I suspect that's where most of my fading knowledge of math history comes from.)

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

haha this is awesome! Euler, omg.

I didn't know anything about Cauchy as a person (obviously I knew he must have been a power in analysis since everything in analysis is named after him, lol), that is great!

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.’

HA! So that's where all those weird counterexamples come from :P

OK, I'm definitely putting this on my (extremely long) list, this sounds awesome!

From:

luzula

Glad you appreciated! : D

I kind of love those counterexamples. Have a function which is continuous only in x=0! Have a function which is continuous at every irrational number and discontinuous at every rational number other than x=0! Etc, it blows your mind. There’s a great book called ‘Counterexamples in Analysis’…

Yes, I do think you would love this book! Yes, my list is long as well...but I regard mine not as a to-read list, but as a list I can look to for inspiration when choosing my next book. So don't feel bad if you don't get around to it. : )