The problem mentioned in the article (a mathematical apparatus that is completely alien to us moderns) is the opposite of that of contemporary translators. Around 1750, Émilie, marquise du Châtelet, made the first translation into French (which her lover Voltaire called "the everyday language of Europe")
As a woman, she didn't have any university education, but that wouldn't have included calculus anyway. Her problem was the modern concepts, not the old "geometric" way of reasoning about quantities. So she had to write a bulky Commentaire that really was her magnum opus
As a woman, she was completely accepted as an equal by scientists of her time, but treated rather dismissively by later historians of science. Clearly, history doesn't obey Newtons's first law: it doesn't progress in a straight line.
It's not three pages long in Clarke (nor in Heath), and either way, the argument is only 10 statements long.
> But the reader who takes the trouble to decode the proposition will see that it is trivial primary school arithmetic.
This is also not super obvious from Clarke or Heath. I'd love to know how the author would render this proposition trivial to a fourth grader.
To be sure, the mathematics of Euclid is a very foreign country from what's commonly taught in schools today. But Newton's thought is firmly rooted in that foreign country.
> I'd love to know how the author would render this proposition trivial to a fourth grader.
Looking at the diagram in your link, and labelling AE x, EB y, CF a and FD b, "Let AE, EB, CF, and FD be magnitudes proportional taken separately, so that AE is to EB as CF is to FD. I say that they are also proportional taken jointly, that is, AB is to BE as CD is to FD.", means :
If x/y = a/b, then also (x+y) / y = (a+b) / b.
To prove this is true, operate on the second equation to make it the same as the first:
I came to the comments to say this, and you beat me to it. Its a better proof than the ones in that link, IMHO. I'm not sure why the author felt the need for all those inequalities!
Right - that argument, I'm suggesting, is way beyond most fourth graders even in its modern algebraic form. It relies on lots of algebraic techniques that aren't generally taught until 7th or 8th grade.
It's not, like, rocket science, but it's not trivial either.
Well, it does seem to me fair to say that "it is trivial primary school arithmetic". But I guess the truth of whether it is or it isn't trivial, isn't really something that can be scientifically tested – it rests on how exactly you define "trivial", what you count as the involved operations etc. Well, we could go further into it if we wanted (e.g. What are these "lots of algebraic techniques" involved in these 2 little steps?!) but having asserted our rivals claims without much evidence feels like a good place to stop – this time. :-)
Edit: Or I suppose "tabooing"[0] the word trivial would have helped – it was the trouble-maker.
Euclid, Book 5, Proposition 18. If magnitudes, taken separately, be proportionals, they shall also be proportionals when taken jointly; that is, if the first be to the second as the third to the fourth, then the first and second together shall be to the second as the third and fourth together to the fourth.
This is from the Todhunter edition of Euclid's elements. the proof occupies precisely three pages. in post-Euclidean notation, if a/b = c/d then (a+b)/b = (c+d)/d. For young children replace the letters by small positive integers.
To convert Euclid's formulation to the symbolic formulation requires turning a magnitude into a real number, and defining the division of two real numbers. Euclid's proof avoids these difficulties. His magnitudes are the lengths straight line sections. He can decode a/b = m/n where m and n are positive integers as meaning na = mb. And similarly he can define a/b > m/n and m/n > a/b. So now the statement a/b > c/d can be decoded as the existence of positive integers m and n such that a/b > m/n > c/d. Finally the statement a/b = c/d is decoded as asserting that both a/b > c/d and c/d > a/b are false.
In modern jargon, Euclid is using the Dedekind cut definition of a real number. Using this definition, and working from first principles, the proof unsurprisingly requires three pages.
Chandrashekhar's Principa For the Common Reader, is a good version for modern audiences, including insightful commentary from a modern physicist looking at it.
I can second this recommendation - it took me a while to figure out the proofs, but it was wonderful to be exposed to a whole new style of reasoning that I had only seen filtered in a very indirect way in uni physics classes. His whole approach is regarded ultimately as a too-hard-to-imitate dead-end [ people regard it as having held back physics/maths research in Oxford/Cambridge for as many decades as his style was taught, as opposed to Leibniz's calculus ], but it's still very flashy and cool.
Maybe off-topic, but Prof. Chomsky always mention how important Newton was in the history of science and how significant his giving up on the mechanical philosophy when he failed to explain gravity in a mechanical way, how "lowering the bar" was important to set a realistic direction for science.
I greatly enjoyed that, thank you. It's much easier to follow Chomsky @ 1.25x speed, lol. I found his assertion and explanation during the questions section about the meaninglessness of "the physical" to be rather illuminating.
You did? Maybe you can explain it to me, because I rather thought it was a pure semantic game. Saying that "material" is no longer considered relevant was just a typical intellectual game of stripping words from their usual definitions.
(I also despise his continual use of the rhetoric device during his lecture of claiming that a particular philosophical debate has been settled and the "evident" correct answer is his own position on the subject. He keeps using that turn of phrase where his opinion is presented as the only tenable one by an intelligent thinker. He always does that. I think it's a weak trick so silence critics.)
19 comments
[ 3.9 ms ] story [ 52.2 ms ] threadAs a woman, she didn't have any university education, but that wouldn't have included calculus anyway. Her problem was the modern concepts, not the old "geometric" way of reasoning about quantities. So she had to write a bulky Commentaire that really was her magnum opus
As a woman, she was completely accepted as an equal by scientists of her time, but treated rather dismissively by later historians of science. Clearly, history doesn't obey Newtons's first law: it doesn't progress in a straight line.
[0] https://www.bbc.co.uk/programmes/m000rvnj
It's not three pages long in Clarke (nor in Heath), and either way, the argument is only 10 statements long.
> But the reader who takes the trouble to decode the proposition will see that it is trivial primary school arithmetic.
This is also not super obvious from Clarke or Heath. I'd love to know how the author would render this proposition trivial to a fourth grader.
To be sure, the mathematics of Euclid is a very foreign country from what's commonly taught in schools today. But Newton's thought is firmly rooted in that foreign country.
Looking at the diagram in your link, and labelling AE x, EB y, CF a and FD b, "Let AE, EB, CF, and FD be magnitudes proportional taken separately, so that AE is to EB as CF is to FD. I say that they are also proportional taken jointly, that is, AB is to BE as CD is to FD.", means :
If x/y = a/b, then also (x+y) / y = (a+b) / b.
To prove this is true, operate on the second equation to make it the same as the first:
1. Expand: x/y + y/y = a/b + b/b
2. Subtract 1 from both sides: x/y = a/b. QED.
Notation really makes all the difference!
It's not, like, rocket science, but it's not trivial either.
Edit: Or I suppose "tabooing"[0] the word trivial would have helped – it was the trouble-maker.
[0] https://www.lesswrong.com/posts/WBdvyyHLdxZSAMmoz/taboo-your...
This is from the Todhunter edition of Euclid's elements. the proof occupies precisely three pages. in post-Euclidean notation, if a/b = c/d then (a+b)/b = (c+d)/d. For young children replace the letters by small positive integers.
To convert Euclid's formulation to the symbolic formulation requires turning a magnitude into a real number, and defining the division of two real numbers. Euclid's proof avoids these difficulties. His magnitudes are the lengths straight line sections. He can decode a/b = m/n where m and n are positive integers as meaning na = mb. And similarly he can define a/b > m/n and m/n > a/b. So now the statement a/b > c/d can be decoded as the existence of positive integers m and n such that a/b > m/n > c/d. Finally the statement a/b = c/d is decoded as asserting that both a/b > c/d and c/d > a/b are false.
In modern jargon, Euclid is using the Dedekind cut definition of a real number. Using this definition, and working from first principles, the proof unsurprisingly requires three pages.
https://greenlion.com/books/NewtonPrincipia.html
He has many lectures about it, I like this one: https://www.youtube.com/watch?v=D5in5EdjhD0, one of the best lectures ever.
(I also despise his continual use of the rhetoric device during his lecture of claiming that a particular philosophical debate has been settled and the "evident" correct answer is his own position on the subject. He keeps using that turn of phrase where his opinion is presented as the only tenable one by an intelligent thinker. He always does that. I think it's a weak trick so silence critics.)
He knows strictly no Polish, but ... he has a Polish dictionary.
Not a Polish to English dictionary, mind you ... a Polish to Polish dictionary.
Historical, religious, scientific, political context carries a ton of unspoken information, and when you try to read a text without the context ...