Richard Carrier: Science Education in the Early Roman Empire

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Re: Richard Carrier: Science Education in the Early Roman Em

Post by Peter Kirby » Mon Oct 24, 2016 1:03 pm

Ben C. Smith wrote:Reminds me of this old way of envisioning the Pythagorean theorem (a2 + b2 = c2):

Image

The combined area of the small and medium squares equals the total area of the large square.
Yes, that diagram is the basis of Euclid's proof of the Pythagorean theorem.

In general, when people solved a quadratic equation, they were solving like this (I couldn't find a better picture easily)...

Image

Not like this...

Image
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Re: Richard Carrier: Science Education in the Early Roman Em

Post by lpetrich » Mon Oct 24, 2016 7:06 pm

andrewcriddle wrote:One issue is that doing Science (producing new scientific results) as distinct from learning the established wisdom seems to come to almost a full stop before 200 CE. This is probably too early to blame Christianity. One possible factor in the 3rd century and later is the hostility of the developing Neoplatonic orthodoxy to natural science.
Richard Carrier has considered that issue also, and in some of his talks, he proposes that the downfall of ancient science was due to the Crisis of the Third Century, a period of civil wars, economic slumps, and galloping inflation. The Empire's leaders tried to keep going by debasing their coinage, but all that did was to cause inflation. The Empire even split into three parts for a while, the main Empire, the Empire of the Gauls, and Zenobia's Middle Eastern conquests.

Looking back, the first hint of what was to come was Commodus's reign (b 161 - r 180 - d 192 CE). He was totally unlike his father, Marcus Aurelius, being megalomaniac and egotistical, sort of like a certain recent US Presidential candidate. He was assassinated, but his successor Pertinax soon suffered his fate, in 193 CE, the Year of the Five Emperors. After the Empire went through three other emperors, Septimius Severus took over and ruled for the next 18 years.

That 3rd-century crisis did it one better, having a year with six emperors: 238 CE.

RC suspects that this strife made it difficult to support higher learning, and he points to sculpting having gone into decline, with some old statues getting reused.

But the Empire eventually recovered, more or less, with Diocletian's decreeing wage and price ceilings and making people's jobs fixed and hereditary.

By then, the most popular philosophical school was the Neoplatonists, with their emphasis on mystical revelation.

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Re: Richard Carrier: Science Education in the Early Roman Em

Post by lpetrich » Mon Oct 24, 2016 7:29 pm

History of algebra, Timeline of algebra -- there was a sort of premodern algebra.

The first way of representing algebraic expressions was rhetorical algebra, writing out equations with words. Something like "A quantity added to one makes two" or maybe "A quantity added to 1 makes 2".

The next step was syncopated algebra, a partially-abbreviated version. The first one known to take that step was Diophantus in his book Arithmetica. It could go "Qty add 1 equal 2".

But it was only in early modern times that symbolic algebra was invented, with expressions like "x + 1 = 2". However, it has been the most common way of representing algebra for the last few centuries.

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Re: Richard Carrier: Science Education in the Early Roman Em

Post by lpetrich » Mon Oct 24, 2016 7:52 pm

There is an interesting recapitulation of this history in the history of computer-programming languages.

Instructions for CPU's have the form (operation code or opcode) (whatever operands for the operation, may be none). Machine language is entering the instructions directly.

But that is very awkward to use, and the next step, in the early 1950's, was assembly language, a symbolic version of machine language. Its statements have the form
(label) (opcode) (operands)

Thus, to add the contents of locations A and B and put the result in C, one does something like
. load a
. add b
. store c
a dcl .
b dcl .
c dcl .
dcl = declare a variable location
This is very simplified version of what one would write for a real CPU, but I hope that it gets the basic idea across.

This can get awkward, so one can do things like create macros, statements that get expanded:
. define add2,x,y,z
. load x
. add y
. store z
. end

then one can do
. add2 a,b,c

to insert the code with appropriate substitutions of labels. After doing a lot of macro writing, one might go a step further and write a programming language that is all macros -- a high-level language.

The first of them was Fortran, FORmula Translator. It expresses this operation as
c = a + b

Another early one was Lisp, LISt Processing. It expresses this operation as
(setq c (+ a b))

It seems a lot like assembly language, it must be noted. That is why some people call Lisp "Lots of Irritating Superfluous Parentheses".

Yet another early one was Cobol, COmmon Business-Oriented Language. It expresses this operation as
add a to b giving c

This was an attempt to make source code intelligible to non-programmers, but it did not work out that way. Someone introduced the "compute" command into Cobol, giving us
compute c = a*b

using Fortran-like syntax. In effect, adding symbolic algebra to Cobol's original rhetorical algebra.

What happened?

Nearly every widely-used programming language has a Fortran-like appearance, despite the languages' numerous differences from each other. So symbolic algebra has won there, with Cobol supporting it alongside its original rhetorical algebra.

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Re: Richard Carrier: Science Education in the Early Roman Em

Post by Peter Kirby » Mon Oct 24, 2016 8:28 pm

Interesting point. You could also say that Algol, Fortran, Lisp, C, Prolog, ML, Basic, and all their descendants have let people see higher levels of abstraction in computer science and software engineering more easily. The amount of bookkeeping you need to do in machine code, without macros, means that it's a rare breed who can work with it fluently and do more than very basic things, and arduously at that.

That was basically the point I was making: full-blown symbolic algebra accelerated mathematical knowledge and research greatly (in a way that the earlier, cumbersome, rigid, and verbose antecedents did not - despite their surface similarities to later symbolic algebra, they did not have whatever it took to get us to see things in new ways). The link between the new symbolic algebra and coordinate geometry gave us a Newton (who read Descartes' Geometry in Latin) and his calculus within 40 years.
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Re: Richard Carrier: Science Education in the Early Roman Em

Post by lpetrich » Tue Oct 25, 2016 9:51 am

As for writing numbers, it was a big mess: Greek numerals, Roman numerals. The Alexandrian Greek system was using letters of the alphabet for 1 to 9, then 10 to 90, then 100 to 900, with ways to extend the system to higher values. A place system was used for sexagesimal numbers, mainly in astronomical calculations, but not anywhere else. Learning to calculate with them seems like a headache. If memorizing addition or multiplication tables seems horrible for us, it must have been very horrible for those people -- if they ever tried to do that.

Fortunately, there was a very helpful device, one that was already centuries old. The abacus. An abacus was sometimes set up by using pebbles on a table or on the ground, and that is where we get "calculate" from.

Greco-Roman mathematics also had no conception of zero or negative numbers as legitimate numbers. The first mathematicians to accept those numbers as legitimate lived in India some centuries later, and Europeans did not learn of that until some centuries afterward. However, they discovered irrational numbers rather early.

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Re: Richard Carrier: Science Education in the Early Roman Em

Post by lpetrich » Tue Oct 25, 2016 10:22 am

Peter Kirby wrote:Interesting point. You could also say that Algol, Fortran, Lisp, C, Prolog, ML, Basic, and all their descendants have let people see higher levels of abstraction in computer science and software engineering more easily. The amount of bookkeeping you need to do in machine code, without macros, means that it's a rare breed who can work with it fluently and do more than very basic things, and arduously at that.
Let's try something more complicated. There is an anecdote which states that when mathematician Karl Friedrich Gauss was a boy, he had a teacher who once tried to keep the class busy by adding up all the integers between 1 and 100 inclusive. According to the story, KFG worked out the answer very quickly: 5050. Here is my simplified assembly code for it:

next load sum # Add the number to the sum
. add num
. store sum
. load num # Increase the number by 1
. add one
. store num
. sub maxnum # Check to see if the CPU has added all the numbers to be added
. br neg, done
. br all, next
done write sum

num dcl 1 # The number's initial value
sum dcl 0 # The numbers' sum
one dcl 1 # What to increase the number by
maxnum dcl 100 # How many numbers to add

br is a branch instruction, with operands (condition), (location to transfer control to)
# is for inserting comments, something that a good programmer is supposed to do

My memory of Fortran is rather rusty, so I'll do a very similar-looking programming language, C:
int num = 1;
int sum = 0;
next: sum += num;
num += 1;
if (num > 100) goto done;
goto next;
done: printf("%d\n",sum);

A more compressed version:
int sum = 0;
for (int num = 1; num <= 100; num+=1) sum += num;
printf("%d\n",sum);

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Re: Richard Carrier: Science Education in the Early Roman Em

Post by lpetrich » Tue Oct 25, 2016 11:46 am

Peter Kirby wrote:That was basically the point I was making: full-blown symbolic algebra accelerated mathematical knowledge and research greatly (in a way that the earlier, cumbersome, rigid, and verbose antecedents did not - despite their surface similarities to later symbolic algebra, they did not have whatever it took to get us to see things in new ways). The link between the new symbolic algebra and coordinate geometry gave us a Newton (who read Descartes' Geometry in Latin) and his calculus within 40 years.
Good progress.

I've seen some people claim that Archimedes had been close to discovering integral calculus. I don't think that he came very close to doing it in general, though he did some of it as special cases.

What he did was to take a circle and find:

(circumference of inscribed n-gon) < (circumference of circle) < (circumference of circumscribed n-gon)
or
2n*sin(pi/n) < 2*pi < 2n*tan(pi/n)

He started with n = 3 and he then increased n by factors of 2 by using half-angle formulas, getting up to n = 96. Those formulas require only arithmetic and square roots. Those formulas:
tan(a/2) = sin(a)/(1 + cos(a))
cos(a/2) = 1/sqrt(1 + (tan(a/2))^2)
sin(a/2) = tan(a/2) * cos(a/2)

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Re: Richard Carrier: Science Education in the Early Roman Em

Post by GakuseiDon » Wed Dec 07, 2016 3:09 am

lpetrich wrote:After noting that science was not a high priority for early Xian theologians, RC pointed out that that can easily explain the decline of science and science education after the 4th cy., when Xianity was made the Roman Empire's official religion. It also did not help that not many people learned Greek in the West after the fall of the Western Roman Empire.

It wasn't until the late Middle Ages that science started to revive again, and it only did so in the Roman Catholic parts of Europe, and not in the Eastern Orthodox parts. It also was over a millennium after the origin of Xianity, so Xianity does not deserve any real credit.
Interestingly, Carl Sagan seems to pin the blame on Pythagoras and Plato. In this 7 min clip from episode 7 of his epic "Cosmos" series, he says:
https://www.youtube.com/watch?v=oYbqboZoYFg
  • Plato expressed hostility to observation and experiment. He taught contempt for the real world and disdain for the practical application of scientific knowledge. Plato's followers succeeded in extinguishing the light of science and experiment that had been kindled by Democritus and the other Ionians.

    Plato's unease with the world as revealed by our senses was to dominate and stifle Western philosophy. Even as late as 1600 Johannes Kepler was still struggling to interpret the structure of the cosmos in terms of the Pythagorean solids and Platonic perfection...

    In the recognization by Pythagoras and Plato that the cosmos is knowable, that there is a mathematical underpinning to nature, they greatly advanced the cause of science. But in the suppression of disquieting facts, the sense that science should be kept for a small elite, the distaste for experiment, the embrace of mysticism, the easy acceptance of slave societies; their influence has significantly set back the human endeavour. The books of the Ionian scientists are entirely lost. Their views were suppressed, ridiculed and forgotten by the Platonists, and by the Christians who adopted much of the philosophy of Plato.
It is really important, in life, to concentrate our minds on our enthusiasms, not on our dislikes. -- Roger Pearse

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Re: Richard Carrier: Science Education in the Early Roman Em

Post by andrewcriddle » Thu Dec 08, 2016 11:02 am

GakuseiDon wrote:
lpetrich wrote:After noting that science was not a high priority for early Xian theologians, RC pointed out that that can easily explain the decline of science and science education after the 4th cy., when Xianity was made the Roman Empire's official religion. It also did not help that not many people learned Greek in the West after the fall of the Western Roman Empire.

It wasn't until the late Middle Ages that science started to revive again, and it only did so in the Roman Catholic parts of Europe, and not in the Eastern Orthodox parts. It also was over a millennium after the origin of Xianity, so Xianity does not deserve any real credit.
Interestingly, Carl Sagan seems to pin the blame on Pythagoras and Plato. In this 7 min clip from episode 7 of his epic "Cosmos" series, he says:
https://www.youtube.com/watch?v=oYbqboZoYFg
  • Plato expressed hostility to observation and experiment. He taught contempt for the real world and disdain for the practical application of scientific knowledge. Plato's followers succeeded in extinguishing the light of science and experiment that had been kindled by Democritus and the other Ionians.

    Plato's unease with the world as revealed by our senses was to dominate and stifle Western philosophy. Even as late as 1600 Johannes Kepler was still struggling to interpret the structure of the cosmos in terms of the Pythagorean solids and Platonic perfection...

    In the recognization by Pythagoras and Plato that the cosmos is knowable, that there is a mathematical underpinning to nature, they greatly advanced the cause of science. But in the suppression of disquieting facts, the sense that science should be kept for a small elite, the distaste for experiment, the embrace of mysticism, the easy acceptance of slave societies; their influence has significantly set back the human endeavour. The books of the Ionian scientists are entirely lost. Their views were suppressed, ridiculed and forgotten by the Platonists, and by the Christians who adopted much of the philosophy of Plato.
The responsibility is probably not with Plato's direct followers but more with the Neo-Platonists (c 200 CE onwards) who interpreted Plato as a Pythagorean mystic.

Andrew Criddle

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