Simone Weil's _Gravity and Grace_
Marko Amnell
In her book _Gravity and Grace_, published in 1947,
Simone Weil makes some interesting comments about
algebra, the pursuit of her famous brother Andr� Weil.
The section entitled "Algebra" is two pages long and
starts out:
"Money, mechanization, algebra. The three monsters
of contemporary civilization. Complete analogy.
"Algebra and money are essentially levellers, the first
intellectually, the second effectively.
"About fifty years ago the life of the Provencal peasants
ceased to be like that of the Greek peasants described by
Hesiod. The destruction of science as conceived by the
Greeks took place at about the same period. Money and
algebra triumphed simultaneously.
"The relation of the sign to the thing signified is being
destroyed, the game of exchanges between signs is being
multiplied of itself and for itself. And the increasing
complication demands that there should be signs for signs...
"Among the characteristics of the modern world we must
not forget the impossibility of thinking in concrete terms
of the relationship between effort and the result of effort.
There are too many intermediaries. As in the other cases,
this relationship which does not lie in any thought, lies in
a thing: money."
You can read the rest here:
Her conception of capitalism, it seems to me, owes
more to Georg Simmel than to Karl Marx. In _The Philosophy
of Money_ Simmel argued that money transforms
social interactions into impersonal relations. Money
can be divided and manipulated precisely, it allows
equivalents to be measured exactly. Money is a symbol
of the modern spirit of rationality, it levels qualitative
differences between things and people. Simone Weil
seems to be saying that algebra performs a similar
function in scientific thought, destroying the concrete
science of the ancient Greeks, i.e. Euclidean geometry,
and replacing it with algebra as abstract symbol
manipulation, in which the relation of the sign to the
thing signified is destroyed, leading to a game of
exchanges between signs, a regression of signs for signs.
Her comments are reminiscent of social critics today
who see capitalism as a system in which physical
labour is replaced by labour as the abstract manipulation
of symbols on a computer screen, which is what most
office workers now.
Jim Kalb
ma> "The relation of the sign to the thing signified is being
ma> destroyed, the game of exchanges between signs is being
ma> multiplied of itself and for itself. And the increasing
ma> complication demands that there should be signs for signs...
That's a great description of derivative financial products.
ma> "Among the characteristics of the modern world we must not
ma> forget the impossibility of thinking in concrete terms of the
ma> relationship between effort and the result of effort."
Sounds like the view that everyone was going to become a 401(k)
millionaire. Financial instruments were going to put everybody on easy
street.
--
Jim Kalb
http://jimkalb.com
Arindam Banerjee
"Marko Amnell" <marko....@kolumbus.fi> wrote in message
news:7nqjbjF...@mid.individual.net...
>
> In her book _Gravity and Grace_, published in 1947,
> Simone Weil makes some interesting comments about
> algebra, the pursuit of her famous brother Andr� Weil.
> The section entitled "Algebra" is two pages long and
> starts out:
>
> "Money, mechanization, algebra. The three monsters
> of contemporary civilization. Complete analogy.
> "Algebra and money are essentially levellers, the first
> intellectually, the second effectively.
I suppose some levelling action occurs when loose change is obtained by
beggars.
Marko Amnell
"Arindam Banerjee" <adda...@bigpond.com> wrote in message
1xjSm.60294$ze1....@news-server.bigpond.net.au...
Heh. She's talking about the levelling away of class
differences. In a society where money measures the
value of everything, an untouchable with a million
dollars need not feel socially inferior to a Brahmin.
The bourgeoisie seize power from the landed nobility.
And anyone with enough money can join the bourgeoisie.
Arindam Banerjee
>
> "Arindam Banerjee" <adda...@bigpond.com> wrote in message
> 1xjSm.60294$ze1....@news-server.bigpond.net.au...
>>
>> "Marko Amnell" <marko....@kolumbus.fi> wrote in message
>> news:7nqjbjF...@mid.individual.net...
>>>
>>> In her book _Gravity and Grace_, published in 1947,
>>> Simone Weil makes some interesting comments about
>>> algebra, the pursuit of her famous brother Andr� Weil.
>>> The section entitled "Algebra" is two pages long and
>>> starts out:
>>>
>>> "Money, mechanization, algebra. The three monsters
>>> of contemporary civilization. Complete analogy.
>>> "Algebra and money are essentially levellers, the first
>>> intellectually, the second effectively.
>>
>> I suppose some levelling action occurs when loose change
>> is obtained by beggars.
>
> Heh. She's talking about the levelling away of class
> differences.
She is a communist, then. Not quite of the Mao/Stalin/Pol Pot variety, I
suppose.
In a society where money measures the
> value of everything, an untouchable with a million
> dollars need not feel socially inferior to a Brahmin.
Why should anyone with a million dollars feel socially inferior to a poor
Brahmin? Chaps with money patronise brahmins
(priest/scholar/chef/minister/teller_of_truth/narrator_of_stories), if they
will be so kind. Brahmins used to go way way way to seek such patronage -
in those days when varna differences meant something. These days, in our
atheistic+hedonistic+materialistic world without heart or soul, idealism or
nobility, kindness or morality, poetry or music - who cares? It is the
Abominable GARG, that rules!
And yes, this raises the question - what is money???
> The bourgeoisie seize power from the landed nobility.
> And anyone with enough money can join the bourgeoisie.
With the landed nobility (and much less) long murdered-off, the present
bourgeoisie is finding increasingly interesting methods to rob the working
classes.
Marko Amnell
>
> "Marko Amnell" <marko....@kolumbus.fi> wrote in message
> news:7nu1p5F...@mid.individual.net...
>>
>> "Arindam Banerjee" <adda...@bigpond.com> wrote in message
>> 1xjSm.60294$ze1....@news-server.bigpond.net.au...
>>>
>>> "Marko Amnell" <marko....@kolumbus.fi> wrote in message
>>> news:7nqjbjF...@mid.individual.net...
>>>>
>>>> In her book _Gravity and Grace_, published in 1947,
>>>> Simone Weil makes some interesting comments about
>>>> algebra, the pursuit of her famous brother Andr� Weil.
>>>> The section entitled "Algebra" is two pages long and
>>>> starts out:
>>>>
>>>> "Money, mechanization, algebra. The three monsters
>>>> of contemporary civilization. Complete analogy.
>>>> "Algebra and money are essentially levellers, the first
>>>> intellectually, the second effectively.
>>>
>>> I suppose some levelling action occurs when loose change
>>> is obtained by beggars.
>>
>> Heh. She's talking about the levelling away of class
>> differences.
>
> She is a communist, then. Not quite of the Mao/Stalin/Pol Pot
> variety, I suppose.
Not at all. The word "grace" in her title refers after all
to the religious sense of the word, not "grace" � la Coco Chanel.
In her book, she defends the role of spirituality in life.
> In a society where money measures the
>> value of everything, an untouchable with a million
>> dollars need not feel socially inferior to a Brahmin.
>
> Why should anyone with a million dollars feel socially inferior to a poor
> Brahmin? Chaps with money patronise brahmins
> (priest/scholar/chef/minister/teller_of_truth/narrator_of_stories), if
> they will be so kind. Brahmins used to go way way way to seek such
> patronage - in those days when varna differences meant something. These
> days, in our atheistic+hedonistic+materialistic world without heart or
> soul, idealism or nobility, kindness or morality, poetry or music - who
> cares? It is the Abominable GARG, that rules!
She is speaking about the dying away of the old social
order with the rise of the bourgeoisie. As she writes:
"About fifty years ago the life of the Provencal peasants
ceased to be like that of the Greek peasants described by
Hesiod." That puts her back to at least 1897. Of course
feeling socially inferior is much less important than the
fact that in the past, the law was different for members of
different estates and castes.
Just to be clear. I'm not endorsing Simone Weil's ideas.
I said I found them interesting. The initial aphorisms are
the best, just as memorable aphorisms about algebra:
"Money, mechanization, algebra. The three monsters
of contemporary civilization. Complete analogy.
"Algebra and money are essentially levellers, the first
intellectually, the second effectively."
Arindam Banerjee
A spiritual communist! How modern.
>> In a society where money measures the
>>> value of everything, an untouchable with a million
>>> dollars need not feel socially inferior to a Brahmin.
>>
>> Why should anyone with a million dollars feel socially inferior to a poor
>> Brahmin? Chaps with money patronise brahmins
>> (priest/scholar/chef/minister/teller_of_truth/narrator_of_stories), if
>> they will be so kind. Brahmins used to go way way way to seek such
>> patronage - in those days when varna differences meant something. These
>> days, in our atheistic+hedonistic+materialistic world without heart or
>> soul, idealism or nobility, kindness or morality, poetry or music - who
>> cares? It is the Abominable GARG, that rules!
>
> She is speaking about the dying away of the old social
> order with the rise of the bourgeoisie. As she writes:
> "About fifty years ago the life of the Provencal peasants
> ceased to be like that of the Greek peasants described by
> Hesiod." That puts her back to at least 1897. Of course
> feeling socially inferior is much less important than the
> fact that in the past, the law was different for members of
> different estates and castes.
Was that a great bother? Or very different from the overall present
situation - taking the different worlds (firstworld, thirdworld,
fourthworld, etc.), religions, skin colouration, etc. ?
> Just to be clear. I'm not endorsing Simone Weil's ideas.
> I said I found them interesting. The initial aphorisms are
> the best, just as memorable aphorisms about algebra:
>
> "Money, mechanization, algebra. The three monsters
> of contemporary civilization. Complete analogy.
> "Algebra and money are essentially levellers, the first
> intellectually, the second effectively."
I don't know what you find interesting about that total bullshit. But then,
I am a professional engineer/computer scientist. Contemporary civilisation
means HDTV and who does not want it? You cannot get that without money,
mechanisation and algebra.
Marko Amnell
So everyone who criticizes capitalism is a communist?
There are no conservative critics of capitalism? They
are all really crypto-communists? G.H. Hardy was a
communist because he wrote: "A science is said to be
useful if its development tends to accentuate the existing
inequalities in the distribution of wealth, or more directly
promotes the destruction of human life."
And Matthew Arnold was a communist because he wrote:
"Wealth, again, that end to which our prodigious works for material
advantage are directed,--the commonest of commonplaces tells us how
men are always apt to regard wealth as a precious end in itself; and
certainly they have never been so apt thus to regard it as they are
in England at the present time. Never did people believe anything
more firmly, than nine Englishmen out of ten at the present day
believe that our greatness and welfare are proved by our being so
very rich. Now, the use of culture is that it helps us, by means of
its spiritual standard of perfection, to regard wealth as but
machinery, and not only to say as a matter of words that we regard
wealth as but machinery, but really to perceive and feel that it is
so. If it were not for this purging effect wrought upon our minds by
culture, the whole world, the future as well as the present, would
inevitably belong to the Philistines. The people who believe most
that our greatness and welfare are proved by our being very
rich, and who most give their lives and thoughts to becoming rich,
are just the very people whom we call the Philistines. Culture says:
'Consider these people, then, their way of life, their habits, their
manners, the very tones of their voice; look at them attentively;
observe the literature they read, the things which give them
pleasure, the words which come forth out of their mouths, the
thoughts which make the furniture of their minds; would any amount of
wealth be worth having with the condition that one was to become just
like these people by having it?' And thus culture begets a
dissatisfaction which is of the highest possible value in stemming
the common tide of men's thoughts in a wealthy and industrial
community, and which saves the future, as one may hope, from being
vulgarised, even if it cannot save the present."
Simone Weil was critical of both capitalism and socialism.
She did support the Republican side in the Spanish Civil War
but called herself an anarchist. She was critical of Marxism.
She was a Christian mystic. She also learned Sanskrit after
reading the Bhagavad Gita.
Arindam Banerjee
One who wishes to do away with existing social classes, is more communist
than anything else. It may be that this attitude is for all other
classes/cultures other that what they relate to. Thus Soviet communism in
practice amounted to Russian imperialism, Chinese communism amounted to
Chinese imperialism, and so on.
One who criticizes capitalism (I define it as the way of the cannibal
swine - it has to grow and grow, anyhow, till it bursts or splits and is
consequently devoured by smaller and hungrier swine who must also grow
similarly) need not be a communist. He could be a romantic, a socialist, an
ascetic, a nature-lover, a non-swine capitalist with a conscience - that is,
he wants growth on moral and environmentally friendly basis ONLY.
> There are no conservative critics of capitalism? They
> are all really crypto-communists?
That I don't know. I suppose socialists are neither capitalists nor
crypto-commmunists, so they are very popular in poor and lazy countries.
G.H. Hardy was a
> communist because he wrote: "A science is said to be
> useful if its development tends to accentuate the existing
> inequalities in the distribution of wealth, or more directly
> promotes the destruction of human life."
No, he was not a communist, just an ass. I wasn't taught anything like that
in the Soviet commune I grew up in my formative years. I was taught that
science and technology liberated humans from oppression.
Will talk more later, Marko, on what you have written below. Lots of work
to do, now. Bye.
Cheers,
Arindam Banerjee.
Arindam Banerjee
>> And Matthew Arnold was a communist because he wrote:
>> "Wealth, again, that end to which our prodigious works for material
>> advantage are directed,--the commonest of commonplaces tells us how
>> men are always apt to regard wealth as a precious end in itself; and
>> certainly they have never been so apt thus to regard it as they are
>> in England at the present time. Never did people believe anything
>> more firmly, than nine Englishmen out of ten at the present day
>> believe that our greatness and welfare are proved by our being so
>> very rich.
The USAns also have (at least, used to have till recently) this very
feeling.
Now, the use of culture is that it helps us, by means of
>> its spiritual standard of perfection, to regard wealth as but
>> machinery, and not only to say as a matter of words that we regard
>> wealth as but machinery, but really to perceive and feel that it is
>> so.
That is a man's view of culture, partially too. Mostly, men would value
wealth as giving them more freedom of choice, more opportunity, more
prestige.
A feminine woman's view of culture is that men should work hard enough to
buy her jewellery, to show her love and regard. Which is why gold is the
ultimate standard of wealth. For as much gold as you may produce, will be
bought up by Indian men for Indian women. As you may know, the very concept
of money (paper money) originated from receipts for gold deposited in safe
keeping (banks).
If it were not for this purging effect wrought upon our minds by
>> culture, the whole world, the future as well as the present, would
>> inevitably belong to the Philistines. The people who believe most
>> that our greatness and welfare are proved by our being very
>> rich, and who most give their lives and thoughts to becoming rich,
>> are just the very people whom we call the Philistines.
I don't know if this statement is correct. To be very rich, without
impoverishing others or destroying the environment, is the most magnificent
thing to do. For one can wipe out the poverty of others with such wealth,
by giving job/business opportunity, good health and hope to so very many.
One can also be very highly cultured, and also very rich. A philistine is
more a snobbish wasteful wannabe rather than a person with talent and
generosity, the truly rich person that is.
Culture says:
>> 'Consider these people, then, their way of life, their habits, their
>> manners, the very tones of their voice; look at them attentively;
>> observe the literature they read, the things which give them
>> pleasure, the words which come forth out of their mouths, the
>> thoughts which make the furniture of their minds; would any amount of
>> wealth be worth having with the condition that one was to become just
>> like these people by having it?'
Intellectual or cultural snobbery is a potent, unpleasant and self-defeating
form of snobbery. The cultured person should not be unkind to the
hard-working money-grubbers, and their flashy ways. On the other hand, he
cannot compromise on cultural standards.
And thus culture begets a
>> dissatisfaction which is of the highest possible value in stemming
>> the common tide of men's thoughts in a wealthy and industrial
>> community, and which saves the future, as one may hope, from being
>> vulgarised, even if it cannot save the present."
So what happens when the rich uncultured vulgarians (being all-rich) take
over the media, the press, the professors, the politicians, the bureaucrats,
public places etc.? The cultured people with no money, what are they to do?
Hmm. Amateur dramatics, poetry reading in pubs, attending small muscial
circles, singing to your dog, posting in Usenet, reading books, watching
videos... quite a lot actually.
>> Simone Weil was critical of both capitalism and socialism.
>> She did support the Republican side in the Spanish Civil War
>> but called herself an anarchist. She was critical of Marxism.
>> She was a Christian mystic. She also learned Sanskrit after
>> reading the Bhagavad Gita.
She sounds confused.
Cheers,
Arindam Banerjee.
Marko Amnell
> "Marko Amnell" <marko.amn...@kolumbus.fi> writes:
> > In her book _Gravity and Grace_, published in 1947, Simone Weil
> > makes some interesting comments about algebra, the pursuit of her
> > pages long and starts out:
>
> > "Money, mechanization, algebra. The three monsters
> > of contemporary civilization. Complete analogy.
> > "Algebra and money are essentially levellers, the first
> > intellectually, the second effectively.
> > "About fifty years ago the life of the Provencal peasants
> > ceased to be like that of the Greek peasants described by
> > Hesiod. The destruction of science as conceived by the
> > Greeks took place at about the same period. Money and
> > algebra triumphed simultaneously.
> > "The relation of the sign to the thing signified is being
> > destroyed, the game of exchanges between signs is being
> > multiplied of itself and for itself. And the increasing
> > complication demands that there should be signs for signs...
> > "Among the characteristics of the modern world we must
> > not forget the impossibility of thinking in concrete terms
> > of the relationship between effort and the result of effort.
> > There are too many intermediaries. As in the other cases,
> > this relationship which does not lie in any thought, lies in
> > a thing: money."
>
> talking about the application of algebra to areas outside of
> mathematics - science, technology, etc. - that would make sense to me.
> (The bit about money also sounds reasonable enough; it's reminiscent
> of Marx on commodity fetishism as well.)
>
> But it seems more likely she's talking about the effect of algebra
> within mathematics. In that case I don't think she's right.
> Algebraicists even in Weil's time didn't usually think of their
> "signs" as signifying anything at all; they were just symbols
> manipulated by formal rules. At least that's what mathematicians
> thought that they thought ("Mathematicians are Hilbertians on Sunday
> and Platonists on workdays"). But that was as true of geometers,
> analysts, topologists, etc. as it was of algebraists. Modern
> mathematics isn't "leveling", it ostensibly denies any intrinsic
> connection of its subject matter to real metaphysical entities. Or
> maybe that's the "destruction of science as conceived by the Greeks"
> that supposedly happened at the turn of the 20th century?
>
> You say that algebra destroys the "concrete science...of Euclidean
> geometry",
That's me trying to interpret Simone Weil. I was not expressing
my personal opinion about geometry there.
> but Euclidean geometry itself was an abstract science which
> superseded the concrete geometry of the Egyptians. (Supposedly the
> Egyptians knew and used the Pythagorean Theorem as technology, but
> were not interested in proving it.) The Pythagorean Theorem, in fact
> the whole Euclidean abstraction of "right triangle", levels all
> concrete instances of right triangles. I think it's more likely that
> Weil would prefer to think of Euclidean geometry not as concrete, but
> as a transcendant reality on which the concrete is metaphysically
> founded.
>
> But I admit I'm confused. I still can't understand how algebra
> triumphed over "science as conceived by the Greeks" at the
> turn of the 20th century.
Simone Weil was not a mathematician. It is not clear how well
she understood mathematics. Her writings are notoriously
ambiguous and difficult to understand. But she apparently held
some mystical religious beliefs concerning Euclidean geometry:
"T. S. Eliot's preface to The Need for Roots suggests that
Simone Weil might be regarded as a modern-day Marcionite,
due to her virtually wholesale rejection of the Old Testament
and her overall distaste for the Judaism which was technically
hers by birth; others have identified her as a gnostic for similar
reasons, as well as for her mystical theologization of geometry
and Platonist philosophy." http://en.wikipedia.org/wiki/Simone_Weil
I don't know exactly what she means by "algegra" in her
aphorisms. I took her to be referring to the effects of
David Hilbert's first rigorous axioms for Euclidean geometry
on mathematics. Hilbert published his axioms in his book
_Grundlagen der Geometrie_ in 1899 so that fits in with
Simone Weil's timing. That means she should really have
said "formal logic" not "algebra". If she did mean Hilbert then
it kind of makes sense. Simone Weil valued intuition and
believed that Euclidean geometry expresses certain
geometrical intuitions that have religious, mystical meaning.
This concrete meaning was destroyed by Hilbert's abstract
axiomatic approach in which the primitive concepts "point",
"line" and "plane" could be replaced with, say, "table", "chair",
and "glass of beer" (as Hilbert famously said). Although we
may have strong intuitions about the objects of geometry,
within the formal system we don't have to assign any explicit
meaning to the primitive concepts. The formal system studies
the relationships between the primitive concepts. That would
be a very unappealing result for someone like Weil and that
is what I took her to mean when she says "algebra" had
"destroyed" "science as conceived by the Greeks." Hilbert's
axioms then led to the formalization of all of mathematics
and the hope that some consistent and complete set of axioms
for mathematics could be found. Then, of course, Goedel proved
that this is impossible.
Marko Amnell
[...]
> But I admit I'm confused. I still can't understand how algebra
> triumphed over "science as conceived by the Greeks" at the
> turn of the 20th century.
There is a sense in which in modern mathematics algebra
replaces geometry as the basis of all of mathematics. Here
is something I wrote about this topic earlier:
For almost 2,000 years, all of mathematics was ultimately
based on geometry and there was no abstract algebraic
notation. Thus, for example, when Cardano pulished
his new solution for the cubic in radicals in 1545, his
argument was rhetorical and verbal, not abstract and
symbolic and took up three pages. A modern symbolic
argument with the same content takes up half a page.
How did we get from everything being rhetorical and
based on Euclidean geometry to modern abstract algebraic
notation? There were two main steps.
The first step was taken in 1591 by Francois Viète in
his book _In artem analyticam isagoge_. Before this book,
mathematicians did not deal with, say, the general
quadratic equation in the abstract, they solved
particular, specific quadratic equations with certain
numerical coefficients using rhetorical, verbal arguments.
The procedrure for solving such quadratic equations
(essentially using the same procedure that students
learn in secondary school today) was well known to the
ancient Babylonians in 1,600 BC, a thousand years
before the first Greek mathematicians.
Viète introduced arbitrary parameters into an equation and
distinguished these from the variables of the equation.
But his notation was only *partly* symbolic and was still
ultimately based on Euclidean geometry. But for the first
time, one could speak of a general quadratic equation,
not just certain particular equations with particular numerical
values. The new method allowed Viète to discover relations
between the coefficients and roots of polynomal equations.
The consequences of being able to study general equations
were enormous. It made possible Descartes' analytic geometry,
the calculus of Leibniz and Newton, and Newton's
mathematical physics.
What did Viète's partly symbolic notation look like?
To express the equation
x^3 + 3 B^2 x = 2 C^3
he would write, (replacing x with A):
A cubus + B plano 3 in A aequari C solido 2
But the justification for this equation was still completely
based on Euclidean geometry. Negative numbers and complex
numbers could not be expressed geometrically, so Viète
only considered real and positive roots of the equations.
(Complex numbers were not accepted as a part of the number
system until 1831 when Gauss expressed them as points in
the plane.)
For Viète, equations could only contain terms of the
same degree. In the example above, the equation is
written so that all the terms are of the third degree.
Following the ancient Greek way of understanding and doing
mathematics, Viète considered a product A x B to mean
the area of a rectangle with sides A and B, and A x B x C
always denoted a certain volume. An expression such as
A X B + C was meaningless because you could not add
an area to a line in geometry.
The second step towards modern algebraic notation was
taken 46 years later in 1637 by Rene Descartes in
his book _La Géométrie_. The notation looks modern
(or rather, modern algebraic notation looks Cartesian).
Descartes denotes variables by x,y,z,... and parameters
by a,b,c,... And Descartes introduced the key innovation
that finally severed the dependence of all of mathematics
on geometry. It is a ironic that he did this in a book
entitled "Geometry". The innovation was his "algebra
of line segments". A x B no longer meant the area of a
rectangle. For Descartes, A x B meant a line segment
of length A times B (division was treated similarly). This
innovation eliminated the need for geometry in algebra.
Starting from this point on, algebra began to become
the universal language of mathematics, replacing geometry
(which had been the universal of mathematics for the
previous 2,000 years).
Jacob Klein discusses Viète's work in his book
_Greek Mathematical Thought and the Origin of Algebra_
(1968).
Marko Amnell
> ..continuing
>
>>> And Matthew Arnold was a communist because he wrote:
>>> "Wealth, again, that end to which our prodigious works for material
>>> advantage are directed,--the commonest of commonplaces tells us how
>>> men are always apt to regard wealth as a precious end in itself; and
>>> certainly they have never been so apt thus to regard it as they are
>>> in England at the present time. Never did people believe anything
>>> more firmly, than nine Englishmen out of ten at the present day
>>> believe that our greatness and welfare are proved by our being so
>>> very rich.
>
> The USAns also have (at least, used to have till recently) this very
> feeling.
Right. Thanks to guys like this:
http://cache.gawker.com/assets/images/gawker/2009/11/madoffauctionflyer.jpg
> Now, the use of culture is that it helps us, by means of
>>> its spiritual standard of perfection, to regard wealth as but
>>> machinery, and not only to say as a matter of words that we regard
>>> wealth as but machinery, but really to perceive and feel that it is
>>> so.
>
> That is a man's view of culture, partially too. Mostly, men would value
> wealth as giving them more freedom of choice, more opportunity, more
> prestige.
>
> A feminine woman's view of culture is that men should work hard enough to
> buy her jewellery,
It worked for Bernie:
http://a.abcnews.com/images/Business/ht_lot_218_091110_ssv.jpg
> to show her love and regard. Which is why gold is the ultimate standard
> of wealth. For as much gold as you may produce, will be bought up by
> Indian men for Indian women. As you may know, the very concept of money
> (paper money) originated from receipts for gold deposited in safe keeping
> (banks).
ObBook. _The Power of Gold: The History of an Obsession_,
by Peter Bernstein, who died earlier this year.
Arindam Banerjee
"The Other" <ot...@address.invalid> wrote in message
news:lytyw2g...@circe.aeaea...
> "Arindam Banerjee" <adda...@bigpond.com> writes:
>
>> She sounds confused.
>
> From Wikipedia:
>
> ...The following year she took a 12-month leave of absence from her
> teaching position to work incognito as a laborer in two factories,
> one owned by Renault, believing that this experience would allow
> her to connect with the working class. Her poor health and
> inadequate physical strength forced her to quit after some
> months....
>
> ...She identified herself as an anarchist and joined the S�bastien
> Faure Century, the French-speaking section of the anarchist
> militia. However, her clumsiness repeatedly put her comrades at
> risk. After burning herself over a cooking fire, she left Spain to
> recuperate in Assisi. She continued to write essays on labor and
> management issues....
Ah well, she at least believed in getting first hand experience of things.
Arindam Banerjee
On Dec 6, 10:55 am, The Other <ot...@address.invalid> wrote:
[...]
> But I admit I'm confused. I still can't understand how algebra
> triumphed over "science as conceived by the Greeks" at the
> turn of the 20th century.
There is a sense in which in modern mathematics algebra
replaces geometry as the basis of all of mathematics. Here
is something I wrote about this topic earlier:
For almost 2,000 years, all of mathematics was ultimately
based on geometry and there was no abstract algebraic
notation. Thus, for example, when Cardano pulished
his new solution for the cubic in radicals in 1545, his
argument was rhetorical and verbal, not abstract and
symbolic and took up three pages. A modern symbolic
argument with the same content takes up half a page.
How did we get from everything being rhetorical and
based on Euclidean geometry to modern abstract algebraic
notation? There were two main steps.
The first step was taken in 1591 by Francois Vi�te in
his book _In artem analyticam isagoge_. Before this book,
mathematicians did not deal with, say, the general
quadratic equation in the abstract, they solved
particular, specific quadratic equations with certain
numerical coefficients using rhetorical, verbal arguments.
The procedrure for solving such quadratic equations
(essentially using the same procedure that students
learn in secondary school today) was well known to the
ancient Babylonians in 1,600 BC, a thousand years
before the first Greek mathematicians.
Vi�te introduced arbitrary parameters into an equation and
distinguished these from the variables of the equation.
But his notation was only *partly* symbolic and was still
ultimately based on Euclidean geometry. But for the first
time, one could speak of a general quadratic equation,
not just certain particular equations with particular numerical
values. The new method allowed Vi�te to discover relations
between the coefficients and roots of polynomal equations.
The consequences of being able to study general equations
were enormous. It made possible Descartes' analytic geometry,
the calculus of Leibniz and Newton, and Newton's
mathematical physics.
What did Vi�te's partly symbolic notation look like?
To express the equation
x^3 + 3 B^2 x = 2 C^3
he would write, (replacing x with A):
A cubus + B plano 3 in A aequari C solido 2
But the justification for this equation was still completely
based on Euclidean geometry. Negative numbers and complex
numbers could not be expressed geometrically, so Vi�te
only considered real and positive roots of the equations.
(Complex numbers were not accepted as a part of the number
system until 1831 when Gauss expressed them as points in
the plane.)
For Vi�te, equations could only contain terms of the
same degree. In the example above, the equation is
written so that all the terms are of the third degree.
Following the ancient Greek way of understanding and doing
mathematics, Vi�te considered a product A x B to mean
the area of a rectangle with sides A and B, and A x B x C
always denoted a certain volume. An expression such as
A X B + C was meaningless because you could not add
an area to a line in geometry.
AB: If C is defined as an area, then yes you can use this expression
correctly. Dimensionality of any mathematical expression is the very first
thing taught in engineering. What a pity that this world is not run by
engineers! Mathematical expressions have to have the same dimensionality
(length, area, volume, force, momentum, dollar, etc.) for operator (+, - ,
etc) usages.
The second step towards modern algebraic notation was
taken 46 years later in 1637 by Rene Descartes in
his book _La G�om�trie_. The notation looks modern
(or rather, modern algebraic notation looks Cartesian).
Descartes denotes variables by x,y,z,... and parameters
by a,b,c,... And Descartes introduced the key innovation
that finally severed the dependence of all of mathematics
on geometry.
AB: Huh? Geometry is always there - it is not limited to line, length and
volume, that is all. The structures are much more glorious. One with the
vision must always sense the marvellous geometry behind it all, which gives
us superior engineering.
It is a ironic that he did this in a book
entitled "Geometry". The innovation was his "algebra
of line segments". A x B no longer meant the area of a
rectangle. For Descartes, A x B meant a line segment
of length A times B (division was treated similarly).
AB: If A and B are line lengths at right angles to each other, then C = A*B
must be an area. If A is just a number with no dimensions, then C is a
length of B multiplied A times, and thus a length. The interesting thing
happnes when we relate mass, time, length units together. You cannot
multiply five apples with four oranges, but in physics you can multiply one
kilogram with one kilometer and divide that by an hour to get some
meaningful entity (momentum).
Going by Weil's and others' poor conception of algebra, it is easy to
understand Einstein's howling blunders resulting in the disastrous e=mcc, a
mistake for which the world has paid very heavily. And till I develop and
prototype the Internal Force Engine, or at least a working model, will
continue to do so.
Cheers,
Arindam Banerjee
This
innovation eliminated the need for geometry in algebra.
Starting from this point on, algebra began to become
the universal language of mathematics, replacing geometry
(which had been the universal of mathematics for the
previous 2,000 years).
Jacob Klein discusses Vi�te's work in his book
Marko Amnell
That's an ahistorical statement. The whole point is that
in the 16th century, it was not possible to conceive of C
as an area. The only meaning that C could have is the length
of a line segment in Euclidean geometry. The only way to
express an area was as the product of two line segments A x B
or as a square, say "C plano" but it was not possible that
C alone was an area. That was something that simply could
not be conceived. Abstract algebra as conceived by us today
did not exist. You cannot apply our 21st century concepts
ahistorically to understand what happened. To understand the
history of mathematics you must try to put yourself into the
mind of a person in the 16th century, and try to see the world
as he saw it. So, in the 16th century, the following expression
was meaningless:
A x B + C
You could not add an area to a line. To express this in the
16th century you would have to write:
A x B + C plano
Arindam Banerjee
So there, they were by no means fools in the 16th century. They were only
giving the complete information in the mathematical equation. Today we
simply drop the plano as it is understood. Like, for A*B + C to be valid, C
must be an area, if A and B were lines.
Anyway, this reminds me of our English teacher of literature, Father Victor
Rosner. He confessed he was absolutely hopeless in maths, especially
algebra, which made no sense to him. I quote him "A is A and B is B, so how
can A be equal to B?" I guess he was not the only one baffled!
Marko Amnell
> "Marko Amnell" <marko.amn...@kolumbus.fi> wrote in message
>
> news:7o91eiF...@mid.individual.net...
>
>
>
>
>
>
>
> > "Arindam Banerjee" <adda1...@bigpond.com> wrote in message
> > AoDTm.61153$ze1.26...@news-server.bigpond.net.au...
>
> >> "Marko Amnell" <marko.amn...@kolumbus.fi> wrote in message
> >>news:682f6d4f-b400-4d46...@20g2000vbz.googlegroups.com...
> >> On Dec 6, 10:55 am, The Other <ot...@address.invalid> wrote:
>
> >> [...]
>
> >>> But I admit I'm confused. I still can't understand how algebra
> >>> triumphed over "science as conceived by the Greeks" at the
> >>> turn of the 20th century.
>
> >> There is a sense in which in modern mathematics algebra
> >> replaces geometry as the basis of all of mathematics. Here
> >> is something I wrote about this topic earlier:
>
> >> For almost 2,000 years, all of mathematics was ultimately
> >> based on geometry and there was no abstract algebraic
> >> notation. Thus, for example, when Cardano pulished
> >> his new solution for the cubic in radicals in 1545, his
> >> argument was rhetorical and verbal, not abstract and
> >> symbolic and took up three pages. A modern symbolic
> >> argument with the same content takes up half a page.
> >> How did we get from everything being rhetorical and
> >> based on Euclidean geometry to modern abstract algebraic
> >> notation? There were two main steps.
> >> his book _In artem analyticam isagoge_. Before this book,
> >> mathematicians did not deal with, say, the general
> >> quadratic equation in the abstract, they solved
> >> particular, specific quadratic equations with certain
> >> numerical coefficients using rhetorical, verbal arguments.
> >> The procedrure for solving such quadratic equations
> >> (essentially using the same procedure that students
> >> learn in secondary school today) was well known to the
> >> ancient Babylonians in 1,600 BC, a thousand years
> >> before the first Greek mathematicians.
> >> distinguished these from the variables of the equation.
> >> But his notation was only *partly* symbolic and was still
> >> ultimately based on Euclidean geometry. But for the first
> >> time, one could speak of a general quadratic equation,
> >> not just certain particular equations with particular numerical
> >> between the coefficients and roots of polynomal equations.
> >> The consequences of being able to study general equations
> >> were enormous. It made possible Descartes' analytic geometry,
> >> the calculus of Leibniz and Newton, and Newton's
> >> mathematical physics.
> >> To express the equation
>
> >> x^3 + 3 B^2 x = 2 C^3
>
> >> he would write, (replacing x with A):
>
> >> A cubus + B plano 3 in A aequari C solido 2
>
> >> But the justification for this equation was still completely
> >> based on Euclidean geometry. Negative numbers and complex
> >> only considered real and positive roots of the equations.
> >> (Complex numbers were not accepted as a part of the number
> >> system until 1831 when Gauss expressed them as points in
> >> the plane.)
> >> same degree. In the example above, the equation is
> >> written so that all the terms are of the third degree.
> >> Following the ancient Greek way of understanding and doing
Sorry, I think I got that wrong. It should be
A x B + C quadratus
(with "quadratus" meaning "square" in Latin) or
A x B + C x C
which amounts to the same thing.
Anyway, this was just Viète's notation in 1591. Before him
there was no abstract algebraic notation at all. Everything
was expressed rhetorically, i.e. using senteces, not equations.
Marko Amnell
As I suspected, the "X" sign for multiplication
was not invented at Viète's time, so even the
expression "A x B + C quadratus" is partly
anachronistic. It seems that William Oughtred
introduced the "X" sign for multiplication only
in 1631. So I think Viète would have written
A B + C quadratus.
Or he would have written something like
a rectangle with sides A and B + C quadratus
I don't think he would have used the verb
"multiplicare" in this context since strictly
speaking you cannot multiply lines together.
But it isn't really clear since the terminology was
not fixed at this time and things were in flux.
The plus sign did exist by Viète's time, having been
introduced by Henricus Grammateus in 1518.
http://en.wikipedia.org/wiki/Multiplication_sign
http://en.wikipedia.org/wiki/Plus_and_minus_signs
Catawumpus
> I don't really see where the stuff she wrote merits much more effort, to tell
> the truth.
Weil can be very, very good, but don't work too hard: you
may not be one of her better readers.
-- Catawumpus