Andrew:
>Brokmeyer links up Marx's critique of mathematicians' method to the
> way human thought and computer algorithms are being equated today.
> And very important, I think, is Brokmeyer's discussion of Marx's
> view that the real process of differentiation is not how Newton (e.g.)
> portrayed it, but is a "negation of the negation."
>
>One major reason this is important is that many have claimed that negation
> of the negation is not part of Marx's concept of dialectics (how they
> manage to ignore or distort Ch. 32 of _Capital_, Vol. I is a story in
> itself), and the Stalinists removed this concept from the list of
> "laws" of dialectics in Russia.
>
Iwao:
I also think this point is important. How to understand the process
of differential calculus depends on how to understand the concept
of infinite and the concept of continuous.
Let's hear from Marx(#1) directly.
"Assume the dependent variable y increases to y1 as long as the independent
variable x increases to x1. Here we consider a simple case that x appears
in first dimension here(sub 1).
1) y=ax; if x increases to x1 then
y1=ax1 therefore y1-y=a(x1-x).
provided diffrentiation is done, in other words we decrease x1 to x,
x1=x; x1-x=0,
therefore,
a(x1-x)=a*0=0.
moreover, y changes to y1 only because x changes to x1, same for y,
y1=y; y1-y=0.
therefore,
y1-y=a(x1-x)
turns to 0=0.
That doing Differentiation at the beginning and then 'aufheben' this
really leads to null. The difficult points in understanding of differentiation
are ascertaining how this operation is distincted from such a simple
procedure and how it leads to the actual result. (similar to the understanding
of a nagation of the nagation generally)."
(Marx's manuscript on differential calculus. '1. about the concept of derived
function',1881 )
First, I think it important to understand what is zero as a number and
distinction of zero from infinitesimal.
In formal system, zero is a number by axiom. But the invention of zero
required much more thinking power than invention of natural number
and even invention of irrational number historically. Zero was invented
as a number in India in early 6th century. Treating zero in arithmetic
was expanded in 7th century. Mathematics in India is said being tied
with commerce and it did not develop as a science. In this period in
India, zero was not distinguished from infinitesimal. Even Newton-Leibniz
did not clearly distinct those when they developed the theory of analysis.
Such distinction of zero from infinitesimal was first clarified by Cauchy
in 19th century. But even in his age, the necessary and sufficient condition
of existence of single solution of differential equation was unknown. This
was found by Okamura at around 1930.
Regarding such development of the theory of analysis, Marx's recongnition
that dy/dx is a process of a nagation of the nagation is excellent. In the age
of Newton and Leibniz, Berkely criticized thier method of differentiation
for the reason that 0/0 derives no definite number. Such misunderstanding
of course came from his lack of understanding of dialectics. But Newton
and Leibniz didn't countered to such argument successfully.
Though Marx's description was not a mathematical solution for the matter,
I find it an important lesson. Mathematicians invented more abstract manner
of treatment of the matter. Can't we, however, interprete this as a contemporary
mathematics way of description of some kinds of dialectical processes?
#1 K. Marx: Matematicheskie Rikopisi, Nuaka, Moskow, 1968
japanese edition is translated by Takashi Sugawara, Otsuki, 1973
Other references:
#2 Shuntaro Itoh, Teikichi Hara, Tamotsu Murata: History of mathematics
Chikuma, 1975
#3 Akira Kobori: History of mathamatics, Asakura, 1956
#4 Arpad Szabo: Sugaku no akebono, selected works, Tokyo-tosho, 1976
#5 Nicholas Bourbaki: Elements d'histoire des mathematiques, Hermann, 1969
japanese edition tranlated by Tamotsu Murata,Tatsuo Shimizu,1970
in OPE solidarity,
Iwao
----------------------
Iwao Kitamura
E-mail: ikita@st.rim.or.jp