[FOM] reply to S.S. Kutateladze (19 Mar)

[S. S. Kutateladze]

3/22/2007, you wrote to me:

Gabriel Stolzenberg>    In his March 19 reply to my "Antonino Drago on Leibniz,"
Gabriel Stolzenberg> S.S. Kutateladze wrote,

>> The history of infinitesimals is much older and deeper than the
>> oversimplified progressivist's views.

Infinitely small and infinitely large are instances of the infinite
and so have a notable history. The evolution of a concept in time
is its content even definition. An infinitesimal is technically the
difference of a real and its definable part,or  a real as it stands
and the real as it seems.

Calculus cannot be purified from the concept of infinitesimal.
Calculus reflects the desire of see the  whole through a monad
or infinitesimal which is the whole of everything in Leibniz's similes.
It is at least "impolite" or "progressivist's to speak of infinitesimals
or monads of Leibniz or the nils or zeros of Euler or the hyperreals of Robinson
as shorthand notes.  Euler was and still is one the best minds in
calculus , but he knew nothing about epsilons and deltas.

There is no single article that displays the whole evolution of the
ideas behind the calculus. In fact such an article is impossible.
The two articles I mentioned are an invitation. In fact the one of them is just  a
short introductory chapter of the book "Infinitesimal Analysis"  by Gordon,
Kusraev, and  Kutateladze.

As regards my implications on progressivism, I am sorry for using this term
since it is slightly offensive in my opinion. However, the views of
what I mean under progressivism are  viler.

My first point is simple: to understand the meaning the idea of
infinitesimal (as any other deep idea) one should get acquaintance with its evolution.
My second point is judgmental of course but there is something in it:
A progressivist considers the current state of affairs   as better
then anything possible ever before since by the definition he or she favors
progress, implying that newer things are   better, and the newer
guys are smarter. This is not so technically since intellectually and
biologically the human beings of today are the  same as hundred
thousand ages  before and the particularities of their views are the same.
We cannot understand the genuine problems of the mind we are addressing on
assuming that we are closer to truth than our predecessors.
Negligence is not an argument but the way of positioning oneself  with
respect to the known and the
unbeknown.

The great achievement of Robinson was not only his formalization of
infinitesimal but the way he did it, reverently reexamining the evolution of
the idea.  Sorry, for the wordy explanation but I fell more sorry for
the fact that we are losing many achievements by neglect.


Best regards,
S. Kutateladze




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Sobolev Institute of Mathematics
Novosibirsk State University
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