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

[joeshipman@aol.com]

Calculus CAN indeed be purified from the concept of an infinitesimal; 
not merely formally, but conceptually too. (We know that Archimedes did 
not need infinitesimals in his proofs though using the concept helped 
him find his results; we know that Leibniz understood how to eliminate 
infinitesimals though he found them useful as a manner of speaking; and 
mathematicians since Cauchy have used Calculus without needing 
infinitesimals even conceptually). It is true that Calculus cannot be 
so "purified" in a HISTORICAL sense, but that is of no philosophical 
consequence.

I agree only that speaking of the infinitesimals of Leibniz, Euler, and 
Robinson as "shorthand notes" does not mean that this is the only 
consistent way to regard them; but I reject your contention that 
Calculus philosophically entails infinitesimals because of the "desire 
to see the whole through a monad or infinitesimal". Calculus sees the 
whole by summing up over the parts, and whether there are uncountably 
many "infinitely small parts" or whether there are merely regularities 
in finer refinements of finite partitions which we represent as a limit 
is not a question with mathematical consequences. (On the other hand, 
to the extent that functions are presumed to be "analytic", so that a 
monad or infinitesimal really does determine the whole, there is a real 
philosophical quandary, which led to the development of the modern 
concept of "function"; but this is not the essence of "Calculus".)

Euler certainly understood what Archimedes was doing, and therefore 
understood "epsilons and deltas" even though that particular 
connotation of those Greek letters was not established until the 19th 
century.

If you are just trying to say that Calculus SHOULD not be purified from 
the concept of infinitesimal, rather than that it CANNOT, I am open to 
hearing philosophical arguments for this.  If you think you have 
already made some, please try to recast them in language which makes a 
prescriptive case -- your absolutism about the *necessity* of 
infinitesimals is obscuring any good points about the *preferability* 
of infinitesimals.

-- JS


Kutateladze:
****
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.
****
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