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

[Robbie Lindauer]

Berkeley - "Quantities infinitely less than the least discernible  
Quantity" (The Analyst)
Hegel - "The said infinite magnitudes, therefore, are not merely  
comparable, but they exist only as moments of comparison, i.e. of the  
ratio." (Science of Logic also applying to infinitesimals)
Leibniz - "well founded fictions"

The list goes on... in response to:

G. Stolzenberg's:

> I did say that there is no philosophical
> concept of an infinitesmal.  But I shouldn't have because I'm in
> no position to make such a blanket statement.  I should instead
> have invited Drago, or anyone else, to teach us such a concept, if
> he knows one.


>    For what it is worth, in constructive mathematics, real numbers
> may be taken to be collections of rational intervals, every pair of
> which intersect and among which are ones of arbitrarily small length,
> where any two of them are defined to be equal if their union is a
> real number, i.e., if they differ by a real number equal to 0, i.e.,
> if they differ by an infinitesmal.

Hegel's view "expanded to make sense".  If you take an infinitesimal  
to be a relation among rational intervals, see above.

>    A curiosity question.  Why is this a better candidate for the
> monad of a real number than, say, the union of all real numbers
> equal to it, which is itself a real number equal to it?  I ask in
> the hope of learning more about what a monad is.

If one defines the real number as a concrete thing "in itself" rather  
than in relation to other things (e.g. the things greater-than or  
less-than it or which it divides, etc.) it is impossible to give a  
constructive definition of them and we are left without any way of  
knowing which thing is being identified.

On this view, "real numbers" (as opposed to real Real Numbers) are  
always only an abstraction from more concrete numerical properties,  
e.g. quantity.

Robbie Lindauer