[kmill]

> Don't confuse the limitations on mathematical notation with a limitation on imagination

Good luck proving or calculating anything.

You can define "infinitesimal/2 == infinitesimal", but nothing good will come out of it. A definition is no good unless it lets you do something.

Letting e=infinitesimal, you have e/2==e, so e==2e so 0==2e-e so 0==e. This definition is inconsistent with being able divide by non-zero integers and subtraction.

[logfromblammo]

That's not the definition, that's just what it does.

The definition of infinitesimal is "the smallest-magnitude number that is greater than zero". If you divide a finite number by infinity, infinitesimal is what you get, but don't go thinking that if you multiply it by infinity again that you will get the same number back, because you won't.

The floating point standard does not include a representation for infinitesimal, but an underflow now hints at its existence, instead of just going to zero.

It's probably easier to think of quantities like zero, one, infinity, and infinitesimal as the base vectors in mutually orthogonal dimensions. Their behaviors can be defined separately, such that whatever rules you choose for them can produce different types of math, perhaps useful for different purposes (or none beyond cranking out the dissertation), in the same way that slightly changing the Euclidian parallel lines property can produce elliptic and hyperbolic geometries.

[kmill]

> The definition of infinitesimal is "the smallest-magnitude number that is greater than zero"

That is hardly a definition. The real numbers are defined either as Dedekind cuts or as equivalence classes of Cauchy sequences of rationals. If you say "e is defined to be a real number such that e>0 and for all c>0, c>e", you would get a contradiction purely from the definition of the reals since "for all c>0, c>e" implies "e <= 0".

The only consistent scenario I can think is that you are actually extending the real numbers with a new element called "infinitesimal." Go ahead, but don't pretend that it is an element of the set of real numbers. Also, don't get the idea that there is some "true" set of numbers that we are trying to approximate with better accuracy. Modern mathematics has blown this idea wide open by introducing a wide array of mutually-inconsistent number systems.

> If you divide a finite number by infinity, infinitesimal is what you get

So you say. This would need to be part of the definition, or at least provable from it. Quoting Timothy Gowers, a mathematical object is what it does. How was I supposed to know that twice infinitesimal is equal to infinitesimal?

Elaborating extension: there is a way to add ("adjoin") an infinitesimal element to the real numbers. Let R(e) be the set of rational functions in e, a formal constant. For instance, 2+3e or 5e^2. I think there is a way to give R(e) a total order by saying 0<e<c for all positive real c. You also have things like 1/e=e^{-1} ("infinity") is greater than all real numbers. Don't make the mistake that R(e) is the real numbers, however.

> It's probably easier to think of quantities like zero, one, infinity, and infinitesimal as the base vectors in mutually orthogonal dimensions

In what way? In a vector space, I can divide by two, and I am apparently not able to divide infinitesimal by two (in the sense that e/2=e implies e=0).

> in the same way that slightly changing the Euclidian parallel lines property can produce elliptic and hyperbolic geometries

At least right now, there is a rather large difference: many interesting theorems follow from hyperbolic geometry.

What interesting things follow from asserting that there is a smallest-magnitude real number greater than zero? (If you say "I never said the number was real," then there has been no point to this discussion, because it started when you claimed the real numbers were countable by multiplying "infinitesimal" by other numbers.)