infinitesmals

I personally like understanding infinitesmals, and knowing what the d in dy/dx means.

And if infinitesmals could not be trusted, how could Calculus be put on a rigorous footing?

At that point any interest in infinitesmals has to come down to curiosity and the claim that it helps intuition.

The axioms behind the established thing are much weaker than the axioms required for infinitesmals.

The infinitesmals are a set identifiable from the construction, but are NOT a nonstandard set.

In fact Fourier's examples were more of a problem for Lagrange's approach than infinitesmals!

Calculus can be rigorously done with limits, continuous functions, little-o notation or infinitesmals.

The reason people like them is that you they contain infinitesmals too, which you can use to formalise an alternative foundation for calculus.

But he went on to show that infinitesmals allowed abandoned proofs to be made rigorous, and claimed that they were good for intuition.

In other words infinitesmals were trying to replace something that had become established.

Suppose that (f(x + dx) - f(x))/dx only varies by an infinitesmal from a non-standard version of a real across all possible infinitesmals.

Cauchy came up with Cauchy sequences while attempting to define infinitesmals rigorously.

Bishop Berkeley's famous comment about "ghosts of departed quantities" was a criticism of fluxions, not infinitesmals.

>The famous quote of "ghosts of departed quantities" was written by George Berkeley as a criticism of infinitesmals.

I've spent many years "pretending to understand" calculus, but things I remember gnawing at me, like limits & infinitesmals, are accompanied with context and history such that you can finally put yourself into the conversation and understand that my confusion is simply due to only getting a fraction of the story.

Infinitesmals as understood by mathematicians then do not correspond to any numbers we use today.

Infinitesmals in differentiation and integration.