3
$\begingroup$

When differential and integral calculus were first discovered, in the 1600-1700s, they were proven to be immensely useful in so many applications it is almost mind boggling. But as far as I know, it was first long into the 1900s before any strict theoretic foundation for infinitesimals was actually established.

In many modern physics and engineering applications, concepts such as multiresolution analysis, scale spaces and coarse-to-fine play an important role. Roughly speaking in those concepts short time (or length) integrals and differential operators are mixed together. Which size portion of each decides the scale / resolution.

Is there some connection between how infinitesimals were first formally defined and these developments?

$\endgroup$
  • 1
    $\begingroup$ Why would you imagine that either multiresolution analysis, scale spaces, or coarse-to-fine methods have anything to do with either 17th-century infinitesimals or Robinson's theory? I strongly suspect that very few if any of the people who developed those methods had any interest in the formal definition of infinitesimals (or even knew a formal definition, other than possibly having heard of it as "something some mathematicians did"). $\endgroup$ – David K Feb 17 '18 at 21:30
  • 1
    $\begingroup$ Exactly. That is why I am asking the question. If they did not look for a connection, maybe there is something to find. $\endgroup$ – mathreadler Feb 17 '18 at 21:36
  • 1
    $\begingroup$ I don't see the relevance of that question. If we never looked for answers to questions we at the time did not have all reasons clear to ask then there is lots we would never learn. $\endgroup$ – mathreadler Feb 17 '18 at 21:53
  • 1
    $\begingroup$ We do not learn by choosing concepts at random and looking for connections between them. $\endgroup$ – David K Feb 17 '18 at 22:59
  • 2
    $\begingroup$ We do not learn by claiming something must be random if we don't see a pattern right away. $\endgroup$ – mathreadler Feb 18 '18 at 7:57
7
$\begingroup$

If multiresolution analysis, scale spaces, or coarse-to-fine methods are branches of applied mathematics (in which I am not an expert) then there is a good chance Abraham Robinson may have had something to say about them, because Robinson was an applied mathematician and in fact published a book called Wing Theory in the 1950s. In fact his book introducing infinitesimal analysis contains several applications in applied mathematics; see this MO post.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.