# What importance, if any, do infinitesimals still have for philosophers?

What importance, if any, do infinitesimals still have to philosophers? It seems like many people are baffled by them. E.g., there's a slew of questions relating to Zeno on this site (not least by myself), and I hear that 0.999 is a common topic on discussion forums.

But it seems like trying to make sense of, for example, whether a point belongs to a line, makes no difference; nothing in mathematics hinges on it. So if the question about 'points' in mathematics is baffling, and after making certain standard assumptions, I find it so, I wondered whether infinitesimals raise problems for contemporary philosophers?

Does anything in philosophy hinge on infinitesimals, perhaps a phenomenology of mathematics? Or is asking about them outisde mathematics just an expression of bafflement?

• A point B along AC does not both end AB and begin BC because AB and BC have no overlapping parts. So a thing can’t both end a line and begin an interval, and movement along a line has only an imagined time after it. Phenomena have no time that they end. But death has a time. Peace!! – user28474 Sep 5 '17 at 22:46
• Could you explain the context, and what "whether a point belongs to a line" has to do with the bafflement, there is a disconnect between the first two paragraphs even with the comment. I am also not sure what you mean by importance they "still" have. The suggestive talk of "infinitesimal" changes when deriving equations is used in informal physical explanations, just as it was for a very long time. There were attempts to put it on firmer footing by using nonstandard analysis, but not too successful. – Conifold Sep 6 '17 at 1:32
• I see the opening question as important. I hope it attracts a bunch of answers. But LUKE, your second paragraph seems to have two run-on sentences. Divide them for clarity. – Mark Andrews Sep 6 '17 at 2:15
• It's a good question. Not sure about this, but it seems that the answer would be 'little or none'. If infinitessimals solved the paradoxes of space-time, motion, change etc. then perhaps they would be important in philosophy. But the topic of infinitessimals is important and philosophers are regularly found wondering how many angels can dance on the head of a pin. – user20253 Sep 6 '17 at 11:06
• @jobermark They no more use nonstandard analysis when doing that than Euclid "used" axiomatic method, or Archimedes "used" calculus. In any case, the issue is terminological, I referred to expositions explicitly built on Robinson's formalism. It is not the only way to incorporate infinitesimals, nor does one need any formalism (or even existence of one) to use them intuitively. – Conifold Sep 6 '17 at 20:26

I think this is one of those places where everyone who should care, just doesn't. Nonstandard Analysis is really commonplace in back-of-the-envelope computations where people happily 'integrate by multiplying by dt on both sides'.

At the same time, various constructions of nonstandard analysis are a very interesting way of looking at the idealization of potential infinity. They raise the question of whether there is a real distinction between actual and potential infinity, especially in constructions like the Hyper-Reals.

Whitehead's defense of the 'organic notion of space' is the fuzzy version of Abraham Robinson's geometrical monads, and it is therefore upheld by the discovery that this notion has enough internal consistency to re-derive calculus based on it without loss of precision.

But the notion comes on the scene too late.

Physicists are used enough to abusing analytic notions that they are not going to bother learning the rules for when one can and when one cannot actually get away with it. They will just rely on their own sense of nonsense and back off to the careful side when things stop cohering. Philosophers who care about actual and potential infinities generally aren't the analytic sorts who can take constructions from math seriously.

I think this is kind of tragic, but that no one really does care.

• You seem to be using "nonstandard analysis" for something other than Robinson's formalism, which is the standard usage. "Back-of-the-envelope computations" with infinitesimals have been done long before Robinson, and certainly have no need for non-standard analysis as justification. Tao's cheap version suffices for most of them, not that one needs even that, as the practice of 17-18th centuries shows. – Conifold Sep 6 '17 at 20:35
• @Conifold I mean the intuitive notion that infinitesimals are internally consistent at some level, which can be approached in dozens of ways from Conway's hyperreals to Cauchy's rules of thumb (which are pretty much what physicists naturally use). If you set 'standard analysis' aside as the form based exclusively on limits and declare that only one thing can be 'non-standard analysis' all the rest of these don't just disappear. I guess I should not capitalize it... – user9166 Sep 6 '17 at 20:51
• As Wittgenstein liked to point out, consistency is not a requirement for a calculus to be useful, and many intuitive notions are used incoherently. So I am not sure that we can make much out of what physicists do on napkins. – Conifold Sep 6 '17 at 20:58
• @Conifold Useful is beside the point. There is philosophical content in the fact that e.g. Leibniz's notation wins in physics over Newton's, because humans who use these notions a lot tend to think infinitesimals make sense. This is an intuition that has content, regardless of whether you pin that notion to arithmetizations, Los theorem, Ultra-filters, surreal numbers, un-namable External Elements, etc. There remains philosophical content in Zeno's paradox and infinitesimals as an embodiment of one way of stepping over it. – user9166 Sep 7 '17 at 18:22
• Useful is precisely the source of philosophical content here, "intuition" is arguably derivative from it, that would be Wittgenstein's position. It is attuned to what proves to be useful, and then refined and reinforced by the use. – Conifold Sep 7 '17 at 19:33