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

3

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!! – None – 2017-09-05T22:46:23.717

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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 – 2017-09-06T01:32:00.183

@Conifold btw your comment makes no grammatical sense, even if it is literally on topic, you should edit it. – None – 2017-09-06T01:57:42.030

@Conifold is it ontopic? you link to a physics site. does the question make sense now? – None – 2017-09-06T02:04:11.993

1I 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 – 2017-09-06T02:15:07.100

hi @MarkAndrews alas i'm poor at grammar, please edit the question, if you can – None – 2017-09-06T02:15:48.323

Well, I gave it a try at expressing what I think you were trying to get across. The edit is now being peer reviewed. I hope it meets with your approval. In any case, LUKE, thank you for the confidence you placed in me. – Mark Andrews – 2017-09-06T04:33:32.470

thanks @MarkAndrews hopefully someone replies. i'll keep checking up on it, tho i deleted my account again, cos i'm scared of sounding over zealous / pushy. cheers (i won't be back, it's not a good hobby of mine) – None – 2017-09-06T07:19:15.007

btw there is an SEP article on them, which suggests to me there could be some good answers

– None – 2017-09-06T08:17:39.047

or e.g. "If the idea is that the philosophical questions about infinitesimals don't exist because derivatives can be analyzed merely in terms of functions, it is deceptive, for that implication does not follow." seems to be a lot out there

– None – 2017-09-06T08:25:07.293

2It'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. – None – 2017-09-06T11:06:31.073

@Conifold The 'not too successful' call is out of order. Physicists constantly use nonstandard analysis when they multiply by dt or mix in a Dirac function. And they know when doing it will and will not get them in trouble. Physics just never signed on to the arithmetization of modern analysis, so they don't care that this has real rules rather than just rules of thumb. – None – 2017-09-06T20:15:32.873

1@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 – 2017-09-06T20:26:38.587

The issue of infinitessimals is huge in philosophy. Our entire view of time and space is affected by our theories about the structure of the number line and the continuum. I don't think it's possible to spend too long considering the logical problems associated with the usual mathematical conception of the number line and the paradoxes that arise for a continuum thought of as a series of points. . – None – 2017-12-06T13:32:29.333

## Answers

1

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.

2

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 – 2017-09-06T20:35:28.227

@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... – None – 2017-09-06T20:51:12.177

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 – 2017-09-06T20:58:13.127

@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. – None – 2017-09-07T18:22:49.917

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 – 2017-09-07T19:33:48.483

@jobermark - Not everybody agrees that infinitesimals solve Zeno's problems and to me it they seem to be the cause of those problems. A discussion for another time. But I'd generally agree with your comments. . – None – 2017-12-07T13:41:02.640

@PeterJ Something that did not exist when the problem arose cannot be its cause. Nor did I really say they are its solution. I just mean there is investigation to be done there that has now been abandoned because philosophy has simply dropped this notion. (You seem to imagine I have said something false, often, without it actually being there.) – None – 2017-12-09T01:38:16.883

Did you not say that infinities are a problem and that physicists and philosophers tend to back-off and avoid the problem? I was suggesting that the problem of infinitessimals is caused not by inventing them but by reifying them. But okay. apologies if I misread you. – None – 2017-12-09T10:41:12.853

@PeterJ I don't see any "problem of infinitesimals", since they have both an intuitive and a formalistic interpretation and those do not conflict. The former is a good excuse for inventing them and the latter indicates we did not necessarily make an error by reifying them. I see a problem with the fact that this is not integrated into anyone's overall view, perhaps excepting Whitehead, since philosophy kind of dropped the question at the same time math did. – None – 2017-12-11T20:30:30.850

It seems to me that the conflict between the continuum of direct experience and that of mathematical formalism is a well-known problem. Dantzig calls it the clash between the 'legato' and 'staccato' views. I thought this was the problem you were suggesting has been forgotten. For me Zeno dealt a death blow to points. I may ask a question about this to gauge current opinion. . – None – 2017-12-12T10:46:05.073