Talk:Infinitesimal

From Wikipedia, the free encyclopedia

You have new messages (last change).

Contents

[edit] Existence of Infinitesimals

A reason to consider that infinitesimals do "exist" in some sense can be found by considerations on density. An ordered set is dense if between any two of its elements is another element of that set; hence between any two elements there are infinitely many elements of the set (meaning arbitrarily many or infinitely many, depending on one's conception of infinity).

In a dense set \left(\mathbb{R}, \mathbb{Q}\right) there is no immediate successor: i.e. for a given number x it is not possible to state which is next. Consider a dense set of points on a line, these points cannot, in some sense, "touch" each other. If a point a were touching a point b with a < b then b would be the immediate successor of a and it would not be possible to find points between them, thus contradicting the fact that the set is dense.

This does not contradict the completeness of the real numbers. Completeness does not mean that there are no holes: this is an interpretation under the unproven assumption that \mathbb{R} is the continuum. Completeness states that any set of reals bounded above has a least upper bound in \mathbb{R}, or that any converging sequence of reals has a real limit. Completeness is a property of the reals which remains true whatever the interpretation.

(Ironically, \mathbb{Z} is also complete in that sense, and it does have "holes")

Then if real number points do not touch each other, there is room for other numbers which would necessarily be infinitesimals or infinitely close to real numbers. The question is then how these are defined - or constructed.

--160.53.250.105 08:09, 21 Jan 2005 (UTC)Richard O'Donovan

[edit] Example?

Could the distance between the graph of y = 1/x and the x-axis (or y-axis, if you prefer) be used as an example of an infinitesimal quantity? This distance, if it is defined at all, must be smaller than any positive real number and yet cannot be zero since the graph does not intersect the axis. Just thinking that an example (apart from calculus) would be nice... - dcljr (talk) 7 July 2005 06:59 (UTC)

Or how about the distance between two line segments obtained by removing a single point from a given line segment? Is there a problem with using the term "distance" for either of these examples? - dcljr (talk) 05:49, 20 December 2005 (UTC)
Indeed, that first example can be considered an infinitesimal. It's a hyperreal number, to be specific. As for the second... well, that's essentially "the number after 0", which can probably be formalized somehow. Neither one is actually a real number--the reals have no infinitesimals, unless you consider 0 to be infinitesimal--but you can call them numbers nonetheless. --Ihope127 18:55, 11 March 2007 (UTC)

[edit] The Law of Excluded Middle

"This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of the excluded middle--i.e., NOT (a ≠ b) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x ² = 0 is true, but x ≠ 0 can also be true at the same time."

If nilpotent infinitesimals deny the law of excluded middle, then so do negative numbers. -1 can be defined as "a number x where x ² = 1 is true, but x ≠ 1 can also be true at the same time." How are nilpotent infinitesimals any different? --Slobad 22:45, 21 December 2005 (UTC)

Good point here about negative numbers.
The comment about excluded middle is about another approach to infinitesimals. Maybe one thing that appears when studying different flavours of nonstandard analysis is that "infinity" is not a very well defined concept. This has led to different formalisations which are not all compatible. The nilpotent infinitesimals come from a theory by J.L. Bell. The more or less intuitive image given above to explain how one can understand that there is space between real points on the geometric line describes the Robinson hyperreals, so the debate about excluded middle, though interesting per se, is slightly out of place. 85.1.144.180 07:51, 23 April 2006 (UTC)
No, it is not a good point. If x ² = 0, but x is not zero, then, as a nonzero element of a field, it is invertible. Hence, multiplying by its inverse 1/x, we obtain x = (1/x) 0 = 0, a contradiction. This is why nilpotent infinitesimal real numbers are incompatible with classical logic. On the other hand applying a similar argument when x² = 1 yields only x = 1/x, for which 1 and -1 are both perfectly valid solutions. Geometry guy 22:26, 11 March 2007 (UTC)
My problem with smooth infinitesimal analysis is that negating the law of the excluded middle (LEM) precludes the powerful method of proof by contradiction (reductio ad absurdum). If the LEM is negated only in certain situations, then it needs to be made clear when it is safe to do so. Moreover, nilsquare infinitesimals are not needed when one can construct "working models" of infinitesimals using polynomial ratios or Robinson's hyperreals. The existence of a model of infinitesimals within the real number system demonstrates that they are as consistent as the real number system itself.Alan R. Fisher 00:35, 25 February 2007 (UTC)
Some people (e.g. constructivists) argue that this is a strength, in that the method of proof by contradiction is confusing and should only be used when it is really needed. For example, I recently had to prove that some vectors were linearly independent. For my first attempt, I assumed that there was a nontrivial linear dependence relation between the vectors, and derived a contradiction. Then I rewrote it: I assumed instead that there was a linear dependence relation and showed that it must be trivial (all coefficients zero). The second proof was much clearer (and would also be valid in the presence of nilpotent infinitesimals). Geometry guy 22:26, 11 March 2007 (UTC)

[edit] Unit Vector Differentiation

In dynamics, when a reference frame rotates we need to find the derivative of unit vectors \vec u_r, \vec u_\theta.

In finding the derivative we arrive at \lim_{\Delta \theta \to 0}\Delta \vec u_r = 2\lim_{\Delta \theta \to 0}[sin(\theta/2) (\vec u_\theta + \Delta \vec u_\theta)].

For small Δθ we can write d\vec u_r = d\theta (\vec u_\theta + d\vec u_\theta)].

As it is mentioned in the article, it is not rigorous to write \vec u_\theta + d\vec u_\theta = \vec u_\theta.

The articles about limit, derivative and differential do not provide insight.

If we expand, we can write d\vec u_r = \vec u_\theta d\theta + d\theta d\vec u_\theta.

It is a common practice (and I think an erroneous one) among engineers to set d\theta d\vec u_\theta=0 since to the multiplication of two infinitesimal numbers is assumed to be zero. Any ideas? Skorkmaz 09:15, 30 October 2006 (UTC)

[edit] Multiplication Of Two Infinitesimal Numbers

Is it possible to proove that multiplication of two infinitesimal numbers is exactly zero by the proof given below?

  • Let's take an infinitesimal number dx
  • Assumption: Assume that dx is an infinitesimal number such that there is no any other infinitesimal number between 0 and dx
  • Obviously 0 < dx < 1 = > dx2 < dx
  • If dx was a Real Number we could write 0 < dx2 < dx but here we can not. If there is a number dx2 such that dx2 > 0 this is in conflict with our assumption.
  • So if there is a number smaller than dx that number must be exactly zero. This is the end of the proof.

Fgeridonmez 15:24, 30 October 2006 (UTC)

Your second assumption is impossible with hyperreal numbers, but I can't rule it out if we're talking about other sorts of systems. Michael Hardy 18:04, 30 October 2006 (UTC)

In Nelson's IST it is also impossible but Bell uses a category definition in which there are nilsquares, nonzero numbers such that their square is zero (as mentioned above).

Sure, but Bell also has a bunch of numbers that you can't prove aren't zero. And a bunch of other *different* numbers that you can't tell apart from zero. Bell also discards the law of excluded middle, so the 4th point doesn't work in Bell's context. Lastly, as long as people are using infantesimals called dx, I'll point out that dydx (and higher order terms) see a lot of use, and that dydx = dy * dx wouldn't be all that usefull if it were always zero. Endomorphic 23:04, 18 March 2007 (UTC)

[edit] Re-writing

Lots of intersting things and questions in the article and discussion, but maybe now would be a good time to re-write the whole article. This I suggets to do as I have been working in the area of nonstandard analysis for some time. I use it for teaching and have met most of the leading researchers of the field. The historical part and philosophical part are also of interest. I think the article needs a new outline so that extensions would be easier.

I would not plan to hope to write The definite article. I just think I could make some things clearer and hope that others would improve.

No offence to the first writer. First version will come up soon.

Odonovanr 20:04, 10 January 2007 (UTC)

[edit] Merger?

I've rewritten the article originally known as Differential (calculus) into an article Differential (infinitesimal). There is some overlap with the article here, although the purpose of the Differential (infinitesimal) article is to give an overview of the historical meaning of differentials such as dx, as well as ways to the notion rigorous. Comments would be welcome. Also if anyone wants to propose some sort of merger or any other way to pool ideas related to infinitesimals, please discuss below. Geometry guy 22:32, 11 March 2007 (UTC)

I think that's a bad idea. Infintesimals and differentials are confused for each other enough already. Infintesimals are numbers smaller than any positive real number; they are elements that might live within a field structure extending the reals. If t is an infintesimal, you can always say t < 3 and be sure of the truth. Differentials in integration and differential equations are very different creatures; within the context of dy/dx = 3x, the equation dx < 3 is not sensible at all. Endomorphic 23:28, 18 March 2007 (UTC)

Why is it not sensible? One can have dy/dx = 3x and of course dy and dx are infinitesimals, and so of course |dy| and |dx| < 1. Michael Hardy 23:32, 18 March 2007 (UTC)

It's not sensible because dx and dy are either measures, differential 1-forms, linear mappings, or whatnot. They're not something you can intuitively extend the reals to include. You can't take the wedge product of 2 and 3, for instance. There are contexts for numbers like 3, such as being an equivalence class of certain Cauchy sequences. There are contexts for differential forms and measures, such as dy = F dx where F = dy/dx. Infintesimals (in the sense of number smaller than all other strictly positive reals) work in the first context, but not in the second. Endomorphic 00:12, 19 March 2007 (UTC)

In some contexts they may by differential 1-forms or the like. But Leibniz thought of dy and dx as being infinitely small incrememnts of y and x, so that dy/dx is the ratio of increments. For integrals, if you think of dx as an infinitely small increment of x, then, for example, if f(x) is measured in meters per second, and x (and so also dx) in seconds, then f(x) dx is in meters, and is an infinitely small distance, and the integral is the sum of infinitely many such infinitely small distances. So we're talking about things that you can "intuitively extend the reals to include". Of course there are other ways of looking at things, in which dx is for example a measure; no one denies that. Michael Hardy 00:53, 19 March 2007 (UTC)

The infintesimals used by Leibnitz and Newton to justify calculus were found to be lacking rigor, hence the developments of forms and measures and suchlike. Calculus from infintesimals is now only of historical interest and as an educational aid prior to limits, but that's all. The idea of derivatives being ratios of infintesimals is like goldfish heaven; a comforting story you tell your kids, not something considered fact. You can differentiate with limits of reals, with linear operators on function spaces, or with connections on tangent bundles, but there aren't any infintesimals anywhere. Endomorphic 01:59, 19 March 2007 (UTC)

Let's see... first you use the word "intuitively", but then you want logical rigor. I disagree that intuitive unrigorous ideas should be used ONLY until rigorous methods are learned. I can write an argument using infinitesimals in an intuitive way, knowing that later if I prepare something for publication, I may need to write out a rigorous rather than intuitive version of the argument, but in the early stage I'm more interested in where the argument will ultimately lead than in what it looks like after I've dotted the last "i". Rigor has its place, and it's an important place, and rigor is important. But rigor isn't everything. Michael Hardy 02:16, 19 March 2007 (UTC)

I didn't want to open with a rigorous discussion of why dy and dx aren't infintesimals and a bunch differential geometry links :)
Your points are good, but the article gives a different impression. It doesn't do nearly enough to point out that modern mathematics uses rather different methods. No mention of measures. No forms. No little-o notation. One link to limits. No mention that epsilon and dx often denote things which are *not* infintesimals, but are easily mistaken for them. Nothing to say that infintesimals are now really only used for brief informal back-of-the-envolope type calculations. The present article makes it sound like infintesimals had a few hiccups with Berkely, but are otherwise cool. They're not. Endomorphic 04:14, 19 March 2007 (UTC)
Oh dear, I seem to have opened a can of worms here. For the record, I agree with Endomorphic that the articles should not be merged, but I also agree with Michael Hardy that differentials and infinitesimals are closely related: the unrigorous arguments are useful intuitively, and there are many ways to make them rigorous. However, I hope you won't be offended if (as a relative novice here), I issue a reminder that this page is for discussing improvements to the article, rather than the pros and cons of infinitesimals. Though, if you continue the discussion on your user pages, let me know, and I'll add them to my watch list, since I find the discussion rather interesting. Geometry guy 19:45, 19 March 2007 (UTC)
Retrieved from "http://en.wikipedia.org../../../i/n/f/Talk%7EInfinitesimal_0021.html"