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If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give to the students? What I am asking for are specific techniques for explaining infinitesimals to students, geometrically, algebraically, or analytically.

Note 1. This page is related as is this.

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    $\begingroup$ This is a political thread, in which you have changed a correct claim (the one you've had trouble contending with in the recent discussions of nonstandard analysis) into a straw-man that is easier to rebut. The true question was, for the framework of Robinson's nonstandard analysis, in which one defines and evaluate (for example) dy/dx not as a limit but as a ratio of infinitesimals, using an infinitely large or small number $H$ in the process, whether you can give students a concrete example of an individual $H$ ,or how to compute $f(H)$, or how to take the standard part of a bounded $f(H)$. $\endgroup$ – zyx Jan 7 '16 at 12:19
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    $\begingroup$ Many examples of taking the standard part can be found in Keisler's book starting with page 37; see math.wisc.edu/~keisler/calc.html $\endgroup$ – Mikhail Katz Jan 7 '16 at 12:25
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    $\begingroup$ I'm not sure. The dual numbers are numbers of the form $a+b\varepsilon$ where $\varepsilon^2=0$; they can be thought of as elements of $\Bbb R[X]/\langle X^2\rangle$ by identifying $X+\langle X^2\rangle$ with $\varepsilon$. (Unfortunately, division by $\varepsilon$ is undefined.) I would call $\varepsilon$ an infinitesimal. A nice feature of these numbers is that, for any polynomial $p$, we have $p(x+\varepsilon)=p(x)+p'(x)\varepsilon$. $\endgroup$ – Akiva Weinberger Jan 7 '16 at 16:30
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    $\begingroup$ An example of an infinitesimal quantity, may be given as the ratio of the weight of an ant divided by the weight of the planet earth. For educational purposes, ratios of known things bring in implicit visualization that could lead to understanding. $\endgroup$ – NoChance Jan 7 '16 at 18:17
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    $\begingroup$ @zyx I did come in late and I am not aware of the history. Anyhow, I am not planning to get dragged into this. I answered the question because I was asked to via email, and I'll leave it at that. Have fun fighting for justice. $\endgroup$ – Andrej Bauer Jan 11 '16 at 5:45

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One example of infinitesimals that has been used historically is that of hornangles, which were particularly popular in the 17th century. A hornangle $\alpha$ can be thought of as the "crevice" at the origin between the $x$-axis and the graph of the parabola $y=x^2$. If a real number $r>0$ is represented geometrically by the angle (in the first quadrant) between the $x$-axis and the line $y=r x$, it is easy to convince oneself that $\alpha$ is less than $r$ because a sufficiently short arc of the parabola $y=x^2$ will necessarily dip below the line $y=rx$ (when $r$ is fixed).

In a more arithmetic vein, Skolem in 1933 used sequences of integers to construct an extended number system incorporating infinite numbers. Here an infinite number is represented by a sequence tending to infinity. Skolem's construction does not rely on the axiom of choice. Using the quotient field of Skolem's integers, one gets a number system where a large fragment of calculus can be treated.

Similarly, Keisler in his book https://www.math.wisc.edu/~keisler/calc.html on page 913 gives an example of an infinitesimal represented by a sequence tending to zero, such as $(\frac{1}{n})$. Here the infinitesimal represented by $(\frac{1}{n^2})$ will be smaller than the infinitesimal represented by $(\frac{1}{n})$, etc.

Classroom experience shows that students find such examples intuitively appealing.

An additional approach is the one using Levi-Civita fields with the lexicographic ordering. These were used by Shamseddine and colleagues to develop computer implementations with infinity; see http://www.bt.pa.msu.edu/index_cosy.htm Needless to say, these "nonstandard models" are completely explicit.

As editor @nombre pointed out in a comment, the transfer principle is the crux of the matter. Depending on the theory one wishes to transfer, the level of difficulty may vary considerably. For example, if the theory is PA then Skolem's construction in ZF (without relying on the axiom of choice) is enough. If one wishes more powerful tools one will need more foundational input. This is a point often overlooked, even by high-profile people like Connes. The issue of constructiveness of Skolem's procedure is probed in more detail in this question: https://mathoverflow.net/questions/227945

Skolem's numbers are relevant because they actually embed in the hyperreals as explained in this article: http://dx.doi.org/10.1007/s10699-012-9316-5

Teaching experience shows that freshmen react well to examples of infinitesimals as represented by null sequences (i.e., sequences tending to zero). They also have some exposure to equivalence relations usually, so they find comprehensible a comment to the effect that an infinitesimal is not exactly a null sequence but rather an equivalence class of those. Of course the hyperreals cannot be constructed in a freshman calculus class any more than the reals.

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The discussion is mostly revolving around Robinson's non-standard analysis, even though the OP has not actually specified what kind of infinitesimals he is looking for. There are definitely good sources on how to teach Robinson's non-standard analysis, but I am not familiar with those, so I cannot say much about that.

Since the purpose is to teach people, it is worth looking at nilpotent infinitesimals, as known in Synthetic Differential Geometry. They quickly give the students methods of calculation that are practically the same as the methods employed by physicists. Synthetic Differential Geometry can be presented axiomatically, without ultrafilters or much attention to the logical language (standard vs. non-standard, internal formulas and). As long as we are interested in concrete computations, we will not even notice the main snag, which is lack of excluded middle.

Here are some references which explain infinitesimals in an accessible way which ought to appeal to students:

  1. I highly recommend John Bell's A primer on infinitesimal analysis, or his shorter An Invitation to Infinitesimal Analysis. If you tone down the stuff about intuitionistic logic and just skip to the axioms and computations, you can get a lot of interesting stuff quickly.

  2. I apologize for blowing my own horn, but I once wrote a blog post about intuitionistic mathematics for physics which has a section about infinitesimal analysis. This was targeted at students of physics. There is a similar exposition by me in the book A Computable Universe.

  3. If you teach computer science students you can motivate infinitesimals through the use of dual numbers in automatic differentiation, a very cool technique for writing programs that automagically calculate derivatives. You don't quite get "true infinitesimals" but it is a start and it can be quite appealing to computer-sciency students.

You asked specifically how to present to the students infinitesimals, or perhaps how to show them "a concrete" infinitesimal. This is always a bit difficult to do, both in Robinson's non-standard analysis and in Synthetic Differential Geometry. In Robinson's theory things revolve about non-principal ultrafilters, which are probably not the sort of thing you want to teach beginning analysis students. In Synthetic Differential Geometry we have (a squre-nilpotent infinitesimal is defined to be an element of the smooth real line $R$ whose square is zero).

  1. $0$ is not the only square-nilpotent infinitesimal: $\lnot \forall x \in R . (x^2 = 0 \Rightarrow x = 0)$.

  2. No square-nilpotent is distinct from zero: $\lnot \exists x \in R . x^2 = 0 \land (x < 0 \lor x > 0)$.

These two statements taken together are quite counter-intuitive, especially for a person who is used to classical logic (so 99.99% of mathematicians). The second statement actually tells us that we cannot ever display a concrete infinitesimal that is detectably different from $0$. Infinitesimals are intrinsically strange! But you can actually take advantage of the oddity and entice students to question some basic assumptions about how their geometric intuitions work and what sort of things are possible in mathematics. It is a lot of fun. I tried to explain the strange status of infinitesimals in my blog post and the paper, so I will not repeat that here.

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  • $\begingroup$ 1-2 are intuitive in algebraic geometry as doubled points and lines, which already in pre-rigorous times were thought of as somehow bigger or heavier than ordinary points and lines. The nontriviality of the algebra of dual numbers shows that classical logic and axioms of commutative rings do not suffice to prove "nonzero = invertible", so distinctions 1 and 2 are not exclusively an intuitionistic phenomenon and can be exemplified in classical situations. $\endgroup$ – zyx Jan 12 '16 at 7:21
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    $\begingroup$ Right. I think the intuitionistic stuff comes in once you are in a field and have $x^2 = 0$, and you wonder whether $x = 0$. Intuitionistic fields are messy. $\endgroup$ – Andrej Bauer Jan 12 '16 at 9:37
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I am teaching a calculus class using infinitesimals, so this question is very relevant for me too, user72694. I am avoiding formal constructions with ultrafilters and such, because I have found that these are too abstract for my students. What I want is something more concrete that will allow the students to build a reasonably robust concept image for infinitesimals -- something they can appeal to as necessary when they start working with dxs.

So the examples I give them are 0.000...1 (infinity 0s followed by a 1), 0.000...2, 0.000...01, etc. The reason I like these is that in my research students are able to independently reason with them and to develop conjectures about what happens when you add them, multiply, take ratios, square them, etc.

Of course, if someone asks me what these really are, then I will provide a more formal explanation for using sequences (keeping in mind that research shows they don't usually even understand what 0.999... is at this point). Then I'll talk about how you can think of these infinitesimal decimals as specific sequences of numbers that converge to 0 (there are others like 1, 1/2, 1/3, 1/4, ... that I don't know how to write as "decimals"). Ultrafilters etc. arise only when a student figures out that it's actually a bit hairy to compare two convergent sequences where one doesn't strictly dominate the other.

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  • $\begingroup$ Nice answer. I understand you have a published article along these lines. It may be good to link this if possible. $\endgroup$ – Mikhail Katz Jan 7 '16 at 18:41
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    $\begingroup$ The examples with 0.00...1 are either at odds with unique infinite extensions in nonstandard analysis, where the sequence would continue as 0 forever if 0 for all finite positions; or question-begging if interpreted as (0.1)^H for infinite integer H (the question being "what is this infinite H?"). What is the book or logical framework for infinitesiemals used in the class? $\endgroup$ – zyx Jan 7 '16 at 22:14
  • $\begingroup$ @zyx, we ultimately appeal to the framework in Keisler's textbook, although that is not the course text. I don't think I am viewing these weird decimals in either of the ways you mentioned. I treat 0.000...1 as (1/10, 1/100, 1/1000, ...) and 0.000...01 as (1/100, 1/10000, 1/1000000, ...) -- two different sequences. I think this avoids the logical trouble you mention, but correct me if I'm wrong. The real issue is that you can combine these decimals and get sequences that don't have such weird decimal expansions, so you ultimately have to move to the sequence notation to be more formal. $\endgroup$ – Rob Ely Jan 8 '16 at 4:51
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    $\begingroup$ I don't want to appeal to anything about ordinals. I regret saying "omega" in the previous comment -- my point was simply that the sequences in question are (1/10)^n vs. (1/100)^n, different sequences. These are not transfinitely indexed. Like any plain-old integer-indexed sequences, they can be treated as infinitesimals (as you point out in your answer to the question below). The important thing is that these strange decimals are ones through which students can build intuition, and which can be formalized unambiguously as sequences, which can then be formalized as infinitesimals if needed. $\endgroup$ – Rob Ely Jan 12 '16 at 17:39
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    $\begingroup$ For instance, the student interviewed in your paper understands that smaller and smaller infinitesimal stuff can be added by moving to the right in the decimal expansion, but throughout the (quoted) discussion does not perceive a need to also have spaces to the left of the "1" within the extra post-finite decimal places. $\endgroup$ – zyx Jan 17 '16 at 6:09
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A (relatively) explicit construction of a field with infinitesimal elements via ultraproducts. The problem with the sequences is that the cartesian product of fields $${\Bbb R}^{\Bbb N} = \Bbb R\times\Bbb R\times\Bbb R\times\cdots$$ isn't a field because has zero divisors: $$(0,1,0,1,\dots)(1,0,1,0,\dots)=(0,0,0,0,\dots).$$ The solution is taking a quotient: let be $\mathcal U$ a nonprincipal ultrafilter on $\Bbb N$. Define $$(a_1,a_2,\dots)\sim(b_1,b_2,\dots)$$ when $$\{n\in\Bbb N\,|\,a_n=b_n\}\in\mathcal U$$ The quotient $\Bbb R^* = {\Bbb R}^{\Bbb N}/\sim$ will be a field with infinitesimals, like the class of equivalence of the sequence $$(1,1/2,1/3,1/4,\dots).$$

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Regardless of what you believe about the axiom of choice, the fact that there are ordered fields with infinitesimals is provably true. e,g. the field Q[t] of rational functions over Q. Comparing infinitesimals to fields with one element is misleading at best: the statement "there is a field with one element" is provably false. Fields with one element are suggestive (for example sets would be just vector spaces over these fields) and if any sense is to be made of the concept it would have to be by considering some larger set of structures.

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    $\begingroup$ Do you seriously think Carl wasn't aware of non-archimedean fields? The question you are purporting to address here was about Robinson's nonstandard infinitesimals/infinities, and you have fudged that by leaving the word "nonstandard" out of the title. @user72694 $\endgroup$ – zyx Jan 7 '16 at 19:16
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    $\begingroup$ (sorry to the author of this answer. You walked into a propaganda campaign by the OP with link web between the different questions. So things can get a bit weird in the comments.) $\endgroup$ – zyx Jan 7 '16 at 19:25
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    $\begingroup$ @user72694: Bad answers sometimes get many upvotes too. I personally upvoted yours because you mentioned hornangles which I didn't know. $\endgroup$ – nombre Jan 7 '16 at 19:57
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    $\begingroup$ @user72694 There are also, currently 4 up and 4 down votes on your question, if that tells you anything you didn't figure out from a moderator's comment (and several upvotes on that) stating that you engage in intentional gross misrepresentation of others' comments. You can check my user profile to verify that I have never cast a downvote on this site. This page is the first time I have seen significant astroturfing on MSE; 3 answers are from your coauthors, new users who registered today and posted at almost the same time. The question as posted is of course a farce. $\endgroup$ – zyx Jan 7 '16 at 22:54
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    $\begingroup$ Mind you, I think it's great if experts join in at whatever number per day per question. What's unusual is the rallying of (astro) support for the cause, as opposed to concretely answering simple questions about infinite $H$ etc, that might have led to an admission that, yes, NSA has its peculiar difficulties and is not necessariy the Leibnizian best of all possible (infinitesimal) worlds. $\endgroup$ – zyx Jan 7 '16 at 23:00
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My pedagogical answer is to explain one over a generic natural number.

We cannot explicitly write down a generic natural number just as we cannot explicitly write a generic (non-constructible) irrational number. Like a non-constructible irrational number, it is an abstraction. We do know that it is larger than any fixed integer. We have no algorithmic method to determine any of its non-trivial properties, such as whether it is even or odd, prime or composite. Indeed, we have no algorithmic method to distinguish two different generic natural numbers.

I feel that this approach is close to the infinitesimals of old, and it's also highly intuitive. The notion of one over a generic natural number as an "example of an infinitesimal" comes from Kauffman's version of Sergeyev's grossone. It also relates to a view I have heard Tim Gowers express online, that a large integer out to be judged by how much we can say about it, and therefore (my words now) that one over a generic natural number is "functionally" an infinitesimal quantity.

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    $\begingroup$ Users interested in some professional assessments of the grossone theory can consult this thread. $\endgroup$ – Mikhail Katz Jan 18 '16 at 15:09
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An infinitesimal is some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity ( http://mathworld.wolfram.com/Infinitesimal.html )

Consider a quantity$=1\:$ for example. Then, the half. Then the half of the half. And so on. Of course in continuing to repeat this operation, the quatity tends to $\:0\:$. But a small and more and more smaller quantity always remains to be again divided in two. It's not exactly zero. It is an infinitesimal on the common sens.

More expanation in the above link and in : https://en.wikipedia.org/wiki/Infinitesimal

Comment :

In fact, "infinitesimal" is an abstract concept. Think also to another abstract concept : "Infinity" is an unbounded quantity that is greater than every real number. Both abstract concepts cannot be represented in real world on a concrete manner. Nevertheless, they can be intuitively understood.

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  • $\begingroup$ Thanks, J. Basically what you seem to be suggesting is that an infinitesimal should be represented by a sequence tending to zero, is that correct? $\endgroup$ – Mikhail Katz Jan 7 '16 at 9:39
  • $\begingroup$ Not exactly. The sequence tending to zero is only an exemple to produce an infinitesimal as requested. The sequence itself is not an infinitesimal. But the result is an example of infinitesimal : It is a quantity that is nonzero and smaller than any real quantity $\endgroup$ – JJacquelin Jan 7 '16 at 9:48
  • $\begingroup$ Thanks for the clarification and I agree, a sequence itself is not an infinitesimal but rather generates one; that's why I wrote "represented". $\endgroup$ – Mikhail Katz Jan 7 '16 at 9:49
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An infinitesimal is a positive number whose absolute value is less than any assignable positive number. Observe the word assignable. The difference between assignable and non-assignable was well understood by the founders of the calculus. It is arrogant to think that the giants like Newton, Lebniz and Euler did not know that there is no nonzero number whose absolute value is less than any positive real. They bear in mind the fact that there are two kinds of reals. Our concept of standard is the ancient assignable in a modern logical disguise.

When one asks for an example of infinitesimal which is the inverse of an infinite and says there are no such, he mocks himself for the numbers are mental constructions, and there are no numbers independent of human beings. That is, he cannot give an example of any number in much the same way as he knows no "real" examples of infinitely large and infinitely small numbers.

If we think about the numbers expressing the quantity of molecules in a room or if we count the grains of sands on a beach, then we see that these numbers are unreachable in contract to the number of fingers. The natural series starts with assignable numbers but there are clearly some numbers that are practically unreachable by successive count. These examples explain the difference between standard or assignable and non-standard or non-assignable numbers on a naïve level.

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    $\begingroup$ Very nicely put, thanks. The technical implementation of these intuitions about infinitesimals is of course Ed Nelson's Internal Set Theory where infinitesimals are found within the ordinary real line itself. $\endgroup$ – Mikhail Katz Jan 7 '16 at 16:39
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To define "infinitesimal" as either

  • a sequence or function that converges to zero

or

  • as an adjective for describing such a sequence or function

is a very old tradition that goes back in some form to Cauchy's Cours de Analyse or earlier, and is consistent with (if not necessarily the same as) statements in writings of old masters since Newton. If supplemented with ideas and notation from asymptotic analysis it can do most of what was accomplished with infinitesimal arguments before the rise of abstract analysis in the 20th century.

Of course it is easy to give examples of things that converge to 0.

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    $\begingroup$ You should see this: meta.math.stackexchange.com/q/22400/166353 $\endgroup$ – Akiva Weinberger Jan 12 '16 at 15:27
  • $\begingroup$ @AkivaWeinberger, I had already informed this user of the post at the time using @ $\endgroup$ – Mikhail Katz Jan 12 '16 at 15:47
  • $\begingroup$ @zyx, this is a bit more complicated than that. Cauchy said that a sequence converging to zero becomes an infinitesimal rather than is an infinitesimal. The idea that he meant a null sequence to generate an infinitesimal is confirmed by the fact that he was interested in rates of growth of sequences and even tried to classify those. In his textbook Cours d'Analyse he formulates and proves a series of results all of which focus on asymptotic behavior. Thus his procedures were distinctly non-archimedean, as were Leibniz's. $\endgroup$ – Mikhail Katz Jan 12 '16 at 19:16
  • $\begingroup$ Since Euler if not earlier there was the more general idea of asymptotic behavior (vs convergence or nullness) and I don't know to what extent those are distinguished by anyone before the mid-1800's. But would Cauchy have objected to the adjectival form "an infinitesimal sequence" as opposed to the words "an infinitesimal is a sequence such that ..."? Apart from what Cauchy might have said or done, using convergent to 0 as a definition of infinitesimal is done in some older books (not older than Cours d'Analyse, old relative to today). $\endgroup$ – zyx Jan 12 '16 at 20:46
  • $\begingroup$ @zyx, I appreciate your calm tone. I am not sure what you are asking, though. As I mentioned earlier, in most cases Cauchy does not write "an infinitesimal is a sequence", but rather that "a sequence becomes an infinitesimal. I would describe this as some kind of reification for lack of a better term. Today we would understand this in terms of equivalence relations but Cauchy did not have those yet. I find Detlef Laugwitz's analysis of Cauchy in his 1987 article in Historia Mathematica very perceptive. $\endgroup$ – Mikhail Katz Jan 13 '16 at 4:04
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For this Q (the existence of a concrete, meaningful infinitesimal) to have a mathematical meaning, one has to specify the structure (a non-archimedean extension of $\mathbb R$) wrt which the Q is considered. If it is a typical Robinson-style ultrapower of $\mathbb R$ then the method described by Martín-Blas Pérez Pinilla gives you a lot of concrete infinitesimals in such an ultrapower. One can also name concrete infinitesimals, in different manner, in typical pre-Robinson non-archimedean fields like Levi-Civita.

In the other direction, one can probably ask if there is a non-archimedean extension of $\mathbb R$ with no real-ordinal definable (ROD) infinitesimals, and I wonder if one can get such an extension in the Solovay model

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    $\begingroup$ Finding proper extensions of the ordered field of the real numbers with "explicit" (at least definable in ZF) infinitesimals is not difficult. (for instance, $\frac{1}{X}$ in $\mathbb{R}(X)$ ordered so that $X$ is infinite) What is more difficult is finding elementary extensions which satisfy the transfer principle over the copy of the set of reals. $\endgroup$ – nombre Jan 7 '16 at 17:01
  • $\begingroup$ @nombre, nice comment. The transfer principle is indeed the crux of the matter. Note that depending on the theory you want to transfer, the level of difficulty may vary considerably. For example, if you want PA then Skolem's construction in ZF (without relying on choice) is enough. If you want more powerful tools you need more foundational input. This is a point often overlooked in this type of discussion, even by leading people like Connes. If you like the answer don't forget to upvote it. $\endgroup$ – Mikhail Katz Jan 7 '16 at 19:03

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