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This question pertains to smooth infinitesimal analysis as presented in the book A Primer of Infinitesimal Analysis by John Bell. The book uses intuitionistic logic.

Let $\Delta$ denote the set of infinitesimal quantities (real numbers which square to zero). It is proved that every infinitesimal is indistinguishable from zero, but not necessarily identical to zero. In other words, if $\varepsilon \in \Delta$, then the relation $\varepsilon \ne 0$ is false, but the relation $\varepsilon = 0$ is not necessarily true.

Many arguments in the book are geometric in nature but use infinitesimals. For example, rectangles are constructed such that one side has an infinitesimal length.


Question. If infinitesimals are indistinguishable from zero, how is it valid to use them in the construction of geometric figures? Isn't a rectangle with infinitesimal width indistinguishable from a line segment of the same height?

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  • $\begingroup$ It seems like to answer this question properly, one needs to be familiar with the book you are referring to. It isn't valid to use them in the case that one were thinking from the perspective of standard real analysis, but if looking through the lens of nonstandard analysis maybe they had resolved the issues. $\endgroup$
    – JMoravitz
    Commented Jun 3, 2020 at 14:57
  • $\begingroup$ Look up synthetic differential geometry $\endgroup$ Commented Aug 20, 2022 at 22:41
  • $\begingroup$ @simple jack Hello. I start learning SIA. You wrote that relation $ε≠0$ is false. Why is it so? Thanks. $\endgroup$
    – Mike_bb
    Commented Dec 4, 2022 at 10:28
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    $\begingroup$ @Mike_bb Since $\varepsilon^2 = 0$. If $\varepsilon \ne 0$ then $\varepsilon^2 > 0$. $\endgroup$ Commented Jan 2, 2023 at 0:32
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    $\begingroup$ @Mike_bb Zero is an infinitesimal, so $\varepsilon = 0$ may be true. However, the statement that $\varepsilon = 0$ for all infinitesimals $\varepsilon$ is false. That is what I meant by saying that $\varepsilon = 0$ is not necessarily true. The comments are not meant for extended discussion so please create a new question if you want to ask something else. $\endgroup$ Commented Jan 2, 2023 at 18:31

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1. You ask about the validity of geometric constructions in Smooth Infinitesimal Analysis, in light of the result that we cannot distinguish an arbitrary infinitesimal from zero in this setting.

Your concerns have merit: geometric reasoning in Smooth Infinitesimal Analysis can go wrong if one refuses to exercise caution. However, the specific examples of geometric reasoning that Bell employed in his book happen to work just fine (not by accident, but because the author has exercised appropriate caution when he wrote the book). To describe how this all works, I'll have to give an overview of what geometric reasoning means in usual, classical, bog-standard real analysis.

First, a reminder: in Smooth Infinitesimal Analysis, one passes to intuitionistic logic, and replaces the classical ordered field of real numbers $\mathbb{R}$ of usual (classical, limit-based) Real Analysis with a "smooth real line object" $\mathcal{R}$ that satisfies algebraic propeties reminiscent of those enjoyed by $\mathbb{R}$.

In this setting, we can say that the subset $\Delta \subseteq \mathcal{R}$ that consists of quantities $d \in \mathbb{R}$ satisfying $d^2 = 0$ contains many infinitesimals in the sense that the statement $\neg \forall x \in \Delta. x = 0$ holds. However, since we passed to intuitionistic logic, this does not logically entail $\exists x \in \Delta. x \neq 0$. Indeed, one cannot get access to any particular infinitesimal apart from zero: defining $\Delta_{nz} = \{ x \in \Delta \:|\: x \neq 0 \}$, one can even prove that $\Delta_{nz} = \emptyset$. In lieu of direct access to individual infinitesimals, we have to quantify over all of them: e.g. one formulates Kock-Lawvere-style axioms (Chapter 1, Principle of Microaffineness) by saying that $g(\varepsilon) = g(0) + b\varepsilon$ for all $\varepsilon \in \Delta$.

Now, usual real analysis identifies geometric objects with certain subsets of $n$-dimensional Euclidean space $\mathbb{R}^n$. For example, one could define line segments in Euclidean 2-space as point sets of the form $\{z \in \mathbb{R}^2 \:|\: \exists t \in [0,1]. z = x+ty \}$ where $x,y \in \mathbb{R}^2$ and $y \neq (0,0)$, polygons as certain unions of these line segments, circles as sets of the form $\{x \in \mathbb{R}^2 \:|\: (x-c)\cdot(x-c) = r^2 \}$ where $c \in \mathbb{R}^2$, $r \in \mathbb{R}$, and so on. The resulting analytic geometry subsumes synthetic (ruler-and-compass and more general diagram-based) geometry: e.g. if one can construct a point of intersection of a line segment and a circle using geometric reasoning, then one will find that the point set representing that line and the set representing that circle intersect as well, and one can prove this by a non-geometric, algebraic/analytic argument. Mathematicians freely use geometric arguments in Real Analysis, confident that we can replace them with rigorous analytic proofs if the reviewers demand it.

Textbook proofs are informal proofs. Nobody writes fully formal proofs, even thoug the fully formal proofs of real analysis are not geometric, but real-analytic. A "geometric proof" is a convenient shorthand employed in informal argument, that helps us remember how to write the formal argument if anyone ever asks for it. While geometric arguments can always be faithfully translated to real-analytic ones (and first-order proofs in real-closed fields of characteristic 0 give rise to geometric arguments),it's a safe bet that less than 10 percent of living mathematicians have any familiarity with how such a translation actually goes. But we don't need to know the faithful translation! We have enough mathematical maturity to just read the informal geometric argument, convince ourselves that the result indeed holds as stated, and to write up a formal argument that replaces geometry with algebra and analysis if required. This is why geometric arguments are admissible. The existence of a formal translation is reassuring, but ultimately not necessary.

A side note: usual real analysis allows us to define the same geometric object in many different ways: for example, instead of defining triangles as unions of line segments, one could have defined a prototype triangle as the convex hull of $\{(0,0),(0,1),(1,1)\}$ and other triangles as images of this prototype under invertible affine transformations.

Now, let's return to your question. At first, it might seem that Smooth Infinitesimal Analysis admits the analogous identification between synthetic and coordinate geometry: one just identifies geometric objects with certain subsets of the smooth plane $\mathcal{R}^n$ instead of subsets of $\mathbb{R}^n$. But upon second thoughts, one starts having doubts. Take any $\varepsilon \in \Delta$. Does the convex hull of $\{(0,0), (0,\varepsilon), (\varepsilon, \varepsilon)\}$ form a triangle under the definition of line segment given above (replacing $\mathbb{R}$ with $\mathcal{R}$)? One can't prove that it does! Indeed, just building the line segment between $(0,0)$ and $(0,\varepsilon)$ would require one to prove that $(0,\varepsilon) \neq (0,0)$, so $\varepsilon \neq 0$. But one cannot prove that, on pain of contradiction (Theorem 1.1. in the book)! (Exercise: Show that this infinitesimal convex hull would not form a triangle under the prototype definition either. Do the union of line segments definition and the prototype definition coincide in Smooth Infinitesimal Analysis?)

For the purpose of carrying out the geometric arguments employed in Bell's book, one would really want a sensible, rigorous definition that encompasses both infinitesimal and appreciable triangles, but not degenerate ones such as the convex hull of $\{(0,0),(0,0),(0,0)\}$. Such a definition cannot be realized, as shown by the indistinguishability results above. So if we wish to have a formal theory of geometric objects, then we have no other choice but to allow completely degenerate instances, such as points and line segments that constitute valid rectangles. Fortunately, a good chunk of geometric reasoning remains valid on these degenerate objects. In fact, the inventors of synthetic differential geometry gave considerable thought to similar questions: to make closed intervals better-behaved, they had to make definition of the order relation $\leq$ (given on page 19) a bit odd.

But one should not try too hard to find rigorous definitions, much less all-encompassing formal correspondences, for the informal geometric reasoning employed in the book and the formalism! Instead, recall how geometric proofs work in usual real analysis: the formal theorem that one tries to prove constitutes the "real deal"; the textbook proof merely an informal argument intended to convince us that the result indeed follows from the assumption, and to give us a mnemonic allowing us to produce a more rigorous proof if needed.

For example, consider Figure 3.2. showing the cross section of a cone. Does ACEB gives rise to a bona fide rectangle, and if so, under what formal definition? One could meditate on this question for a long time. But if a step in some proof asserts that the area of ACEB is so-and-so, one should forget about rectangles, and realize that the informal term area of ACEB is really just a convenient shorthand for some actual, formal object under consideration, and that formal object is just a sum of two integrals. And integrals on "infinitesimal intervals" make perfect sense: the Integration Axiom (Chapter 6, Integration Principle) provides the appropriate substrate, asserting that for any $f: [0,1] \rightarrow \mathcal{R}$ we can find a unique $F: [0,1] \rightarrow \mathcal{R}$ satisfying $F' = f$ and $F(0) = 0$. This function allows one to define "areas" under $f$, e.g. $\int_0^\varepsilon f(x) dx$ as the value $F(\varepsilon)$. One can even prove the "trapezoid formula" purely algebraically using microaffinity, without ever mentioning geometry or ever using the word trapezoid.

The arguments in Bell's book are perfectly valid and appropriate in this second sense: by reading the geometric arguments, you should be able to reconstruct rigorous, analytic arguments proving the same conclusions, without undue difficulty.


2. As for your second question, regarding the indistinguishability of a rectangle with infinitesimal width from a line segment of the same height. This question has no good answer, as everything strongly depends on how you define rectangles and line segments. For example, consider Figure 1.4 of the book and take the "rectangle" situated between the origin and the turning point of the parabola. Would you consider this a rectangle with infinitesimal width (and if so, does it even have a width)? What about the set $\{(x,y) \:|\: x=0 \wedge y \in [0,f(0)] \}$? Would you consider that a line segment? These two can be distinguished trivially: you can prove that they are not equal, by assuming that they are equal, then concluding that $\Delta = \{0\}$ for a contradiction.

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"Question. If infinitesimals are indistinguishable from zero, how is it valid to use them in the construction of geometric figures?"

There is an aspect of this that didn't come out sufficiently clearly in the other answers, so it may be worth pointing out the following.

The term indistinguishable is not being used in its generic meaning, but rather in a technical meaning, namely: $\epsilon$ is neither provably zero nor provably nonzero. What makes such entities useful is the cancellation lemma: all the nilsquare infinitesimals work together (with an effect as if they were nonzero).

As such, the notion of "nilsquare infinitesimal indistinguishable from zero" is an idealisation which provides a possible formalisation of historical infinitesimals, and can certainly be used to capture the idea of a "very thin rectangle" used in the calculus, as can other notions of infinitesimals (such as those of nonstandard analysis).

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I read "A primer of Infinitesimal Analysis" John Bell and I found something.

1.If infinitesimals are indistinguishable from zero, how is it valid to use them in the construction of geometric figures?

As you know, "indistinguishable from zero" doesn't mean that infinitesimals coincide with zero. There is "infinitesimal neighbourhood of 0" that comprises all points indistinguishable from zero. Thus it's valid to use infinitesimals in the construction of geometric figures.

2.Isn't a rectangle with infinitesimal width indistinguishable from a line segment of the same height?

Such rectangle is nondegenerate with base (width) of infinitesimal length $\varepsilon$.

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