3

I've been learning synthetic differential geometry, and while the infinitesimals and microlinearity axiom make sense to me, I have not been able to grasp the geometry of these infinitesimals. In smooth infinitesimal analysis and synthetic differential geometry, the ring $R$ replaces the real line, so we can imagine it as a line. In illustrations of the K-L Axiom, I have seen many pictures that look like this

enter image description here

that make the set of infinitesimals $D$ appear to be an "infinitesimal line segment". I got this idea not only from the illustrations from SIA/SDG that I've seen online, but in more formal textbooks that explain the infinitesimal world in similar geometric terms, like this:

As $D$ is not an ideal, $D(2) \not = D \times D$. We remark that $D(2)$ includes the axes $$ \{ (d,0) \ | \ d \in D \} \text{ and } \{ (0,d) \ | \ d \in D \} $$ but also the "diagonal" $$ \{ (d,d) \ | \ \in D \} $$ and indeed $$ \{ ( \alpha d, \beta b) \ | \ d \in D \} $$ for all $ \alpha, \beta $ of $R$.

Following this cue, I have tried to picture zooming in on a point in $R$ and coming to find another line of infinitesimals centered around it. I first encountered the issue with my visualization when I was trying to determine whether there is always some $k\in R$ (non-infinitesimal) such that $d_2 = kd_1$ for any pair $(d_1, d_2)\in D\times D$. Such a theorem would make sense of to me if $D$ can be conceptualized as a microscopic line. If this were the case then, $d_2\cdot d_2 = kd_1\cdot d_2 = 0$, and the invertibility of $k$ implies that $d_1d_2 = 0$. Hence, $D\times D = D(2)$, which I know is not true, as $D$ is not an ideal of $R$. This broke my picture, and I'm having trouble finding a new one.

Perhaps the fundamental issue with this picture of the "infinitesimal world" around a point is that it carries the notion of distance and an order. As I understand it, these are incompatible with the nature of infinitesimals as neither equal to zero nor distinct from it. It does not make sense to describe one infinitesimal as closer to zero than another.

Could I have some help on how best to conceptualize these new additions to the number line?

Noah M
  • 345

1 Answers1

4

When I want to visualize or conceptualize the real line object of Synthetic Differential Geometry, I usually think of an infinitely long chain built from tiny metallic links. You may think of these tiny links as the nilsquare infinitesimal neighborhoods of ordinary numbers.

Applying a function to the real line

Applying a function allows us to twist and turn the chain into seemingly arbitrarily complicated curves.

However, when we focus on a single link, we see that the transformation behaved linearly: its overall shape of the link remains the same, even though the orientation changed. The Kock-Lawvere axiom allows us to inspect these orientations.

Kock-Lawvere axioms allow us to inspect linear orientation

This visual metaphor is far from perfect: the real line object behaves how it behaves, its behaviors need not correspond exactly† to the behavior of any physical object we interact with. Fortunately, the metaphor does account for several intuitive properties of the real line object:

  • We can define smooth curves, but not jagged ones: to create the function $x \mapsto |x|$, we must "remove a link" around $0$.

  • Zooming in on a chain, we don't find that the links consist of smaller chains. These small links are quantitatively different objects from the whole line. This explains what went wrong with your first intuitive picture (microscopic clone of the full line).

  • Interpreting the preorder relation $x \leq y$ as "the link containing the point $x$ lies to the left of the link containing the point $y$, unless they lie on the same link" motivates the observation that $0 \leq \varepsilon$ and $\varepsilon \leq 0$ both hold for any $\varepsilon \in D$. (Keep in mind that this is not the only type of preorder $\leq$ used in the literature.)

† Notice that the usual analytic real line $\mathbb{R}$ does not correspond exactly to any physical object either.

Z. A. K.
  • 11,359
  • I really like this visualization, thank you! I'm surprised the analogy gets as much mileage as it does, even including the preorder. As a follow-up, how do you think of subsets of $D$ like $D(2)$, just as existing somewhere inside each link? That might be stretching the metaphor too much, without even mentioning manifolds, tangent bundles , etc. – Noah M Jun 15 '20 at 00:46