I’ve been writing a post explaining the practical difference between the synthetic and nonstandard notions of infinitesimals. It was getting a bit long, so I’m splitting it into two posts, of which this is the first.

Synthetic differential geometry (or smooth infinitesimal analysis) is a way to add infinitesimals to the reals; one that is an alternative to the nonstandard analysis approach. In SDC, infinitesimals can be nilpotent: their square or some higher power can be zero. This allows you to formalize arguments such as “this quantity is so small, we can treat its square as if it is zero”. You can also formalize these arguments in nonstandard analysis, but with more care (you can’t actually set the square to be zero, but you can treat it as an even smaller infinitesimal). Nonstandard analysis cannot have nilpotent infinitesimals directly, because it is required to preserve first-order theorems about the reals (which includes the theorem that the only nilpotent element of the reals is zero).

You can formalize infinitesimal arguments at the level of calculus equally well using either approach, so why would you ever want nilpotent infinitesimals? Here are some examples with a differential geometric flavor. Consider two points on the real line, and move them together so that they coalsce into one point. In ordinary differential geometry, that’s all they are — one point. In the synthetic approach, you can treat this as a double point, with defining equations x² = 0. You can imitate this in nonstandard analysis; I’ll explain how in the next post.

Here’s an example that’s considerably harder to simulate in nonstandard analysis. Consider four lines in the plane, two horizontal, and two vertical. Collectively they intersect in four points. If we let the two horizontal lines move towards each other, they become a double line, which intersects each of the two vertical lines in a double point. If we now let the two vertical lines degenerate into a double line, we have two double lines intersecting in a quadruple point. But not just any quadruple point (there’s more than one kind), they intersect in the quadruple point with defining equations x² = 0 and y² = 0.

J. L. Bell’s A Primer of Infinitesimal Analysis (an intro to synthetic differential geometry) begins with a series of quotes to motivate why we should think of the reals as containing infinitesimals. The quotes all involve the idea that philosophically speaking the continuum is an indivisible whole. This quote from Hermann Weyl is a typical example:

A true continuum is simply something connected in itself and cannot be split into separate pieces; that contradicts its true nature.

I find this line of reasoning completely unconvincing as motivation for allowing the reals to have nilpotent infinitesimals. I can grant, for the sake of argument, that maybe its unnatural that our model of the line can be split cleanly into two or more parts, but to me this is an argument for constructivism, not infinitesimals.