Time representation in Synthetic Differential Geometry

Question

I found in John Bell's book "A Primer of Infinitesimal Analysis" such interpretation of time representation:

But I can't fully understand this interpretation.

Can someone explain easily what John Bell means here?

Thanks.

Answer 1

Bell mentions Zeno's paradox of the arrow. This is a paradox of motion. Bell is interested in time as a parameter with respect to which one differentiates to obtain the velocity of motion. As you know, the principle of microstraightness enables one to calculate derivatives using only the plurality of nilsquare infinitesimals, even though we cannot prove that any individual nilsquare infinitesimal is nonzero.

In the classical framework, the line (including the time line) "dissolves" into points. More precisely, $\mathbb R$ in the classical framework is the union of the points on the line. This is not the case in Synthetic Differential Geometry, based as it is on category theory and intuitionistic logic (instead of set theory and classical logic).

The upshot is that in SDG, the time line does not dissolve into points in the sense above. A more accurate image is that the time line is a superposition of what Bell refers to as "timelets", each of which is of nilsquare-infinitesimal length (rather than being a point).