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adrian_b 3 days ago

I wonder what ultrafinitists do about topology.

Topology, i.e. the analysis of connectivity, is built upon the notion of continuity and infinite divisibility, which seems to be difficult to handle in an ultrafinitist way.

Topology is an exceedingly important branch of mathematics, not only theoretically (I consider some of the results of topology as very beautiful) but also practically, as a great part of the engineering design work is for solving problems where only the topology matters, not the geometry, e.g. in electronic schematics design work.

So I would consider any framework for mathematics that does not handle well topology as incomplete and unusable.

Ultrafinitist theories may be interesting to study as an alternative, but the reality is that infinitesimal calculus in its modern rigorous form does not need any alternatives, because it works well enough and until now I have not seen alternatives that are simpler, but only alternatives that are more complicated, without benefits sufficient to justify that.

I also wonder what ultrafinitists do about projective geometry and inversive geometry.

I consider projective geometry as one of the most beautiful parts of mathematics. When I encountered it for the first time when very young, it was quite a revelation, due to the unification that it allows for various concepts that are distinct in classic geometry. The projective geometry is based on completing the affine spaces with various kinds of subspaces located at an "infinite" distance.

Without handling infinities, and without visualizing how various curves located at infinity look like (as parts of surfaces that can be seen at finite distances), projective geometry would become very hard to understand, even if one would duplicate its algorithms while avoiding the names related to "infinity".

Similarly for inversive geometry, where the affine spaces are completed with points located at "inifinity".

Such geometries are beautiful and very useful, so I would not consider as usable a variant of mathematics where they are not included.