# Ordinals vs. Cardinals in N? and Beyond

If you have a question about this talk, please contact Paul Taylor.

Special seminar arranged by Aaron Sloman

I will present a preliminary abstract framework of object-representation (in terms of humans’ interaction with objects), that makes room for the representation of mathematical objects on par with that of non-mathematical (mainly physical) ones. This framework serves as a bridge, through which the vast scientific knowledge concerning the latter can guide our understanding concerning the former. Using this framework, I will provide a novel account of the intricate interaction between ordinals, cardinals, and natural numbers, in-between the finite and the infinite. The account will dispute the classic mathematical tale of what actual infinity supposedly did to our concept of numbers and substantiate a different view: From the very beginning, even in the finite, ordinals and cardinals are inherently “mathematically” distinct; different types of objects. The restricted domain of the finite, however, allows for merging the two and bringing about the compound objects that numbers actually are and automating the handling of the finite ordinals and cardinals themselves and the interaction between them. But the finite is simply the implicit context in which these mathematical objects happen to first be experienced and discovered or taught (akin to a piece of geometric reasoning that inconspicuously depends on the drawn triangle being acute-angled). It is a contingent, empirical statistic rather than an absolute mathematical necessity that the cognitive system nonetheless, by its nature, comes to process equivalently. The introduction of actual infinity, much later on, merely forces us to notice the notions’ distinctness. Foundationally, that distinction is already at the heart of things—and of their mental representation too.

This talk is part of the Theoretical computer science seminar series.

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