[FOM] Why Voevodsky was concerned about the foundations of the natural numbers?

Dr A. Koutsoukou-Argyraki ak2110 at cam.ac.uk
Wed Aug 8 13:13:49 EDT 2018

Dear Jos?,

Jos? Manuel Rodriguez Caballero wrote:

> I guess that Voevodsky was motivated to worry about the foundations of 
> the
> natural numbers by his intuition from elementary topos theory, where a
> number object does not always exist. Indeed, he was motivated by
> the formalisation of category theory in UniMath.
> Cultures without numbers, or with only one or two precise numbers, 
> include
> the Munduruku and Pirah? in Amazonia. Researchers have also studied 
> some
> adults in Nicaragua who were never taught number words. So, it seems 
> that
> natural numbers are a cultural phenomenon of some civilizations rather 
> than
> a knowledge given a priori. In Jean B?nabou's language, the word "very" 
> is
> a sort of fossil from a time when the Western civilization didn't have
> natural numbers.

I would like to point out the following:
The fact that certain cultures  have no concept of natural numbers 
should tell us nothing about the well-foundedness (or not) of natural 
numbers as we know them. In the same sense that the fact that the 
ancient Greeks had not conceived
non-Euclidean geometries tells us nothing about the well-foundedness of
non-Euclidean geometries.
Your comment tells me that you are a platonist who believes in universal 
and I have nothing against platonism, but be careful here: even the most 
platonist would never claim that because an individual/a group of 
have not attained a certain universal truth, then this truth is not 
universal or even problematic. And indeed, survival without having 
attained certain universal truths is absolutely possible.


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