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

José Manuel Rodriguez Caballero josephcmac at gmail.com
Thu Aug 9 11:45:15 EDT 2018

> Angeliki said:
> 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.

Yes, I agree. The fact that some cultures have not natural numbers just
means that natural numbers are not as obvious as Kronecker said, I quote
him: "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist
Menschenwerk" (God made the integers, all else is the work of man). My
concern is not about natural numbers in themselves, but on the motivations
of Voevodsky to claim that:

CIC [...] has this induction definition scheme which allows you to do
> things parametrized by natural numbers. I think this is wrong,
> foundationally speaking, in this sense that CIC is not cleanly defined,
> because it is an initial model of a theory which itself requires natural
> numbers to be specified

I'm trying to reconstruct his thinking in a systematic way, as someone who
is studying a philosopher. So, it is not about the foundations of
mathematics in general, but about Voevodsky's approach to this problem. I
cannot take a Platonist point of view, because he was not a Platonist.

Jose M.
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