One obvious feature is the central importance of lack of an upper bound in
true number competence (in something like the way that our sentence
understanding and sentence generating competence does not intrinsically
have a maximum length bound, though various practical problems may impose
length bounds. (This is essentially a generalisation of Chomsky's old
distinction between grammatical competence (potentially infinite) and
grammatical performance (bounded by short term memory, brain size, etc.).
The first competence could be possesed by a child with a grasp of an
unbounded iterative or recursive mechanism for generating number names.
Without various competences related to the uses of those names, the child
would not have number competences. I gave an introduction to some of those
competences and ways in which they could be implemented in a computer
capable of generating parallel synchronised systematic actions in chapter 8
of my 1978 book
http://www.cs.bham.ac.uk/research/projects/cogaff/crp/#chap8
Unfortunately, in 1978 I did make clear enough that I was assuming
another requirement, known to David Hume, Gottlob Frege, Bertrand Russell,
Jean Piaget and many others, but I later found was ignored by many
researchers into number competences: namely an understanding that one-one
correspondence (bijection) is a necessarily transitive relation. (It is
also symmetric and reflexive, but thos are simpler conditions.)
Frege and Russell/Whitehead tried to show that using pure logic we can
define a concept of number that includes 1-1 correspondence relations that
are transitive etc. But that's analogous to Descarte showing that Euclidean
geometry an be modelled in logic and arithmetic. As Frege pointed out to
Hilbert: the model was not the original. What he failed to notice was that
the same objection applies to his logicisation of the ancient
cardinal/ordinal (non-fractional) number competence.
Anyhow, Piaget had studied those logicians and knew that a grasp of
transitivity of 1-1 correspondence was required. Unfortunately he
mistakenly labelled it 'conservation' by analogy with conservation of
volume, or mass, or length, which was a serious error, since the second
case was a meta-property of a continuously varying property, whereas 1-1
correspondence (except in very advanced mathematics not relevant to finite
cardinal or ordinal numbers) is restricted to discrete items.
Piaget's research (as I understand it) indicated that children did not
grasp that 1-1 correspondence is a transitive relationship until they were
five or six years old (though not everyone interprets his experimental
results that way).
May main point is that I don't think there is any evidence that any
non-human animal understands the general fact that 1-1 correspondence is
transitive. I don't think any *current* robot does either though they may
have 'compiled in' implementations of their designers' understanding!
Also I don't think any neuroscientist that I have encountered can explain
what needs to change in a brain in order to produce an understanding of the
transitivity -- not just as an empirical generalisation from examples, but
as a necessary feature of 1-1 mappings.
This leads me to suspect that during the first 5-6/7 years typical human
brains use forms of information processing that transform themselves so as
to grasp that transitivity, and other features of 1-1 mappings. But nothing
I have heard can explain how brains do that.
(I think the Canadian psychologist Lance Rips has made similar points, but
he seemed to conclude, wrongly, that for humans that transitivity was
somehow innate knowledge.)