Husserl on the number twelve specifically
I looked up the thing I poked fun at yesterday where twelve is authentic and thirteen is inauthentic, and the real version in Philosophy of Arithmetic is a bit more sensible. Twelve is his upper bound on what we're able to perceive in one go, rather than the largest number we can always perceive:
Only under especially favorable conditions can we represent authentically concrete multiplicities of approximately a dozen elements. Authentically: that is, factually (and as intended) grasping each of their terms as something separately and specifically noticed, and together with all of the others, in one act. Accordingly, twelve (or a lower number close to it) is also the ultimate limit for the conceptualization of authentic number concepts.
And later in the book:
The symbolic representations of groups form the foundation for the symbolic representations of numbers. Had we only the authentic representations of groups, then the number series would at best end with twelve, and we would not even have the concept of a continuation beyond that. Along with the obvious lack of restriction on the symbolic expansion of groups, the same is also given for numbers, as we will soon see.
I suppose this is a bit better. I looked up the Wikipedia article on subitizing and twelve seems to be at least roughly the right limit:
In the 1990s, babies three weeks old were shown to differentiate between 1–3 objects, that is, to subitize. A more recent meta-study summarizing five different studies concluded that infants are born with an innate ability to differentiate quantities within a small range, which increases over time. By the age of seven that ability increases to 4–7 objects. Some practitioners claim that with training, children are capable of subitizing 15+ objects correctly.
I haven't read much more yet, but I did come across a useful footnote where Husserl says he got this authentic/inauthentic distinction from Brentano:
In his university lectures Franz Brentano always placed the greatest of emphasis upon the distinction between "authentic" and "inauthentic" or "symbolic" representations. To him I owe the deeper understanding of the vast significance of inauthentic representing for our whole psychical life, which before him, so far as I can tell, no one had fully grasped.