The fact that we can reliably count to 152 and distinguish it from 153 does not mean that we have a “sense” of either of those numbers. In fact I know of no-one who does. But they are not “social constructions” for us either.
Our understanding of the distinct features of 152, while not directly built into our brains, is an inevitable consequence of certain simpler features that are built in – namely the tendency to clump aspects of our experience together into distinct objects (including the finding of smaller clumps within larger ones and conversely identifying groups of clumps as new bigger clumps), to identify pairs or groups of clumps as being somehow the “same” as one another in various ways (eg as having equivalent elements, as having the same small number of elements, or as having similar structure in space or time, or…), to have some idea of “relationships” between different things with the possibility that a putative such relationship may be “true” or “false”, and to have rules of “logic” that allow us to relate the truth and falsehood of various such relationships. Any entity with these capacities, even if alone in the universe, might well learn to distinguish 152 from 153, to factor both of them, to recognize 151 as prime, and even to prove Fermat’s Theorem and wonder about the Riemann Hypothesis. There is definitely nothing social or cultural about any of that (apart from the names with which we label the various concepts).
Of course that doesn’t make it all a “real” feature of the universe (though I guess it would be real by definition as a feature of our minds if those minds were really capable of implementing it to all levels), as it depends on those inherited means of processing data which may or may not suffice to describe and predict things at all levels of accuracy.
This is basically just a rehash (or messed up complication) of what I said last time, but the second half of Ball’s article brings up another important point.
Although the arithmetic of large numbers can be analyzed in terms of mental tools which only include built in models for very small ones, we do have (and share with other animals) other ways of dealing with quantity. These involve intuition about relative magnitudes which allow us to compare them without the use of discrete numbers. These comparisons often seem based on geometry (and sometimes fail us – especially as children – when the physical scales conflict with the quantities we are asked to compare). They also often seem based on ratios, and so involve a logarithmic relationship to the additive scales we sometimes use for the same quantities (perhaps also related to the physiological structure of some of our sensory apparati).
Whatever the reason, I think it is important to recognize that our built-in mental apparatus for recognizing and comparing quantities may have (at least) two completely distinct components, which may have evolved to meet different environmental pressures, and may be located in different parts of the brain with different types of implementation. A deeper understanding of this may well be extremely useful in mathematics education, and also perhaps in many other situations where quantitative information needs to be communicated to a wide variety of human types.