There are three independent properties of a mode of presentation (m.p.) of a number: being specific; being recognitional; and being canonical. A perceptual m.p. of the form that many Fs is specific although it is neither recognitional nor canonical. The literature has not distinguished noncanonical from nonspecific m.p.s of numbers. Ratios are fundamentally ratios of magnitudes.

We agree with Clarke and Beck that the approximate number system represents rational numbers, and we demonstrate our support by highlighting the case of the empty set – the non-symbolic manifestation of zero. It is particularly interesting because of its perceptual and semantic uniqueness, and its exploration reveals fundamental new insights about how numerical information is represented.

There is ongoing debate about whether number is perceived directly. Clarke and Beck suggest that what plagues this debate is a lack of shared understanding about what it means to perceive number in the first place. I agree. I argue that the perception of number is held to a different standard than, say, the perception of objecthood; considering this, I explore what it might mean for the number system to represent rational numbers.

To investigate mechanisms of rational representation, I consider (1) construction of an ordered continuum of psychophysical scale of magnitude of sensation; (2) counting mechanism leading to an approximate numerosity scale for integers; and (3) conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system (ANS) represents numbers or numerosities, and why the ANS represents rational (but not irrational) numbers.