Clarke and Beck (C&B) contrast the representations of the approximate number system (ANS) with precise number concepts. Now, humans can employ a perceptually based way of thinking of a number that many Fs, as in that many circles, that many toys. That many Fs refers, in context, to a particular natural number. That many Fs, employed in a given context, does not itself supply information about which number it refers to as given in Arabic or any other canonical system for referring to natural numbers. That many Fs is, nevertheless, specific. It does not leave it open whether the reference is possibly a range of numbers, or a probability or confidence distribution over them. Similarly, that length, a mode of presentation (“m.p.”) of a particular length as given in perception, does not give information about the length in any particular units. That many Fs and that length are specific, but not canonical.
A subject thinking of a number as that many Fs may or may not be able to recognize that number, so given, again. If the subject is a normal human, and the reference is to the number four, there will be a corresponding recognitional capacity. But there does not need to be for that many Fs to refer to in a given perceptual context.
This is an instance of an entirely general phenomenon in perception. A person without absolute pitch may hear a note, given under a perceptual m.p. that note. This is a specific m.p., but the subject may not be able to recognize the note again 10 min later. The distinction between discrimination and recognition applies to that many Fs just as it applies in elsewhere in perception. Therefore we must also distinguish the specific from the recognitional.
In the case of the perception of pitch, we can conceive of someone who can recognize the same pitch again over longer intervals but does not use any particular canonical system of representations (as middle C, D, E, F,…). Hence, we should further distinguish the recognitional from the canonical. Similarly for numbers: A subject may be able to recognize the number in question in uses of that many Fs, when the numbers are small, without employing a particular system such as the Arabic.
By contrast with C&B's claims about the m.p.s of an ANS, use of an m.p. that many Fs is sensitive, in matters of discriminability, to what individuates that number. Any natural number n is individuated by the condition for there to be n Fs, for arbitrary F. The successor relation in which a number stands is consequential upon this condition, rather than being fundamentally individuative of the number itself. To be sensitive to the condition for there to be n Fs does not at all require the subject to think of n as the successor of some other number.
Therefore, we need to distinguish an ANS from a noncanonical number system (NNS) which uses m.p.s of the form that many Fs. The modes of presentation employed in an ANS, as opposed to an NNS, are those expressed by roughly that many Fs. M.p.s of the form roughly that many Fs are genuinely unspecific, unlike those of the form that many Fs.
It is an empirical question whether a creature is using an ANS or an NNS. The two hypotheses are often not clearly distinguished. In both cases, the mental representations of number may involve analog elements. I hope for further empirical and conceptual work on the issues.
C&B also see subjects as “deriving ratios (hence rational numbers)” from representations of natural numbers in the ANS. A rival approach – originating in classical Greece – regards ratios as fundamentally ratios of magnitudes (such as distances, durations, and areas), of which ratios of natural numbers are one special case. Under this rival treatment, the “initial identification” (sect. 7.1, para. 1) of √2 is not quite what constituted a major discovery. There is clearly a ratio that is the ratio of the length of the hypotenuse of a unit right-angled triangle to the length of one of its other sides. That was not the major discovery. The discovery was rather that this ratio is not the ratio of any pair of natural numbers. It was not an option to say that there is no such ratio. The conception of ratios as ratios of magnitudes serves all comers. The discovery is that some such ratios are irrational.
Subjects can perceive the ratio of the lengths of one pair of objects as the same or as different from, the ratio of the length of a second pair of objects. The mental representation of ratios is important not only in the probabilistic examples the authors cite, but also in such decisions as foraging (ratio of food covering in a given area), and choice of a mate (ratio of size of body parts in a healthy conspecific). In such examples as these, the ratios may or may not be the ratios of natural numbers of entities of a given kind.
To think of something as a ratio, even a ratio of natural numbers, is one thing. It is another to represent this ratio mentally as a single rational number. The “converging evidence” cited by the authors (p. 42) does support the claim that organisms compare ratios of positive integers. It is a further claim, and would need additional evidence, to support the thesis that these ratios are coded as a single rational number, integrated into a single rational number line. Appreciation that 6:4, 12:8, and 3:2 are all the same ratio is not yet encoding that ratio as a single number 1½. That ratio is also a perceptually based way of thinking of a ratio, and, analogously to that many Fs, may not involve thinking of its referent as having a certain position in a rational number line.
Clarke and Beck (C&B) contrast the representations of the approximate number system (ANS) with precise number concepts. Now, humans can employ a perceptually based way of thinking of a number that many Fs, as in that many circles, that many toys. That many Fs refers, in context, to a particular natural number. That many Fs, employed in a given context, does not itself supply information about which number it refers to as given in Arabic or any other canonical system for referring to natural numbers. That many Fs is, nevertheless, specific. It does not leave it open whether the reference is possibly a range of numbers, or a probability or confidence distribution over them. Similarly, that length, a mode of presentation (“m.p.”) of a particular length as given in perception, does not give information about the length in any particular units. That many Fs and that length are specific, but not canonical.
A subject thinking of a number as that many Fs may or may not be able to recognize that number, so given, again. If the subject is a normal human, and the reference is to the number four, there will be a corresponding recognitional capacity. But there does not need to be for that many Fs to refer to in a given perceptual context.
This is an instance of an entirely general phenomenon in perception. A person without absolute pitch may hear a note, given under a perceptual m.p. that note. This is a specific m.p., but the subject may not be able to recognize the note again 10 min later. The distinction between discrimination and recognition applies to that many Fs just as it applies in elsewhere in perception. Therefore we must also distinguish the specific from the recognitional.
In the case of the perception of pitch, we can conceive of someone who can recognize the same pitch again over longer intervals but does not use any particular canonical system of representations (as middle C, D, E, F,…). Hence, we should further distinguish the recognitional from the canonical. Similarly for numbers: A subject may be able to recognize the number in question in uses of that many Fs, when the numbers are small, without employing a particular system such as the Arabic.
By contrast with C&B's claims about the m.p.s of an ANS, use of an m.p. that many Fs is sensitive, in matters of discriminability, to what individuates that number. Any natural number n is individuated by the condition for there to be n Fs, for arbitrary F. The successor relation in which a number stands is consequential upon this condition, rather than being fundamentally individuative of the number itself. To be sensitive to the condition for there to be n Fs does not at all require the subject to think of n as the successor of some other number.
Therefore, we need to distinguish an ANS from a noncanonical number system (NNS) which uses m.p.s of the form that many Fs. The modes of presentation employed in an ANS, as opposed to an NNS, are those expressed by roughly that many Fs. M.p.s of the form roughly that many Fs are genuinely unspecific, unlike those of the form that many Fs.
It is an empirical question whether a creature is using an ANS or an NNS. The two hypotheses are often not clearly distinguished. In both cases, the mental representations of number may involve analog elements. I hope for further empirical and conceptual work on the issues.
C&B also see subjects as “deriving ratios (hence rational numbers)” from representations of natural numbers in the ANS. A rival approach – originating in classical Greece – regards ratios as fundamentally ratios of magnitudes (such as distances, durations, and areas), of which ratios of natural numbers are one special case. Under this rival treatment, the “initial identification” (sect. 7.1, para. 1) of √2 is not quite what constituted a major discovery. There is clearly a ratio that is the ratio of the length of the hypotenuse of a unit right-angled triangle to the length of one of its other sides. That was not the major discovery. The discovery was rather that this ratio is not the ratio of any pair of natural numbers. It was not an option to say that there is no such ratio. The conception of ratios as ratios of magnitudes serves all comers. The discovery is that some such ratios are irrational.
Subjects can perceive the ratio of the lengths of one pair of objects as the same or as different from, the ratio of the length of a second pair of objects. The mental representation of ratios is important not only in the probabilistic examples the authors cite, but also in such decisions as foraging (ratio of food covering in a given area), and choice of a mate (ratio of size of body parts in a healthy conspecific). In such examples as these, the ratios may or may not be the ratios of natural numbers of entities of a given kind.
To think of something as a ratio, even a ratio of natural numbers, is one thing. It is another to represent this ratio mentally as a single rational number. The “converging evidence” cited by the authors (p. 42) does support the claim that organisms compare ratios of positive integers. It is a further claim, and would need additional evidence, to support the thesis that these ratios are coded as a single rational number, integrated into a single rational number line. Appreciation that 6:4, 12:8, and 3:2 are all the same ratio is not yet encoding that ratio as a single number 1½. That ratio is also a perceptually based way of thinking of a ratio, and, analogously to that many Fs, may not involve thinking of its referent as having a certain position in a rational number line.
Financial support
This research received no specific grant from any funding agency, commercial, or non-profit organization.
Conflict of interest
None.