Despite our title (Gallistel & Gelman, Reference Gallistel and Gelman2000), I agree with Clarke and Beck (C&B) that the approximate number system does not represent irrational numbers. Most of them cannot be represented in any way, because they are unidentifiable (Chaitin, Reference Chaitin2005). None can be represented exactly by any physically realized system.
Gelman and I argue that human quantitative reasoning is founded on a prelinguistic system for representing both discrete and continuous quantity, which is phylogenetically and ontogenetically primitive. It provides concepts to which toddlers map the count words as they learn to count, and it supports adult reasoning about quantities. We called the neurobiologically realized symbols in a brain's system for representing quantities numerons. Gelman and Gallistel (Reference Gelman and Gallistel1978) stressed that numerons both refer and are the object of the arithmetic operations by which brains draw conclusions about the referents.
C&B don't like numerosity. Gelman and Gallistel (Reference Gelman and Gallistel1978) used numerosity to refer to the property of the easily counted sets that are conventionally represented by a number. A number, as we understand it, is a player in the game of arithmetic, defined by the rules of that game. In the section of our Chapter 11 headed “The Laws of Arithmetic and the Definition of Number,” we quoted the mathematician Knopp as follows: “Every system of objects for which this is true is called a number system, because to put the matter baldly, it is customary to call all those objects numbers with which one can operate according to the fundamental laws we have listed” (Knopp, Reference Knopp1952, p 5; italics his; “this” refers to any object manipulated in accord with the axioms of arithmetic). Having defined number conceptually, we could not define it referentially, which seems to be the only form of definition recognized by C&B and many others (e.g., Carey and Barner, Reference Carey and Barner2019). A fortiori, we could not define number by the fact that a number sometimes refers to the property of a set that we denoted by numerosity.
We defined the numerosity of a set operationally as the number you get when you correctly count it, thereby, explicitly rejecting set-theoretic definitions (Frege, Reference Frege1884). In my view, this usage is both unproblematic and necessary, because, in experimental work on perception, one needs one word for the percept (e.g., “brightness”) and another for the corresponding distal stimulus (e.g., “luminance”). In work on number perception, “number” most gracefully denotes the percept – or, in many contexts, the concept. Therefore, we need another word for the distal stimulus. That word has long been – and likely will continue to be – “numerosity.” Why some philosophers think there is something dodgy about this usage is a mystery. If they did psychophysical experiments, they would find it unsatisfactory to say “the number sense represents number” (C&B, sect. 6, para. 1); it's equivalent to saying brightness represents brightness.
A coherent discussion of the psychology of number by a convinced materialist like myself requires vocabulary that makes at least three distinctions: (1) number qua arithmetically defined concept; (2) number qua property of a finite set; and (3) number as a symbol in a computing machine like the brain. A number symbol in a computing machine sometimes refers to the property of a set measured by counting it. More often, however, it refers to a continuous quantity, such as a duration. And, perhaps even more often, it is just the name the machine uses for something, for example, the ASCII names for the symbols on a keyboard. Gelman and Gallistel have been tolerably consistent in denoting (1) by number, (2) by numerosity, and (3) by numeron. They did not, for example, title their book, The Child's Understanding of Numerosity.
In 2000, Gelman and I suggested that numerons were noisy magnitudes. We subsequently disavowed that hypothesis (Leslie, Gelman, & Gallistel, Reference Leslie, Gelman and Gallistel2008). If the numerons that represent distance traveled in animal navigation had 10% noise, path integration would be impossible (see Fig. 1 in Gallistel, Reference Gallistel2017). Path integration is well developed even in ants. They count their 13 mm steps over distances of at least 1,300 m (Buehlmann, Graham, Hansson, & Knaden, Reference Buehlmann, Graham, Hansson and Knaden2014; Wittlinger, Wehner, & Wolf, Reference Wittlinger, Wehner and Wolf2006), a count that rises to 100,000.
The symbols in physically realized systems for representing quantities and manipulating them arithmetically make only approximate reference to the computable numbers. When efficiency, speed and low energy consumption are strong considerations, engineered number symbols are a fixed-point data type. They have two parts, the exponential part, which specifies the scale, and the significand, which specifies the number of subdivisions distinguished at any given scale. If, for example, 3 binary digits constitute the significands, then a fixed-point binary symbol system distinguishes 23 = 8 different magnitudes at any scale. The number of bits in the exponent specifies the scale. Thus, for example, 0e0 denotes 0 × 20 = 0; 1e0 denotes 1 × 20 = 1; and 101e11 denotes 5 × 23 = decimal 40. (See Gallistel, Reference Gallistel2017 for details, including the explanation of why this system may represent any signed integer – and rational numbers that approximately represent quantities such as rates and probabilities.)
The numbers of binary digits in the significands of numerons may be estimated by the reciprocals of the Weber fractions. Weber fractions, generally, fall in the range from 0.0625 to 0.25, which implies 2–5 binary digits in the significands of most numerons.
The small number of binary digits in numeron significands bespeaks the sophistication of basic brain mechanisms: Numerons convey into computations the limited precision with which a brain's measurement operations generate the symbols that carry forward in time information about empirical quantities. These measurement operations, which Gelman (Reference Gelman and Reese1972) called estimators, rarely deliver a precision better than ±10%, whether the quantity measured is discrete or continuous (Cheyette & Piantadosi, Reference Cheyette and Piantadosi2020; Cordes, Gelman, Gallistel, & Whalen, Reference Cordes, Gelman, Gallistel and Whalen2001; Durgin, Akagi, Gallistel, & Haiken, Reference Durgin, Akagi, Gallistel and Haiken2009; Gallistel, Reference Gallistel2017; Gibbon, Malapani, Dale, & Gallistel, Reference Gibbon, Malapani, Dale and Gallistel1997; Halberda, Reference Halberda, Barner and Baron2016). Representing empirical quantities with more bits in the significands would imply a misleading precision. That can be disastrous, as any navigator should know.
Despite our title (Gallistel & Gelman, Reference Gallistel and Gelman2000), I agree with Clarke and Beck (C&B) that the approximate number system does not represent irrational numbers. Most of them cannot be represented in any way, because they are unidentifiable (Chaitin, Reference Chaitin2005). None can be represented exactly by any physically realized system.
Gelman and I argue that human quantitative reasoning is founded on a prelinguistic system for representing both discrete and continuous quantity, which is phylogenetically and ontogenetically primitive. It provides concepts to which toddlers map the count words as they learn to count, and it supports adult reasoning about quantities. We called the neurobiologically realized symbols in a brain's system for representing quantities numerons. Gelman and Gallistel (Reference Gelman and Gallistel1978) stressed that numerons both refer and are the object of the arithmetic operations by which brains draw conclusions about the referents.
C&B don't like numerosity. Gelman and Gallistel (Reference Gelman and Gallistel1978) used numerosity to refer to the property of the easily counted sets that are conventionally represented by a number. A number, as we understand it, is a player in the game of arithmetic, defined by the rules of that game. In the section of our Chapter 11 headed “The Laws of Arithmetic and the Definition of Number,” we quoted the mathematician Knopp as follows: “Every system of objects for which this is true is called a number system, because to put the matter baldly, it is customary to call all those objects numbers with which one can operate according to the fundamental laws we have listed” (Knopp, Reference Knopp1952, p 5; italics his; “this” refers to any object manipulated in accord with the axioms of arithmetic). Having defined number conceptually, we could not define it referentially, which seems to be the only form of definition recognized by C&B and many others (e.g., Carey and Barner, Reference Carey and Barner2019). A fortiori, we could not define number by the fact that a number sometimes refers to the property of a set that we denoted by numerosity.
We defined the numerosity of a set operationally as the number you get when you correctly count it, thereby, explicitly rejecting set-theoretic definitions (Frege, Reference Frege1884). In my view, this usage is both unproblematic and necessary, because, in experimental work on perception, one needs one word for the percept (e.g., “brightness”) and another for the corresponding distal stimulus (e.g., “luminance”). In work on number perception, “number” most gracefully denotes the percept – or, in many contexts, the concept. Therefore, we need another word for the distal stimulus. That word has long been – and likely will continue to be – “numerosity.” Why some philosophers think there is something dodgy about this usage is a mystery. If they did psychophysical experiments, they would find it unsatisfactory to say “the number sense represents number” (C&B, sect. 6, para. 1); it's equivalent to saying brightness represents brightness.
A coherent discussion of the psychology of number by a convinced materialist like myself requires vocabulary that makes at least three distinctions: (1) number qua arithmetically defined concept; (2) number qua property of a finite set; and (3) number as a symbol in a computing machine like the brain. A number symbol in a computing machine sometimes refers to the property of a set measured by counting it. More often, however, it refers to a continuous quantity, such as a duration. And, perhaps even more often, it is just the name the machine uses for something, for example, the ASCII names for the symbols on a keyboard. Gelman and Gallistel have been tolerably consistent in denoting (1) by number, (2) by numerosity, and (3) by numeron. They did not, for example, title their book, The Child's Understanding of Numerosity.
In 2000, Gelman and I suggested that numerons were noisy magnitudes. We subsequently disavowed that hypothesis (Leslie, Gelman, & Gallistel, Reference Leslie, Gelman and Gallistel2008). If the numerons that represent distance traveled in animal navigation had 10% noise, path integration would be impossible (see Fig. 1 in Gallistel, Reference Gallistel2017). Path integration is well developed even in ants. They count their 13 mm steps over distances of at least 1,300 m (Buehlmann, Graham, Hansson, & Knaden, Reference Buehlmann, Graham, Hansson and Knaden2014; Wittlinger, Wehner, & Wolf, Reference Wittlinger, Wehner and Wolf2006), a count that rises to 100,000.
The symbols in physically realized systems for representing quantities and manipulating them arithmetically make only approximate reference to the computable numbers. When efficiency, speed and low energy consumption are strong considerations, engineered number symbols are a fixed-point data type. They have two parts, the exponential part, which specifies the scale, and the significand, which specifies the number of subdivisions distinguished at any given scale. If, for example, 3 binary digits constitute the significands, then a fixed-point binary symbol system distinguishes 23 = 8 different magnitudes at any scale. The number of bits in the exponent specifies the scale. Thus, for example, 0e0 denotes 0 × 20 = 0; 1e0 denotes 1 × 20 = 1; and 101e11 denotes 5 × 23 = decimal 40. (See Gallistel, Reference Gallistel2017 for details, including the explanation of why this system may represent any signed integer – and rational numbers that approximately represent quantities such as rates and probabilities.)
The numbers of binary digits in the significands of numerons may be estimated by the reciprocals of the Weber fractions. Weber fractions, generally, fall in the range from 0.0625 to 0.25, which implies 2–5 binary digits in the significands of most numerons.
The small number of binary digits in numeron significands bespeaks the sophistication of basic brain mechanisms: Numerons convey into computations the limited precision with which a brain's measurement operations generate the symbols that carry forward in time information about empirical quantities. These measurement operations, which Gelman (Reference Gelman and Reese1972) called estimators, rarely deliver a precision better than ±10%, whether the quantity measured is discrete or continuous (Cheyette & Piantadosi, Reference Cheyette and Piantadosi2020; Cordes, Gelman, Gallistel, & Whalen, Reference Cordes, Gelman, Gallistel and Whalen2001; Durgin, Akagi, Gallistel, & Haiken, Reference Durgin, Akagi, Gallistel and Haiken2009; Gallistel, Reference Gallistel2017; Gibbon, Malapani, Dale, & Gallistel, Reference Gibbon, Malapani, Dale and Gallistel1997; Halberda, Reference Halberda, Barner and Baron2016). Representing empirical quantities with more bits in the significands would imply a misleading precision. That can be disastrous, as any navigator should know.
Financial support
This research received no specific grant from any funding agency, commercial, or not-for-profit sectors.
Conflict of interest
None.