Conceptual engineering and number cognition

Discussions in research cognition often concerns terminological inconsistencies and conceptual abuses. As example consider the discussion between two worldly recognised specialists in number cognition: neuropsychologist Stanislas Dehaene and psychologist and education scientist Brian Butterworth.

Stanislas Dehaene claims that there exist an innate system specialised in processing discrete quantities in approximate manner. He calls it “number sense” and claims that this system is the proper number cognition.

Brian Butterworth highlights that “number sense” means only that humans (and animals) are equipped in a cognitive system to process  discrete quantities. According to Butterworth another cognitive module, called “number module”, guarantees that the number concept arises. Butterworth explicitly claims that Dehaene’s “number sense” is not a sufficient (neither necessary) condition to arise.

In this project we are not aiming at solving these important empirical questions. However, we will offer an insight into the conceptual structure of the concept used in both “number sense” and in “number module” paradigm, and also in other. In consequence we will justify why enthusiastic passages as the one coming from one of the students of Dehaene:

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. (Izard et al. 2008, 491)

In the name of conceptual clarity, should be reformulated as follows:

Humans possess two nonverbal systems which enable humans to react appropriately to quantitative information and provide conceptual content for numerical expressions, both limited in their representational power: the first one provides content of quantity in an approximate fashion, and the second one conveys information about small quantities only.

Dehaene’s framework leads most importantly to such conceptual abuses as Matsuzawa referring to achievements of chimpanzees subject he works with as “knowing the number”. As this type of discourse seems serving as a great research motivation in empirical research, it seems to us that lack of further scrutiny might importantly disable us to make progress. We strongly believe that a “terminology hack” is necessary.

This work will be done with the full scrutiny of the method of conceptual engineering. Conceptual engineering is a powerful multi-tool philosophical method aiming at sharpening, reviewing and/or improving concepts, which was recently shown highly effective in various philosophical contexts. (Eklund 2015) risked even a thesis that “Philosophy should rather be thought of as conceptual engineering. We should consider what the best concepts to employ are like.”

The content of the concept of natural number constantly invokes vivid discussions and causes many disagreements: should we rather define individual numbers and think about the structure they form only in the second place, or – at the contrary – start by defining what is the structure that natural numbers form and care about the properties of individual elements occupying places in the structure – if ever – only at the second place. In addition, the most recently, philosophers started to be interested in the first concepts related to natural numbers that humans acquire in the ontogenetic development. The objective of this research is double: from the one hand side, it is interesting to understand what really are concepts that children form in the first years of life, but most importantly, we want to disclose how this earliest arithmetic relates to full blooded arithmetic of accountants or even professional mathematicians.

Other examples 

(Hauser et al. 2002) make a lot conceptual abuses. Here we point (in red) at some of them (page 1577). Next step will consist in indicating consequences of keeping these this way, proposing conceptual adjustments, and then indicating subsequent possible conceptual improvements.

“More than 50 years of research using classical training studies demonstrates that animals can represent number, with careful controls for various important confounds (80). In the typical experiment, a rat or pigeon is trained to press a lever x number of times to obtain a food reward. Results show that animals can hit the target number to within a closely matched mean, with a standard deviation that increases with magnitude: As the target number increases, so does variation around the mean. These results have led to the idea that animals, including human infants and adults, can represent number approximately as a magnitude with scalar variability (101, 102). Number discrimination is limited in this system by Weber’s law, with greater discriminability among small numbers than among large numbers (keeping distances between pairs constant) and between numbers that are farther apart (e.g., 7 versus 8 is harder than 7 versus 12). The approximate number sense is accompanied by a second precise mechanism that is limited to values less than 4 but accurately distinguishes 1 from 2, 2 from 3, and 3 from 4; this second system appears to be recruited in the context of object tracking and is limited by working memory constraints (103). Of direct relevance to the current discussion, animals can be trained to understand the meaning of number words or Arabic numeral symbols. However, these studies reveal striking differences in how animals and human children acquire the integer list, and provide further evidence that animals lack the capacity to create openended generative systems.

Boysen and Matsuzawa have trained chimpanzees to map the number of objects onto a single Arabic numeral, to correctly order such numerals in either an ascending or descending list, and to indicate the sums of two numerals (104 –106 ). For example, Boy- sen shows that a chimpanzee seeing two oranges placed in one box, and another two oranges placed in a second box, will pick the correct sum of four out of a lineup of three cards, each with a different Arabic numeral. The chimpanzees’ performance might suggest that their representation of number is like ours. Closer inspection of how these chim- panzees acquired such competences, however, indicates that the format and content of their number representations differ fundamentally from those of human children. In particular, these chimpanzees required thousands of training trials, and often years, to acquire the integer list up to nine, with no evidence of the kind of “aha” experience that all human children of approximately 3.5 years acquire (107). A human child who has acquired the numbers 1, 2, and 3 (and sometimes 4) goes on to acquire all the others; he or she grasps the idea that the integer list is constructed on the basis of the successor function. For the chimpanzees, in contrast, each number on the integer list required the same amount of time to learn. In essence, although the chimpanzees’ understanding of Arabic numerals is impressive, it parallels their understanding of other symbols and their referential properties: The system apparently never takes on the open-ended generative property of human language. This limitation may, however, reveal an interesting quirk of the child’s learning environment and a difference from the training regime of animals: Children typically first learn an arbitrary ordered list of symbols (“1, 2, 3, 4 . . . ”) and later learn the precise meaning of such words; apes and parrots, in contrast, were taught the meanings one by one without learning the list. As Carey (103) has argued, this may represent a fundamental difference in experience, a hypothesis that could be tested by first training animals with an arbitrary ordered list.”


David Papineau, Philosophical Devices, Proofs, Probabilities, Possibilities, and Sets, OUP 2012

Talks and publications

Talk at “Numbers in Mind: The Philosophy of Numerical Cognition”

Talk in Montreal