[FOM] on numerical cognition

[catarina dutilh]

Related to the previous thread on integers and symbols, I thought it might be 
useful to provide some references on recent empirical work on numerical 
cognition. There has been a vast number of articles published on the topic in 
the last decades, and even though there are still quite a few open questions, 
some robust results on numerical cognition in infants, children and untrained 
adults have emerged. One of them is that, while we seem to have an almost 
'innate' understanding of low cardinals, or at least to be able to judge 
correctly and exactly the number of objects in a collection at a glance IF the 
number is low (usually up to three), after that we seem to operate with 
estimates and numerosities rather than with exact numbers. 

There is quite some work done on this in populations whose languages lack words 
for numbers larger than 2 or 3. Beyond 3, a child needs to be taught to count in 
order to be able to perform simple arithmetic operations. One interesting result 
is that both young children and untrained adults seem to have a logarithmic 
representation of the sequence of the natural numbers, rather than a linear one. 
There is a very influential article on this which appeared in Nature fairly 
recently:

http://www.sciencemag.org/content/320/5880/1217

The first author of this article is Stanislas Dehaene, who is one of the main 
researchers in the field. He also has a survey book, called 'The Number Sense' 
(and some have remarked that it would have been more accurate to call it 'The 
Numerosity Sense', given that untrained adults seem to operate with numerosities 
rather than with exact numbers beyond very small quantities), but was published 
in 1997, and in the meantime many more new results have emerged.

For recent and comprehensive overviews on the topic, I suggest two papers by 
Helen de Cruz and collaborators:

http://kuleuven.academia.edu/HelenDeCruz/Papers/165885/The_innateness_hypothesis_and_mathematical_concepts


http://kuleuven.academia.edu/HelenDeCruz/Papers/227628/The_cognitive_basis_of_arithmetic


Some of the researchers who have done important work in the field are: Alan 
Leslie, Rochel Gelman, Randy Gallistel (all three at Rutgers), Elizabeth Spelke 
and Susan Carey (both at Harvard), among many others. In other words, there are 
solid results in the field, so there is no need for us to speculate and treat 
issues which are empirical as non-empirical. The data are there for those who 
are interested.

This being said, Let me add that to approach the foundations of mathematics from 
the point of view of the actual ontogeny of mathematical concepts in humans is 
but one possible approach. I am personally very sympathetic to such a 
naturalistic approach, by the way, but the determination of the most primitive 
concepts within mathematics need not necessarily be grounded  in how humans 
actually develop these concepts. For example, the fact that we don't seem to 
have a natural grasp of the concept of 'successor' does not mean that PA or any 
other axiomatization of arithmetic relying on this notion is completely flawed: 
the concept of successor may be a conceptually primitive element even if it is 
not a cognitively primitive element (and in any case, the very notion of the 
sequence of the natural numbers as linear is not something that we seem to grasp 
intuitively without at least being trained to count, as the paper by Dehaene et 
al. in Nature suggests). In other words, all this does not imply that the 
foundations of mathematics should become an empirical, psychological enterprise 
exclusively.

Regards,

Catarina