Some cognitive scientists have done an experiment on some people in Papua New Guinea to test the hypothesis that the number line is based on an in-built intuition that all humans share. They concluded that it isn’t, and that you can use cardinal numbers without placing them mentally on a line.
The experiment was performed on 26 Yupno people, of whom 20 were “unschooled” while 6 had “middle-school education”, and a control group of 10 adults from San Diego. It looks like the Yupno are the go-to people for cognitive scientists wanting to find out how the “uneducated” human mind understands things — a Google Scholar search for ‘Yupno’ turns up lots of experiments to do with counting, classifying, and spatial awareness.
The discussion section of the paper sums up the result quite concisely:
Our results show that adults from the isolated and largely unschooled Yupno community in the remote mountains of Papua New Guinea, despite having precise cardinal number concepts, do not spontaneously exhibit number line intuitions when presented with an external line. First, unlike schooled Yupnos, unschooled participants had serious difficulties understanding the fundamental endpoint anchoring required by the number line task. Importantly, more explicit instructions showing the mapping of the intermediate number stimulus 5 onto a location between the anchors 1 and 10 did not help. Second, in the cases where unschooled Yupno participants did establish the endpoint anchors in the training trials, the resulting mapping exhibited a bi-categorical pattern with intermediate values mapped onto the segment’s endpoints, thus violating fundamental metric properties of the number line.
This agrees with my experience: after being astonished to hear someone tell me that the number line goes vertically, I once asked a roomful of PhD students which way the number line goes. Opinion was divided between “vertically” and “horizontally”, while one person said it doesn’t go in any particular direction and went on to dispute the very existence of the number line. As far as I could tell, the orientations of people’s number lines seemed to depend on where the free space was around their primary school classroom’s blackboard.
They also mention, tantalisingly, that no depictions of the number line seem to exist from before the 17th Century. I must revise all my Eratosthenes/Omar Khayyam fanfic!