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!

Paper: Number Concepts without Number Lines in an Indigenous Group of Papua New Guinea, PLoS ONE

I talked to my girlfriend about this one day. Her number line bends at 10, at 20 and around 50. It changes directions in 3-dimensional space. I read about it on the internet, and it seems that some people have the first 10 numbers together, since they are learned first. Then you learn to count to 20, then to 50, and then pretty much the rest of the numbers come.

> 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.

Putting aside the confirmation bias (re. “the go-to people”), there do seem to have been many experiments on the Yupno people involving counting, going back at least as far as the 1980s (Yupno Number System and Counting, 1994, refers to Wassmann carrying out “extensive fieldwork in Papua New Guinea, in particular among… the Yupno (1986-1988)”). The press release says there are around 5,000 Yupno people in the area. You have to wonder, even though some of the people in the study might not have been formally schooled, what effect decades of counting experiments might have had! Also, the abstract for that 1994 article refers to a “culture of an elaborate number system”, which is presumably learned informally.

Yes, that’s a good point. I don’t think it affects this study though, because they showed that the participants

didn’tthink the conventional way, which isn’t what you’d expect if they’d been acquiring knowledge from previous studies.Yes agreed, although the effect of many years of experiments may not be to indoctrinate our number line system, exactly, but may have unplanned effects.

Relatedly, I was thinking that <doesn’t_really_know_what_he’s_talking_about>I think there are studies of young children that show our way of thinking about numbers is not particularly intuitive either (like we’re naturally inclined to log thinking) and so that too indicates the way we think about numbers is not intuitive but learned.</doesn’t_really_know_what_he’s_talking_about>

I am also astonished that people have vertical number lines, although I’ve been shown similar mathematical thinking tricks (e.g. by John Mason) where I’m astonished that not everyone in the room agrees with my way of looking at a problem. Though David’s girlfriend’s 3D bendy number line seems especially strange!

They actually mention the logarithm thing in the paper:

I meant to put that in the post, must have forgotten.

In cultures that write from right to left, does the number line ever go the other way?

And for children who have been taught about complex numbers, is it partly intuitive or purely learned to visualise the complex number line as orthogonal to the number line thus forming a complex plane?