The point at issue was communicating about higher mathematics with people who have no mathematical training, rather than people who have some mathematical training.
Remember, the original point concerned communicating about enlightenment. “Some mathematical training” may be analogous to “partially but not fully enlightened.” “No mathematical training” is analogous to “never effectively practiced meditation.”
I still believe with high probability that you think “higher mathematics is impossible to communicate about to people without any mathematical training” is true. A good place to find someone without mathematical training would be a member of a hunter-gatherer tribe.
Aren’t numbers a human universal? Sure, it’s hard to talk about curves without defining “curve” first, but if I can just draw in the sand and say “that’s a curve,” we’re back to the option of communicating the gist of things without handing the person a textbook. Could I communicate any sort of higher math I know in this way? This is tricky because I can’t think of anything, but that’s hardly a general proof. Maybe quaternions would be hard to communicate to a hunter-gatherer, but again “hard” is a far cry from impossible.
The last time I looked this up, all results on the Piraha language are due to a single anthropologist, Daniel Everett. There’s been some debate in the literature about whether or not he was actually correct about their innumeracy; see the “Further Reading” section on the wikipedia page for some examples.
On the object level, your belief-as-stated is not conclusively known. Everett sub 1986 believed that there were words for “one”, “two” and “many”; this belief was updated in 2008 when one speaker in an n=4 study used the word for “one” when there were six things presented to them.
On the meta-level, none of Everett’s results (as far as I know) have been replicated by an independent anthropologist, which means that your belief-as-stated has one point of failure. Given the surprising nature of his results, we should demand strong evidence that his results are true and not due to, e.g., cultural/linguistic misunderstandings. In fact, the linguistics community has indeed questioned the data closely.
The point at issue was communicating about higher mathematics with people who have no mathematical training, rather than people who have some mathematical training.
Remember, the original point concerned communicating about enlightenment. “Some mathematical training” may be analogous to “partially but not fully enlightened.” “No mathematical training” is analogous to “never effectively practiced meditation.”
I still believe with high probability that you think “higher mathematics is impossible to communicate about to people without any mathematical training” is true. A good place to find someone without mathematical training would be a member of a hunter-gatherer tribe.
Aren’t numbers a human universal? Sure, it’s hard to talk about curves without defining “curve” first, but if I can just draw in the sand and say “that’s a curve,” we’re back to the option of communicating the gist of things without handing the person a textbook. Could I communicate any sort of higher math I know in this way? This is tricky because I can’t think of anything, but that’s hardly a general proof. Maybe quaternions would be hard to communicate to a hunter-gatherer, but again “hard” is a far cry from impossible.
No. The Pirahã, for example, have no concept of exact numbers, only of smaller and larger amounts.
The last time I looked this up, all results on the Piraha language are due to a single anthropologist, Daniel Everett. There’s been some debate in the literature about whether or not he was actually correct about their innumeracy; see the “Further Reading” section on the wikipedia page for some examples.
I see nothing there that contradicts what I said, but it does seem most of the links are dead.
On the object level, your belief-as-stated is not conclusively known. Everett sub 1986 believed that there were words for “one”, “two” and “many”; this belief was updated in 2008 when one speaker in an n=4 study used the word for “one” when there were six things presented to them.
On the meta-level, none of Everett’s results (as far as I know) have been replicated by an independent anthropologist, which means that your belief-as-stated has one point of failure. Given the surprising nature of his results, we should demand strong evidence that his results are true and not due to, e.g., cultural/linguistic misunderstandings. In fact, the linguistics community has indeed questioned the data closely.