An Example of Wittgenstein's Private Language

Wittgenstein's idea of private language (as I have interpreted/learned it from [1]) is that language must rely on a shared basis for it to be useable, but that it is possible that we share no fundamental thing that all language is based upon. I saw this report on a different approach to math education, during the New Math era (an era of education that focused more on self-discovery of math, and less on rote memorization), [2]. I didn't realize it at the time, but it really seems to support Wittgenstein's concept of private language. Basically, in this paper (based off their summary/conclusion) they talked with a person, Benny, who had been doing good with the IPI system. This system was one that made individualized problems to address an individual's weaknesses while being mostly self-paced. But instead of converging to the modern math system, Benny created a his own system of rules that would produce right answers, but not be conceptually correct. Furthermore, there did exist counter-examples and inconsistencies in his system that he wasn't know how to fix (but did realize). Thus the instructors wanted Benny to observe the modern math system and what was correct, and produce by himself the correct rules. Thus Benny's math system can be seen as the private language, and as Benny didn't know what the shared aspects that formed the basis of the modern math language, he wasn't able to actually use math properly.

[1] https://iep.utm.edu/wittgens/#H6
[2] https://people.wou.edu/~girodm/library/benny.pdf