Kenneth Chang

For sure they used it in new ways, but if you read what you wrote (from Labourie inter alia) about the history of these developments, the claims in the article are incorrect, as several historians of 20c math could have told you

Nobody expects you to write a complete history. I think it's reasonable to expect you to want to be correct about the parts of the history you do use.

What specifically was happening in the mid-20th century that Labourie was ignoring?

The specific question you're objecting to appears to be this: "…most mathematicians in the middle of the 20th century did not think much of probability, which was at the bottom in the hierarchy of mathematics, below number theory, algebra and differential geometry."

yes, probability was entirely mainstream in mid-century, embraced as an important method by prestigious folks in many places, including in Labourie's France. It was important in both pure and applied work (a distinction that was being actively re-negotiated in this period).

Yes, and Labourie said that most mathematicians looked down on it, just as physicists think of engineering as a lesser academic discipline because it's just "applied physics." Probability is and was of course of great use, but it wasn't for most mathematicians.

yes, I, a professional historian of 20th century mathematics, am telling you that Labourie's claim is wrong. He has no evidence that "most mathematicians looked down on it" because most did not. The comparison to physics/engineering is also flawed, but that's another story

Feller book: "[In 1940s] few mathematicians outside the USSR recognized probability as a legitimate branch of math" And Doob "Feller & 20th century probability" seems to make a handful of references to probability and 'respectable established topics' being at odds in 30s & 40s ?