As for human languages, being bilingual myself (acquired over 30ish years (am 37), not innate), I can only testify that this is as close to having "two different minds" as it gets— the sheer experience of life is different depending on which language you think it in, you speak it or write it to others.
That's as close as I can get experientially (qualia) to the notion that language is culture, civilization, thus psychological mindset (both individual and collective).
It's a very weird experience of life to have this ability to switch between modes. Doesn't make you any smarter I guess, but certainly opens one's horizons, helps thinking 'out of the box' (i.e. perspective), reach more balanced views perhaps.
I'd like to write a short piece about that someday. Hopefully later if #19 hasn't disposed of me by then.
Note that there is little to no overlap with programming languages or math. You can think these domains in any human language, the translation in symbols/code remains identical, ideally (that's formalism to me, requiring mathematical rigor, not literature skills).
As such, it is interesting and enriching to learn both— several human languages, and math (or at least one programming language). You gain things from each that do not exist in the other subset no matter how deep you look.