The polyglots I know all agree with a moderate form of the Sapir-Whorf hypothesis, which says that the language(s) you use affect what and how you think. In fact, when I say that some linguists find it controversial they look at me as if I'm demented. They just take it for granted that it's true.
BTW, this is often confused/conflated with "linguistic relativity"[0] and I'm not going to get into where one ends and the other starts. More, it seems that the effects are different for coordinate multilinguals versus compound multilinguals.[+]
Pinker says in "The Language Instinct"[1] that careful experiments show that it's nonsense. On the other hand, more recent work suggests that the effect is real, and that speaking a different language can result in measurable differences in your modes of thought, perception of time, and even your personality.[2][3][4][5][6][7]
So I was curious, but I didn't have time to learn an entire new language. But then I heard of LogLan, and subsequently Lojban, which was originally designed specifically for the purpose of testing this idea. The reasoning goes like this:
* Design a language that has something specific that is not present in any known "natural" language.
* Learn to speak it, and preferably raise some children with it as (one of) their cradle language(s)
* Measure their thinking and see if there are differences
* Use that to design a careful experiment to test the hypothesised effects.
So I learned (some) lojban in the hope that I would see some changes in my thinking. And I did. Specifically, after the weekend I spent learning the tense (spatial and temporal) system I found myself using tenses that I don't believe I'd ever used before. Someone asked if I'd enjoyed a party, and I started my reply with:
"I would have had been having a good time,
except that ..."
So I strongly believe that studying another language affects how you perceive things, and how you think about things.
[+] It is suggested that coordinate multilinguals[8] do not experience the same effects as compound multilinguals[8], even if they are equally fluent. A friend of mine, now sadly deceased, never completed his PhD, but his work showed an extremely clear split between coordinate and compound bilinguals (English/German) wherein the compound bilinguals showed a clear change of personality as they shifted language, but the coordinate bilinguals showed no measurable change. He never published, I only have a copy of his thesis without the data, and I promised I would never release it.
[0] http://en.wikipedia.org/wiki/Linguistic_relativity
[1] http://en.wikipedia.org/wiki/The_Language_Instinct
[2] http://www.communicaid.com/business-language-courses/blog/bi...
[3] http://www.slate.com/articles/news_and_politics/uc/2014/06/c...
[4] http://lera.ucsd.edu/papers/language-time.pdf
[5] http://www.dailymail.co.uk/sciencetech/article-3004943/Being...
[6] http://www.economist.com/blogs/prospero/2013/11/multilingual...
[7] http://www.newrepublic.com/article/117485/multilinguals-have...
[8] http://www.blackwellreference.com/public/tocnode?id=g9780631...
Edit: Note to self - extract this as a blog post.