Rashi (Nazir 29b) writes that the earliest age at which the Torah refers to someone as an “ish”, being a mature adult, is 13. The Torah in Parshas VaYishlach writes about Shimon and Levi, who we know b’mesorah were 13 at the time of their fight with Shechem(and Rashi adds that if you want to calculate it, go ahead), “VYikchu ish charbo”. This is the source for the age of bar mitzvah. The Rosh in a tshuvah disagrees and says that the source is a halacha l’Moshe m’Sinai.
In a letter to my wife’s grandfather, R’ Dov Yehudah Shochet, on the occasion of one of her uncles becoming bar mitzvah, the Lubavitcher Rebbe cites this machlokes and writes that a nafka mina between Rashi and the Rosh would be the age of gadlus for a ben Noach, a question raised by the Mishne l’Melech and other Achronim. If the age of gadlus is simply a measure of maturity, as Rashi seems to indicate, there is no reason to distinguish between a yisrael and a ben noach. But if it is simply an arbitrary number set by halacha l’Moshe m’Sinai, as the Rosh writes, then it would apply only to a yisrael because shiurim are part of Torah and were never given to bnei noach (Chasam Sofer Y.D. 317). (I actually suggested this same nafka mina in a post last year, so baruch shekivanti.)
The Rebbe then adds a bit of derush. Since the age of kabbalos ol mitzvos, at least according to Rashi, is a matter of reason and not an arbitrary number, one might mistakenly think that the kabbalos ol itself is dependent upon reason. Therefore, precisely where the textual hook of the word “ish” appears, the Torah continues with the word “charbo”, an allusion to mesirus nefesh, a quality which transcends the bounds of reason and logic. (I wonder how my wife’s grandfather, who was steeped in the mesorah of Telz known for emphasis on da’as (e.g. Shiurei Da’as), would have taken this.)
We actually find a hint to the idea of gadlus starting from age 13 earlier in the Torah. In Parshas Toldos, Rashi comments on "VaYigdilu hane'arim" (25:27) that Eisav and Ya'akov were 13 when they each went their own way in life.
On a related note, interestingly the gemara (Nazir 62a) says that m’doraysa from the age of “mufla ha’samuch l’ish” nedarim made by a ben noach are binding. However, the same may be true only m’derabbanan with respect to a yisrael. Why is there a distinction in this shiur with respect to age between a ben noach and yisrael? Rashi explains that this is simply a chumra that applies to ben noach, as we find in other areas. The Rogatchover (see R’ Kasher’ into to Rogatchover al haTorah, Braishis, p. 60) offers a deeper insight. The concept of tziruf, which binds together individual units into a larger whole, is a legal fiction which exists only because we have a legal framework of Torah that gives it reality. For a ben noach, no laws that depend on tziruf exist. The Rogatchover poses the following chakira: Is becoming a gadol a matter of reaching a specific point in time, or is it a matter of having lived through 13 years worth of hours, minutes, and seconds? The Rogatchover suggests that for a ben noach, it is a matter of reaching a point in time, such as mufla ha’samuch l’ish. For a yisrael, gadlus is measured by shtei sa’aros, not reaching a specific point like mufla ha’samuch l’ish. It is the passage of time, not a specific point in time, which leads to the growth of shtei sa’aros – the tziruf of previous years together make a person into a gadol.
Can you use the Rogatchover’s chakira to explain the machlokes Rashi and Rosh? I think maybe you can: According to the Rashi, “ish” is a point of time at which maturity is reached, no difference if you are a yisrael or a ben noach. According to the Rosh, gadlus is a product of the tziruf of the passage of previous years and therefore a halacha l’Moshe m’Sinai is needed to be mechadesh it as a conceptual construct.