I remember hearing from Rabbi Yosef Mizrachi that HaShem created two Torahs, one the Oral and Written and the other the natural world. My question is, when is it a positive mitzvah to study science and philosophy? I know that it can be justified vis-a-vis earning a parnassah, or saving a life. That is, in order to fulfill a primary mitzvah. I have also heard that it is a mitzvah when used for a holy purpose, e.g. to defend and support Torah ideas. But is that where the obligation to learn science, i.e. HaShem's creation, ends? Sometimes, learning science for its own sake helps to acquire greater depth in understanding.
According to Moses Maimonides, as is discussed in Mishnah Torah, Sefer HaMada, Hilchot Yesodei HaTorah, chapter 2, it is essential to study science in order to have a place in the world to come. Maimonides was an advocate for the improvement of science for humanity's sake. He taught that the Torah was revealed for three essential reasons: to improve society, improve the self, and share some truth, since in every generation there are some non-Jews who hold more knowledge than us.
While it is perfectly fine to study science and improve our lives here on earth, as is our duty to repair the world, it is futile to study mysticism as it is baseless in the Torah. This is a hard fact for most people to accept but an essential one.
This is why the rabbis were cautious about the study of esoteric study and called it dangerous. They wrote in the Mishnah Chagigah 2:1, that “Whoever ponders on four things, it is better for him if he had not come into the world: what is above, what is below, what was before, and what will be hereafter.” These were the four subjects that interested all mystics. What is certain is that the rabbis were afraid of the public learning these things that were mysticism. Because the Torah does not prohibit what is true; but what is false. See Ibn Ezra commentary.
My answer addresses the question of whether studying science can be a Mitzvah on its own:
Studying medicine to cure people or studying science to resolve a Halachic issue can be called הכי תמצי (an instrument) for other Mitzvos (Rambam's view).
Studying "pure" science cannot be a Mitzvah on its own or as a branch of learning Torah because it contradicts its essence: traditionally, Torah's knowledge comes from above - it was given to Moses and passed on by Rabbis, but the scientific knowledge comes from below, from gathering empiric facts and formulating theories.
This is the reason Sages prohibited studying science on its own, stressing the difference between the Jews that are dedicated to studying the "Eternal" and "absolute" truth and Gentiles that are dedicated to the Earthy and relative one.
Another important distinction is that studying the Torah aims to improve the higher Jewish soul of ours in order to promote us in the world to come, while studying science aims to improve our bodily experiences, our standards of living in this world only.