Divine Providence, in theological terms, refers to G-d's intervention and guidance in the universe. It's the belief that G-d is actively involved in sustaining and directing creation, including human affairs, towards a specific purpose. This can be understood as both a general, ongoing governance of the universe and specific, extraordinary interventions in the lives of individuals.
In any case, as a consequence, I have a new chavrusa! We are learning a two volume set on Faith, Divine Providence, and Free Will of שפתי חיים. (אמונה/Faith runs through both volumes, but the first volume focuses on Divine Providence, while the second on Free Will; within a Torah framework.) These volumes are based on the monthly talks given by R' Chaim Friedlander, z"tzl over many decades. The sefer includes a paraphrase of the introduction with which R' Chaim would begin these lectures. In short, he says that these topics are obviously at the core of the philosophical outlook of the Torah. These matters are intellectually deep, broad ranging, and encompass every dimension of Torah philosophy and מוסר/discipline. However, they also touch and are relevant to הלכה למעשה/practical observance of normative, day-to-day halacha.
As if that were not enough, the שפתי חיים adds: [these matters are relevant to הלכה למעשה] as do all the words of Chazal, the Rishonim, and the Acharonim. We were startled and read that again. How are all these matters relevant to practical halacha? Every gemara? There are some wild gemaras.
First, though, my second favorite topic, physics! By the mid-1800s, the laws of electricity and magnetism had been studied and elegantly summarized in four laws, known by the names of the main scientists in their development: Gauss, Coulomb, Faraday, and Ampère. A mathematician was bothered by a certain detail. They just didn't look balanced. So he added a term, known as the displacement current, that didn't change much as far as practical science, engineering, technology of the day. But it made the laws so beautiful and symmetrical. It did, though, mean that the electricity and magnetism could produce waves; theoretically speaking, of course. Some years later, another scientist, Hertz, actually made some waves in the laboratory. Then more people started looking. That led to radio, television, radar, X-rays, cell phones. You can argue how good that is; you can't argue how practical that is. So 100 years (give or take) after a mathematician fixed the beauty of some theoretical physics equations, we have oodles and oodles of practical applications.
Think about this: 100s and 1000s of students over decades learned about those equations. If you had asked any one of them why they needed to know that; of what practical use was that information? None of them would have known. Maxwell himself could not have imagined the impact his work would have on future generations.
So when you learn a gemara like Avoda Zara 34b that discusses the value of excrement from an animal that has gored and killed a human being versus that from an animal that has been dedicated to avoda zara... who knows why that information is vital? But it is Torah, therefore an integral part of the structure of reality.
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