Hume's problem of induction

The library had this very slim "Philosophy of Science - A Very Short Introduction" by Samir Okasha, and of course, I had to pick it up.

A quick scan of the book seems to indicate that the practice of science has rather loose philosophical foundations. A bigger mystery than any in the book then is why is it that science works? A second question is - are there ways of doing science that we have not yet imagined (this thought because perhaps the incompleteness of our knowledge is what leads to this philosophical mess).

Hume's problem of induction: Inductive reasoning is essentially the generalizing we do from a set of particular instances (e.g., having observed the sun to rise in the east every day, how do we justify that it will do so tomorrow? The process we use is induction). Hume argued that it is impossible to provide a rational (deductive) argument to justify induction. When we try to do so, we find we have to make additional assumptions, such as the uniformity of nature (which begs the question, because we cannot justify the uniformity of nature without induction).

From my point of view, errors of induction abound. A most common one is when someone justifies their prejudices against an entire people based on their experience with a handful of them. But it extends to science, too. Induction would lead us to believe that all matter is made of atoms (or various arrangements of atomic constituents). And yet when we look into deep space, we find that to explain what we see there is that there must be dark gravitating matter that is unlike any matter we are familiar with; consistency with known laws requires this matter to be non-baryonic (i.e., not the familiar atomic constituents).

The answer that, yes, science gives us provisional knowledge, and the laws (or models) that we have are of limited scope still does not answer the problem of induction; because even within the limited scope of the law we have to assume nature behaves uniformly.

Now, I'm not a philosopher, let alone a competent one. Still, here goes. From the point of view of the Indic traditions, before addressing the problem of induction, the focus would be on this rational creature that we are trying to convince from some self-evident premises and logical reasoning that induction is justifiable. My intuition is that the very existence of such a being is knowable to us only by induction. Why does language have any meaning? because we can connect it to the world - the real world, or the mathematical world - through induction. "Cat", "gas", "point", "Euclidean geometry" only make sense because there is some stability of memory and experience. "I", "I think" are concepts arrived at by induction. Perhaps only the God of the Bible or Quran can apprehend the world without induction; we humans cannot. Therefore we cannot properly pose Hume's question. "Convince this creature that knows of itself solely through induction that induction is a valid means of knowledge".

PS: Descartes' "I think, therefore I am" is inductive, not deductive.