I'm looking for recommendations for works that present roughly (or at least parts of) the following perspective on epistemology in mathematics. I hope having access to similar perspectives will allow me to be able to refine my own thoughts, and eventually present them in a coherent manner, interconnected with other works in the field and with correct vocabulary. I am a mathematical logician with no formal training in philosophy, so even telling me what the official names of various perspectives I'm describing would be helpful.
First off, when I talk of "knowledge" I'm interested in reasons why people assert things: is there a mathematical proof? is it the result of a scientific experiment? did they hear it from someone? is it from an official source? did they experience it directly? And how, given an assertion, it could be otherwise. This forms a nuanced version of "levels of knowledge".
As a formalist, within pure mathematics, I believe we "know" something (within a particular formal "term rewriting" system) if there is a sequence of steps following the symbolic manipulation rules in the formal system deriving that statement (or more practically, a less formal argument to convince us that such a derivation exists). As such, we have unparalleled levels of certainty: to question a pure mathematical result requires questioning the formal system itself or the concept of deduction (which is allowed, but it's telling that one must go this far to question a mathematical result). In exchange we forsake the ability to say anything about the real world: we can only make assertions about systems whose foundations we are certain of, i.e. only formal systems that we have created to have those particular foundations.
(The actual practice of mathematicians simply convincing each other that deductions exist adds an extra level of knowledge, where published mathematical results could be wrong because a mathematician made a mistake somewhere.)
Applied mathematics takes the other side of this trade: we get the ability to talk about the real world, but at the cost of dropping to a completely different level of certainty. A real-world situation is translated using a mathematical model to formal mathematics, where we can do some pure mathematical reasoning, and then a result is translated using the model to a real-world prediction. To question this result, one now also gets the ability to question the model being used. As far as we know, all mathematical models contain flaws or simplifications, so we lose a great deal of certainty in our results. The results of mathematical models could either be telling us things about the real world or about the limitations of our models.
I'm not particularly interested in a philosophical discussion or debate: I am already well aware of a wide variety of contrasting perspectives on mathematical philosophy. However, my research into formalism has mostly been turning up older works from the era of Hilbert or earlier. I imagine that with the modern practice of formalizing concepts for computers, there might be more sophisticated formalist perspectives. I'm hoping there is modern mathematical philosophy work that will help me refine and put into context my perspective, and would greatly appreciate any pointers in the right direction.