Gettiers in software engineering (2019)

Gettier cases tell us something interesting about truth and knowledge. This is that a factual claim should depict the event that was the effective cause of the claim being made. Depiction is a picturing relationship: a correspondence between the words and a possible event (eg a cow in a field). Knowledge is when the depicted event was the effective cause of the belief. Since the paper mache cow was the cause of the belief, not a real cow, our intuitions tell us this is not normal knowledge. Therefore, true statements must have both a causal and depictional relationship with something in the world. Put another way, true statements implicitly describe a part of their own causal history.

Mathematicians already explored exactly what you describe: this is the difference between classical logic and intuitionistic logic:

In classical logic statements can be true in and of themselves even if there as no proof of it, but in intuitionistic logic statements are true only if there is a proof of it: the proof is the cause for the statement to be true.

In intuitionistic logic, things are not as simple as "either there is a cow in the field, or there is none" because as you said, for the knowledge of "a cow is in the field" to be true, you need a proof of it. It brings lots of nuance, for example "there isn't no cow in the field" is a weaker knowledge than "there is a cow in the field".