Problems imply solutions (whether as proofs).The only way that I know a formula works is via a proof. I might be able to verify by hand that it works for a number of cases, but that doesn't mean it always works; see here and here for some examples of this. This is not to say that experimental evidence is worthless - quite the contrary. But the special role of a proof is something which cannot be ignored.
Now, there are subtleties here. In order to prove something, I need to begin with axioms. What axioms are "acceptable?" The standard axiomatic foundation of mathematics is ZFC (but see here), but there are some "concrete" problems which can't be proved using these axioms alone (see e.g. here). The existence of such problems - and Goedel's theorem more generally - shows that ultimately, the notion of "proof" is more nuanced than we might think at first. For example, there could be a formula that always "works" for a given concrete problem, yet can't be proved to always work inside ZFC.
However, this situation tends to be the exception rather than the norm. And the answer to your question is no - formulas and proofs are quite different!
https://math.stackexchange.com/quest...la-and-a-proof
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