I assume implication is the operator that is most of the times used in mathematics, am I right? (i.e. whenever if/then construct is used)
If yes, then I think it is known that it has some philosophical problems
Merely that implication is true if premise is false. Which doesn't make sense in real world because I could say: if pigs can fly, then we can cure any illness. But this is not true in the way we understand if/then in real life, isn't it? Because we can't cure any illness so far.
So my question is more philosophical in nature, if we agree that mathematics is using an operator implication which is problematic from philosophical point of view and doesn't very well align with the real world meaning of if/then, then doesn't this cause problems for mathematics when we try to apply some results of mathematics to real world? More particularly because the operator (implication) that was used in math may not be the one that reflects the one in real life? This means we can't use some results from math which relied on implication, in real life?
Actually, your reasoning isn't right here: just because the statement 'if pigs can fly, then we can cure any illness' would be (mathematically!) regarded as true because pigs don;t fly, does not mean that we can cure all illnesses. In fact, what this statement is doing is that because we can't cure all illnesses, it must be false that pigs fly ... which is really the point of the statement 'if pigs can fly, then we can cure any illness'
That said, however, I agree with you on the mismatch between the mathematical implication and the real-life conditional. Consider this:
'If Bob lives in Los Angeles, then Bob lives in Florida'. (and to be clear: by 'Los Angeles I mean that big city on the West coast of the U.S., in the state of California, and by 'Florida, I mean that state on the South-East of the U.S.)
Well, any normal person would consider this a false statement: If Bob lives in Los Angeles, then Bob lives in California, not Florida! However, if we treat the 'if ... then ...' mathematically, and if it turns out that Bob does not live in Los Angeles, then suddenly the statement becomes true.
It turns out that real life conditionals often do not have the truth-functional property: knowing the truth-values of its parts does not mean we automatically know the truth-value of the whole. This mismatch between the way we normally think about conditionals in real life and the way we use the mathematically defined operator of material implication sometimes leads to very counterintuitive results which collectively are called the Paradoxes of Material Implication.
And yes, what this means is that we should be very careful in our application of mathematics to the real world ... just as with any other branch of mathematics or scientific idealizations: Euclidian geometry works ... as long as we don't deal with massive objects or very high speeds. Same for Newtonian mechanics. So in the end, these are all tools and, like all tools, we should know their scope and limitations.