Positivism was actually debunked by mathematics, not philosophy. In fact, this is why Platonism is accepted in mathematics and formalism scoffed at.
Take a look at Godel's incompleteness theorems - they hold that it's impossible to provide a proof for everything (formalization of "this statement is not provable", which is arguably both true and not true, results in a proof that the statement is true but not provably so).
A system containing first-order arithmetic cannot be both consistent and complete because it cannot prove its own correctness itself, as it will result in the paradox mentioned above - it must resort to a higher-order system to do so (metamathematics). Thus, one can never have a firm axiomatic system from which to deduce all non-empirically-derivable aspects of reality, and as a result, logical positivism doesn't work.
But positivism claims knowledge comes only from a posterior knowledge not a priori. So it does not require completeness in any formal system, just consistency.
And even if a deductive system is inconsistent in an interesting way, observation (a posterior knowledge) will catch it and force us to devise another system.
Take a look at Godel's incompleteness theorems - they hold that it's impossible to provide a proof for everything (formalization of "this statement is not provable", which is arguably both true and not true, results in a proof that the statement is true but not provably so).
A system containing first-order arithmetic cannot be both consistent and complete because it cannot prove its own correctness itself, as it will result in the paradox mentioned above - it must resort to a higher-order system to do so (metamathematics). Thus, one can never have a firm axiomatic system from which to deduce all non-empirically-derivable aspects of reality, and as a result, logical positivism doesn't work.