Irrationality is no hindrance for something being true

"For 100 years after its invention calculus was shown to be irrational ie contradictory but it still worked if working is a sign of being true then Irrationality is no hindrance for something being true"Bishop Berkeleys allegations are disputed and several modern mathematicians consider the formulation of Leibniz to be free of contradiction[1], but that really doesn't matter. Even if a mathematical theory seems to produce true results, it obeys aristotelian logic. In aristotelian logic contradictions always imply that the theory can prove any statement. If one could find an explicit contradiction then one could use it to prove that the true results obtained are actually false, or vice versa. Then the theory fails to be useful. The proof that one contradiction implies that all statements are true is quite simple:1: assume A AND NOT Athen2: Aand3: NOT Athen introduce an arbitrary statement B, and make a disjunction introduction4: A OR Bbut from (3), A is false so B must be true, by (4):5: Bthen we have seen that the assumption that bot A and NOT A are true, implies ANY statement!Applied to calculus, this means it is possible to prove any result, not just the correct ones."I gave statements from scientist that quantum mechanics is irrational but produces results thus Irrationality being no hindrance for something being true"There are two widespread meanings of the word paradox. One is the logical definition, that incompatible statements are true simultaneously. But another one commonly used is to call results that are incompatible with intuition paradoxes, sometimes referred to as apparent paradoxes. It is in this sense the word is used in the twin paradox [2] for instance. It is not a logical paradox, just an unintuitive result. The same applies to most uses of the word paradox by scientists, since a theory containing an actual logical paradox would be considered worthless by them and there is thus not much said about such theories.Your citations about quantum mechanics do not exactly specify which paradoxes they are talking about. The specific example you mention is:"one clear example is the dual nature of light ie both wave and particle"This is not a logical paradox, it is a result that contradicts intuition. The duality disappears in a rigorous mathematical formalism, where particles are replaced by wave packets or excitations of fields. The information on both wavelike and particlelike properties is encoded in the wavefunction even in early quantum mechanics. Even in a less rigorous formalism, the wave- and particle interpretations are used for studying different aspects of a phenomenon and are thus not really contradicting each other. When postmodern thinkers cite scientists they usually do not differentiate between logical paradoxes and paradoxes in the sense of nonintuitive results.Also mentioned in the text by Dean is superposition. But this is again only an intuitive paradox. Superpositions of states is a mathematical construct used to predict the world, they can never be directly observed and thus no contradiction with measurements arise. As mathematical objects, they follow all laws of mathematics. Schrödingers cat will always be observed as either dead or alive, not both. The superposition is only a mathematical construct to calculate the probabilities of the two outcomes.[1] http://arxiv.org...[2] http://en.wikipedia.org...