fevereiro 11, 2010

Axiomas I

"Ask a beginning philosophy of mathematics student why we believe the theorems of mathematics and you are likely to hear, "because we have proofs!" The more sophisticated might add that those proofs are based on true axioms, and that our rules of inference preserve truth. The next question, naturally, is why we believe the axioms, and here the response will usually be that they are "obvious", or "self-evident", that to deny them is "to contradict oneself" or "to commit a crime against the intellect". Again, the more sophisticated might prefer to say that the axioms are "laws of logic" or "implicit definitions" or "conceptual truths" or some such thing.

Unfortunately, heartwarming answers along these lines are no longer tenable (if they ever were). On the one hand, assumptions once thought to be self-evident have turned out to be debatable, like the law of the excluded middle, or outright false, like the idea that every property determines a set. Conversely, the axiomatization of set theory has led to the consideration of axiom candidates that no one finds obvious, not even their staunchest supporters. In such cases, we find the methodology has more in common with the natural scientist's hypotheses formation and testing than the caricature of the mathematician writing down a few obvious truths and preceeding to draw logical consequences. [...] The fact that these few axioms [of set theory] are commonly enshrined in the opening pages of mathematics texts should be viewed as an historical accident, not a sign of their privileged epistemological or metaphysical status." Believing the Axioms, Penelope Maddy (1987)

1 comentário:

Anónimo disse...

Em axiomas não cremos. Simplesmente os adotamos, e sejam quais forem, haverão afirmações indecidíveis. Provar ou demonstrar é uma tarefa que nunca se esgota. É claro que isso implica que não se está a buscar verdades, mas se está a buscar clareza. Alguma clareza.