The example which came up in the seminar was that Pythagoras believed there were two non zero integers such that the ratio between them was the square root of 2. During his life time this was demonstrated not to be the case. So therefore Pythagoras believed something that was impossible.
But my contention is that this is just another way of saying that Pythagoras believed something false. Mathematical truths are timeless. So it will never be and was never true that the square root of 2 is the ratio between two integers, and we know that now. But I guess it is a fairly standard view that all propositions are timeless in this way. So it will never be and was never true that I had kippers for breakfast this morning. And from my perspective now, it is impossible that I did. But yesterday evening it was possible. My point is that just because something is timelessly true doesn't give it the special status of necessarily.
But maybe mathematically proven truths are necessary, not because they are timeless, but because they are mathematical. What is mathematically proven is true in all mathematically possible worlds. But what about mathematical theorems that have yet neither been proven nor disproven? Are they possible in mathematically possible worlds? There is no way of telling. If there is no way of telling, then introducing the notion of mathematically possible worlds doesn't help in elucidating the meaning of possibility. In a possible world I prove that Goldbach's conjecture is false. Is this a mathematically possible world? It seems like we have to find out whether Goldbach's conjecture is false before we can answer this question.
Some say it is impossible to find a needle in a haystack, but this is false. Given enough time and a good methodology, it is possible to find a needle in a haystack. So Pythagoras sets about searching for a needle in a haystack. He believes that he will find the needle in the haystack. It is possible that his belief is true. After Pythagoras has been searching for many years someone invents a metal detector and demonstrates that there never was a needle in the haystack. So in a sense it was impossible that Pythagoras would find one. But is this to say anything mataphysically different from simply that Pythagorus's belief that he would find a needle was false? Or his belief that he could find a needle was, in this case, false? I can't bring myself to believe that his belief was necessarily false in any non epistemic sense. Of course it is impossible to find something that isn't there. Once you realise it isn't there you stop looking.
One good rule seems to be that if something is true then it is impossible that it is false. But another equally valid rule seems to be that it is possible that something is true, were it not for the inconvenient fact that it is false. So we are tempted to think of a more absolute impossibility where something is necessarily false when, even if it were true, it would still be false. The only thing I can think of that fits into this category are self denials like "This sentence is false". It is impossible that this is true, because even if it is true it is false. But unfortunately this just means that it is necessarily true as well as necessarily false. I contend that the only propositions that are metaphysically necessarily false are also necessarily true. I am generalising from one case and my own lack of imagination. So counterexamples please: Necessarily false timeless propositions.