In his unpublished notes from 1615, commenting on Cardinal Bellarmine’s Letter to Foscarini, the Carmelite Father, Galileo wrote as follows:
It is of the highest prudence to believe that there is no demonstration of the mobility of the earth until such a proof has been given. And we do not ask anyone to believe this point without a demonstration. Rather we ask, for the good of the Holy Church, that what the followers of this doctrine know and can offer be examined with the greatest care, and that nothing be admitted unless it has a force which is greatly superior to the reasons on the other side. If the proponents were to have no more than ninety percent of the arguments on their side, they would be rebutted. But when everything offered by the philosophers and astronomers on the other side is proven for the most part false and without any importance, then the position of the proponents should not be scorned and be considered to be a paradox because of the fact that it cannot be demonstrated conclusively.
He went on to add that “[n]o greater truth can be, or ought to be, sought for in a position than that it corresponds to all the particular appearances.”
There are several strands of thought involved here that are worth unravelling, of which I consider two. Firstly, we have to bear in mind that Galileo was working within the Aristotelian context wherein science was thought to be capable of certainty (unlike the fallibilist epistemology we adopt today). It is interesting, then, to see Galileo adjusting this understanding slightly by appealing to a balance of probabilities, in the form of “a force greatly superior”. Although he still sets the bar high (ninety percent is not yet enough), we can see that his requirement for the abandonment of one theory in favour of another (in this case the Ptolemaic system for the Copernican) has changed from a certain demonstration of truth to almost a precursor of a falsificationist fallibilism, wherein the Ptolemaic view is to be rejected as refuted while the Copernican is not proven thereby but nevertheless is no longer to be scorned. In short, it represents another possibility that must now be considered instead of being dismissed as absurd.
Secondly, it is tempting to go on to read his additional remark as indicative of an instrumentalism typically held to be prevalent at that time and prior to it, wherein the aim of a theory or system was only to save the appearances, and this is indeed the way Bellarmine took it (in public, at least). However, the context is far more subtle: if this advice is adhered to, in the form of a global instrumentalism, it follows that the requirement that astronomical theories not clash with Scripture is also superseded (and hence there could be no subsequent injunction against Copernicanism). So far as I can tell, this implication escaped Bellarmine and perhaps even Galileo himself.