Anachronism and infinity

William Connolley (aka our commenter 'Belette') is discussing the problems of characterising early scientific thinking - in this case, Galileo's thinking about infinity. Everyone who has tried this is familiar with the problem of anachronism: mistakenly characterising the thoughts and ideas of early thinkers in a way that they would not have recognised or understood. This is particularly difficult when, as usually happens you are translating their work from another language. Clearly you cannot use exactly the terms  they would have used, since they were writing a different language. So you have to use terms with the same meaning, while avoiding meanings they may not have understood. For example 'one to one correspondence' or 'set'.

A further wrinkle is terms in modern mathematical and scientific and philosophical language that are directly inherited from early writers. Most scientific language before the twentieth century was imported from Latin or Greek. Thus, we have the word 'continuum'. In Latin this just means 'the continuous'. Do you translate it as 'continuum' - running the risk of connoting ideas probably alien to medieval and early modern writers on mathematics? Or 'the continuous', which may wrongly imply that the Latin word had no technical meaning. Similarly 'vacuum', which would be wrongly translated as the modern 'vacuum', i.e. airless, when the Latin writers didn't just mean without air, but without anything at all, 'the void'. On other hand, it is clearly correct to ascribe concepts like 'concentric', which simply means 'having the same centre', and is derived from 'concentricus' which entered the Latin language in about 1260.

Concerning Galileo's problem, of explaining how the points of a circle can be put into 'one to one correspondence' with a smaller concentric circle, here is a chapter from Ockham I am working on, which addresses a similar issue.  He writes (my translation)

Likewise, it is of the thinking of Aristotle (as is clear in Physics IV) that air can be condensed without all or some of its qualities, changing. Hence, when air is condensed, it does not have to lose any quality, or at least it does not have to lose every quality which it had before. From which I argue that when air is condensed, either the whole preceding quantity remains, and precisely that which [was there] before, or not. If so, then the same quantity is now less than before only because the parts of quantity lie closer now than before. Therefore since the parts of the substance are in the same way lying closer now than before, and quantity is not supposed to exist for any other reason, it seems quantity is superfluous. But if the whole quantity which was there before does not remain, therefore some part is lost, and since from the corruption of the immediate subject there some accident of it is corrupted, it follows that not every quality remains, which is against Aristotle.

This clearly has an affinity with Galileo's problem of "a small ball of gold expanded into a very large space without the introduction of a finite number of empty spaces, always provided the gold is made up of an infinite number of indivisible parts".