Sunday, April 01, 2012

Scotus on the actual infinite

Cantor thought that the Scholastics strictly followed Aristotle in rejecting the actual infinite. Aquinas certainly did, but Scotus seemed to have come pretty close to accepting. Here is a passage from Quodlibet 5, where he argues that we can conceive of an actually infinite being. He says that Aristotle ( Physics book III 207a8) defines the infinite as that which for those taking any quantity (i.e. any quantity however large) there always remains something else to take*.

But why can’t we imagine all the parts that could be taken to be actually taken together, so that then we would have an actually infinite quantity, something as great in actuality, as it was potentially? Furthermore, if we can imagine the actual infinite in respect of mere quantity, why not something actually infinite in respect of being? This would be absolutely perfect. For, while an actually infinite quantity has parts which are imperfect (for example, the series of even numbers is imperfect because it lacks the series of odd numbers to make the whole number series), a being that was infinite in being would have perfect parts. According to Scotus anyway, but I didn’t follow the argument. Why would any part of anything be perfect, given that it lacked precisely the remaining bits of the whole in virtue of which it is a part. Isn’t that in the very meaning of the word ‘part’?

Nonetheless, the passage is historically interesting.

*The Latin formulation is “cuius quantitatem accipientibus”, which you can find in at least four places in Aquinas’ Summa: Part I q14 a12 and q86 a2, part IIa q30 a4 and part III q10 a3.


khadimir said...

Your question about Scotus presumes that comparing parts makes sense. What if we asked if the part was perfect in its being without regard to another? Would then we not compare it to its own essence or telos? Any imperfection must be fulfilled if its being were infinite perfect, for then it would have no limit to any perfect of which it partook.

Just a suggestion, as I am not well conversant in this field.

It does some that qualifying "infinite" as spatial, temporal, or of "being" leads to interestingly different places.

Edward Ockham said...

>>What if we asked if the part was perfect in its being without regard to another?

Well then I would question whether the word 'part' was correct.

David Brightly said...

Not really relevant but your post put me in mind of the Banach-Tarski paradox.