Actual Infinity - Scholastic Philosophers

Scholastic Philosophers

The overwhelming majority of scholastic philosophers adhered to the motto Infinitum actu non datur. This means there is only a (developing, improper, "syncategorematic") potential infinity but not a (fixed, proper, "categorematic") actual infinity. There were exceptions, however, for example in England.

It is well known that in the Middle Ages all scholastic philosophers advocate Aristotle's "infinitum actu non datur" as an irrefutable principle. (G. Cantor )

The number of points in a segment one ell long is its true measure. (R. Grosseteste )

Actual infinity exists in number, time and quantity. (J. Baconthorpe )

During the Renaissance and by early modern times the voices in favor of actual infinity were rather rare.

The continuum actually consists of infinitely many indivisibles (G. Galilei )

I am so in favour of actual infinity. (G.W. Leibniz )

The majority agreed with the well-known quote of Gauss:

I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction. (C.F. Gauss )

The drastic change was initialized by Bolzano and Cantor in the 19th century.

Bernard Bolzano who introduced the notion of set (in German: Menge) and Georg Cantor who introduced set theory opposed the general attitude. Cantor distinguished three realms of infinity: (1) the infinity of God (which he called the "absolutum"), (2) the infinity of reality (which he called "nature") and (3) the transfinite numbers and sets of mathematics.

A multitude which is larger than any finite multitude, i.e., a multitude with the property that every finite set is only a part of it, I will call an infinite multitude. (B. Bolzano )

There are twice as many focuses as centres of ellipses. (B. Bolzano )

Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. (G. Cantor )

One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor )

The numbers are a free creation of human mind. (R. Dedekind )