The radical approach to infinitesimals in historical perspective

It is shown that the Jacobson radical of a field or division ring, when suitably expressed in a non-Aristotelian language, consists of the elements not unequal to zero, the same elements which Jacques Penon had chosen to call infinitesimals. His definition may be replaced by a theorem: in a normed division ring, the infinitely small elements are precisely the Penon infinitesimals. Already the ancient Greeks were intrigued by the opposition between the discrete and the continuous, as well as by that between rest and motion. To resolve the latter contradiction, the idea of infinitely small stretches of space and time seemed to be necessary and, in the seventeenth century, led Cavalieri, Newton and Leibniz to resort to infinitesimals, which were supposed to be infinitely small without being zero. Many attempts have been made to justify this concept. Of special interest is the proposal by Lawvere and Kock to interpret infinitesimals as nilpotent elements in a non-Aristotelian local ring, where they turned out to be not unequal to zero. (Already, Charles Sanders Peirce had observed that the admission of infinitesimals would require the abandonment of the principle of the excluded third.) More generally, Jacques Penon suggested that one admit all elements not unequal to zero as infinitesimals, even in a non-Aristotelian field. While the nilpotent elements of a commutative ring form its prime radical, the Penon infinitesimals turn out to constitute the Jacobson radical of an intuitionistic field, or even of a division ring. Moreover, one can prove that, in a normed division ring they are precisely the elements which are infinitely small. Let me remind the reader that the prime radical of a commutative ring is the intersection of all prime ideals (ideals which contain the product of two elements only if they contain at least one of them). It may also be characterized as the set of nilpotent elements. On the other hand, the Jacobson radical is the intersection of all (proper) maximal ideals. To prove the existence of enough maximal ideals one requires the axiom choice; but, fortunately, the Jacobian radical can also be described as the set of all elements a such that 1−ua is invertible for all elements u. For non-commutative rings the definition of the prime radical becomes a bit more complicated, but that of the Jacobian radical remains the same, provided we talk about maximal left ideals and left invertible elements. (As it turns out, either occurrence of “left” may be replaced by “right” and we may even replace 1−ua by 1−au.) We will return to this notion more formally in Section 10. 1. Everything is water (Thales). We say “much water” and “many beans”. Beans can be counted, but water must be measured. Much of pre-Socratic Greek philosophy was concerned with the question: do things in