John Penn Mayberry

Of the significant figures in the 19th century, only Dedekind appears to have stood against the Kantian consensus. In Was sind und was sollen die Zahlen, he coolly writes:

Mayberry's position is that all of this, right from Book V of Euclid, constitutes an aberration from the true spirit of mathematics as exemplified in Euclid Books I-IV. The central purpose of his book is to explain his position and to show that it is not corrosive of the essential content or the modern practice of mathematics, but, in his recommendation of a clearer Aristotelian understanding of what mathematics is about and the standard of rigour appropriate to his more exigent understanding of meaning, he is following in a tradition initiated by Cantor of restoring meaning to mathematics after three centuries of formalism. However, in Mayberry's eyes a modern platonically inspired doctrine that holds that say, proper classes, objectively exist is as much a departure from good sense and probable truthfulness as, say, the early 19th century formalistically inspired doctrine, Peacock's “Principle of the equivalence of Permanent Forms”.

Mayberry's positive philosophical views flow from his determined adherence to a small number of philosophical doctrines inspired partly by Aristotle and partly by reflection on the almost two and a half millennia of mathematical experience, particularly that of the 19th century.

Mayberry's fifth core doctrine is that, broadly in analogy with Euclid's postulates for Geometry, postulates for Arithmetic can be laid down, making a good a defect in the Elements which, contrary to the expectations created by the structure Common Notions and Postulates for Geometry, do not contain any such postulates. Mayberry carries out this program in Chapter 4 of his book. His postulates follow, to some degree Euclid in form, but the axiomatic ideas about sets issuing from the 19th and early 20th centuries, in content. Broadly analogous to Euclid's postulates on the construction of a circle given a point and a line or the construction of a unique straight line given two points are the postulates to do with Union, Power Set and Cartesian product which posit global constructions producing new arithmoi from one or more given ones. Somewhat different however are his postulates on Replacement and Comprehension. These do not set out individual constructions which simply have to be grasped but rather make affirmations about all possible constructions and all conceivable properties. In a sense one can understand them as affirming the existence of general bridges from thoughts to things. Both however can, like the postulates concerning specific constructions, be understood as "finiteness principles" affirming the existence of new arithmoi. Mayberry's “corrected” Euclid would thus underpin the sister disciplines of Geometry and Arithmetic with Common Notions, applicable to both, supplemented by two sets of Postulates, one for each discipline. Indeed, in so far as Geometry does rely on the notion of arithmos – it does so even in defining triangles, quadrilaterals, pentagons etc., but more exigently in some Propositions, e.g.. Book VI Prop. 31, which make affirmations about general polygons—the “corrected” Euclid would place the study of Arithmoi before that of Geometry.

In Book VII Euclid introduces, as another type of magnitude alongside his geometric ones of line, angle and figure, the concept of “arithmos”. This is to be understood as “a multitude of units” where a unit is “that by which we call something a one”. With some reservations about the status of singletons and the empty set, the Greek notion of “arithmos” is thus essentially the modern notion of “set”. Mayberry notes that it struck him with the force of a revelation that the significance of Euclid's Common Notion 5,—“the whole is greater than the part”—when applied to arithmoi is that an arithmos cannot be congruent, where this word is understood following Heath as “can be placed upon with an exact fit”, to any proper part of itself, or, in other words, that a set is finite in the modern sense of there being no 1-1 correspondence between the set and a proper subset of itself. The fact that Greek arithmetic, and in particular Euclid Books VII-IX, is really the study of finite sets has been obscured by the ubiquitous translation of “arithmos” as “number” and the transformation in the notion of number from its original “arithmos” meaning to "ratio" that occurred in the 17th century. The transformation in meaning was given clear expression by Newton in his Lectures.

From the middle of the 17th to the 19th century the natural numbers and the notion of unlimited iteration on which they rely acquired foundational status in mathematics, both pragmatically and philosophically. On the philosophical side, Kant classified arithmetic propositions as synthetic a priori knowledge and, in parallel with a similar analysis of geometric theorems which he traced to our intuition of space, traced their compelling nature to our intuition of time. Kant's general position with regard to Arithmetic received the endorsement of the greatest practising mathematicians of the 19th century. Even Gauss, though in dissent from Kant's position on the status of geometry, endorsed his position on Arithmetic.

John Penn Mayberry (18 November 1939 – 19 August 2016) was an American mathematical philosopher and creator of a distinctive Aristotelian philosophy of mathematics to which he gave expression in his book The Foundations of Mathematics in the Theory of Sets. Following completion of a Ph.D. at Illinois under the supervision of Gaisi Takeuti, he took up, in 1966, a position in the mathematics department of the University of Bristol. He remained there until his retirement in 2004 as a Reader in Mathematics.

The second of Mayberry's core philosophical doctrines is that things and arithmoi of things objectively exist and are part of the fabric of external reality. The ontological credentials of an arithmos are exactly those of its constituent units. It is not however the task of the mathematician to investigate or speculate whether things falling into a species – such as clouds in the sky, shades of red, human emotional states, men of the 22nd century – are sufficiently clearly individuated to constitute units of possible arithmoi or whether the boundaries of pluralities of things—e.g. should we count centaurs and mermaids as falling into the species “human kind”? is it exactly determined when shades of red end and shades of purple begin ?—are sufficiently clearly delineated as to constitute an arithmos. The work of the arithmetician can begin with the simple assumption that there are objective clearly individuated things which he can take as units and definite pluralities of such things that he can take as arithmoi. Mayberry writes: