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.
Philosophical work
Mayberry's philosophy rejects the Platonic tradition, which holds mathematics to be a transcendental science concerned with discovering truths about immaterial, but intelligible, objective entities, as metaphysically conceited. This stance sets him apart from what probably is the "silent majority" view among practising mathematicians. Roger Penrose eloquently expresses a typical Platonic position.
"The natural numbers were there before there were human beings or indeed any other creatures here in earth, and they will remain after all life has perished. It has always been the case that each natural number is the sum of four squares and it did not have to wait for Lagrange to conjure this fact into existence."
On the other hand, Mayberry also vehemently rejects any understanding of mathematics tainted, as he would think of it, by operationalism. He writes:
"I take operationalism in mathematics to be the doctrine that the foundations of mathematics are to be discovered in the activities (actual or idealised) of mathematicians when they count, calculate, write down proofs, invent symbols, draw diagrams and so on. …… Considerations of human activities and capacities, actual or idealised, have no place in the foundations of mathematics, and we must make every effort to exclude them from the elements, principles and methods on which we intend to base our mathematics."
The most archetypal and most universally promulgated of such operationalist doctrines is that the natural numbers can be constructed beginning with 1, adding 1 to get 2, adding 1 again to get 3 and continuing indefinitely. This is expressed by the notation N = 1, 2, 3 ..., where the dots denote the indefinite replication of "adding 1". In accepting these dots of ellipsis, one accepts the intelligibility of indefinite iteration. Mayberry does not believe that a definition of this type is sufficiently clear and sufficiently disentangled from naïve and possibly misguided intuitions about the nature of time as to warrant its inclusion in mathematics without further justification. He writes: "When the natural number system is taken as the primary datum, something simply "given", it is natural to see the principles of proof by mathematical induction and definition by recursion along that system as "given" as well. ... The natural numbers are thus seen as what we arrive at in the process of counting out: 1, 2, ..., where the dots of ellipsis "..." are seen as somehow self-explanatory – after all, we know how to continue the count, no matter how far we have taken it. But these dots of ellipsis contain the whole mystery of the notion of natural number! ... Nor are the operations of counting out or calculating to be taken as primary data: they must be analysed in terms of more fundamental notions. We are thus led to reject the operationalism that all the anti-Cantorian schools share. For us moderns, numbers take their being from what we can do with them, namely count and calculate: but Greek "numbers" (arithmoi) were objects in their own right with simple intelligible natures. Our natural numbers are things we can (in principle) construct (by counting out to them): Greek numbers were simply "there", so to speak. ....... I am convinced that this operationalist conception of natural number is the central fallacy that underlies all our thinking about the foundations of mathematics. It is not confined to heretics but is shared by the orthodox Cantorian majority."
Views on Euclid His stance puts him at odds not just with pedagogic practice over the last few centuries but also with a tradition going back to antiquity. In Definition 4 of Book V of his Elements, Euclid defines two magnitudes of the same type, A and B, to "have a ratio to one another" as follows: "Magnitudes are said to have a ratio to one another which are capable when multiplied of exceeding one another"
In other words, if the repeated addition of one of them, say A, to itself results in a magnitude that exceeds the other, say B, i.e. for some natural number n, nA > B. Conversely A and B are do not have a ratio to one another if the indefinitely repeated addition of one of them to itself never produces a magnitude exceeding the other. In Book V Euclid develops a general theory of ratios and in Book VI demonstrates the power of the ratio concept both greatly to simplify derivations given in Books I – IV and to extend the scope of some of the theorems of Books I – IV. Particularly notable examples are Book III Prop. 35, where a much simpler proof using similar triangles is immediately available, and Book VI Prop. 31 where he extends Pythagoras’ theorem from squares to general similar figures.
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.
"By number I mean not so much a multitude of unities as the abstracted Ratio of any Quantity to another Quantity of the same kind which we take for Unity."
Dedekind Mayberry's convictions as to the true historical sequence of events in the development of key mathematical concepts are central to his philosophical orientation. He was led to these by his reading of Jacob Klein's "Greek Mathematical Thought and the Origin of Algebra". and Richard Dedekind's memoir "Was sind und was sollen die Zahlen".
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.
"I am coming ever more to the conviction that the necessity of our geometry cannot be proved, at least not by human comprehension for human comprehension. Perhaps in another life we will come to other views on the nature of space which are currently obtainable for us. Until then one must not put Geometry in the same rank as Arithmetic, which stands a priori, but rather in the same rank as, say, Mechanics."
Almost a century later Poincaré writes:
"In this domain of Arithmetic we may think ourselves very far from the infinitesimal analysis, but the idea of mathematical infinity is already playing a preponderating role, and without it there would be no science at all, because there would be nothing general. …… We cannot therefore escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of r contradiction ... This rule, inaccessible to analytical proof and to experiment, is the exact type of the à priori synthetic intuition."
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: "In speaking of arithmetic, (algebra, analysis) as a part of logic, I mean to imply that I consider the number concept entirely independent of the notions or intuitions of space and time."
Dedekind, whom Mayberry greatly admired, showed that the natural numbers could be established without any dependence on a Kantian intuition of time or reliance on indefinitely repeated operations. He did so however on the basis of an explicit acceptance of Cantor's Axiom of Infinity, which, as Mayberry points out, is best understood as simply a contradiction of Euclid's Common Notion 5 as applied to arithmoi. Dedekind's work did not however cause the view that the natural numbers and iterative processes have a special foundational status to lose credit among most mathematicians. The Intuitionist movement, while sharing with Mayberry a rejection of a Platonist understanding of the meaning of mathematics, resorted to an operationalist understanding of the subject, driving an acceptance of indefinitely protracted iterative processes to the very heart of their thinking. The Formalist movement, following Hilbert's program of saving the mathematical fruits of Cantor's Axiom of Infinity via finitary consistency proofs, likewise, in the very definitions of formal systems and the establishment of their properties, accorded a special status to indefinite iteration and associated definitions by recursion and proofs by induction.
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.
Aristotelian realism He is an Aristotelian realist in basic agreement with Aristotle's opinion that mathematics, and in particular the study of arithmoi, is a natural science taking its place alongside other special interest scientific subjects such as entomology or ornithology and dealing with objectively existing this-worldly things. Aristotle writes: