Number Conservation Principle

There can be no doubt that mathematics was developed historically for very pragmatic and unromantic reasons. It was the Middle Eastern hierarchic societies, especially Babylon and Egypt, who not only started arithmetic as we understand it but took it to a surprisingly high level of development : the former, for example, gave a good estimate for the square root of 2 while the Egyptian scribes were adept at handling fractions. Such societies could not exist without efficient centralized planning, standardized weights and measures, effective methods of taxation and ‘fair’ remuneration of officials : indeed, they are not so very different from the EEC today !  This is not to say that the scribes and State officials did not have a ‘pure mathematical’ interest as well : like Civil Servants doing Sudoku in the morning break, their distant predecessors seemed to have enjoyed mathematical puzzles, as the Rhine papyrus shows.

The Greeks cast this mass of sporadic data, methods and formulae into a rigorous axiomatic mould with the results we know. However, in his Elements of Geometry, Euclid still has his eye on figures that can actually be drawn and many of the so-called ‘theorems’ (which Heath, Euclid’s best English translator, calls ‘propositions’) are better described as ‘Procedures’. For example, the very first ‘Proposition’ of Book I is “[how ] On a given straight line to construct an equilateral triangle”.

The success of Euclid’s Elements and the axiomatic method in general meant that many post-Renaissance early scientists attempted to cast their subject in the same mould and contemporary works on (classical) Mechanics and Thermo-dynamics are still written somewhat in this style. Newton in particular loosely imitated Euclid in his Principia but it is important not to see this as a retreat into some transcendental Platonic realm of pure mathematics. Notwithstanding the complicated mathematical formalism, Newton’s system of Mechanics was rooted in human sense experience, universal human experience, although it systematised, extended and idealised this experience in various ways. His audience, even those who were illiterate, could be expected to know what a ‘solid body’ was, what a ‘force’ was and even the somewhat metaphysical notion of ‘mass’ was not so very far fetched when defined in Newtonian terms as “the quantity of matter within a body”. Children know that if you push an object it usually moves in the direction of the push (‘applied force’), though they do not perhaps “know that they know this”. And although bowls on a bowling green and bar billiards are modern games, people have been playing around with balls from time immemorial and know from experience that if you strike something from the side rather than bang on from the back, it moves off at a slant. As far as we know, it was Leonardo da Vinci who first gave us the well-known diagram of the parallelogram of forces (in his Notebooks), and Newton who put the problem on a truly scientific footing by the notion of ‘resultant’ force — but this abstract treatment was entirely intelligible and by and large convincing to anyone who had mucked around with solid bodies.

The accepted procedure in such subjects, following Euclid, is to start off with certain ‘Axioms’, ‘Postulates’ and ‘Definitions’ and then proceed to derive conclusions, the ‘theorems’. For Euclid there was a difference between an ‘Axiom’ and a ‘Postulate’ : the former was an entirely general principle (Heath translates the Greek as ‘Common Notion’), while a ‘postulate’ had a more technical and constructive character. Thus, Euclid takes as one of his Axioms, Things which are equal to the same thing are equal to each other (Heath’s translation). In modern terms, Euclid is asserting the ‘transitivity’ of the ‘equality relation’ : something that is in a sense ‘obvious’ once it has been stated, but is well worth stating nonetheless. But the Postulates are introduced by “Let the following be postulated” and the first one is : 1. To draw a straight line from any point to any point. This doesn’t sound very good English but it is, I think, evident what Euclid has in mind. Practically, it may well be that I cannot ‘draw a straight line from point A to point B’ because the ‘points’ are too far apart, or one or both are inaccessible. But from a mathematical point of view, we need to assume that we can do this, and this assumption needs to be stated. In other words, any geometric conclusions we draw remain valid because the only reasons stopping us actually testing a particular claim are purely technical. This is all very sensible and, from a mathematical point of view, necessary. It does not mean that we can, or believe that anyone ever could, ‘draw a line from here to the Sun’, but that is not sufficient reason to stop us drawing certain conclusions which, hopefully, we can test in more mundane cases : it does not necessarily involve us in any supposed Platonic belief in a timeless world of Forms. Nonetheless, it was this gap between the observed and the imagined that, when it widened still further, started dissociating mathematics from the physical world, a process that has now gone the whole way in a manner that even Plato might not have approved.

Though this aspect is not so evident in Euclid, as the natural sciences developed in the West, it became necessary to make it very clear to what sort of entities the ‘principles’ and deductions therefrom applied. For example, Newtonian Mechanics only applied to ‘objects’, not (necessarily) to human beings in their entirety, and in modern times it became necessary to go even further and make it clear that Newtonian Mechanics only applied to relatively large massive bodies moving at modest speeds relative to each other (modest compared to the speed of light).

Most people would be surprised to learn that Euclid devoted four books of his Elements to Number Theory (Books VII – X). Though containing many important theorems, these Books are not quite so rigorous as the strictly geometric ones and they strike the modern reader as being quite perverse in their presentationof numbers as line segments instead of collections of discrete objects, blobs or squares say. Euclid was building on the earlier work of more ‘primitive’ Number theorists who actually worked with stones and pebbles, hence the interest in the visual appearance of numbers, in ‘square numbers’, i.e. collections of objects that can be made into a square, ‘triangular numbers’ and so forth. Also, there are, as Heath remarks, certain important ‘Common Notions’ (Axioms) that are not expressed such as the ‘transitivity’ of divisibility, as modern mathematicians would put it, i.e. if a number ‘goes into’ another exactly (‘measures it’), then it also ‘goes into’ any multiple of that number. Again, this is something we take for granted but which, for all that, is worth mentioning.

Euclid’s treatment of Numbers is, thus, already abstract and geometrical compared to what we surmise was the earlier approach. Today, ‘numbers’ are defined in a completely abstract way, so abstract that they are unrecognizable as such to the ordinary person. The only person I know of who in fairly recent times dared to treat mathematics, or at least arithmetic, in an empirical manner was John Stuart Mill, with the result that he has been pilloried ever since by Frege,  Russell and more or less everyone else who has written on the foundations of mathematics. The philosopher Mackie once asked disingenuously, “Why cannot we have an empirical mathematics?” but, as far as I know, made no attempts to create one.

I believe that arithmetic and the theory of numbers can, and should, be presented as a science. So, what does this science depend on? Today, a large amount of classical physics is made to depend on conservation laws, themselves extensions of Newton’s Laws, thus we have the Conservation of Momentum, the Conservation of Angular Momentum, and even in an era where very little can ne taken for granted in physics, the Principle of the Conservation of Mass/Energy is still just about standing up. Now any completely generalised ‘principle’, though it can be shown to be ‘wrong’ or at least inappropriate in certain circumstances, can obviously never be justified completely : the validity of such principles is, firstly that they fit a good deal of the data we already have, simplify and make more intelligible the world around us and permit prediction which can in special cases be tested. Sometimes, the ‘principles’ are simply necessities, sine qua nons,m without which we just could not get started at all. For example, in physics, we usually have to assume that, given equivalent conditions, we will get equivalent results in a particular experiment, even though this is by no means self-evident and, if Quantum Mechanics is believed, is not strictly true!

Is there a key principle on which everything about numbers relies?  Yes, I believe there is. It is what I call the Number Conservation Principle and it is made up of two sub-Principles, the Principle of Replacement and the Disordering Principle :

Principle of Replacement

The numerical status of a collection of objects is not changed if each individual object is replaced by a different individual object.

Disordering Principle

The numerical status of a collection of objects is not changed by rearrangement so long as no object is created or destroyed.

I think most people would agree, if they can accept the somewhat portentous language, that this is how things are, that the Principles are true. You think there are ‘seven’ objects on the table. I tell you to close your eyes and if, when you open them again, every previous object has been replaced by a different one, you will nonetheless (I hope) still say there are ‘seven’ objects on the table. Similarly, if a completely change the arrangement, scattering the objects around (while taking care that none falls off the table), there will still be the same ‘amount’ of objects. Also, I can do these two operations in an y order and as many times as I like, and still ‘something’, what I call the ‘numerical status’ of the collection has not changed.    

The formulation is open to the objection that both sub-Principles are stated in the negative: “is not changed”. However, it is perhaps impossible to avoid this since ‘number’, whatever it is, results from an “ignoring of differential qualities” as Piaget and Imfeld put it so well. ‘Number’ is what ‘is left’ when you have thrown away all distinctions of size, colour, race, weight, attractiveness, gender and so forth, and still have something left that is worth having or stating.

It should be emphasized that it is only when children in the Primary School have ‘understood’ the Number Conservation Principle that they are considered to have begun to be numerate : if they do not accept it, they will be classed as children with special needs. Neither the children, nor most likely the Primary Schoolteacher herself, have heard of the Peano Axioms, or the Axioms of Zermelo-Fraenkel Set Theory, but that does not stop them having made a beginning in ‘understanding number’. Zermelo and Fraenkel themselves had to go through this particular mill.
        In other cultures different bases were used depending on the different objects being counted. Flat objects like cloths were counted by the Aztecs in twenties, while round objects like oranges were counted in tens.  The use of classifiers obviously marks an intermediary stage between the era when numbers were completely tied to objects and the era when they became contextless as now. We still retain words like ‘twin’ and ‘duet’ to emphasize special cases of ‘twoness’, note also ‘sextet’, ‘octet’ &c.  The complete dissociation of verbal and written numerals from shape and substance is today universally seen as ‘a good thing’ especially by mathematicians. But classifiers were doubtless once extremely useful because they emphasized what people at the time felt to be important about certain everyday objects and activities, and they remain both a picturesque reminder of the origins of mathematics in the world of objects and our sense-perceptions. The removal of all such features from mathematics proper seems to be a necessary evil but at least let us recognize that it is in part an evil : the banning of contextual meaning from mathematics, the language of science and administration, is typical of the depoeticization of the modern world. In particular, there was strong cultural resistance to using the same set of words or sounds for divinities as for people, or for dead and alive people, for grains of corn and beetles. The Mayas found it necessary to have three sets of numerals where the first two, dealing principally with periods of time, were used exclusively by priests and only the third set was used by ‘ordinary people’. Likewise, a Vth century B.C. Athenian tribute list uses different symbols for the number 2 when the amount is respectively “2 talents”, “2 staters” or “2 obols” showing that even at this late date numbers were still at least partially tied to particular sets of objects (Menninger, Number Words and Number Symbols p. 268-9).

Note 1   Thus, for example, Japanese has the classifier hon for all cylindrical objects and Chinese t’iao for all elongated ones. The class word is placed between the numeral and its application in much the same way as we say (or used to say) “ten head of cattle”. Turkish has two classifiers ‘human’ and ‘non-human’.
  Tribal languages used many classifiers or other methods of distinguishing between objects being counted. The Ojibway, a tribe from Northern Ontario, “classify objects according to their hardness. flexibility and dimensionality….while there are also classifiers for counting the two most important artefacts made within the traditional economy, the house and the boat. (…) All of these numeral classifiers for concrete objects ensure that, when counting, expression is given to essential aspects of the object counted, especially those that affect the handling of the object” (Denny, Cultural Ecology of Mathematics in Closs, pp. 148-9).          The Nootkans used different terms for counting or speaking of          “a.) people, men, women, children, salmon, tobacco;          b.) anything round in shape such as the moon, clothing (except trousers), birds, vessels &c.          c.) an object containing many things such as a block of matches, a herd of cattle, a bale of blankets &c., and several other classes of things.”          (Folan, Calendrical and Numerical Systems of the Nootka, in Closs, Editor, p. 106)