Number Conservation Property

To recapitulate, the Number Conservation Principle combines two subsidiary Principles, the Disordering Principle and the Principle of Replacement:

Disordering Principle
The numerical status/cardinal number of a collection is not changed by rearrangement of the collection so long as no object is created or destroyed.

Principle of Replacement
The numerical status/cardinal number of a collection is not changed if each individual object of the collection is replaced by a different individual object.

Taken together they give us Cantor’s characterisation of cardinal number as based on a ‘Double Negation’ ─
“We will call by the name ‘power’ or ‘cardinal number’ of M the general concept which, by means of our active faculty of thought, arises from the aggregate M when we make abstraction of its various elements m and the order in which they are given.”

For the moment I am not so much concerned about what cardinal number is ─ Cantor says it is a ‘general concept’ ─ as to what one needs to do (or to have) in order to create a number system, either individually or as a society. Cantor gives us a definition rather than a cognitive procedure or strategy, but his ‘definition’ is an extremely enlightening one. He is absolutely right to emphasize the ‘negative’ aspects of the path towards number; as Piaget puts it memorably, “Number results from an ignoring of differential characteristics”. It is almost as if we are invited to throw away as much information as possible and see what we are left with; in particular we must discard all distinction by type such as colour, weight, distance, kinship, gender &c. &c. The reason why ‘primitive’ peoples were so often reluctant to acquire numerical skills from  missionaries or explorers was because they, rightly, sensed it would probably weaken their sensitivity to type distinctions, deemed to be more important.
To speak, as I do, of a number conservation principle makes arithmetic sound more like a natural science than a logical system or an exercise of the creative imagination, but this is intentional. Friedrich Gauss is reported as saying, “Mathematics is the queen of the sciences and number theory is the queen of mathematics”. But how can this be since all the sciences typically begin with observation and are subject to incessant and rigorous reality testing, whereas mathematics is supposed to be a self-contained logico-deductive system? But I believe Gauss was right: arithmetic is not just an indispensable aid to physics, it actually is physics, the most basic type of physics. Specific numerical properties of whole numbers, such as primality, divisibility and so forth, do not exist because of abstract rules laid down by 20^th century mathematicians: they are given once and for all by Nature. A dishevelled heap of stones of approximately the same size either can be arranged as a rectangle or it cannot, and that is that. If I have a set of bags that will hold 7 pebbles of standard size, no more and no less, a mound of such pebbles can either be bagged up using these bags without any remainder, or it cannot be. The man-made rules for manipulating numbers  are so designed as to lead to a system that mimics Nature and so can inform us about features of the physical world including those that are neither obvious nor readily observable. Number theory, as Gauss claimed, qualifies as the ‘queen’ of the sciences because the vast majority of its findings are not ‘approximately true’, ‘statistically true’, like practically all the propositions of physics but are completely true, 100 % true, and even, arguably, ‘true in all possible worlds’¹.
An empirical/imitative theory of arithmetic downplays the role of man-made rules (since they are subsidiary) but, in return, finds itself obliged to appeal to notions such as ‘inherent capacity’ or ‘potential’. A given collection of objects, by its very existence, ‘has the capability’ of being arranged or divided up in a particular manner, whether we know this to be the case or not. A heap of stones, for example, can either be evenly arranged in two matching rows or columns, or it cannot be.  And this ‘property’ is  surely ‘out there’ in the real world, not in here in my mind. Such a middle course avoids the Scylla of mathematical Platonism and the Charybdis of Formalism. For this so-called ‘inherent capacity’ or ‘potential’ is not something that transcends the world of sense or is ‘eternally true’ but nonetheless goes beyond what is immediately observable².

One can in point of fact envisage the Number Conservation Principle as a special case of the physical Space/Time Homogeneity Principle :
“The homogeneity of space implies that the same experiment carried out at different places on the earth (for example) gives the same result. The homogeneity of time means that all instants of time are physically equivalent e.g. the laws of buoyancy discovered by Archimedes many centuries ago can be reproduced today by recreating appropriate conditions for the observation. If these seemingly obvious properties had not existed, it would have been meaningless to carry out scientific observations. The absence of homogeneity of space would mean that the laws of physics would be different at different places and the lack of homogeneity of time would, in turn, have implied that the laws discovered today would not be valid tomorrow.”             Saxena, Principles of Modern Physics   p. 2.2

The Space/Time Homogeneity Principle (STHP) is undoubtedly required for the practice of science as we understand it today³, and the lack of it was precisely what held back ‘primitive’ societies from developing the natural sciences. From a magical perspective time and place are not irrelevant, not neutral, hence the importance given to performing rituals ‘at the right time of day’ (or year) and in the ‘proper place’ (temple, sacred site, altar &c.).
We obviously need STHP in arithmetic as well. A numerical experiment which arranges a collection N as a rectangle side c implies that N can always be arranged in this way, i.e. if some feature of a collection has been shown to be numerically true once, it is always going to be true. This is a very strong result. And the truth involved  is not an empty logical truth like “All bachelors are unmarried males” but is a generalization from experience that it is more often than not impossible to doubt⁴.

The Number Conservation Principle, moreover, does not seem to have any possible exceptions ─ provided only that we are dealing with collections of discrete objects that do not fuse or merge when brought into close proximity. So it is, if anything, more fundamental than the Space/Time Homogeneity Principle. The strictly numerical properties of a heap of pebbles are not changed by putting them in a suitcase and carting them around in a jet plane, even loading them onto an accelerating rocket bound for the Moon. Short of turning to dust, such a group of pebbles retains the same divisibility properties even if it was originally been buried in a tomb in Egypt thousands of years ago. This shows how fundamental strictly numerical physical properties are⁵.
Moreover, the apparently innocuous Number Conservation Principle has certain important consequences both general and specific. The general or ‘metaphysical’ consequence is that, as stated, it allows us to advantageously sidestep both Platonism, which makes too much of mathematics, and Formalism which makes too little. Simply by bringing into existence more and more discrete objects, Nature is manufacturing ‘concrete numbers’ that really do have certain ascertainable properties, such as being prime or rectangular or triangular, even though Nature hasn’t the faintest idea of what it is doing. These ‘potentialities’ are, so I claim, out there in the world, not in my head or in a Platonic Wonderland. The rules we invent pertaining to numbers and their manipulation are thus not free creations of the human mind: they are constrained creations if they are to model accurately what actually is, or can be, the case. And the main reason why consistency is rightly prized in a mathematical model is because Nature actually does seem to be remarkably consistent, at least at the macroscopic level.
It is true that, ultimately, we are probably going to have to take on board certain ‘properties’ without which the whole system (system of Nature, not mathematics) would collapse. Such properties by rights ought to be the most elementary physical facts ascertainable, or if not, properties that clearly underlie the physical facts and point to them, as it were. They should  inform us about how discrete objects combine together, form certain admissible configurations and not others for down to earth physical reasons. Basic mathematics should (and does) show us how and why certain fragments can, or cannot, be made to fit neatly together. Euclid begins the first of his three books on Number Theory with the so-called Euclidian Algorithm, an infallible  procedure for distinguishing between numbers that are relatively prime (have no common factor except the unit) and those which do have a common factor greater than the unit. And this procedure at one and the same time tells us whether a collection can, or cannot be made into a proper rectangle, i.e. one with a side greater than the unit⁶.
It would seem that Euclid (or rather his unknown predecessors) were on to something. Even though the distinction between line numbers (= our primes) and rectangular numbers (= our composites) does not at first sight seem particularly significant, primality (or not) has enormous consequences in practice, so much so that it would hardly  be going too far to say the distinction is as it were, ‘hard-wired’ into the universe⁷.
The Number Conservation Principle can be used as a step towards establishing Unique Prime Factorisation. But, first, it is necessary to say more about the alleged ‘numerical properties’ of heaps of pebbles, or any collections of discrete objects. There are two types of such properties, external or exogamous and internal or endogenous properties, to use scientific jargon. The chief exogamous property is the capability a collection has of being joined up to another collection of discrete objects, giving us the operation we call addition. Then there is the capability of a given collection of objects to be copied or replicated as many times as we desire,  giving us the operation of multiplication.

The chief and in a sense only endogenous property of a collection of discrete objects is its capability of being divided up into numerically equivalent sub-collections with or without remainder ⁸. Now, the Number Conservation Principle (NCP) states that the divisibility properties of a collection, such as having a possible rectangular side greater than the unit (or not having one), are not destroyed, or miraculously brought into existence, by laying out the items of a collection in a particular way, then disarranging them and then laying them out in a different way.

In other words. if N = ♦ ♦ ♦ ♦ ♦ ♦ can be arranged as a rectangle side ♦  the collection N is not going to lose this ability.
♦ ♦ ♦
♦ ♦ ♦
And, alternatively, the collection M = ♦♦♦♦♦♦♦ which, try as I will, I find cannot be laid out as a true rectangle, is not going to miraculously gain this ability after I disarrange it, try again two hours later and, to my surprise, succeed.
Now, suppose we have a rectangle N = p × q where p, q are line numbers (primes) and we reduce this rectangle to a structureless heap, and then try to arrange it again as a rectangle. We have not gained any new numerical properties such as a new possible side different from those we already know about, namely 1, p, q and (p × q). Therefore, if we are told that some non-unitary number labelled t is a possible side of N, we conclude immediately that t must be either p or q (or perhaps N itself). Why are we so sure of this? Because any other outcome would be a violation of the Number Conservation Principle. It is more or less in this manner that Euclid establishes Unique Prime Factorisation with a little help from earlier theorems.

One would, however,  like to go a little further than the bare NCP in order to justify including certain cases and excluding others that look at first sight as if they violate the factorisation rule. To this end, I suggest the following Corollary to the Number Conservation Principle :

Corollary If a rectangular arrangement cannot be transformed into a different rectangular arrangement directly, then no other rectangular arrangement (apart from the strip) is possible.

        This distinguishes between collections such as A and B

♦♦♦♦                        ♦♦♦♦♦♦♦♦♦
   A  ♦♦♦♦                  B    ♦♦♦♦♦♦♦♦♦
        ♦♦♦♦                       ♦♦♦♦♦♦♦♦♦
                                          ♦♦♦♦♦♦♦♦♦
since B can be immediately rearranged as C
♦♦♦♦♦♦♦♦♦ ♦♦♦♦♦♦♦♦♦
♦♦♦♦♦♦♦♦♦ ♦♦♦♦♦♦♦♦♦
whereas A cannot be re-arranged as a different (proper) rectangle.

Why the difference? Because B has sides that can be broken up into numerically equivalent non-unitary sub-collections while A can only be broken up into units. But why should this matter? Because the ‘sub-division’ of a side, provided it is repeated so many times exactly, can become the side of a new rectangle. There is, then,  a sort of two-way equivalence in the ‘capability of being divided up into equal sub-collections’ and the ‘ability to become a side of a rectangle’. This equivalence is natural enough, but should perhaps be introduced as a Postulate. Once accepted, it can be used again and again. SH   

Notes
¹ The claim that “The sum of consecutive odd numbers starting at 1 can be arranged as a square” (e.g. 1 + 3 + 5 = 9 = 3²) is not just ‘statistically true’ but seemingly admits of no possible exceptions. Yet this claim is telling me something about the external world that is far from obvious. Very few physical claims today are made with this sort of confidence. Even such an ‘obvious’ truth as “Heat always flows from a warmer to a cooler body and not vice-versa” is today said to admit of very unlikely but nonetheless possible exceptions. In modern parlance, the flow of heat from a warm body to a cooler, though highly improbable, is not ‘forbidden by the laws of physics’.
² It is much to be regretted that Aristotle, who was an empiricist and seems to have known at least as much mathematics as Plato, never had anything like the influence on the evolution of Western mathematics that Plato had.
³ One could argue that the Space/Time Homogeneity Principle is not strictly true: carrying out an experiment such as Foucault’s Pendulum (where the results vary with latitude) near the equator would not give the same results as carrying it out the near the north pole. And it is today thought that certain constants, such as Hubble’s Constant or even the Constant of Universal Gravitation, may well have varied over time.
However, one understands what is intended: other things being equal, one position in ‘space’ is as good as another, one moment as propitious as another. The Space/Time Homogeneity Principle follows directly from the modern (post-Copernican) belief (or dogma) that there no ‘special spots’ from which to view the unfolding universe.
⁴ Descartes’ criterion of ‘inconceivability’, though it has completely gone out of fashion, strikes me as valid despite its subjectivity. The Cartesians rejected Newton’s theory of gravitation because they considered attraction at a distance to be ‘inconceivable’ without a mechanism that Newton was unable to provide. This feature remained a fundamental weakness of Newton’s world-view and it was precisely concern about this point that ultimately led to Einstein’s relativistic schema that replaced it.
⁵ Only if we have before us a continuous and continuously varying flux in which no discrete elements are ever permitted to emerge even for an instant, would the Number Conservation Principle not apply. And even in this case, it would not be violated: the conditions necessary for it its meaningfulness would not exist (since there would be no discrete objects).
⁶ The Euclidian Algorithm looks to me as if it is extremely ancient, pre-dating the golden age of Greek mathematics and maybe even the Pythagoreans. For note that:
(1) the procedure which consists in seeing whether such and such a quantity ‘goes into’ another so many times without or with a remainder, makes much more sense envisaged in terms of bundles of beans or pebbles than in terms of the lengths of line segments (as Euclid gives it); and
(2) it is a numerical procedure which is not restricted to any particular ‘base’ thus quite possibly pre-dating the invention of numerical bases, or for that matter written arithmetic.
⁷ One is sometimes tempted to consider Unique Prime Factorisation not as a theorem (which requires proof) but as a basic ‘given’, i.e. as an axiom.

⁸ The essential operation of arithmetic is, so I maintain, not addition but division. Arithmetic was, so I would guess, originally developed for the purpose of sharing out a band’s resources, especially food, and doing so equally. Hunter-gatherers seem to have been extremely egalitarian which is one reason, along with their limited amount of possessions, why they didn’t need much arithmetic. Egyptian temple officials, however, were, in the earliest times, paid in beer and bread because money (gold, jewels &c.) was used only for state transactions with the exterior. It is surely no accident that the Egyptians invented fractions and not only that, their scribes were, to judge by the problems covered in extant papyruses, literally obsessed with fractions. Furthermore, Egyptian society was hierarchical, so there was an extended need for, and concern with, unequal division which obviously requires a much more complex arithmetic. The curious use of a string of ‘unit fractions’ (fractions with a unit numerator) to express quantities such as 4/7 or 6/19 is perhaps a relic from  to an even earlier era when everyone was remunerated in the same way.