Concrete Number Theory

Most people do not realize that Euclid devoted three books (VII – IX) of his Elements to Number Theory and established the fundamental theorems and procedures which are still in use today. This misconception about the Elements is understandable because Euclid’s presentation of ‘numbers’ as continuous line segments makes these books barely distinguishable from the books dealing exclusively with geometry. It would seem that there was an earlier Pythagorean Number Theory which envisaged numbers as discrete entities and even employed actual objects such as pebbles or beans to illustrate basic arithmetic properties. This is why we still talk of ‘square’ numbers, also ‘cubes’, though we have lost the vital distinction between ‘line’ numbers and ‘rectangular’ numbers — line numbers are numbers that can only be arranged in a line, i.e. our primes. The discovery of ‘incommensurables’, geometrical quantities such as the diagonal of a unit square, which could not be ‘measured’ exactly by the rational numbers opened up a rift between geometrical and numerical quantity which has never been satisfactorily bridged to this very day. Partly because of the influence of Plato, geometry and the study of continuous quantity displaced the study of the discrete: from the 4th century BC onwards ‘higher’ mathematics was almost entirely geometrical and was what the future philosopher needed to study whereas mere calculation was “the affair of craftsmen and tradesmen”, as Plato put it.
Today, of course, algebra has itself taken the place of geometry and mathematics has become something entirely removed from the contagion of the physical world, “a game played according to fixed rules with meaningless pieces of paper” as Hilbert famously described it.  Mathematicians today shun the contagion of the real  and indeed remind one of the early Christians who found it shameful to be born with a body. This, of course, is why mathematics today only appeals to professionals while lay people generally regard it with fear and loathing. I myself do not think it is either healthy or fruitful to remove oneself entirely from the delights and constraints of the actual world which is why I have formed the ambitious plan of evolving a concrete number theory which founds arithmetic on the possibility of actual operations with standard objects. It is completely ridiculous to introduce numbers, which we are all familiar with almost from the cradle, via the abstract axioms of fields and rings, axioms which Newton and Euler and Gauss did not know (since they were only elaborated in the latter 19th century). Arithmetic and the greater part of mathematics still relies on the natural numbers, i.e. the positive whole numbers + zero, and these do not constitute a group or a ring or a field or any such fancy structure. My plan is not just to return to Euclid but to attempt to recover the even earlier, largely lost, number theory of the Pythagoreans on which Euclid drew, a pre-Platonic number theory rooted in the real.
Nonetheless, some attempt at rigour must be made. My aim is by and large to establish most of the important theorems of Euclid VII – IX without any appeal to geometry but nonetheless using the Euclidian framework of Axioms, Postulates, Definitions and Theorems (or Propositions as Heath calls them) with however a different emphasis. If a theorem or proposition is true, this means that such and such operations can actually be performed or that certain arrangements are actually observable in nature, or could be, given very reasonable expectations or technical extensions of our senses.
Where to start? The best definition of (cardinal) number is that given by Cantor of all people :
“We will call by the name ‘power’ or ‘cardinal number’ of a set M the general concept which, by means of our active faculty of thought, arises from the aggregate M when we make abstraction of the nature of its various elements m and the order in which they are given.”
      Since my approach is pragmatic, I do not intend to waste time on the sterile question of whether ‘number’ is a human concept (as Cantor implies) or is embedded in nature. What Cantor tells us is what we need to do to elaborate a workable number system, any number system worthy of the name no matter how primitive. As he points out, there are two principles involved the Disordering Principle and the Principle of Replacement.

Disordering Principle
The numerical status/cardinal number of a collection is not changed by rearrangement 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 is replaced by a different individual object.

Together these two principles make up a sort of Number Conservation Principle since whatever ‘cardinal number’ is, this ‘something’ persists throughout all the drastic changes a set of discrete objects may undergo (provided no object is destroyed) ― just as, allegedly, a given amount of mass/energy persists throughout the various complicated interactions between molecules within a closed system.
It is not clear whether these two principles should be viewed as Definitions i.e. they tell you what we mean by the term ‘cardinal number’, or as Postulates since, after all,  they are generalisations based on actual or imaginable experiments (the pairing off of random sets of objects with a chosen standard set). They are not ‘logical truths’ and not strictly speaking ‘common notions’ (Heath’s term for axioms).
We also need a third Principle, the Principle of Correspondence . It has a somewhat different status and is more like a true Axiom, i.e. something which we have to take for granted to get started but which is not directly culled from experience.

                                                                  Principle of Correspondence
Whatever is found to be numerically the case with respect to a particular set A, will also be numerically the case for any set B that can be put in one-one correspondence with it.
       By  ‘what is numerically the case’ I mean such  a property as divisibility which clearly has nothing to do with colour, size, weight and so forth. We certainly do assume the Principle of Correspondence all the time, since otherwise we would not gaily use the same rules of arithmetic when dealing with apples, baboons or stars: indeed, without it there would not be a proper science of arithmetic at all, merely ad hoc rules of thumb. But, though the Principle of Correspondence is justified by experience, I am not so sure that it originates there : it is such a sweeping assertion than it is more appropriate to call it an Axiom than anything else. It is certainly not testable in the way a scientific claim is, and not really a definition either.
I pass now to the basic Definitions which in Euclid precede the Postulates and Axioms — as a matter of fact Book VII does not contain any specific  postulates or axioms for number theory though, as Heath notes, some are needed.

              DEFINITIONS 

1. Object: Continuous solid.
2. Discrete object: object surrounded by unoccupied space.
3. Unit: Chosen standard discrete object that cannot be split.
4. Collection. Plurality of units in close proximity to one another but which do not merge or adhere to one another.
5. Number: Collection in the above sense, or the unit itself.
6. Numerical equivalence: If a collection A can be made to cover B completely unit for unit with nothing left over, it is said to be numerically equivalent to B, or we write A =n= B.
7. Smaller than: If a collection A fails to cover B counter for counter it is said to be less than B, and we write A < B.
8. Greater than: If a collection A covers B counter for counter, with at least one counter left over, it is said to be greater than B, and we write A > B.
9. Joining: If a collection A is pushed up against B so that they form a single collection C, we say that A is joined to B producing C and write (A + B) ð C , or C = (A + B).
10. Subtracting: If, from a non-unitary collection A, a smaller collection B (which may be unitary or multi-unitary) is taken aside leaving a collection C, we say that B has been subtracted from A, and write A – B ð C and C = (A – B).
11. Replication: If a flat (two-dimensional) collection A is covered exactly unit for unit by a collection B, then B, once removed, is a replica of A. And if A is a three-dimensional collection with base a and number of layers h, then if B is built up on a numerically equivalent base with equivalent height, B is a replica of A.
12. Reduction: If a collection B is separated out into so many numerically equal collections, possibly unitary ones, then any of these collections A is called a reduction of B. (Note. The same result would be obtained if we had a collection B laid out with base A and uniform height h, and then we ‘reduce’ it by compression to a single level. But since the units cannot, by definition, be squashed together, this operation cannot strictly speaking be performed.)
13. Rectangular Collection: Collection that can be arranged as so many numerically equivalent rows or columns.
14. Linear (or prime) collection: Collection that can only be arranged as a line, i.e. with the unit as side.
15. Common Side: If collections A and B can be arranged as rectangles with a pair of numerically equivalent sides, they are said to have a common side. Alternatively, if A and B can both be fitted between two parallel lines width a units where a > 1, they have a common side.
16. Relatively linear or prime to each other: If collections A and B have no possible side in common apart from the unit, they are said to be relatively prime and we write (A, B) = 1.
17. Common Multiple: If every possible side of A and B is a possible side of C, then C is said to be a Common Multiple of A and B.
18. Complete Common Multiple or Product: The complete common multiple of A and B is the rectangle side A × side B .
19. Least Common Multiple: Smallest collection which is a Common Multiple of A and B.
20. Result Equivalence: If, given some arrangement of units B, it is impossible to say which of two or more operations evolved it from an initial arrangement A, then the operations are result-equivalent. And if the final configuration is evolved from two different starting points using the same or different operations, then the two ‘starting-points + operations’ are result-equivalent.
21. Ratio Equivalence: If a pair of collections (A, B) can be transformed into (C, D) by replication and/or reduction alone, at all times keeping the levels the same, then we say that the ratio of A to B is the same as the ratio of C to D and write A:B = C:D      

In the next Post I will detail the basic Postulates required for Concrete Number Theory and establish the first theorems. The more fundamental theorems, which are those the most closely related to physical operations, will be singled out and named Basic Assertions. They are ‘proved’ by a combination of appeals to sense-impressions, the more direct the better, combined with some rational argument. But, whereas in ‘normal’ mathematics all arguments are strictly deductive, in Concrete Number Theory, inductive arguments are perfectly valid provided (1)a concrete example is given and (2) a possible generalization to cover all, or a multitude of cases, seems to have nothing going against it. Inductive ‘logic’ lacks the finality of deductive logic but it has the advantage that it can never degenerate into complete meaninglessness. One cannot have one’s cake and eat it: the inductive approach does mean that at least you taste a morsel even if you are not entirely sure that you will get hold of the entire cake. SH