Friday, May 9, 2014

Deriving the additive properties of the integers from their multiplicative properties

Usually formal arithmetic begins with additive properties, such as 0 and the +1 operator. Along the way, multiplication n*m gets defined as the successive addition of n, m times. This leads to the notion of primes and the topic of the distribution of primes. 
Instead I want to look at the integers in terms of their multiplicative properties and see if additive properties (in particular the standard ordering of the positive integers) can be derived. So begin with a set of prime numbers, given in order 1,2,3,...... with 1 as multiplicative identity and other numbers formed by stringing together primes with multiplications. The question is: how might these these composites be ordered? So here is a very incomplete thought.

We will try to derive an ordering using the symbol “n” to mean immediate neighbor, so that if B is the immediate neighbor to the right of A we can write:

A n B

If there is a sequence of zero of more numbers C1,C2,C3, … such that AnC1nC2nC3……..nB then we write:

A nn B
Note that  A n B implies A nn B and that A nn B nn C implies A nn C, by definition.

We assume the arithmetic  properties of multiplication: presence of identity “1”, associativity, commutativity.  We also assume the standard ordered sequence or primes 2, 3, 5, 7, 11, etc.

Simple  Axioms
  1. The relation ‘n’ and the relation  ‘nn’ are no not reflexive (A n A is false, A nn A is false)
  2. The relation ‘n’ is not transitive (but ‘nn’ is transitive, as noted above)
  3. If A n B then C*A  nn C*B (and not C*A n C*B unless C==1)
  4. [added] Every A has a B such that A n B

Key Axioms
  1. (ISOPERIMETRY) If A n B and C n D where A nn C we must have A*D nn B*C
  2. (PARSIMONY) Composites occur as early in the sequence as possible without violating (4)
Assume
1 nn 2 nn 3 nn 5 etc.

Axiom (4) is like the isoperimetric concept: of all rectangles with the same perimeter, the square is the one with the largest area. Axiom (5) says that composites are packed as closely together as possible; alternatively, that primes are introduced as infrequently as possible.

Partial Theorem
 1 n 2 n 3 n 4 n 5 n 6 n 7 n 8 n 9 n 10 n 11 follows from the axioms and the order of the primes.

Theorem (unproved)
The conventional additive order of the (composite) integers can be derived from these axioms and the order of the primes.

Proof of the partial theorem is something like this: 1 has a neighbor but it cannot be a composite using just 1, so it must be 2, hence 1 n 2.
Now axiom (5) says we should try to put 2*2 as soon as possible after 2 but that would give us  1 n 2 n 2*2.
 
If we apply axiom (4) this says that 1*(2*2) nn 2*2 which is false by axiom (1). Hence we need another prime next to 2, call it “3” so we have 1 n 2 n 3. Now we must have 3 n 2*2 because of axiom (5), so we have 1 n 2 n 3 n 2*2. We cannot have 2*2 n 2*3 without violating (4) and all other composites are even larger than 2*3, so we must have another prime “5” after 2*2. Now we have 1 n 2 n 3 n 2*2 n5. (The argument continues???)

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