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 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 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

4. (ISOPERIMETRY) If A n B and C n D where A nn C we must have A*D nn B*C
5. (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???)

Word clouds for product reviews

In case anyone wants to claim they thought of this, especially when the words are hyperlinks.

Why geometry of language?

The theory of best model classification is essentially geometric and it seemed so successful in that domain that there was a strong desire to apply it to other things. And an impression it could be applied to reading text. Luckily I work at a place that needed language automation as well as geometry automation and I got to try out an approach informed by those ideas in the narrow world of custom part design. 

But at home, I am trying to really understand what is involved. This requires pursuing the analogy in more details. What is the space of points, and data fitting that can apply a best model approach to text? That question drove the formulation of a proto semantics and its definition of narrative fragments, the "geometric objects" of this linguistics. I have yet to carry out the complete program including a goodness-of-fit metric (something to do with  number of slots filled in a fragment).

Even in college, after reading Bertrand Russell, I became convinced that the way ideas merge together was often more intrinsic to the type of idea than to the grammar that combined words for the ideas. So "or" and "and" took there meaning largely from whatever was being juxtaposed. I was fascinated by how I cannot think of a square and a circle as the same object, but can easily have a square and a redness as the same object. If I try to make a single object red and green, it splits the color in two. I still don't understand how color and shape channels are what they are.

But although I do not understand this built-in logic, that comes with our use of words and thoughts about things, there is a reasonable, practical way to use data fitting ideas with text, provided you narrow the topic enough. Then the word definitions are what you make of them, and reside in your dictionaries and C++ class designs. Language recognition built in this way is real because the class definition is concrete and shared.

Linguistics in the 20th century

Reading first some Chomsky and then, with diminishing respect, looking for something better at the dear old Concord Library and discovering Whorf, I find it interesting that so much energy was spent by the former defeating academic adversaries rather than addressing the topics at hand. 

It is too bad they lived in an age before it was possible to imagine teaching a computer to understand some limited part of a natural language. In narrow world language processing the question is how to capture meaning and be able to extract it automatically from examples of natural language. The idea is that if you have a sufficiently well defined "world" object, its member variables can set themselves. So you have to put your money where your mouth is and write software that fills in data objects from language samples. Then do something with the objects. You have to confront the notion of word meaning at a mathematical level to do it successfully. I am trying to do that with best models and the proto semantics.

But the controversies of the 20th century did not go away. I hope Whorf would approve my scheme of proto semantic shapes, filled with words having native context for each speaker of the language. I hope he would be sympathetic to my view that many of our primary abstract words are developed from games in childhood which are culture specific. I wonder if he would also approve of my idea that the cultural differences get built on top of simpler meaning entities that maybe are universal.

For example, do not all cultures include: "person", "place", "thing", "want", "ask", and action verbs, and things like transformation, sequence, and grouping? Note, it does not really matter if it is universal or not in terms of programming for a single language.

But there is something to what Chomsky is saying that I feel is very true. The words in my mind are encoded with musculature - what you could call the phonemes [or is it morphemes?]. As I dozed off last night, the word "only" split off an "-ly", which brought a sensation of deep meaning. It would not be surprising at all if the concepts that used "ly" where physically manifested in my brain as a thinking mechanism that includes those muscles. But that is the implementation of meaning not the form or content of it. So yeah, Chomsky, meaning is there in the physical implementation of language, and the use of such muscles is critically important. The same way my computer program uses silicon dioxide and larger objects like transistors to perform the operations I want in a computer program. But it is the computer program which is of most interest with the elusive "meaning" we want to understand.

Separately, the algebraic rules of grammar are necessary for extracting subtleties of meaning. They apply during parsing, and ordering of the input into narrative structures. But how important is it? Aren't single concepts usually described with adjacent words? It is a tough subject.
Update: My son David mentions that Sasurre (sp?) wrote that the form and sounds of words cannot be connected to their meaning. I agree completely. At the same time, a system that stores meanings in efficient way will use a strong degree of parallelism between the word forms/sounds and the word meanings.

How do words get their meanings?

[Obviously not a complete answer (from a doc I am writing):] No discussion of semantics can escape the central mystery of how words acquire their meanings and contexts. The narrative structures [of proto semantics] are simple by comparison and define roles for words independent of the native meanings. So let us acknowledge two kinds of word meaning: native meaning context and role within a narrative structure. For the sake of discussion, I assume words acquire their native meaning context through repeated use in a single role within multiple occurrences of a narrative structure in varying physical contexts. That single role is a word’s native role.

The “mystery” of word learning is why some parts of the context are perceived as constant and become associated with the word, while other parts are perceived as varying and become part of the expected narrative usages. For example to use the word “sun”, for me, creates a vague picture of an outdoors with a sky and a specific picture of a sun. The vague picture is waiting to be clarified.

The Turing Test Asks the Wrong Question

The Turing Test, as I understand, asks if a computer program could ever fool a person in a blind exchange of statements and replies. I believe the answer is clearly "yes" such a program could exist. Even if the computer had much thinner context-less definitions for words, real communication might still be possible. But that is the wrong question.

Instead let's ask if a computer program, dressed as an android, could be here with me so as to fool me in every way as to it being such. If this includes my being able to mate with it, then as far as I am concerned it is a person not a robot, and I am not fooled. [I guess that is the point they are making in "Bladerunner".]

My Proto Semantics is nearing a final form

Took a look at Wikipedia's definition of "semantics" today. It is a huge mish-mash of form and content, syntax, grammar, vocabulary, semiotics, linguistics, and language differences - all supposedly in pursuit of the subject of "meaning".
Very little of it seemed to have to do with what I consider narratives, or stories. Sure a Chinese speaker may have some different narratives than me, but we also must have many that are the same and are about the world around us and the insides of our minds. It is precisely those worlds that we share and the common narratives about them that ought to be the proper subject of semantics.
So I slice it and dice it differently from Wikipedia. Form is represented by a proto semantics, as per below. Content is represented by word meanings and the larger mystery: how do words get their meanings?

A PROTO SEMANTICS

Nouns

There are three kinds
 - person (me, or things I lend me-ness)
 - thing
 - location 

These are denoted by single letters or groups of things in parentheses. X, Y, etc.

Adjectives

Two kinds

-        feeling (attributed only to persons)

-        attribute  

These are denoted by A,B, etc. To express that a noun X has or feels an adjective A we write:

X__/A

Verbs

These involve a pair of nouns called actor and target:

actor\target	person	thing	location
person	love understand	want assign_value see	go indicate find
thing	cause_to	act_on compare_to	in at
location	affects	contains on	connect_to

To express that a noun X acts on a noun Y we write:

X-->Y

To express the idea that the same verb occurs in more than one part of a narrative, superscript the arrow like this ‘-->^a‘.

Note that noun and adjective types are automatically converted by usage. To say “the dog loves his owner” or “Niagara Falls loves to see tourists at all seasons” lends personhood to these non-person nouns. Similarly we will be able to put attribute words in the locations of nouns (e.g. “red is shirt”). Although almost nonsense, such constructs do carry slight meaning.

Narrative Fragments, Connectors, and Grouping

Narrative fragments are:

·       noun

·       noun_/adjective

·       noun-->noun

·       two narrative fragments joined by a comma ‘,’. This is a connector that means ‘consecutive’.

·       two narrative fragments joined by a ‘::’. This is a connector that means ‘becomes’.

·       Any narrative fragment in parentheses. This means ‘treated as a noun’ or ‘treated as an adjective’ depending on the usage.

·       Any narrative fragment in square brackets. This means ‘implicit’ noun or adjective depending on usage.

Rule of precedence

For simple expressions: ’__/’   ‘-->’  '::'  ‘,’ .   
Otherwise use parentheses to avoid ambiguity. 

Update: Most of the arguments in semantics seems to have to do with whether words can be used in narrative roles that do not match the words' natural definitions. Duh! I realize that there is such a cacophony of nonsense out there about this subject, good new ideas will never be heard, unless they become the basis of commericially successful applications. This seems like a reasonable test, if evaluated in the long term.