Re: [ontolog-forum] CNL's and ConLangs

[Kingsley Idehen <kidehen@xxxxxxxxxxxxxx>]

On 10/29/14 12:39 PM, Bruce Schuman wrote:

“It seems to me that there should be a taxonomy that includes all languages and what I call "language shorthands" such as chemical, math, Feynman diagrams, etc”

 

“I don’t see the grist for an ontology in math”

 

 

Just to postulate a slightly contrarian view – I think I DO see “grist for an ontology in math”.

 

The very simple and perhaps obvious statement from Peter Wegner’s Programming Languages, Information Structures, and Machine Organization might point in this direction, where he says (p.4) that “Any specific instance of an information structure must have a physical existence in some information-storage medium. The information storage medium contains primitive information-storage devices for storing primitive information units.”

 

So, we ordinarily think of “the real number line” as the foundation of mathematics – the bedrock of the “ontology” for math.  What is the constructivist mapping from the real number line to the symbolism of math, as actually instantiated somewhere?

 

Wegner, p.4:

 

“In order to build up information structures it is necessary to introduce one or more primitive components, out of which more complex information structures are constructed.  Information structures may be constructed in terms of a single primitive unit of information called the binary digit or bit.  A bit is characterized by the fact that it can take on one of two states, which will be here represented by  0 and 1.”

 

“Information structures are constructed from bits by grouping ordered sequences into fields and by grouping fields into successively larger information units.  Specific fields of an information structure are usually interpreted as identifiable components of the information structure as a whole.  A specification of an information structure in terms of lower-level components is referred to as a structure definition.”

 

This is a basic and perhaps obvious description of the build-up of any computer language, through a succession of layers or levels, as “bits” are combined into “bytes” and bytes become an alphabet and – in the case of natural language -- an alphabet becomes words, and words become sentences or paragraphs or books.  This form is “absolutely linearly taxonomic”.

 

It seems appropriate to me to insist that mathematical symbolism be “constructed” in the same way, as we define what “A” means, and what “A + B” means, and what “=” means – all given direct mechanical/electronic interpretation at the machine level, defined in bits.  Anything else, I am suggesting, is floating in an imaginary cognitive/visual space in an individual human mind, in a way that cannot be objectively tested or independently confirmed.

 

Seen this way, all these so-called “primitive” and “indefinable” mathematical elements could indeed and in fact be given hard-core mechanistic and compositional definitions, in a form that ought to make engineers happy.  Stephen Wolfram’s “Mathematica” might be pointing in this direction.

 

At an abstract level, we can argue that “all concepts are constructed assemblies of distinctions” – in an exact analogy with the device/state bit/byte hierarchical assembly of any language in a machine.  Just as a “cut” in the real number line is the foundational distinction of all abstract concepts defined in dimensions, bits (0 or 1) are the fundamental distinction at the bottom of the machine representation.

 

All logic, sometimes defined today in terms of “unknowable or indefinable or irreducible primitives”, ought to be analyzed and decomposed to a deeper level – recognizing that in most or all cases, these objects are not truly “primitive”, but are in fact constructed constituent assemblies of sub-components.

 

Mathematics – and the theory of language – should be built up from this kind of bedrock – rather than presuming that somehow a term like ”A” has an irreducible meaning in some Boolean or algebraic _expression_.

 

And – as regards the question asked in a different context as to “whether it is useful to do this” – my guess is, it is essential to do this.  And doing it – I am guessing, opens an explosive and integral analytic power that is simply not available when so-called “primitive elements” are defined in highly composite (and hence lumpy and incommensurate and highly confusing) terms.

 

What we actually want, in our analysis of natural language, is a comprehensive model of semantic structure that is 100% linearly recursive from top to bottom – such that every level in a descending (taxonomic) stipulative cascade of abstractions is defined in the exact same terms, with zero loss of data in the decomposition cascade until it terminates at some acceptable lowest level (number of decimal places).

 

Define all the elements in this descending cascade with the simple precision outlined by Wegner, and there might be some hope of revolution and “the great simplification”.  My instinct is: enforce this simplicity on system definitions.

 

This, it seems to me, begins to point to the “pure form” – the immaculate ideal of “the neats” and the bane of “the scruffies”.  Get all the internal structures of all these interdependent cascading objects defined in the same immaculate mathematical/mechanical terms, in 100% linear form, and the thing will run at the speed of light.

 

http://en.wikipedia.org/wiki/Neats_vs._scruffies

 

SeeAlso: http://linkeddata.uriburner.com/c/8GFQRD -- An Ontology that describes Math  .

I am sure there are other ontologies out there pursuing similar goals :) 

-- 
Regards,

Kingsley Idehen	      
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