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[ontolog-forum] Taxonomies, cuts, and the decimal system

To: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Bruce Schuman" <bruceschuman@xxxxxxx>
Date: Sat, 10 Aug 2013 18:04:07 -0700
Message-id: <001501ce962e$ae9a68a0$0bcf39e0$@net>

Pardon me for experimentation, if you can – but I’ve got something kinda by the tail here, and I thought I would float this idea past this group and see if anybody can point out something obviously wrong – or possibly suggest something constructive.


My instinct for semantics tends to be absolutely linear and decompositional.  I want to factor everything in ways that can be perfectly mapped to a matrix of rows and columns.  This is basic to the notion of “ad hoc top-down stipulation” as I understand it.




Generally, the “taxa” (plural of “taxon”) in a taxonomy (genus, species, family, whatever) can be understood as “rows” – like a row in a database table.  A taxon can be understood as a single row with a series of “cells” with boundaries between them, and the item being categorized is “inside the cell”.  Thus, if the taxon is “mammals”, the cells might contain things like “cows”, “humans”, “dogs”, “horses”, “seals”, “pigs”, “wolves”, “goats”, “zebras”, “giraffes”, etc.


Think of that row as a Y axis defined in a cellular format – as if drawn on graph paper with a length of as many items as are included (in this case, 10 cells), and a height of one cell (the height of one unit, where the unit is some bounded condition that defines “mammals”).  Essentially, the taxon is a “class”, and if we can order the class by some criteria in the definition of the objects it contains, it’s an “ordered class”.  Since these objects at a minimum are “words”, we do have a legitimate minimum sorting criteria: alphabetical order.  Other possible orderings might include median numeric values in various dimensions, such as average weight or height or cost or life expectancy, etc.  There are many other possibilities.


The “differentia” that divides the taxon (genus) into species can be some value that distinguishes the elements within the cells.  In a rows and columns model, the species becomes a column (an X axis), relative to the genus/row (Y axis).  All the horses in the system stack up inside that column, ordered in some way (size, price, age, attitude, sub-species, etc.), and amenable to all the same principles that govern the genus/row (“mammals”), of which they are a parsing or “cut” – and the “width” of the column is one unit, where the (multi-dimensional/abstract) unit is “horse”.


More can be said – and the issue of what happens at the boundaries of these cells is an interesting question  -- because these boundaries are related to “lower and upper boundary values” that might define a range of values within which the object must be defined, or it’s not that object (if a “cow” is 3 inches tall, is it a cow?).  Or – if an “unborn fetus” is 4 months old – is it a “human being”?  Sparks fly around these issues of boundary values.  In the Trayvon Martin/George Zimmerman case, what are the boundary values distinguishing first-degree murder from second degree or manslaughter?  Boundary values are critically important in the real world, and these things are stipulated all the time.


But for right now – I just want to ask one question.




This is an idea that fascinates me and feels powerful – because it seems “perfectly recursive” and amazingly parsimonious.  To me, this idea feels like Occam’s Razor personified – so tell me why I’m wrong…


Measurement of anything in the real world is defined to a finite number of decimal places in some unit (whether inches or feet or “mammals”).  http://en.wikipedia.org/wiki/Measurement -- http://en.wikipedia.org/wiki/Observational_error


So here’s the idea: in a rational number, in a finite number of decimal places, it looks to me like every subsequent decimal place is a “species” of the previous decimal place.


So, in the number 6.7837 – the trailing digit “7” is a species – one of 9 possibilities – of the genus (previous decimal place) “6.783”.  Or put another way, 6.7837 is a “species” of 6.783.  In the same way, up one level of recursion, 6.783 is a species of 6.78.    And 6.78 is a species of 6.7.  And 6.7 is a species of 6.


In my simple-minded way, that looks to me like pure “recursive descent”.


Plus, it’s a perfect representation of a generalized model of taxonomy.  And who knows, maybe the entire concept of “number” itself can be usefully defined this way. 


So – if this idea is right, and all this makes sense – it’s actually amazingly simple (maybe totally obvious?) – we are starting to look at a linear parsing principle that is


1)      Perfectly linear

2)      Perfectly recursive and “self-similar” at any level

3)      Totally natural and intuitive, and the way we actually do things, and something anybody can understand

4)      Fits exactly into measurement theory as it actually works in the real world

5)      Takes exactly the same form as our basic ideas on taxonomy at any level of abstraction – which suggests that what we might have here (??) is a perfect form of linear recursion (or “recursive descent”) that extends from any high-level abstraction (“mammal”) to any particular instance (my pet cow Margie).


So the question is – isn’t it true that decimal places are a perfect example of the genus/species/differentia relationship?


And if it is true – aren’t we starting to look at a very sensible way to parse any top-down stipulative cascade, such as


“Mammals / cows / Guernsey cows / young Guernsey cows / young female Guernsey cows / my pet Guernsey cow Margie”


I call that model a “cut on a cut on a cut on a cut” – where each subsequent level is a “species” of the previous level.  And cuts like this are “stipulated” from the top-down by somebody in some context at some moment for some reason….




I am asking this – because I want to assemble this model in algebraic row/column diagrams building up from the definition of continuity as the real number line.  So what we are talking about is a kind of “digital-to-analog conversion” at the bottom of our analytic cascade – where rational numbers meet the real number line ( http://en.wikipedia.org/wiki/Real_number_line )  If we are measuring something to within 6.7837 units of something, the “next decimal place” is a mystery – that’s where total uncertainty (and the concept of continuity) enters the picture.  What we seem sure of is – the value is bounded somewhere between 6.7837 and 6.7838, or somewhere close to there – and that “boundary value range” (that “acceptable error tolerance”) just has to be good enough, and we’ll shake hands on it.  There almost certainly IS a “next decimal place” – but we just don’t know what it is for sure, so we settle for a bounded approximation.


Ok, thanks for your patience, apologies for the extreme tedium of this simple-minded thinking.  What I need to do is – within the context of stipulative definition, build up this cascade from the mysterious unknowable/inconceivable perfection of the real number line to boundary value specifications at any level of abstraction (“when is a mammal not a mammal?”).   Just as the last digit in a series creates a discontinuous boundary-value “width” in the real number line, every higher level cut or distinction in the chain also carries “width” in the axis it cuts. In the ad hoc context-specific environment of local immediacy, we should be able to build a perfect cascade of intended meaning from any abstraction to any specific level of measurement.  Maybe our entire legal system should be defined this way.


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