Monday, October 10, 2011

Pattern and attribute

As Peter Geach puts it:
            There is something which Solomon has to David as David has to Jesse.

We call this thing “paternity”, and linguistically we reify it -- substantivize it, specifically.  Solomon and David thus share paternity, virility, and Judaism. Apparently, three things.  They also shared a flagon of wine, a loaf of bread and a zither -- three more things.  Yet thingy in different ways.  For all we originally meant by this thing ‘paternity’, is that both sired. (Note the noun-shorn verb.)
            All this fits so comfortably into our language (there is even a synonymous saxonism, fatherhood), that the plain man is satisfied.  But hold on: both sired a son.  So they share something more than mere paternity (once again reifying), call it virpaternity.  Those who are happy of two sons, share duovirpaternity. Or consider the sire of Romulus and Remus, and that of Tweedledum and his twin: these share homoduovirpaternity.  And so forth.  John sired six sons, and pledged them in a tankard of ale; James sired nine, and pledged them in a tankard of wine.  They share the attribute of having sired a number of males which is a multiple of three, and of having pledged them in a tankard of some blushful stuff.  You could even coin a word for it, along the abundantly lexicogenic lines of the compounds of organic chemistry.

By now we (plain men  all) begin to suspect: This attribute sits on no throne in Platonic heaven;  somehow we have gone too far -- or possibly even set off in the wrong direction (pessima in principiis corruptio -- Geach). John and James really do have this in common, but there exists no “this” which they have.  The relation is real, not fancied; there are facts of the matter, and the facts fit patterns.  Patterns are logical, often mathematical.  You can even perceive them (with the mind’s eye), but you can’t sit on them, or eat them.  Has our penchant for reification  led us astray ?
            And yet  we seem naturally to quantify over such patterns:  “Paternity is a blessing”, that is, each case of paternity is a blessing; yet really this has the sense:  For all men x, if there exists a person y  such that x sired y, then x is blessed.  -- The same is true for more artificial and complex predicates: “What a blessing it is, to have sired twin sons, and to have pledged them in a tankard!”  -- Or, with a sort of second-level quantification, “Among the virtues, courage and paternity stand out.”

Thanks to the work of many recent philosophers and logicians, attributes have been de-fanged, though not quite banished.  But have we here perhaps shifted our ontological longings onto patterns?

The principle quarry in this series of essays has been mathematical objects;  and here is a neat way of seeing them:

Michael Resnick, “Proof as a Source of Truth”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 326:

     Mathematical objects are positions in patterns.

A good slogan, and it seems to relieve us of any excessively literal ascription of thinginess.
As it happens, he takes his own metaphor too literally, going on and on about how “square” numbers can be seen to be a bunch of dots actually arranged in a visual square.  Actually this is to represent the thing, not as a position in a pattern, but as an actual pattern, such as one might print on wallpaper; but let that pass.  Even with some tidying, this approach is idle and over-literal;  it doesn’t scale up. Stone–Čech compactifications are also “positions in patterns”, if you will, but usually impossible to visualize.  Or, more tellingly:  even in cases where you can visualize a space T that is, in fact, the Stone–Čech compactification of a space S, merely inspecting the space in its birthday suit, as it were, does not tell you that it is a Stone–Čech compactification.  To see that, you have to take in T’s position in a pattern that contains S, along with the function or process of compactifying in the Stone–Čech manner.  In other words, mathematical objects are relational terms, like brother, rather than more immediately empirical terms like man.  (You cannot tell that a given man is a brother by any sort of inspection of the mere man himself.)  Some mathematical objects are named in a way that draws attention to this feature, e.g. covering space, quotient space, and compactification (as against merely a compact space).


This business of the status of attributes may seem airily abstract, but you can grab it with a practical handle:  namely,  their role in substantive moral debate.   Their philosophical ontological debility smooths the way for the eliminativists who, for dark reasons of their own, wish to deny the existence of (the “existence” of) things like consciousness and free will.   Theirs is a work of deflation;  and it is easier to deflate something if it is a bit puffed-up to begin with.  Rhetorically, it sounds more plausible to say that consciousness “does not exist” or “is an illusion” or “an epiphenomenon”, than to say plainly:  You are not conscious.   Or to say, “free will is a myth, a figment of your misanalysis”, than to say: You never do anything intentionally.

Perhaps the plain man is not yet alarmed;  but it does get worse.  From just the other day:

The End of Evil?  Neuroscientists suggest there is no such thing. Are they right? 

It is not difficult to detect the ultimate upshot of such claims, undermining our society like a lichen crumbling rocks.   The thing gets ratified in the courtroom, when some scoundrel gets off with some ludicrous Twinkie Defense;  and spreads to the populace in general, fostering a climate of irresponsibility and excuses.

For a polemic counterblast against the eliminativists, click here.

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