Sunday, October 27, 2013

Reification: Substantive and Trivial

We have been upholding a Realist (Platonist) account of mathematics;  and while this position is best understood in terms of patterns and implications rather than “the existence of mathematical objects”, let us relax our vigilance for an instant, and consider the Ontology of Things.

Thus, consider that feline prototype of a statement,

~ The cat is on the mat. ~

Life is simplified if we admit that cats and mats exist, as sets of objects.  Likewise mammals;  and since the set of cats (real ones, we mean, not Putnam’s perverse pet) is a subset of the set of mammals, we can conclude from the above that “A mammal is on the mat,”  and from that deduce the non-emptiness of the set of mat-borne mammals.   All quite convenient.

But what of “on”:  Is that too  an object, or an entity?  Or, onness, maybe, or onning (as in: This cat ons this mat).
If you like, you can talk that way (and there are academics who have been paid to do so). The set “Onness” consists of all the ordered pairs of tops and bottoms;  exempli gratiâ:  <that cat; this mat>.  Nothing hinges upon this.

In mathematics, by contrast, where relationships, and the implicational interrelations among such relationships, are key (while objects not so), such nominalization comes naturally.
Thus, a topological space is second-countable (adjective) if its topology has a countable basis.   And:  A topological space is regular (adjective) if … (yadda yadda, look it up).
From this, we quite smoothly segue into a nounier style of speech, which often brings greater concision (rather than less, as with "Onness").  Thus this statement of the Urysohn Metrization Theorem:

In 2-countable spaces, regularity is equivalent to metrizability.
-- James Dugundji, Topology (1965), p. 195.

Or, going the whole hog into the nomina substantiva style of formulation:

For a topological space:  Second-countability plus regularity equates to metrizability.

Or consider:

Compactness and separability are each more fruitful properties than connectedness or paracompactness in terms of further implications.

Such a proposition would be cumbersome to word in a de-nominalized style.


Quine, who likes his whisky neat and his universes spare, takes an opposite tack of “entities explained away” (with the aid of variables and quantifiers), abolishing those unaesthetic singular terms in favor of quasi-verbal predicates.  Here Pegasus disappears in a puff of “pegasizing”.  (Caveat:  Do not yourself attempt to pegasize at home;  only Pegasus himself can do that.)   Ontologically, perhaps, we are back where we began, though psychologically it is quite otherwise:   You can fall in love with Juliet;  you cannot fall in love with that x such that x julietizes  and, for all y, if y julietizes, then y = x.

Pegasus explained away


Since we shy away from the dry well of “the existence of mathematical objects” (being interested in the Platonist account  more in the context of epistemology and theology, than in pure ontology), it isn’t really a theme of these essays;  but we touch upon the matter tangentially here:

For a deconstruction of the dubious conjunction “language and math”, cf. this note:

For terrific detective stories (hey, gotta support my family in retirement), cf. this:

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