Saturday, 3 August 2013

It's not just Platonism but "Plenitudinous Platonism"

I'm willing to admit that I'm pretty verbose. But I hope that this verbosity is not like Conrad Black's or Rex Murphy's verbosity (who seem to say big, obscure adjectives just for the sake of it); that there is "a method to my madness", a theoretical consistency that allow one to anticipate what comes next in my discourse (even if at times that logical path is unclear).

Discourse - rather than just words - is my thing. As a linguist, I've come to realize that de Saussure (1857-1913), though a great thinker and considered one of the fathers of modern linguistics, was ultimately a closet Platonist: one word, one signifier...sort of thing. When I was studying linguistics at Memorial University of Newfoundland I took semantics, and I couldn't stand the guy (nor the idiot professor and his condescending "English" accent that gave off airs of reified sophistication - unintended perhaps but nonetheless irritating and annoying because it came out so contrived in that land that set the standard for kitchen table talk).

See, I had already discovered Rorty and Dewey and Wittgenstein by then and came to realize that language is less about words and meanings (though, to be sure, it is those) than it is about ideas, and that language must have some discernible coherency and completeness about it ("it" being the discourse) at some psychological level in order to be satisfying.

Now where did I get this "plenitudinous Platonism"? Mathematical philosophy - and more specifically, mathematical epistemology - is less a field than a ragtag collection of brush and bramble, if you ask me. And, that's where I came across this term. Here is the gist of it from the online Stanford Encyclopedia of Philosophy:

A version of platonism has been developed which is intended to provide a solution to Benacerraf's epistemological problem (Linsky & Zalta 1995; Balaguer 1998). This position is known as plenitudinous platonism. The central thesis of this theory is that every logically consistent mathematical theory necessarily refers to an abstract entity...By entertaining a consistent mathematical theory, a mathematician automatically acquires knowledge about the subject matter of the theory. So, on this view, there is no epistemological problem to solve anymore.

In Balaguer's version, plenitudinous platonism postulates a multiplicity of mathematical universes, each corresponding to a consistent mathematical theory. Thus, a question such as the continuum problem does not receive a unique answer: in some set-theoretical universes the continuum hypothesis holds, in others it fails to hold. However, not everyone agrees that this picture can be maintained. Martin has developed an argument to show that multiple universes can always to a large extent be “accumulated” into a single universe (Martin 2001).

In Linsky and Zalta's version of plenitudinous platonism, the mathematical entity that is postulated by a consistent mathematical theory has exactly the mathematical properties which are attributed to it by the theory. (http://plato.stanford.edu/entries/philosophy-mathematics/)

-The "multiplicity of mathematical universes" is a signal (to me) that the folks don't really know what they're talking - much like the superstring and many worlds theorists in physics. It's a catch-all phrase that really doesn't mean anything because it is, first of all, aesthetically ugly and intellectually lazy to try and jump gaps rather than admit that there is a gap in the thinking. It is disingenuous, at best.

I find this whole thing fascinating and my "meanness" is rather more an admonition than tsk-tsk.

I been thinking about this for years (actually) - since, that bad experience I had at MUN semantics course. There is something deeply unsatisfying about platonic forms: cold, austere, inert...they just stand there useless and pretty, pretty useless. What I've been thinking about is that there is a missing part in the discourse. Naturally as a linguist I think that missing part is "linguistic" insight.

What I mean is that mathematical philosophy, indeed the whole of mathematics, is really a study of human languages. Being as advances in mathematics has largely been a European achievement (a historical accident, no doubt) and that it's been largely an exercise in futility as de Saussure's program, perhaps taking a step or two back could clarify some of the confusion.

Mathematics, as the orthographical grammar now stands, is really an SVO language (ie, Subject Verb Object): S = V operating on O. The Subject, in this line of reasoning, is really the resulting integer; the Verb is the function/operation on the integer(s), O.

Now, as in linguistics, there are well-formedness (rigor) rules to the generation of these "phrases". The Russell's paradox is largely a question of what the barber really is (a verb) and where he rightfully belongs in this linguistic rendering of set theory (not a noun). And, Gödel's incompleteness theorem is really just a meaningless phrase like: "little, green ideas dream furiously" - it is a linguistic accident rather than an undecidable construct.

There are other utterly banal and boring questions posed by "structuralism" and "nominalism", such as "what numbers could not be" (again, from the Stanford Encyclopedia of Philosophy):

"There exist infinitely many ways of identifying the natural numbers with pure sets. Let us restrict, without essential loss of generality, our discussion to two such ways:
I:
{∅}
{{∅}}
{{{∅}}}
… 
II:
={∅}
{∅, { ∅}}
{∅, {∅}, {∅, {∅}}}
… 
The simple question that Benacerraf asks is: Which of these consists solely of true identity statements: I or II?"

-This, to my linguistic bent, is equivalent to obsessing on the phonological differences between the words: "dog" and "qimmiq" (which exactly amount the same thing, dog). Putting on my "formalist" hat for the moment, I'd say: decide which orthographical convention will prevail and get on with it.

There are similar but utterly fascinating set of problems as these. It is called "jurisprudence" and "constitutional governance" (of the Greco-Roman tradition, and reaching its pinnacle in the American founding fathers/French revolution where it briefly flowered then faded away slowly and sadly) where every textual rendition of an idea has real-world consequences. These, to me, are equivalent to the Riemann hypothesis (which links imaginary number system with the real system with the Cartesian (or Argand) coordinates with statistical analysis with the...). These ideas, to me, link axiomatic systems with politics, with spiritual/philosophical principles. Truly beautiful things to contemplate.

Jay

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