[Inquiry] Re: Doctrine Of Individuals -- Discussion

[Jon Awbrey]

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DOI.  Discussion Note 4

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CG = Clark Goble
JA = Jon Awbrey

Re: DOI-COM 1.  http://stderr.org/pipermail/inquiry/2005-January/002327.html
In: DOI-COM.    http://stderr.org/pipermail/inquiry/2005-January/thread.html#2327

Clark,

Just trying to get organized again,
after some time away from the list,
pre-occupied with other business.

CG, quoting JA:

JA: It's been a long time since I looked into the Monadology,
    and I'm not sure that I got what it was all about, but it
    seems like it had something to do with rationalizing the
    doctrine of pre-established harmomy.

CG: The pre-established harmony is what makes it all work together.
    Basically Leibniz' version of the Greek notion of Logos.  However
    it's kind of a "fudge" in a way as well since it basically just says
    God makes everything behave in a proper fashion "as if" mechanically.

CG: However that's not what' interesting in Leibniz.  (Indeed I find
    his appeal there somewhat embarrassing, although understandable given
    the times.)  Rather the monads allow things to be predicated of individuals.

CG: Without the monads there is no identity which allows his entire
    logical structure to work.  It is very elegant and seemed quite
    pertinent to your discussion.  He misses a lot, of course, not
    the least of which is "feeling" in the sense Peirce discusses.
    (Indeed Peirce critiques Leibniz for this.)  However in terms
    of logic and predication I think it is important and gets to
    why we think in terms of generals -- one of the themes of
    your post.

CG, quoting JA:

JA: In a similar connection, we need to remember that a logical atom is a term
    not capable of logical division, not a thing that's materially indivisible.

CG: That's true, however the question is then *why* it is not capable of
    logical division.  I think your caution is apt in that we ought not
    confuse ontology and logic.  But at the same time that question of
    why is quite important given Peirce's notion of continuity.

Not sure there's a question why.  The logical atom or
absolutely individual term is just defined that way.

CG, quoting JA:

JA: Typically, we need to set up a definite universe of discourse, in which
    the elementary objects of discussion and the primitive predicates used
    to describe them are bounded in some fairlydefinite ways.

CG: If the discussion are existing entities and ascribing predicates to them
    then the elementary objects must be elementary existing things which have
    some indexical relation.

That does not suffice to bound the discussion --
we need something more definite to say what
sorts of things qualify as discussables
and what sorts of descriptions qualify
as predicates.  Otherwise it's just
a lot of colliding impressions and
uncollective unconscious axioms.

CG: The reason I think your earlier caution is apt, however, is that I
    think Leibniz moves towards a kind of nominalism I think Peirce would
    object to.  For instance an organism functioning as a whole entity
    clearly can't be logically divided and still be the organism, even
    though it can be divided in terms of other entities making it up.
    Yet this is the point Leibniz would object to.  As I understand
    him he'd argue that there must be something fundamental not open
    to any logical division of any sort: the monad.  It is this that
    we predicate of and that gives life to the whole aggregate entity
    in a fashion analogous to a Cartesian mind and matter.

CG: I think your reference to "On a Limited Universe of Marks" was quite helpful.
    In a sense Peirce allows many universes of discourse whereas Leibniz (and
    perhaps all nominalists) do not.  That's a powerful point to keep in mind.
    Further it relates directly to my comments from a couple of weeks ago on
    the influence or at least parallels between Peirce and Davidson.  Davidson's
    anomalous monism basically is the recognize that one can't translate between
    modes of discourse the way a nominalism would wish to.

CG: BTW -- you gave the wrong link to "On a Limited Universe of Marks".  It was:

    http://suo.ieee.org/ontology/msg03204.html

CG: Very helpful passage I'd not read before.

CG, quoting JA:

JA: I guess I've never understood what that was all about.
    Can you explain what "actual infinity" means to you?

CG: Infinity taken not as potential.  Consider the way we might count the set of integers.
    At any time will have counted only a finite number but can always continue to count.
    Thus infinity is an ideal limit.  This is infinity as considered by Aristotle and
    pretty well adopted by most philosophers since.  The opposing view was actual
    infinities or the idea that one can have an infinite set as something *actual*
    as opposed to purely potential.  A lot of Platonists adopted that view,
    especially the neoPlatonists.  Leibniz is somewhat unique among modern
    philosophers for adopting it.

CG: In modern mathematics one can see the distinction still raging between
    mathematical platonists and then Constructivists.  The latter often have
    various rules about what kinds of infinities or infinite steps are allowed
    in a proof because they are always potential.  Thus a mathematical platonist
    would allow one to do an infinite process as part of a proof and then an other
    infinite process.  A constructivist will typically allow only one infinite process
    in a proof.  (With some adding more qualifications)

CG: There an excellent paper by Richard Arthur up at McMaster on
    the difference between Leibniz and Cantor on actual infinities.

    http://www.humanities.mcmaster.ca/~rarthur/papers/LeibCant.pdf

CG: It appears Leibniz embraced some medieval notions
    of actual infinity which may have influenced Peirce
    as well.

CG: This paper also by Arthur is quite relevant as well
    and seems quite relevant to the topic at hand.

    http://www.humanities.mcmaster.ca/~rarthur/papers/Zenonists.pdf

CG: As you note, however, we ought keep discussions of substance and
    discussions of logic separate -- however wrapped up together they
    were for Leibniz.

CG: A few quotes from Leibniz:

    | I am so much in favor of the actual infinite, that rather than admit nature
    | abhors it, as one says vulgarly, I hold that nature exemplifies it everywhere,
    | in order to display better the perfections of her author.  Thus I believe that
    | there is no part of matter which is not, I do not say divisible, but actually
    | divided; and consequently the least particle ought to be considered as a world
    | full of an infinity of different creatures.  (Phil. I, p. 416)

CG: Oddly though he didn't admit infinite numbers:

    | I do not at all admit any true infinite number,
    | though I concede that the multitude of things
    | surpasses every finite number, or rather every
    | number.  (Phil., vi, p. 629)

CG: This is also related to magnitudes and thus he rejects infinite magnitudes.
    However he accepts infinite entities as his monadology requires.  (As does
    his treatment of the calculus -- an approach that bothered mathematicians
    for quite some time)

CG: BTW -- you might find the following discussion
    on potential vs. actual infinities interesting.

    http://www.cs.nyu.edu/pipermail/fom/2003-January/006137.html
    http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Infinity.html

I had a year-long flirtation with intuitionism as an undergraduate,
but I'm afraid it merely diverted me from doing any actual math,
though I do like constructive proofs when I can get them,
sometimes an excluded-middle proof is just plain quicker
and avoids many irrevelant constructions that everybody
does differently according to taste.  There is a bit
of this way of thinking though in computer science,
where we restrict ourselves to what's computable,
and then there's a contrast between the recursive
and the recursively enumerable that's reminiscent
of the contrast between actual and generative
infinities.  On the other hand, the issue
does not come up in every consideration
of infinity, but only in certain places,
and the common part of mathematics is
far larger than the frayed fringes.

Jon Awbrey

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