SUO: Re: Hypostatic And Prescisive Abstraction
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HAPA. Discussion Note 6
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BM = Bernard Morand
Bernard,
The genealogy of this circle of thoughts goes a bit like this:
| Bentham's "Theory of Fictions" begat (paraphrastically)
| Schönfinkel's "Bausteine" and this begat (independently)
| Church's "Lambda Calculus" and this begat (in good time)
| McCarthy's "Lisp" and all the rest is AI and IEEE ...
It is no accident, at least not from the right "state of information" (SOI),
how lambda abstraction got its tale, as it is truly most pertinently tagged.
It is said that the lambda came from Russell('s) and Whitehead's employment
of a caret (^) to mark a cousin operation of relational conversion, but let
me try to look that up later. At any rate, the main idea has been stock in
trade of mathematics for as long as anybody can remember, and in philosophy
more generally (or vaguely, I can never remember which) the laurel is often
placed on Bentham for his idea of paraphrasis. Here's a general/vague link:
http://www.utm.edu/research/iep/b/bentham.htm
What we see here is the very same thing going on
in the colloquial homilies that Peirce attempted
to use to adumbrate the spirit of abstraction in
the formal sciences.
BM: I wonder whether Peirce is refering here to second
intention or namely to hypostatic abstraction (HA).
BM: If we take as a starting case:
(1) "Opium puts to sleep",
in order to transform it by HA, we get:
(2) "Opium has a dormitive virtue".
Here is the diagram that I drew for the analogous case
of "virtus dulcitiva", in lay terminology, "sweetness".
Referring to a few of Peirce's standard discussions
of "hypostatic abstraction" (HA), the main thing
about HA is that it turns an adjective or some
part of a predicate into an extra subject,
upping the arity of the main predicate
in the process.
For example, a typical case of HA occurs in the transformation
from "honey is sweet" to "honey possesses sweetness", which we
could choose to represent in several different ways as follows:
Sweet(honey) ~~~> Possesses(honey, sweetness)
S(h) ~~~> P(h, s)
S P
o o
| ~~~> |
o o
h <h,s>
^
[S] ~~~> /P\
| o->-o
| | |
o o o
h h s
Figs. Are Sweet. If served in season. At just the right temps.
The chief thing about this form of grammatical transformation is that we
abstract the adjective "sweet" from the main predicate, thus arriving at
a new, increased-arity predicate "possesses", and as a by-product of the
reaction, as it were, precipitating out the substantive "sweetness" as a
new subject of the new predicate.
BM: I see it as the transformation of a fact into a more abstract concept, or
say something like "opium has the general property of putting to sleep".
Sticking, sweetly, if you will, to the notion that a concept is a mental symbol,
some might say that a sufficently "precise" abstract concept is already present
in the predicate "is_sweet", but HA takes a step beyond that, as some would say,
onto the flypaper of "abstract but substantial objects" like 'virtus dulcitiva'.
is just a menat or otherwise, to the Some would say
BM: It is hypostatic in the sense that it requires no further
proposition than (1) and that the transformation relies on
an "ens rationis".
Yes, this is the critical observation.
BM: But from (2) we can also get for example:
(3) "this discourse has a dormitive virtue",
which requires a second subject (a fact about discourse).
This is known as "application of the abstraction to another argument",
and it is analogous to the other half of the lambda calculus paradigm.
BM: I would be tempted to call this latter transformation
second intention, and it seems to fit with your quote
before. But going from (2) to (3) doesn't seem to be
an hypostatic abstraction stricly speaking.
As for the matter of intentional orders, I foggily peirceive
but the clue that it has something to do with the operations
that I throw together under the name of "reflection", and by
this plurality of reflection to say I abstract some fraction
of my action's contentious tensor that here-to-fore had been
too obsistently refractory to all of my previous reflections.
BM: Thanks for throwing some light on this if possible.
And thank you for a very peirceptive set of questions.
Jon Awbrey
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