List:
In his 2012 book, *Peirce's Logic of Continuity: A Conceptual and
Mathematical Approach*, Fernando Zalamea specifies three "global
properties" of a Peircean continuum, along with three corresponding
implications. The text itself does not provide succinct definitions, but I
have gleaned the following from the longer discussions that it does include.
- Genericity - "free of particularizing attachments, determinative,
existential or actual … a law or regularity beyond the merely individual"
(p. 11); which implies
- Supermultitudeness – " its size must be fully generic, and cannot
be bounded by any other actually determined size" (p. 14).
- Reflexivity - "any of its parts possesses in turn another part similar
to the whole … the whole can be reflected in *any* of its parts" (p.
16); which implies
- Inextensibility - "cannot be composed by points … not possessing
other parts than themselves" (p. 16).
- Modality - "while points can 'exist' as discontinuous marks … the
'true' and steady components … are generic and *indefinite*
neighborhoods" (p. 20); which implies
- Plasticity - allowing "the 'transit' of modalities, the 'fusion' of
individualities, the 'overlapping' of neighborhoods" (p. 22)
In his 2015 presentation, "Modeling Mathematically Peirce's Continuum: Theme
and Variations," Francisco Vargas retains supermultitudeness, reflexivity,
and inextensibility, but replaces the other three properties with
potentiality and infinitesimals. Both of these scholars are
mathematicians, and thus understandably employ the nomenclature of their
discipline; but because my own interests are more *philosophical*, I find
them a bit inscrutable and even misleading to a degree. In particular,
"supermultitudeness" and "inextensibility" seem like vestiges of the
bottom-up/analytical/collection-theoretic approach that Peirce ultimately
abandoned in favor of the top-down/synthetical/topical conception.
After reviewing again the relevant passages in Peirce's writings (quoted
below), I suggest instead that the following five properties--arranged and
numbered to match up loosely with the four identified in R 144, as
discussed previously in this thread--are jointly necessary and sufficient
for a true Peircean continuum.
- 1 - Regularity - every portion conforms to one general law or Idea,
which is the final cause by which the whole calls out its parts.
- 2 - Divisibility - every portion is an indefinite *material* part,
unless and until it is deliberately marked off with a limit to become a
distinct *actual* part.
- 3a - Homogeneity - every portion has the same dimensionality as the
whole, while every limit between portions is a topical singularity of lower
dimensionality.
- 3b - Contiguity - every portion has a limit in common with each
adjacent portion, and thus the same mode of immediate connection with
others as every other has.
- 4 - Inexhaustibility - limits of any multitude, or even exceeding all
multitude, may always be marked off to create additional actual parts
within any previously uninterrupted portion.
Respectively, these definitions roughly correspond to Zalamea's genericity,
modality (or Varga's potentiality and infinitesimals), reflexivity and
inextensibility, plasticity, and supermutitudeness. However, in
conjunction with the distinction between portions as (material/actual)
parts and limits as (immediate) connections, I believe that my proposed
terminology is more consistent with the common-sense notion of continuity
that Peirce persistently sought to capture.
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Professional Engineer, Amateur Philosopher, Lutheran Layman
www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
CSP: It is already quite plain that any continuum we can think of is
perfectly regular in its way as far as its continuity extends. No doubt, a
line may be say an arc of a circle up to a certain point and beyond that
point it may be straight. Then it is in one sense continuous and without a
break, while in another sense, it does not all follow one law. But in so
far as it is continuous, it everywhere follows a law; that is, the same
thing is true of every portion of it; while in the sense in which it is
irregular its continuity is broken. In short, the idea of continuity is the
idea of a homogeneity, or sameness, which is a regularity. On the other
hand, just as a continuous line is one which afford room for any multitude
of points, no matter how great, so all regularity affords scope for any
multitude of variant particulars; so that the idea [of] continuity is an
extension of the idea of regularity. Regularity implies generality … (CP
7.535; 1899)
CSP: Efficient causation is that kind of causation whereby the parts
compose the whole; final causation is that kind of causation whereby the
whole calls out its parts. (CP 1.220; 1902).
CSP: Rationality is being governed by final causes. (CP 2.66; 1902).
CSP: On the whole, therefore, I think we must say that continuity is the
relation of the parts of an unbroken space or time. The precise definition
is still in doubt; but Kant's definition, that a continuum is that of which
every part has itself parts of the same kind, seems to be correct. This
must not be confounded (as Kant himself confounded it) with infinite
divisibility, but implies that a line, for example, contains no points
until the continuity is broken by marking the points. In accordance with
this it seems necessary to say that a continuum, where it is continuous and
unbroken, contains no definite parts; that its parts are created in the act
of defining them and the precise definition of them breaks the continuity.
(CP 6.168; c. 1903-1904)
CSP: In the first place, then, I do not call a line, or a surface, or
anything else, continuous unless every part of it that is homogeneous in
dimensionality with the whole and is marked off in the simplest way is, in
respect to the connexions of its parts, precisely like every other such
part; although, if the whole has but a finite number of interruptions, I do
call it "continuous in its uninterrupted portions." In the next place, I
conceive that a Continuum has, IN ITSELF, no definite parts, although to
endow it with definite parts of no matter what multitude, and even parts of
lesser dimensionality down to absolute simplicity, it is only necessary
that these should be marked off, and although even the operation of thought
suffices to impart an approach to definiteness of parts of any multitude we
please.*
*This indubitably proves that the possession of parts by a continuum is not
a real character of it. For the real is that whose being one way or another
does not depend upon how individual persons may imagine it to be. It shows,
too, that Continuity is of a Rational nature. (R S-30 [Copy T:5-7]; c. 1906)
CSP: In a continuum there really are no points except such as are marked;
and such interrupt the continuum. It is true that the capability of being
marked gives to the points the beginnings of *potential being*, but only
the beginnings. It should be called a *conditional being*, since it depends
upon some will's being exerted to complete it. (R 1041:13; 1906)
CSP: Whatever is continuous has *material parts*. I begin by defining
these thus: The *material parts* of a thing or other object, *W*, that is
composed of such parts, are whatever things are, firstly, each and every
one of them, other than *W*; secondly, are all of some one internal nature
(for example, are all places, or all times, or all spatial realities, or
are all spiritual realities, or are all ideas, or are all characters, or
are all relations, or are all external representations, etc.); thirdly,
form together a collection of objects in which no one occurs twice over
and, fourthly, are such that the Being of each of them together with the
modes of connexion between all subcollections of them, constitute the being
of *W*. Almost everything which has material parts has different sets of
such parts, often various *ad libitum*. Nothing which has an Essence (such
as an essential purpose or use, like the jackknife of the celebrated poser)
has any material parts in the strict sense just defined. (CP 6.174; 1908)
CSP: A perfect continuum belongs to the genus, of a whole all whose parts
without any exception whatsoever conform to one general law to which same
law conform likewise all the parts of each single part. *Continuity* is
thus a special kind of *generality*, or conformity to one Idea. More
specifically, it is a *homogeneity*, or generality among all of a certain
kind of parts of one whole. Still more specifically, the characters which
are the same in all the parts are a certain kind of relationship of each
part to all the coördinate parts; that is, it is a *regularity*. The step
of specification which seems called for next, as appropriate to our purpose
of defining, or logically analyzing the Idea of continuity, is that of
asking ourselves what kind [of] relationship between parts it is that
constitutes the regularity [of] a continuity; and the first, and therefore
doubtless the best answer for our purpose, not as the ultimate answer, but
as the proximate one, is that it is the relation or relations of
*contiguity*; for continuity is unbrokenness (whatever that may be,) and
this seems to imply a *passage* from one part to a contiguous part. (CP
7.535n6; 1908 May 24)
CSP: If in an otherwise unoccupied continuum a figure of lower
dimensionality be constructed--such as an oval line on a spheroidal or
anchor-ring surface--either that figure is a part of the continuum or it is
not. If it is, it is a topical singularity, and according to my concept of
continuity, is a breach of continuity. If it is not, it constitutes no
objection to my view that all the parts of a perfect continuum have the
same dimensionality as the whole. (Strictly, all the *material*, or *actual*,
parts, but I cannot now take the space that minute accuracy would require,
which would be many pages.) That being the case, my notion of the essential
character of a perfect continuum is the absolute generality with which two
rules hold good, first, that every part has parts; and second, that every
sufficiently small part has the same mode of immediate connection with
others as every other has. (CP 4.642; 1908 May 26)
>
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L
to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . To
UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu with the
line "UNSubscribe PEIRCE-L" in the BODY of the message. More at
http://www.cspeirce.com/peirce-l/peirce-l.htm .
