Jon,
I think all you're missing here is the difference between
metaphysical theory and the practice of scientific inquiry.
There are of course mathematical adventures in nonstandard
analysis and element-free categories that vie with anything
dreamt of in the metaphysical imagination, but the humdrum
fact is that all the sciences set out from, and rarely if
ever find occasion to venture beyond just such a concept
of continua as instanced by homilies like “the real line
is a continuum consisting of individual points”.
The logical utility of concepts and terms of general reference and
the logical utility of concepts and terms of individual reference
does not depend on such high-falution notions of continuity but
only on a simpler logical sort. You may define a human being
as a featherless biped if it pleases you, thereby committing
yourself to admitting an indefinite number of featherless
bipeds that may ever exist in the cosmos to human status.
But a mortal human's anthropology will still be based on
finite samples drawn from the generality of that genus.
Regards,
Jon
On 2/6/2017 2:24 PM, Jon Alan Schmidt wrote:
Jon A., List:
It does not seem right to me to say that, from Peirce's perspective, a
continuum is "constituted of individuals" or that a truly continuous line
"consists of individual points." My impression is that instead he saw the
continuum or line as the more fundamental entity, such that its parts are
not individuals or points, but themselves also continua; hence the notion
that all of reality is general (i.e., continuous) to some degree. Between
any two *actual *individuals in a continuum or points on a line are *potential
*individuals or points exceeding all multitude. A continuum or line thus
far outruns any collection of individuals or points within it. The
direction of "composition," so to speak, is then from the continuum or line
to individuals or points, rather than the other way around. Am I missing
something?
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Professional Engineer, Amateur Philosopher, Lutheran Layman
www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
On Mon, Feb 6, 2017 at 1:10 PM, Jon Awbrey <[email protected]> wrote:
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
On 2/6/2017 9:31 AM, Jon Alan Schmidt wrote:
JAS:https://list.iupui.edu/sympa/arc/peirce-l/2017-02/msg00043.html
Is it right to say that “generals are constituted of individuals”?
For Peirce, generality is continuity, and my understanding is that
no continuum is “constituted of individuals”, since no collection
of individuals is truly continuous.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Jon, List,
I would be content to say that, in the same vein I might say,
“the real line is a continuum constituted of individual points”.
If it's merely the word “constituted” that is causing difficulty
then I would substitute “consisting” and it would mean the same
thing for all practical mathematical and scientific purposes:
“the real line is a continuum consisting of individual points”.
Continuity is a matter of the relations among individual points
not a matter of their ontologies per se.
An adequate discussion of mathematical continua and their relation
to physical continua and whether there really are such things would
make for a long and diverting digression at this point, but it's not
really called for since the concept of continuity that Peirce relates
to logical generality does not demand the full power of those sorts of
continua but only a logical sort of continuity that is more general or
simply weaker, depending on your point of view.
I know I've remarked on this point before ... so let me go hunt that up ..
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
JA:https://inquiryintoinquiry.com/2014/11/09/continuity-gene
rality-infinity-law-synechism-1/
The concept of continuity that Peirce highlights in his
synechism is a logical principle that is somewhat more
general than the concepts of either mathematical or
physical continua.
Peirce’s concept of continuity is better understood as
a concept of lawful regularity or parametric variation.
As such, it is basic to the coherence and utility of
science, whether classical, relativistic, quantum
mechanical, or any conceivable future science that
deserves the name. (As Aristotle already knew.)
Perhaps the most pervasive examples of this brand of continuity
in physics are the “correspondence principles” that describe the
connections between classical and contemporary paradigms.
The importance of lawful regularities and parametric variations
is not diminished one bit in passing from continuous mathematics
to discrete mathematics, nor from theory to application.
Here are some further points of information, the missing of which
seems to lie at the root of many recent disputes on the Peirce List:
It is necessary to distinguish the mathematical concepts of
continuity and infinity from the question of their physical
realization. The mathematical concepts retain their practical
utility for modeling empirical phenomena quite independently of
the (meta-)physical question of whether these continua and
cardinalities are literally realized in the physical universe.
This is equally true of any other domain or level of phenomena —
chemical, biological, mental, social, or whatever.
As far as the mathematical concept goes, continuity is relative
to topology. That is, what counts as a continuous function or
transformation between spaces is relative to the topology under
which those spaces are considered and the same spaces may be
considered under many different topologies. What topology
makes the most sense in a given application is another one
of those abductive matters.
--
inquiry into inquiry: https://inquiryintoinquiry.com/
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