Gary R., List,
My aim in raising a different sort of question was to move the recent
discussion about the conceptions of continuity and the continuum in a different
direction. The passage Michael copied into the email is one that I've spent
some time trying to make out--and I find that it is not a simple point that
Peirce is making. As such, I was trying to return to the starting point in our
recent discussion of continuity--which began with a question about the
relationship between generality and continuity and the motivations behind
Peirce's synechism.
Gary R., would you clarify your last remark, where you say: "I don't see that
it offers any particular difficulties in answering (Peirce is quite explicit),
and Michael in passing did offer a remark clearly articulating this distinction
(I didn't quickly find it)." My initial hunch reading this line was that you
are referring in both clauses to Michael's response to Jerry. Is this hunch on
the right track? Or, are you responding to me by saying that Peirce is being
quite explicit about what he means in CP 4.642--and that it shouldn't take much
work on our part to clarify what is being said? If you are saying that latter,
then I disagree.
At the risk of defeating my own aim in trying to move the discussion in a
different--and hopefully more productive--direction, let me consider the
question: "How are the conceptions of continuity, continuous and continuum
connected to each other?"
Let me say that I am working on the assumption that the root conception we use
in talking about things that appear to be connected in the way that our
experience of something like time is connected is to say, "time is continuous."
What is the function of the monadic predicate: "-is continuous"? I tend to
think that it often functions like an adjective. That is, we say about our
common experience of moving from what we were thinking a few seconds ago to
what we are thinking now is "of a continuous series of changes."
The term "continuity" is the result of taking this adjective and turning it
into an abstract noun. As such, it is the result of hypostatic abstraction.
How is the conception of the continuum connected to the conception of
continuity? One place where we use the conception of "the continuum" is in
mathematics. If we move from one area of mathematical inquiry to another, we
come across a number of different examples of things that seem--at least to
some mathematicians--to be cases of continua. For example: there is the line
in geometry; and the field of real numbers in analysis; and the power set of
the real numbers in set theory. After considering a number of such examples,
the mathematicians ask: do these different cases that appear to involve
continuity all have something in common? That is, can we arrive at a
conception of "The continuum" that is at the root of the plurality of examples
of things that appear to be continuous? That, I take it, is the kind of thing
Gödel's was talking about when he posed the question: “What is Cantor’s
continuum problem?” (1947)
Peirce also uses the term 'continuum' in philosophy to talk about things that
appear to be continuous. He is following the lead of other philosophers who,
in the mid-1700's, appear to have coined the term by taking the Latin
"continuus" and the English "continuous" and then used the new term to refer
the thing that has the property of being continuous. In this way, Peirce asks
questions about the nature of the continuum of feelings, and the continuum of
space, etc. We might, if we were thinking like mathematicians, ask if these
different examples all have something in common. Is there a philosophical
conception of "The continuum" that is at the root of these different continua?
It might appear that there is little difference between the adjectival use of
continuous, and the conception of continuity that we arrive at by hypostatic
abstraction, and the thing we refer to by calling it a continuum. But, we
should keep in mind that Peirce labored over the differences between the
conceptions of relative, relation and relationship. I think we could probably
draw on the points he makes about the conceptions of relative, relation and
relationship in order to get clearer about the conceptions of continuous,
continuity and continuum. But that is just my hunch. I’ll leave that task to
another time.
With this much having been said, let me ask a simple-minded question about what
Peirce seems to be saying in CP 4.642. In this passage, he makes a distinction
between a perfect continuum and an imperfect continuum. Elsewhere, he offers
the following definition about the construction of Cantor's set: “but now I
define a pseudo-continuum as that which modern writers on the theory of
functions call a continuum. But this is fully represented by, and according to
G. Cantor stands in one-to-one correspondence with, the totality of real
values, rational and irrational; and these are iconized, in their turn,
according to these writers [by the] entire body of decimal expressions carried
out to the right to all finite powers of 1/10 without going on to Cantor's
{ö}th place of decimals.” (CP 6.176) Peirce uses this definition in an
argument that the pseudo-continuum is not really continuous. So, here is my
simple-minded question: is an imperfect continuum continuous, or is it not
continuous?
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Gary Richmond [[email protected]]
Sent: Monday, November 17, 2014 10:39 PM
To: Peirce-L
Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc.
Michael, Jeff D., Jerry, list,
Michael wrote: I part with this very late gem [ca. 1908], where I think the
continuity-continuum seamlessness is quite evident – continuous, if you will.
I quite agree, and if one could say of a quotation offered to prove a point
that it did, one might say of this one: Q.E.D.
Jeff asked: What might we do to arrive at greater clarity about what the
distinction between the conceptions of a perfect and an imperfect continuum
consists in and what the basis of the distinction rests on?
As Michael noted earlier, this is a very different question than the one he was
arguing with Jerry. I don't see that it offers any particular difficulties in
answering (Peirce is quite explicit), and Michael in passing did offer a remark
clearly articulating this distinction (I didn't quickly find it).
Best,
Gary R
[Gary Richmond]
Gary Richmond
Philosophy and Critical Thinking
Communication Studies
LaGuardia College of the City University of New York
C 745
718 482-5690<tel:718%20482-5690>
On Mon, Nov 17, 2014 at 11:53 PM, Michael DeLaurentis
<[email protected]<mailto:[email protected]>> wrote:
Lest you lost sight of my sample citations -- and for your further musement --
I part with this very late gem [ca. 1908], where I think the
continuity-continuum seamlessness is quite evident – continuous, if you will.
In going over the proofs of this paper, written nearly a year ago [1907], I can
announce that I have, in the interval, taken a considerable stride toward the
solution of the question of continuity, having at length clearly and minutely
analyzed my own conception of a perfect continuum as well as that of an
imperfect continuum, that is, a continuum having topical singularities, or
places of lower dimensionality where it is interrupted or divides… . 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)
With that, I will rest my case.
From: Michael DeLaurentis [mailto:[email protected]]
Sent: Monday, November 17, 2014 7:56 PM
To: 'Jerry LR Chandler'; 'Peirce List'
Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc.
Jerry – A lot of words, but no explication whatsoever of any distinction CSP
makes between continuity and its instantiation in continua. After some
irrelevancy re the Continuum Hypothesis, you make some statements about
continuity and the philosophy of the natural sciences, continuity and
chemistry, and “continuity in term of units, individuals and collections” --
three sets of comments about continuity, none involving the continuum.
Where’s the purported distinction between continuity and the continuum – i.e.,
between the continuity exhibited in continua and any other alleged continuity?
Where are CSP’s words that indicate such a distinction? What has this claim –
“One notion of continuity was constructed by CSP from units, individuals and
collections” -- contrasted with this claim – “A second notion of usage of
continuity emerges from Cantor's view of the number line as a closed interval
that could be separated into two notions of distance” – got to do with a
purported distinction PEIRCE makes between continuity and the continuum, other
than that the former is exhibited in, and only in, continua? And how do these
meandering musings confirm that “Kirsti’s intuition was spot on”? Nothing you
say even begins to addresses this.
I cited the late articles and passages where, to the contrary, CPS, as even
Kirsti has now acknowledged, moves seamlessly between the two, in just the
manner I have described. You have splashed a bunch of disconnected comments
around, but have cited nothing in Peirce to the contrary.
I don’t see the point of continuing this thread if you’re just going to toss a
hodge-podge of unrelated statements around. And with this post, I will
therefore close end my responses to these aimless meanderings.
From: Jerry LR Chandler [mailto:[email protected]]
Sent: Monday, November 17, 2014 7:26 PM
To: Peirce List
Cc: Michael DeLaurentis
Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc.
List, Michael, John, Kirsti:
On Nov 14, 2014, at 11:41 AM, Michael DeLaurentis wrote:
Jerry – All due respect, but my post concerned the distinction Kirsti claimed
to find, not anything in your post. So I don’t see the relevance..
The immediate relevance of my post is that this is a listserve for CSP writings
and the recent correspondence relate to his writings, although I do not think
that is what was of concern to you.
Your post leave me puzzled about your penultimate post which in turn was
puzzling, so I reviewed the tread from its beginning and read more widely.
The immediate motivations for my contributions and, as I understand it, yours
also, was the issue raised by Kirsti with respect to the possible distinction
the continuum and continuity in philosophy, mathematics and CSP's writings.
Your post (Nov 12) expresses this perspective:
Continuity is simply the unique quality which continua, and only continua,
exhibit.
Does this assertion close the philosophical issue that Kirsti raised?
In response to John Deely's questions (post 18 in the listing)
"Deely, John N." [[email protected]<mailto:[email protected]>] kirjoitti:
Kirsti, would you mind clarifying for me, if possible (but not necessarily)
with some specific ref. to a Peirce text(s), your remark re the difference
between "continuity" and "continuum":
I believe that it is relevant to cite the specific texts 4.172-176 from CSP in
part of the answer to John's and Kristi's questions.
These paragraphs brought to mind the Cantor's famous "Continuum Hypothesis" .
A couple of citation from the web place the concept of the Continuum is a
completely different context that that of mere continuity.
[Introduction. Arguably the most famous formally unsolvable problem of
mathematics is Hilbert's first problem: Cantor's Continuum Hypothesis:
{The proposal originally made by Georg Cantor that there is no infinite set
with a cardinal number<http://mathworld.wolfram.com/CardinalNumber.html>
between that of the "small" infinite set of
integers<http://mathworld.wolfram.com/Integer.html> [aleph_0] and the "large"
infinite set of real numbers<http://mathworld.wolfram.com/RealNumber.html>
(the "continuum<http://mathworld.wolfram.com/Continuum.html>"). Symbolically,
the continuum hypothesis is that [aleph_1=c] . Problem 1a of Hilbert's
problems<http://mathworld.wolfram.com/HilbertsProblems.html> asks if the
continuum hypothesis is true.]
Another aspect of this issue was how did CSP relate his views on continuity to
the philosophy of the natural sciences?
And these to synechism?
4.584 (1906) It is that synthesis of tychism and of pragmatism for which I long
ago proposed the name, Synechism
Yet, with respect to chemistry and continuity, he writes CP1.62 (1896?)
Now it enters into every fundamental and exact law of physics or of psychics
that is known. The few laws of chemistry which do not involve continuity seem
for the most part to be very roughly true. It seems not unlikely that if the
veritable laws were known continuity would be found to be involved in them…
This is to be contrasted with his statements in 4. 173 where he justifies the
origin of continuity in term of units, individuals and collections, strongly
implies a consistency with the legisigns of chemistry with atoms as units,
individuals as proper names of elements and collections becoming continuous.
Thus, my conclusion from these readings is that Kristi's intuition was spot on.
One notion of continuity was constructed by CSP from units, individuals and
collections.
A second notion of usage of continuity emerges from Cantor's view of the
number line as a closed interval that could be separated into two notions of
distance, as shown in his well know "removal of the middle third" argument to
construct infinite numbers of continuous closed intervals from a line of UNIT
length. The "Continuum Hypothesis" is a proposition about Cantor's
mathematical philosophy. It is not an extension of CSP's notion of continuity.
On another topic, I think it is important to support Stefan's quote of CP 5.131:
"Man makes the word, and the word means nothing which the man has not made it
mean, and that only to some man. But since man can think only by means of words
or other external symbols, these might turn round and say: ”You mean nothing
which we have not taught you, and then only so far as you address some word as
the interpretant of your thought.“ In fact, therefore, men and words
reciprocally educate each other; each increase of a man‘s information involves
and is involved by, a corresponding increase of a word’s information."
I was not aware of this quote, but have had a similar thought in mind for
decades from my sensory experiences in the world. The observation that meaning
is individualized is true for all individuals as a consequence of their
antecedent sensory experiences. It is also true of language usage among
disciplines. It is particularly important for those who love knowledge.
Cheers
Jerry
From: Jerry LR Chandler [mailto:[email protected]]
Sent: Friday, November 14, 2014 12:33 PM
To: Peirce List
Cc: Michael DeLaurentis; John N. Deely; Määttänen Kirsti
Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc.
List, Michael, Kirsti, John:
On Nov 12, 2014, at 11:47 AM, Michael DeLaurentis wrote:
I don’t find any such distinction, implicit or explicit, in Peirce’s late
writings.
Motivated by your assertions, I re-read 4.172 and later paragraphs, searching
for distinctions between CSP logic and set theory logic.
In contrast to your assertion, I certainly find numerous critical philosophic
distinctions between CSP logic and Cantorian/Russellian logic with respect to
inquiry into the mathematics/logic of the continuum.
Although a large number of texts could be cited, availability of time and
energy restrict my rhetoric principally to 4.172 to 4.176.
1. 4.173 introduces with the notion of a collection.
A collection is a consequence of "bring or gather together", parts of a whole.
CSP bases his notion of relation on collections as parts of a whole. It
requires activity to bring together a collection.
Thus, CSP is grounding his argument, among other mathematical concepts, on the
theory of numbers, the collectability of numbers, and the antecedent parts
being brought together to construct a whole.
This is clearly distinct from Cantor / Russell views which pre-supposes a
geometric line.
2. 4.174 (and 4.172) introduces the notion of a relative of a part versus the
relative of a whole, drawing on the statistical example in 4.172 and the
concept of a unit of a partition of a role of a pair of dice. Each role of the
pair of die generates a relative value among all possible roles of the pair of
six-sided die, exactly 36.
This is clearly distinct from Cantor / Russell views.
3. 4.175 "But when the units lose there individual identity because the
collection exceeds every positive existence of the universe, the word multitude
ceases to be applicable. I will take the word multiplicity to mean the
greatness of any collection discrete or continuous."
I infer from this, in light of 4.172-175, that individual identity is related
to parts of a whole such that parts, as units, can be collected into whole,
generating the NOUN, collection. The "bringing together" of a collection is of
the nature of a sublation. The quality of the collection, is, presumable for
CSP, a matter of sensory experience, as one perceives from the usage of the
term "because" in this sentence, inferring causality.
(And qualities are an aspect of sensory experiences, are they not?)
This is clearly distinct from Cantor / Russell views of memberships and
classes.
Yes, set theory, as a dominant force in modern mathematics, has ignored the
logical basis of CSP notion of multitude and his terminology for distinguishing
between parts and wholes, points and lines, and sensory experiences.
But, CSP’s philosophy expressed in 4.172-4.175 is consistent with many aspects
of chemical logic; modern mathematics is not consistent with chemical logic for
very specific reasons of the non-transitivity of the mathematics of chemical
sublations of individual identities. Non-transitivity is illustrated, for
example, by the handedness of chemical isomers.)
I conclude that although many many aspects of CSP logic and set theory logic
are consistent with one another, the distinction between them (modes of
constructions) at the rhetorical and semantic levels differ in mathematically
profound ways.
The basic conundrum of the nature of distinction between discrete and
continuous mathematics remains alive and open. Indeed, a very active subfield
of mathematics is the Brouwer School of intuitionism.
( http://en.wikipedia.org/wiki/Intuitionistic_logic )
Parenthetically (or perhaps metaphorically) I conclude that studying CSP texts
without an in-depth knowledge of the state of the science in the 2nd half of
the 19 Th Century is like attempting to solve a crossword puzzle with only the
superficial "across" clues. The depth of his thought corresponds with
knowledge of mathematics and the natural sciences and the natural propositions
in his time, that is, the "down" clues.
Extending the metaphor, the sensory experiences of the American cultural
milieu of the late 19 Th Century are interwoven into the very fabric of CSP's
text.
Cheers, Jerry
(BTW, Thanks to Gary F. for suggesting a puzzle analogy for hermeneutics.)
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