Re: [PEIRCE-L] Cyclical Systems and Continuity

[Benjamin Udell]

or a denumeral multitude of points, or an 
abnumeral multitude, or any multitude whatsoever? What might, from one 
point of view, appear to be disjointed might, from another point of 
view, appear to be continuous—just like the relations between the 
dimensions in space. From the perspective of a person living on a one 
dimensional line, a point is a kind of discontinuity. Similarly, from 
the perspective of person living in a two dimensional space, a one 
dimensional line appears as a kind of discontinuity…and so on.


Having said that, it still isn't clear to me how the illustration 
helps to clarify the puzzling features of the amendments he is making 
to the conception of continuity. What is more, it isn't clear what 
role the point about the continuity time being a sort of standard for 
measure is supposed to play in the account.


The insights that seem to have prompted the revisions in his account 
of continuity appear to have grown from reflections on the character 
of cyclical systems and the light that such systems shed on the 
relations between a perfect continuum and those that are imperfect. 
Following this line of thought, I tend to think that the continuity of 
time can, on his account, serve as a standard for measuring the degree 
to which different sorts of systems are more or less perfect in their 
continuity. His point, I take it, is that it doesn't really matter 
whether one or another thing (the connections between the parts of 
space, the connections between shades of the hue of a color, the 
connections between parts of time etc.) are entirely perfect as 
continua. Instead, time is like oxygen in the scale of atomic weights 
in that it supplies us with a sufficiently reliable standard that we 
are thereby enabled to make relative comparisons--even if we don't 
(yet) have an absolute standard.


Matthew makes a further remark to the effect that Peirce moves in this 
addendum from an account of continuity that is based on the size of 
collections to an account that is grounded on topological 
relations—and that this seems to represent a dramatic shift in the way 
he is thinking about continuity. The last lecture of RLT makes it 
clear, I think, that Peirce has been reflecting in rather deep ways on 
the relationships between these two different mathematical approaches 
to understanding different aspects of the conception of continuity. By 
the time that he is writing the addendum, he has been reflecting on 
the relationship between more arithmetic approaches that start with 
what is discrete and more topological approaches that start with what 
is continuous. As such, I tend to think that the insights that sparked 
the revisions in the conception of continuity in the year that 
intervened between the time that he received the proofs for the "First 
Curiosity" of "Some Amazing Mazes and the addendum might stem from 
something that can been seen when one thinks about the topological 
character of different systems of number--especially when one 
experiments with diagrams involving cyclical systems.


My hunch is that Peirce was drawing on cyclical systems in his 
exploration of different sorts of multitudes, and that he was thinking 
quite deeply about the different sorts of formal relations (e.g., 
symmetries) that hold between the different systems of numbers. In 
doing so, he was working in the same spirit as contemporary 
topologists when they use what is called the Farey diagram to explore 
the relations between the number systems of the rationals, the reals, 
the imaginaries, etc. See, for example, Allen Hatcher's /The Topology 
of Numbers./


( https://www.math.cornell.edu/~hatcher/TN/TNbook.pdf 
<https://www.math.cornell.edu/%7Ehatcher/TN/TNbook.pdf> )


Pursuing this line of thought further would take considerable time and 
space, so I'll stop here.


--Jeff

Jeffrey Downard
Associate Professor
Department of Philosophy
Northern Arizona University
(o) 928 523-8354
____

From: g...@gnusystems.ca [g...@gnusystems.ca]
Sent: Wednesday, February 22, 2017 12:16 PM
To: peirce-l@list.iupui.edu
Subject: RE: [PEIRCE-L] Cyclical Systems and Continuity

Ben, you’re right, the addendum is Selection 27 in Matthew Moore’s 
collection, and his commentary on it goes in part like this:


Peirce rightly points out that even if there is an upper bound on the 
multitude titude of points that can be placed on a line, it does not 
follow that a line can be filled with a point set of the appropriate 
multitude; and he appeals once again to our consciousness of time (in 
particular, to memory) to argue the need for a "more perfect 
continuity than the so-called `continuity' of the theory ory of 
functions"; as in his supermultitudinous theory, "a line [with this 
more perfect continuity] does not consist of points."


By the time he received the proofs of the article, Peirce thought he 
could do better, and wrote three ve

RE: [PEIRCE-L] Cyclical Systems and Continuity

[gnox]

Jeff, your post seems to head in directions I'm unable to follow, so I'll
just mention this: the final two selections in Moore's "Philosophy of
Mathematics" collection are probably the best tools for "filling in the gap"
in Peirce's thinking between arrival of the proofs of the article and the
addendum that was printed was printed it. Both of those selections were
written before the published version and show the train of thought Peirce
was following as he abandoned his "supermultitudinous" view of continuity.
The second one (the last selection in Moore) is especially interesting, to
me anyway, although both of them break off at the end, as presumably Peirce
decided to start over with the next draft.

 

I'm happy that Matthew Moore's collection is available relatively cheaply,
because it's very helpful for understanding Peirce's philosophy (not just
his mathematics). I might like it even better if it were presented in
chronological order . 

 

Gary f.

 

From: Jeffrey Brian Downard [mailto:jeffrey.down...@nau.edu] 
Sent: 23-Feb-17 04:39



 

Ben, Gary F, Jon S, List,

Thanks to Ben for reminding me that Matthew provides some comments about
this addendum in his collection, and also to Gary F. for supplying the
relevant passage. I'd like to respond to a concern that Matthew raises about
the way Peirce is explaining what he is trying to accomplish:  "It is
perhaps an ominous sign that Peirce devotes too much space to what appears
to be a somewhat manufactured objection: since he does not explain what he
means by `immediate connection,' it would hardly have occurred to the reader
that time was bound up with such connection, had Peirce himself not brought
it up."

Here is a hypothesis that Peirce puts forward in "A Guess at the Riddle"
about the real nature of time as it first evolved in the cosmos:

Our conceptions of the first stages of the development, before time yet
existed, must be as vague and figurative as the expressions of the first
chapter of Genesis. Out of the womb of indeterminacy we must say that there
would have come something, by the principle of Firstness, which we may call
a flash. Then by the principle of habit there would have been a second
flash. Though time would not yet have been, this second flash was in some
sense after the first, because resulting from it. Then there would have come
other successions ever more and more closely connected, the habits and the
tendency to take them ever strengthening themselves, until the events would
have been bound together into something like a continuous flow. We have no
reason to think that even now time is quite perfectly continuous and uniform
in its flow. The quasi-flow which would result would, however, differ
essentially from time in this respect, that it would not necessarily be in a
single stream. Different flashes might start different streams, between
which there should be no relations of contemporaneity or succession. So one
stream might branch into two, or two might coalesce. But the further result
of habit would inevitably be to separate utterly those that were long
separated, and to make those which presented frequent common points coalesce
into perfect union. Those that were completely separated would be so many
different worlds which would know nothing of one another; so that the effect
would be just what we actually observe. 

But Secondness is of two types. Consequently besides flashes genuinely
second to others, so as to come after them, there will be pairs of flashes,
or, since time is now supposed to be developed, we had better say pairs of
states, which are reciprocally second, each member of the pair to the other.
This is the first germ of spatial extension. These states will undergo
changes; and habits will be formed of passing from certain states to certain
others, and of not passing from certain states to certain others. Those
states to which a state will immediately pass will be adjacent to it; and
thus habits will be formed which will constitute a spatial continuum, but
differing from our space by being very irregular in its connections, having
one number of dimensions in one place and another number in another place,
and being different for one moving state from what it is for another.

Pairs of states will also begin to take habits, and thus each state having
different habits with reference to the different other states will give rise
to bundles of habits, which will be substances. Some of these states will
chance to take habits of persistency, and will get to be less and less
liable to disappear; while those that fail to take such habits will fall out
of existence. Thus, substances will get to be permanent.

In fact, habits, from the mode of their formation, necessarily consist in
the permanence of some relation, and therefore, on this theory, each law of
nature would consist in some permanence, such as the permanence of mass,
momentum, and energy. In this respect, the theory suits the facts admirably.


The 

Re: [PEIRCE-L] Cyclical Systems and Continuity

[John F Sowa]

Jeff and Gary,

JBD

I'm wondering if anyone can explain in greater detail what Peirce
is suggesting in this passage in making the comparison between the
atomic weight of oxygen and the continuity of Time


GF

I think the claim is that our experience of time is the prototype
for all conceptions of a perfect continuum. The analogy with atomic
weight is misleading if we think of time as a metrical space. Rather
time is a continuum because there are no points in real time (as
opposed to representations of ‘distance’ between events), and that
pointlessness is the ‘essence’ of continuity, so to speak.


I agree with Gary.  CSP is trying to break the circular definition
by taking one concept as the standard and defining the others in
terms of it.  Note the term in italics:

CSP

Now if my definition of continuity involves the notion of immediate
connection, and my definition of immediate connection involves the
notion of time; and the notion of time involves that of continuity,
I am falling into a /circulus in definiendo/.  But on analyzing
carefully the idea of Time, I find that to say it is continuous is
just like saying that the atomic weight of oxygen is 16, meaning
that that shall be the standard for all other atomic weights. The
one asserts no more of Time than the other asserts concerning the
atomic weight of oxygen; that is, just nothing at all.


For more discussion about Peirce's notion of continuity, see
the introduction to RLT by Ketner & Putnam.  Cantor, like Zeno,
assumed that a continuous time interval is identical to a set
of time points.  That assumption led to Zeno's paradox.

Aristotle's solution to Zeno's paradox is to assume that the proper
parts of a continuous line are shorter line segments.  Points are
not parts of a line, but markers on a line.

Peirce adopted Aristotle's solution.  But then he took a further
step by saying that you could have infinitesimal markers on a line.
But all those markers are purely imaginary, since you could never
observe them or draw them in actuality.  Once you add infinitesimal
markers, there is no stopping point, since you can keep imagining
(or postulating) infinitely many orders of imaginary infinitesimals.

John

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RE: [PEIRCE-L] Cyclical Systems and Continuity

[gnox]

Jeff, list,

 

I was struck by that passage too, but I don’t think Peirce’s claim is “that the 
continuity of our experience of time can serve as a kind of standard for 
measure.” Rather I think the claim is that our experience of time is the 
prototype for all conceptions of a perfect continuum. The analogy with atomic 
weight is misleading if we think of time as a metrical space. Rather time is a 
continuum because there are no points in real time (as opposed to 
representations of ‘distance’ between events), and that pointlessness is the 
‘essence’ of continuity, so to speak.

 

This however is a point (if you’ll pardon the expression) that Peirce has made 
before, so it wouldn’t explain Peirce’s suggestion that he is saying something 
new here — if that’s what he is suggesting in the addendum.

 

Gary f.

 

} The perfect knot needs neither rope nor twine, yet cannot be untied. [Tao Te 
Ching 27] {

http://gnusystems.ca/wp/ }{ Turning Signs gateway

 

 

From: Jeffrey Brian Downard [mailto:jeffrey.down...@nau.edu] 
Sent: 22-Feb-17 00:07



 

List,

I've been trying to sort through the points Peirce is making about topology and 
the mathematical conception of continuity in the last lecture of RLT. In the 
attempts to trace the development of the ideas concerning the conceptions of 
continua, furcations and dimensions in his later works, I've been puzzled by 
some later remarks he makes about cyclical systems in "Some Amazing Mazes" 
(Monist, pp. 227-41, April 1908; CP 4.585-641).

In a short addendum, Peirce indicates that he has, in the year since writing 
the paper,  "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." (CP, 4.642)

Here is a passage that has caught my attention:   

 

Now if my definition of continuity involves the notion of immediate connection, 
and my definition of immediate connection involves the notion of time; and the 
notion of time involves that of continuity, I am falling into a circulus in 
definiendo. But on analyzing carefully the idea of Time, I find that to say it 
is continuous is just like saying that the atomic weight of oxygen is 16, 
meaning that that shall be the standard for all other atomic weights. The one 
asserts no more of Time than the other asserts concerning the atomic weight of 
oxygen; that is, just nothing at all.  

I'm wondering if anyone can explain in greater detail what Peirce is suggesting 
in this passage in making the comparison between the atomic weight of oxygen 
and the continuity of Time--or if anyone knows of clear reconstructions of what 
he is doing in the secondary literature? The claim that the continuity of our 
experience of time can serve as a kind of standard for measure is, I think, 
quite a remarkable suggestion.  

 

--Jeff

Jeffrey Downard
Associate Professor
Department of Philosophy
Northern Arizona University
(o) 928 523-8354




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[PEIRCE-L] Cyclical Systems and Continuity

[Jeffrey Brian Downard]

List,

I've been trying to sort through the points Peirce is making about topology and 
the mathematical conception of continuity in the last lecture of RLT. In the 
attempts to trace the development of the ideas concerning the conceptions of 
continua, furcations and dimensions in his later works, I've been puzzled by 
some later remarks he makes about cyclical systems in "Some Amazing Mazes" 
(Monist, pp. 227-41, April 1908; CP 4.585-641).

In a short addendum, Peirce indicates that he has, in the year since writing 
the paper,  "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." (CP, 4.642)

Here is a passage that has caught my attention:

Now if my definition of continuity involves the notion of immediate connection, 
and my definition of immediate connection involves the notion of time; and the 
notion of time involves that of continuity, I am falling into a circulus in 
definiendo. But on analyzing carefully the idea of Time, I find that to say it 
is continuous is just like saying that the atomic weight of oxygen is 16, 
meaning that that shall be the standard for all other atomic weights. The one 
asserts no more of Time than the other asserts concerning the atomic weight of 
oxygen; that is, just nothing at all.

I'm wondering if anyone can explain in greater detail what Peirce is suggesting 
in this passage in making the comparison between the atomic weight of oxygen 
and the continuity of Time--or if anyone knows of clear reconstructions of what 
he is doing in the secondary literature? The claim that the continuity of our 
experience of time can serve as a kind of standard for measure is, I think, 
quite a remarkable suggestion.

--Jeff



Jeffrey Downard
Associate Professor
Department of Philosophy
Northern Arizona University
(o) 928 523-8354

From: Jon Awbrey 
Sent: Wednesday, February 8, 2017 1:26 PM
To: Peirce List
Cc: Arisbe List
Subject: [PEIRCE-L] Re: The Difference That Makes A Difference That Peirce Makes

A Flash From The Past ⚡⚡⚡
=

The Difference That Makes A Difference That Peirce Makes
https://inquiryintoinquiry.com/2013/02/07/the-difference-that-makes-a-difference-that-peirce-makes-1/

Being one who does not view Peirce's work as a flickering
foreshadowing of analytic philosophy, logical whatevism,
or anything else you want to call it, but leans more to
thinking of the latter philosophies as fumbling fallbacks
losing what ground Peirce had gained for our understanding
of logic, mathematics, science, not to mention the life of
inquiry in general, I am dropping this thread anchor toward
the end of remembering the critical insights Peirce gave us,
as they come to mind.

cc:
http://web.archive.org/web/20130212171424/http://permalink.gmane.org/gmane.science.philosophy.peirce/9562
http://web.archive.org/web/20130212171340/http://stderr.org/pipermail/arisbe/2013-February/thread.html#3911
http://web.archive.org/web/20130217050656/http://stderr.org/pipermail/inquiry/2013-February/thread.html#4054

o~o~o~o~o~o~o

Peircers,

My mind keeps flashing back to the days when I first encountered
Peirce's thought.  It was so fresh, it spoke to me like no other
thinker's thought I knew, and it held so much promise of setting
aside all the old schisms that boggled the mind through the ages.

I feel that way about it still but communicating precisely what I find
so revolutionary in Peirce's thought remains a work in progress for me.

Many readers of Peirce share the opinion that there is something truly
novel in his thought, a difference that makes a critical difference in
the way we understand our thoughts and undertake our actions its light.
The question has arisen once again, just what that difference might be.

So I'll make another try at answering that ...

Regards,

Jon

--

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