Jeff, list,
By "immediate connection" in his 1908 addition, I took Peirce to mean
the connection that two pieces have via a positive intersection or an
immediate adjacency, such that nothing mediates said connection. In the
text that you quote from "A Guess at the Riddle", I see that he says
"Those states to which a state will immediately pass will be adjacent to
it [...]." I remember that somewhere Peirce says that he took
reaction/resistance, rather than sheer contiguity, as Secondness,
because it's in virtue of reaction/resistance that we ascribe
contiguity. Peirce wants to understand the connectedness of time (and of
space). (The current notion that spacetime is an emergent phenomenon
based on quantum correlations is another track in that question, and
that's a case where spacetime hangs together beyond immediate
interactions because of the joint pasts of things in it.) Peirce's
objection does not seem so manufactured to me as it does to Matthew but,
on the other hand, Matthew understands these things generally rather
better than I do. In the past I've read, I think in Peirce's Logic
Notebook, or somewhere in the early volumes of _Writings_, objections
that I doubt most people would make, drawn up by Peirce to his own ideas.
I remember that Ketner's 1994 anthology of essays on Peirce _Peirce and
Contemporary Thought_ contains an essay "Peirce's Continuum" by Hilary
Putnam discussing (among other things) Peirce's way of dealing with
adjacent parts of a continuum. Unless it was Havenel in the more recent
article! My memory is getting rusty. Anyway, if you divide a discrete
nowhere-dense set, say of six eggs, into two halves, you can do so
without cutting AT an egg, but instead between two eggs, so that each
egg ends up in one or the other half, and zero intersection. But when
you cut a continuous line into halves, the point at which the cut is
made ends up in just one half or just the other, or ends up as the point
of intersection, or ends up as an "extra" point in neither half. You
don't conventionally get two co-exhaustive non-intersecting halves of
the continuous line without arbitrarily giving the cutting point to just
one half or the other. Peirce got into the idea of "exploding" points
from the cut's point so that both halves end up with it/them. I
understand little of this sort of thing, of course.
Best, Ben
On 2/23/2017 4:38 AM, Jeffrey Brian Downard wrote:
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 substances carrying their habits with them in their motions
through space will tend to render the different parts of space alike.
Thus, the dimensionality of space will tend gradually to uniformity;
and multiple connections, except at infinity, where substances never
go, will be obliterated. At the outset, the connections of space were
probably different for one substance and part of a substance from what
they were for another; that is to say, points adjacent or near one
another for the motions of one body would not be so for another; and
this may possibly have contributed to break substances into little
pieces or atoms. But the mutual actions of bodies would have tended to
reduce their habits to uniformity in this respect; and besides there
must have arisen conflicts between the habits of bodies and the habits
of parts of space, which would never have ceased till they were
brought into conformity. (W, Vol. 6, 209-10; CP 1.412-416)
The idea that I find striking here is that Peirce is offering an
explanation of how things might occur simultaneously--at the same time
but in two different places--by suggesting that the habits governing
such dyadic relations evolved from the habits governing a different
sort of dyadic relation that is involved in ordered relations in time.
He says: "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." The idea of "common points
coalescing into a perfect union" would seem to be the root of the idea
of "immediate connection," would it not?
One thing I find particularly puzzling about the addendum is the
rapidity with which Peirce moves from more phenomenological points
about our experience of time to more metaphysical points about the
real nature of time and then back again. The key idea in making such a
rapid movement back and forth is to draw a contrast between our /idea/
of time and the /facts/ we seek to explain by introducing and using
such an idea.
How does the suggestion that time can function as a kind of standard
for measuring degrees of continuity akin to way that oxygen can serve
as a standard for the measurement of atomic weights help us to better
understand how the /idea/ of time first arose as a hypothetical
explanation of some sort of surprising set of /facts/? The puzzling
phenomena that are offered in the example he provides are the
disjointed experiences associated with looking out of the window of a
steamboat at night at the shore as the lightning periodically
illuminates scenes on the riverbank.
He claims that the surprising and apparently contradictory character
of phenomena we observe can be explained if we "suppose them to be
mere aspects, that is, relations to ourselves, and the phenomena are
explained by supposing our standpoint to be different in the different
flashes."
The point of this illustration, I take it, is to help us see more
clearly what otherwise might be obscure in the newly amended
conception of continuity that he is trying to articulate. As such, how
does the illustration help to clarify how it is possible for a
continuum to have room for 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: [email protected] [[email protected]]
Sent: Wednesday, February 22, 2017 12:16 PM
To: [email protected]
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 versions of an addendum for the
published essay. The latest of the three, written on 26 May 1908, is
included in this selection; it is the one that was completed and
published. Peirce announces a new theory ory of continuity, based in
topical geometry rather than the theory of collections. tions. A true
continuum obeys the (corrected) Kantian principle that every part has
parts, and is such that all sufficiently small parts have the same
mode of immediate connection to one another. Moreover, Peirce asserts,
all the material parts (cf. selections 26 and 29) of a continuum have
the same dimensionality. Rather than explaining the central idea of
immediate connection, tion, he notes that the explanation involves
time, and answers the objection that his definition is therefore
circular. It is perhaps an ominous sign that Peirce devotes to much
space to what appears to be a somewhat manufactured tured objection:
since he does not explain what he means by `immediate connection,'
nection,' it would hardly have occurred to the reader that time was
bound up with such connection, had Peirce himself not brought it up.
(In selection 29, the involvement of time in `contiguity' is made
clearer.) The excessive attention tion to side issues, when the main
ideas are still so underexplained, would be less worrisome if Peirce
had explained himself more fully elsewhere; but so far as we know, he
did not.
* Charles S. Peirce. Philosophy of Mathematics: Selected Writings
(Kindle Locations 3293-3304). Kindle Edition.
Gary f.
From: Benjamin Udell [mailto:[email protected]]
Sent: 22-Feb-17 12:48
To: [email protected]
Subject: Re: [PEIRCE-L] Cyclical Systems and Continuity
Jeff D., list,
I agree with John S. and Gary F. about Peirce's not very detailed
analogy between time regarded as continuous and oxygen's atomic weight
regarding as 16 in Peirce's addition (beginning "_Added_, 1908, May
26.") of "Some Amazing Mazes (Conclusion), Explanation of Curiosity
the First". The addition is rather important, as it happens, because
of what Peirce winds up saying in it.
Jérôme Havenel (2008): "It is on May 26, 1908, that Peirce finally
gave up his idea that in every continuum there is room for whatever
collection of any multitude. From now on, there are different kinds of
continua, which have different properties." I don't remember whether
Havenel gets into the analogy of continuity with atomic weight.
Havenel, Jérôme (2008), "Peirce's Clarifications on Continuity",
_Transactions_ Winter 2008 pp. 68–133, see 119. Abstract
http://www.jstor.org/pss/40321237
I think Matthew Moore also discusses the addition in his Peirce
collection _Philosophy of Mathematics: Selected Writings_
http://www.iupui.edu/~arisbe/newbooks.htm#peirce_moore , but I don't
have it handy at the moment. The addition itself is there. You might
also look into the collection, edited by Moore, of essays on Peirce,
_New Essays on Peirce's Mathematical Philosophy_
http://www.iupui.edu/~arisbe/newbooks.htm#moore
Other links for interested peirce-listers:
Peirce (1908), "Some Amazing Mazes (Conclusion), Explanation of
Curiosity the First", _The Monist_, v. 18, n. 3, pp. 416-64, see 463-4
for the addition.
Google link to p. 463:
https://books.google.com/books?id=CqsLAAAAIAAJ&pg=PA463
Oxford PDF of article:
http://monist.oxfordjournals.org/content/monist/18/3/416.full.pdf
Reprinted CP 4.594-642, see 642 for the addition.
Best, Ben
On 2/22/2017 12:06 AM, Jeffrey Brian Downard wrote:
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 <[email protected]><mailto:[email protected]>
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
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