Jon S., Jeff,
Mathematical singularity theory may be relevant here. I've just now read
some things on it including Bézout's theorem. If it is relevant, then I
would wonder why Peirce didn't mention Bézout's theorem (he does mention
Bézout a few times) or the like. Decades ago a topological singularity
theorist told me that the theory provided ways to distinguish points
that one would think were not distinguishable (or discernible). For
example, think of a string that loops over itself on the floor, or a
curve that loops through itself on a plane. The point of
self-intersection is a point with "multiplicity two", since it
corresponds to two different points along the curve. They both
correspond to a single point on the plane (or floor). They are ordered,
one coming earlier or later than the other, depending the direction in
which the points along the curve are ordered. Another case of
multiplicity two is that of the factor 3 in the equality 3×3=9.
Deforming the string or curve so that it does not pass across itself
does not result in our needing to attribute the former intersection
point to just part of the loop or the other. Has the intersection point
"exploded" into two points? But it always had multiplicity two. Peirce's
example involves neither such a loop nor a cusp, but a simple circle.
The circle gets cut by a line segment that is not a part of the circle.
The loop cuts itself by intersection of two parts that "locally" are not
part of each other. The multiplicity of the circle's cut point is not
the multiplicity of an intesecting line segment's point coinciding with
a point on the circle. Instead, Peirce cuts the circle show that any
point on it is a kind of potential multiplicity of ordered points. He
cuts the circle and widens it into a C, and if one closes it again,
there is nothing to stop us from considering the rejoined end points as
distinct but coinciding at a point on the plane, as long as we can find
them again. But in making it back into a simple circle, we have melted
them into each other. Instead, some re-interruption of the circle, some
decision as to how to cut it, is required in order to revive the
singularity and determine its actual multiplicity or, to put it another
way, to determine what multiplicity has been actualized.
I hope that makes some sense. It's still kind of early in the morning
for me.
Best, Ben
On 2/23/2017 10:28 PM, Jon Alan Schmidt wrote:
Jeff, List:
JD: 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?
Kelly Parker discusses the same passage on pages 116-117 of The
Continuity of Peirce's Thought.
The continuum of time is a species of generality, and is present
in any event whatever. Moreover, time is the most perfect
continuum in experience. Accordingly, when Peirce defined a
continuum as that which, first, has no ultimate parts, and second,
exhibits immediate connection among sufficiently small neighboring
parts, he appealed to the experience of time to illustrate the
notion of immediate connection. Time serves as the experienced
standard of continuity, through which we envisage all other
continua (CP 6.86).
There is an apparent circularity in using time to define
continuity. The definition of continuity involves the idea of
immediate connection, immediate connection is clarified by an
appeal to the concept of time, and time is conceived as continuous
(CP 4.642). In chapter 4, I sought to clarify the notion of
immediate connection /without/ an appeal to time, so as to break
this circle. With the mathematical account of immediate
connection in hand, though, we can see that Peirce's appeal to
time identifies it as the experiential standard of continuity.
Peirce asserts that to say time is continouos is "just like saying
the atomic weight of oxygen is 16, meaning 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" (CP 4.642).
Time, the standard of continuity, is the "most perfect" continuum
in experience, but should not be taken as an absolutely perfect
continuum. The perfect /true continuum/ is only described
hypothetically in mathematics. Peirce observed that time is in
all likelihood not "quite perfectly continuous and uniform in its
flow (CP 1.412). Phenomenological time does exhibit the properties
of infinite divisibility and immediate connection, but is probably
/not/ best conceived as an unbroken and absolutely regular thread.
The only constant we have noted in time is the regularity of
development or change, but change is not smooth. Changes differ
from one another. The "regular" phenomenon of change consists, on
closer examination, of numerous (perhaps infinite) parallel
courses of development with different patterns and histories that
interweave and diverge.
Here is what Parker says in chapter 4, page 89, about "immediate
connection."
Immediate connection is a kind of relation, but a rather peculiar
kind. That the connection is /immediate/ suggests that there is
no need of a third, C, which mediates the relation of A to B. Not
only does this exclude the possibility that there is anything
located between A and B, it also excludes the possibility that A
and B are related only in virtue of belonging to some class of
objects. In the case where A and B are connected /only/ in
co-being, for example, the third is that which brings the two
independent things together in a collection. My dog and my hat
are "connected" in the sense that both are elements of the set of
things in the house as I write this sentence.
Suppose that two things, A and B, have some part in common. If
that part is the /region of intersection/ of A and B, then the
region acts as a third and the relation is mediated by that
third. Now suppose that A and B have /everything/ in common.
This means that A and B are /the same/. In fact, the relation of
immediate connection appears to be tied up with the relation of
identity. Roughly speaking, the "mode of immediate connection"
between sufficiently small neighboring parts of a continuum is
that such parts are in some sense identical.
If I read Peirce correctly, he is here flirting with what would
appear to be a disastrous contradiction: /neighboring parts of a
continuum may be the same in every respect, including location/.
But if all neighboring parts of a continuum are thus identical,
there seems to be no way to explain how they make up a continuum
rather than collapsing into a single point. How is it that we get
from the identity of neighboring points to a line with a left hand
and a right hand region? Peirce escapes the paradox by denying
that the relations of identity and otherness strictly hold in
respect to sufficiently small neighboring parts of a continuum
(NEM 3:747). This assertion demands some elaboration.
Parker goes on to cite "a loophole in Leibniz's Law," since
"difference is equated with discernable difference, that is, a
difference that /in principle/ could become apparent to a mind." If
"A and B are neighboring parts of a continuous line" that "are
sufficiently small to be immediately connected," they must be "shorter
than any specifiable positive length"; i.e., infinitesimal. "The only
difference between A and B must be in their location on the continuous
line of which they are parts. Because they are neighboring parts and
are connected, they have parts in common; because they are immediately
connected, they have /all/ their parts in common." Parker then
invokes Peirce's illustration from the third RLT lecture of cutting a
line at a point, such that it becomes two points. "It is clear that
if the original 'point' thus admits of being divided, it must be
divisible into parts of the kind we have been discussing. This
example illustrates that the immediately connected neighboring parts A
and B (which are at the same 'point' before the cut) must be ordered."
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Professional Engineer, Amateur Philosopher, Lutheran Layman
www.LinkedIn.com/in/JonAlanSchmidt
<http://www.LinkedIn.com/in/JonAlanSchmidt> -
twitter.com/JonAlanSchmidt <http://twitter.com/JonAlanSchmidt>
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