Jeff, Clark, list,
I needed to look around till I found that you meant "The Logic of
Mathematics: An Attempt to Develop My Categories from Within," and the
three questions posed near its beginning. Here's an online version (sans
italics, unfortunately)
http://web.archive.org/web/20090814011504/http://www.princeton.edu/~batke/peirce/cat_win_96.htm
In an earlier message you wrote,
[Begin quote]
1. What are the different systems of hypotheses from which
mathematical deduction can set out?
2. What are their general characters?
3. Why are not other hypotheses possible, and the like?
Drawing on Peirce’s way of framing these questions about the
starting points for mathematical inquiry, I’ve framed an analogous
set of questions about inquiry in the phenomenological branch of
cenoscopic science. How might the normative sciences help us answer
the following questions about phenomenology.
1. What are the different systems of hypotheses from which
phenomenological inquiry can set out?
2. What are the general characters of these phenomenological hypotheses?
3. Why are not other phenomenological hypotheses possible, and the
like?
[End quote]
I like that idea. I'm one for trying in an area to apply, in lockstep
analogy, a proceeding taken from another area.
Yet - pure-mathematical deduction starts out from hypotheses, but does
phaneroscopic (and, by extension, cenoscopic) analysis start out from
hypotheses? Off the top of my head, and maybe I'm wrong about this, it
seems to me that phaneroscopy a.k.a. phenomenology starts out from some
sort of discernments, noticings, of positive phenomena in general. These
discernments are not hypothetical suppositions or theoretical
expectations. I'm not sure what to call the formulation of such a
noticing or discernment, in the sense that a hypothesis formulates a
supposition and a theory formulates expectations.
Still I'll try a revision of the three questions in order to apply them
to phenomenology by lockstep analogy _/mutatis mutandis/_.
1. What are the different systems of discernments from which
phenomenological inquiry can set out?
2. What are the general characters of these phenomenological discernments?
3. Why are not other phenomenological discernments possible, and the like?
Does that make sense? Does it seem at all promising?
Best, Ben
On 10/29/2015 6:14 PM, Jeffrey Brian Downard wrote:
Hi Ben, Clark, List,
I'm working on an essay for the conference on Peirce and mathematics that Fernando has
organized in Bogota, and the topic is those three questions at the start of "The
Logic of Mathematics." In order to provide a coherent interpretation of what Peirce
is trying to do, my efforts are focused on writings from that same time period. So, I'm
drawing on the explanations of the relations between the parts of geometry in the last
lecture in Reasoning and the Logic of Things and the definitions he provides of
generation and intersection, uniformity and the like in his work on topology in the New
Elements of Geometry and Elements of Mathematics. If I am not mistaken, most of this of
this is from the same basic timeframe (around 1896-1898).
The discussion of the fundamental properties of space in the introduction to the latter
work was rejected by the editor as being too "philosophical" in character. It
looks to me like Peirce is drawing directly from William Benjamin Smith's Introductory
Modern Geometry of Point, Ray, and Circle. Peirce's copy of the text is available
through Google Books online. In the annotations in the introduction, Peirce fills in
missing words, so we know he was reading this section. It is interesting to compare
Smith's account of the fundamental properties of space with Peirce's account in the New
Elements. Here are some features that stand out when making the comparison. Both are
explaining how the mathematical conceptions of continuity, uniformity and the like are
drawn from common experience by a process of abstraction. In addition to refining the
explanations of those two properties, Peirce's account lays emphasis on the perissad
character of the mathematical space that is drawn from experience. Both characterize the
introduction of such things as a ray in terms of relations between the homoloids in the
space. When one set is taken to be dominant, we move from projective to metrical spaces.
The key idea for understanding the character of the hypotheses that lie at the
bases of both number theory and topology is that Peirce starts with a set of
precepts that tell us what to do in constructing a figuring and then putting
the parts into relation with one another. As the hypotheses are formulated,
additional precepts are derived that tell us what we are and are not allowed to
do next. I wonder: what lessons can we learn about the relationships that
hold between math and phenomenology by reflecting on the character of these
precepts? In what sense does the analysis of common experience involve
precepts that govern what we should and shouldn't do by way of making
observations?
Here is a particularly interesting passage (from a different time period) that
appears to bear on this kind of question:
We have, thus far, supposed that although the selection of instances is not
exactly regular, yet the precept followed is such that every unit of the lot
would eventually get drawn. But very often it is impracticable so to draw our
instances, for the reason that a part of the lot to be sampled is absolutely
inaccessible to our powers of observation. If we want to know whether it will
be profitable to open a mine, we sample the ore; but in advance of our mining
operations, we can obtain only what ore lies near the surface. Then, simple
induction becomes worthless, and another method must be resorted to. Suppose we
wish to make an induction regarding a series of events extending from the
distant past to the distant future; only those events of the series which occur
within the period of time over which available history extends can be taken as
instances. Within this period we may find that the events of the class in
question present some uniform character; yet how do we know but this uniformity
was suddenly established a little while before the history commenced, or will
suddenly break up a little while after it terminates? Now, whether the
uniformity observed consists (1) in a mere resemblance between all the
phenomena, or (2) in their consisting of a disorderly mixture of two kinds in a
certain constant proportion, or (3) in the character of the events being a
mathematical function of the time of occurrence--in any of these cases we can
make use of an apagoge from the following probable deduction:... (CP, 2.730)
This provides a really nice example of what it is to observe something like a
uniformity. It also provides some sense of how an analysis of the phenomena
might enable us to sort out--as competing hypotheses--the possibilities
represented in 1-3. What is more, the elements provide us with guidance (they
support the development of the precepts) needed to imagine the kinds of
experiments that could be run to sort through the competing explanations.
Stepping back from the particularities of the examples considered in this
passage, I think we get a nice articulation of how a phenomenological account
of the categories might supply us with the tools necessary to analyze the
observations necessary to support, via an abductive argument, a set of
conclusions in the normative theory of logic about what fair sampling really
requires under different kinds of conditions.
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Benjamin Udell [bud...@nyc.rr.com]
Sent: Thursday, October 29, 2015 1:10 PM
To: peirce-l@list.iupui.edu
Subject: Re: [PEIRCE-L] Peirce's categories
Jeff D., Clark, list,
I think it's important in this to get the quotes and dates. I recall
Peirce's views as changing, and partly it's his acceptance of changing
terminology. Earlier, he had regarded geometry as mathematically applied
science of space; later he accepted the idea that geometers were not
studying space as it is, but instead studying spaces as hypothetical
objects. Digging those quotes up is another little research project.
Best, Ben
On 10/29/2015 3:20 PM, Jeffrey Brian Downard wrote:
Clark, List,
You ask: I wonder how we deal with things like quasi-empirical methods in
mathematics (started I think by Putnam who clearly was influenced by Peirce in
his approach). Admittedly the empirical isn’t the phenomenological (or at least
it’s a complex relationship). I’m here thinking of mathematics as practiced in
the 20th century and less Peirce’s tendency to follow Comte in a fascination
with taxonomy.
Peirce draws on the distinction between pure and applied mathematics. When it
comes to geometry, for instance, only topology is pure mathematics. Both
projective geometry and all systems of metrical geometry import notions that
are not part of pure mathematics, such as the conception of a ray, or a rigid
bar.
When it comes to pure mathematics, he is just as concerned about getting
straight about the the kinds of observations we can draw on as he is concerned
about getting straight on this question for the purposes of a pure science of
cenoscopic inquiry. He makes the following point:
The first is mathematics, which does not undertake to ascertain any matter of
fact whatever, but merely posits hypotheses, and traces out their consequences.
It is observational, in so far as it makes constructions in the imagination
according to abstract precepts, and then observes these imaginary objects,
finding in them relations of parts not specified in the precept of
construction. This is truly observation, yet certainly in a very peculiar
sense; and no other kind of observation would at all answer the purpose of
mathematics. CP 1.240
So, I wonder, what kind of observation is it when a person observes the
relations between the parts of the imaginary (or diagrammed) objects and learns
something about the system that was not evident from the hypotheses and
abstract precepts that the reasoning took its start?
--Jeff
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