Hi, Jeff D., list,
As regards phaneroscopy, I can see how the _/application/_ of
phaneroscopic findings to posterior fields could be taken as initially
hypothetical or initially tentatively inductive. I recall Peirce
somewhere saying that he found his three-category theory borne out and
supported by success in applications. Viewing the phaneroscopic
observations (or discernments) as hypothetical even within phaneroscopy
seems to involve some sort of inductive development within phaneroscopy
whereby one extends ideas found through such observations. Peirce said
some striking things about reasoning in phaneroscopy, Joe Ransdell said
some years ago at peirce-l that he didn't see how phaneroscopy could be
a science, and I remember arguing that Peirce didn't mean that one could
not reason about the phaneroscopic observations. Peirce says that the
phaneroscopic student sedulously avoids hypothetical explanations of any
sort http://www.commens.org/dictionary/term/phaneroscopy , but seem to
mean explanations in terms of physiology, physics, and that sort of
thing. In that quote Peirce also talks about combining minute accuracy
with the broadest possible generalization, and also says that
phaneroscopy "proves, beyond question, that a certain very short list
comprises all of these broadest categories [...]." So I think that
you've grounds in Peircean terms to talk about hypotheses in
phaneroscopy since you've stopped calling them the starting points.
I got myself to worrying, however, just how comparable pure hypothetical
mathematical assumptions are to explanatory hypotheses in other fields.
The hypotheses from which pure mathematics draws conclusions are not
explanatory hypotheses. Peirce says in CP 4.238 in the paper "The
Simplest Mathematics" (1902):
[....] Even the framing of the particular hypotheses of special
problems almost always calls for good judgment and knowledge, and
sometimes for great intellectual power, as in the case of Boole's
logical algebra. Shall we exclude this work from the domain of
mathematics? Perhaps the answer should be that, in the first place,
whatever exercise of intellect may be called for in applying
mathematics to a question not propounded in mathematical form [it]
is certainly not pure mathematical thought; and in the second place,
that the mere creation of a hypothesis may be a grand work of
poietic genius, but cannot be said to be scientific, inasmuch as
that which it produces is neither true nor false, and therefore is
not knowledge. [....]
[End quote]
The case of the pure-mathematical hypothesis seems to be that of "in the
second place." In Peirce's drafts of an intellectual autobiography in
1904 (see Ketner's chapter in _The Logic of Interdisciplinarity_),
Peirce says:
[....] Mathematics merely traces out the consequences of hypotheses
without caring whether they correspond to anything real or not. It
is purely deductive, and all necessary inference is mathematics,
pure or applied. Its hypotheses are suggested by any of the other
sciences, but its assumption of them is not a scientific act. [....]
[End quote]
On the other hand, Peirce in some places discusses the use of abductive
or hypothetical inference within mathematics. I wish I could dig one of
those quotes up right now but I'm sure they exist. Mathematicians often
use guesswork in the process of finding a proof, but usually don't
mention the guesses. Such abduced hypotheses, usually informal, seem the
mathematical counterpart of explanatory hypotheses, often
more-formalized, in other fields. The hypothetical assumptions from
which mathematics deduces, and which are pure, unfounded hypotheses,
seem something else. On the other hand, people arrive at such pure
hypotheses somehow by mental action, so it must involve inference
somehow. This question is murky to me. Even if Peirce ought to allow
that mathematical postulates are true in some realm of possibility (or
can at least be shown to be consistent-if-arithmetic-is-consistent,
etc.), and then allow that their formation or assumption is, after all,
some kind of scientific act, I'm not sure what such postulates explain,
as if one already had the mathematical conclusions and were looking for
postulates to entail them. Come to think of it, in "On the Logic of
Drawing History from Ancient Documents" (1901) EP 2:96
http://www.commens.org/dictionary/term/corollarial-reasoning , Peirce
says that a new abstraction introduced in the course of proving a
theorem ought to be supported with a proper postulate. That seems a case
of a postulate helping explain the truth of the thesis. But the only
proof related to supporting that postulate would be a proof that it
leaves the system as consistent (i.e., as possible) as it was before. So
again it seems different than a hypothesis outside mathematics. On the
other hand, it is because mathematics deduces from pure, unfounded
hypotheses that it can be applied in deducing from founded explanatory
hypotheses in other fields. The hypothesis founded outside mathematics
seems correlated to the hypothesis unfounded within mathematics in
standing at the start of deductions. Now, mathematical postulates should
generally be simple, independent but consistent, and so on, but should
lead to nontrivial results, and in that sense they should _/be/_
nontrivial, i.e., not so simple ("You can't get a ten-pound theorem out
of a five-pound set of postulates," or however the saying goes). If
we're to consider these postulates as hypotheses abduced somehow, how
does the value placed on natural simplicity in abductive conclusions fit
in here?
Well, I'm afraid I've brought more murk than light to this subject.
Best, Ben
On 10/31/2015 4:52 AM, Jeffrey Brian Downard wrote:
Hello Ben, List,
I was particularly interested in the prospect of making a comparison between
the hypotheses that we are working with in mathematics and the hypotheses that
we are working with phenomenology. There are good reasons to point out, as you
have, that the hypotheses in phenomenology are based on something that is, in
some sense prior. Call them, if you will, particular discernments.
Having searched around a bit, I don't see a large number of places where Peirce uses this
kind of language when talking about phenomenology. Having said that, here is one:
"Philosophy has three grand divisions. The first is Phenomenology, which simply
contemplates the Universal Phenomenon and discerns its ubiquitous elements...." (CP
5.121)
There are interesting differences between the ways that we arrive at the hypotheses that serve as
"starting points" for mathematical deduction, and ways that we arrive at the hypotheses
that are being formulated in phenomenology. One reason I retained the language of "starting
points" that was in the original questions that Peirce asked about mathematics is that
hypotheses are, at heart, quite closely related to the questions that are guiding inquiry. We
normally think of hypotheses as explanations that can serve as possible answers to some questions.
In some cases, I think it might be better to think of the formulation of the questions were trying
to answer as itself a kind of hypotheses..
We can ask the following kinds of questions about hypotheses in math, phenomenology,
normative science and the like. What are we drawing on when we formulate these
hypotheses? How should we develop the hypotheses from the "stuff" that we are
drawing on so that the hypotheses we form will offer the greatest promise as we proceed
in our inquiries.
With these kinds of issues in mind, let me rephrase the questions about
phenomenology so as to respond to the concern you've raised:
1. What are the different kinds of hypotheses that might be fruitful for
phenomenological inquiry?
2. What are the general characters of these phenomenological hypotheses?
3. Why are not other phenomenological hypotheses possible, and the like?
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Benjamin Udell [bud...@nyc.rr.com]
Sent: Friday, October 30, 2015 7:14 AM
To: peirce-l@list.iupui.edu
Subject: Re: [PEIRCE-L] Peirce's categories
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 dra!
wn 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 bu!
t this u
niformity 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<mailto:bud...@nyc.rr.com>]
Sent: Thursday, October 29, 2015 1:10 PM
To: peirce-l@list.iupui.edu<mailto: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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