Jeff D., list,
I re-read my post below and realize that my main two points probably got
lost in the chatter, and they were in support of you. I meant
1. that it does make sense to speak of hypotheses in phaneroscopy since
it makes sense (in Peirce's later opinion and FWIW in mine) to speak of
reasoning in phaneroscopy, and
2. that (although I had raised loosely Peirce-based doubts about it in
earlier messages) mathematical hypotheses are comparable, in regard to
their potential for shedding explanatory light, with hypotheses in other
fields, so your desire to compare phaneroscopic hypotheses with
mathematical ones makes sense.
* * *
I didn't go far enough in saying how imaginary and complex numbers are
hypotheses that help simplify (and indeed explain) things about real
numbers. An example that I vaguely remembered from some book but didn't
dig up till the other day:
The impetus to study complex numbers proper first arose in the 16th
century when algebraic solutions for the roots of cubic and quartic
polynomials were discovered by Italian mathematicians (see Niccolò
Fontana Tartaglia, Gerolamo Cardano). It was soon realized that
these formulas, even if one was only interested in real solutions,
sometimes required the manipulation of square roots of negative numbers.
[End quote from https://en.wikipedia.org/wiki/Complex_number ]
As regards non-Archimedean numbers, I didn't know till yesterday that
p-adic numbers are non-Archimedean. They were regarded, like the
nonstandard reals later on, as playthings by some, but are now regarded
as co-equal to the reals, according to (peirce-l member) Joseph W.
Dauben in "Abraham Robinson and Nonstandard Analysis: History,
Philosophy, and Foundations of Mathematics" in _History and Philosophy
of Modern Mathematics_. If nonstandard reals are just another way of
talking about certain limits, why not vice versa? That idea is probably
riskier than I know, since I don't know enough about math. And then of
course there are further things, such as superreals and surreals.
Best, Ben
On 11/4/2015 9:41 AM, Benjamin Udell wrote:
Jeff D., Jon A., list,
Jeff, you wrote,
Peirce points out that inquiry in phenomenology is different in a
number of respects from inquiry in the other parts of philosophy.
He says: It can hardly be said to involve reasoning; for
reasoning reaches a conclusion, and asserts it to be true however
matters may seem; while in Phenomenology there is no assertion
except that there are certain seemings; and even these are not,
and cannot be asserted, because they cannot be described. CP
2.197 Having said that, he makes a very interesting remark:
“Phenomenology can only tell the reader which way to look and to
see what he shall see.” This remark makes me think that one of
the tasks of phenomenology might be to articulate precepts that
can guide us in making and analyzing our observations. As such, I
have a hunch that we might learn something by drawing a more
detailed comparison between the precepts that guide us in doing
math and the “precepts” that might guide our observational activities.
[End quote]
That remark by Peirce
http://www.commens.org/dictionary/entry/quote-minute-logic-chapter-ii-section-ii-why-study-logic-1
was made circa 1902 in "Minute Logic." Even then he goes on to say
"The question of how far Phenomenology does reason will receive
special attention." Two or so years later, circa 1904 he does allow of
proof (a kind of reasoning) in phaneroscopy
http://www.commens.org/dictionary/term/phaneroscopy. I'm not sure how
he arrived from the first position to the second; maybe he decided
that reasoning in phaneroscopy asserts a conclusion as true, not
_/despite/_ how things seem but instead _/on the basis of/_ how things
seem. Maybe reasoning isn't automatically supposed to assert a
conclusion as holding in spite of appearances. Reasoning could still
assert a conclusion as holding in spite of something. If the
question's other elements are brought back in, then the idea seems
that phaneroscopy asserts a conclusion as true, not _/despite/_ how
things seem, much less _/on the basis of/_ physical, psychological,
metaphysical, etc. explanations of the seemings, — but instead _/on
the basis of/_ how things seem _/even despite/_ various (physical,
psychological, metaphysical, etc.) explanations of the seemings.
On the comparability of mathematical hypotheses with hypotheses in
other fields, something that has occurred to me is that maybe Peirce
went too far in regarding mathematical hypotheses not as scientific
acts and leaving it at that. A context of this, in the history of
mathematics, has finally come back to me. Various new abstractions
were introduced, not quite as explanations, but still as simplifiers.
As Peirce points out, mathematicians dislike exceptions. In real
numbers without imaginary numbers, each positive number has two square
roots, one cube root, two fourth roots, etc., while each negative
number has no square roots, one cube root, no fourth roots, etc.
Thanks to imaginary numbers, every real number except zero has _/n/_
_/nth/_ roots, and so does every imaginary number and every complex
number, and a solution exists for every polynomial equation of one
degree or higher. Mathematicians resisted introducing or accepting
such abstractions unless they shed light on abstractions already
accepted. Some debate about non-Archimedean numbers is about how
seriously they should be taken if they don't help solve problems in
Archimedean numbers that can't be solved just with Archimedean
numbers. Dieudonné in his math article (in some version of the
Encyclopedia Britannica's 15th edition) talks about abstractionism as
a somewhat controversial movement in mathematics toward free-wheeling
abstraction. Maybe Peirce took anti-abstractionism as tending to block
inquiry and, in that light, thought that judging a mathematical
hypothesis, from the start, as a scientific act misses something about
the nature and role of mathematical hypotheses. Maybe he's right, but
that shouldn't stop us from seeing what a mathematical hypothesis,
postulate, etc., has in common with explanatory hypotheses in science
generally.
From "F.R.L. [First Rule of Logic]" CP 1.136
https://web.archive.org/web/20120106071421/http://www.princeton.edu/~batke/peirce/frl_99.htm
.
Although it is better to be methodical in our investigations, and
to consider the economics of research, yet there is no positive
sin against logic in trying any theory which may come into our
heads, so long as it is adopted in such a sense as to permit the
investigation to go on unimpeded and undiscouraged.
[End quote]
Best, Ben
On 11/2/2015 10:28 AM, Jeffrey Brian Downard wrote:
Hi Jon, Ben, Lists,
Jon seems to be suggesting that the relations of dependence between
math and the parts of philosophy that he sketches in a diagram (he
provides a link to a blog entry) helps to show where I might be
heading down the wrong track as I explore the relations between math,
phenomenology and the normative sciences for the sake of trying to
clarifying what phenomenology is and how we might use it for the sake
of doing math or philosophy. Quite a lot has been written on Peirce's
classification of the sciences. I won't try to review the literature.
Let me point out, however, that Jon's sketch is at odds with a number
of things Peirce says. Let me draw on some points that Beverley Kent
develops in her monograph on these classificatory matters. Here are a
few points to consider:
Phenomenology draws on mathematics. The relations between the
sciences are not simple. Rather, they are relatively complex. In
order to understand Peirce's classification, we have to consider the
different kinds of relations that obtain. Ben shared a table drawn
from Peirce's classification that can be used to help sort out a
number of the different kinds of relations. For the sake of brevity,
let's narrow things down to a relatively small number of questions:
a) Does one science appeal to another for its principles? If so, then
which?
b) Does one science appeal to another for its questions and problems?
If so, then how?
c) Does one science draw on the subject matter of another science for
some of its data?
d) Does one science depend upon another for assistance in analyzing
the data, identifying possible sources of observational error, or
correcting for such observational errors.
e) Does one science depend upon another for assistance in refining
its methods, or for ins truction in how to draw its inferences? If
so, then what kind of assistance is needed?
f) Does one science provide some of the tests needed to help confirm
the results arrived at in another science. If so, then what kinds of
tests are needed?
If Jon were to correct his diagram by showing that phenomenology
draws on math for its conceptions and principles, and then make the
lines of connections arrows that are going up, then he would have a
simple but helpful picture that we could use to help answer question
(a). Having said that, the diagram doesn't capture the relations
needed to answer questions b-f. In order to answer those questions,
we'd need to examine the following distinction that Peirce
emphasizes. As science can be thought of as a body of results. It can
also be thought of as a living community engaged in inquiry. The
answers we give to a-f will depend upon which of these two aspects of
a science we're talking about.
So, let me ask, what does phenomenology "draw" from the other
sciences--especially from math and the normative sciences? What does
phenomenology offer to the other sciences--especially math and the
normative sciences? Let us consider what phenomenology draws from
math. Mathematics studies the formal relations that obtain between
hypotheses concerning idealized states of affairs. Theorematic
reasoning in math relies, crucially, on the construction of diagrams
and experiments that we conduct by manipulating the parts of the
diagrams. Phenomenology can and should draw on conceptions from math
for the purposes of clarifying the kinds of formal relations that
might possibly obtain between the elements in our experience. This
shouldn’t be too surprising given the fact that the elementary
relations that comprise the hypotheses for mathematics are, of
course, drawn from experience.
Peirce points out that inquiry in phenomenology is different in a
number of respects from inquiry in the other parts of philosophy. He
says: It can hardly be said to involve reasoning; for reasoning
reaches a conclusion, and asserts it to be true however matters may
seem; while in Phenomenology there is no assertion except that there
are certain seemings; and even these are not, and cannot be asserted,
because they cannot be described. CP 2.197 Having said that, he makes
a very interesting remark: “Phenomenology can only tell the reader
which way to look and to see what he shall see.” This remark makes me
think that one of the tasks of phenomenology might be to articulate
precepts that can guide us in making and analyzing our observations.
As such, I have a hunch that we might learn something by drawing a
more detailed comparison between the precepts that guide us in doing
math and the “precepts” that might guide our observational activities.
Peirce concludes by adding this rejoinder: “The question of how far
Phenomenology does reason will receive special attention.” Any
suggestions for where in the text we might look to see where Peirce
gave the matter special attention? This passage is from the Minute
Logic, which is from 1902. I'm guessing that Peirce was intending to
take up that question more directly in another part of that work. In
chapter 3 of the Minute Logic (CP 4.227 following), Peirce is
developing the simplest parts of mathematics, which consist in the
dichotomic and trichotomic systems of mathematical logic. He is
drawing on a number of other areas of mathematics in the development
of these systems, including graph theory, various parts of algebra,
and some interesting analysis of different ways of putting things
into relations of correspondence that bear a striking similarity to
what has come, in the 20th century, to be called the functor relation
in category theory. Most of the discussion is just about math and
mathematical logic. Late in the chapter, however, he steps back at
4.318 and asks "What is experience?" This question of phenomenology
has, I believe, been bubbling beneath the surface of the discussion
throughout the entire chapter. What might this chapter teach us about
the kind of reasoning that that is needed in phenomenology?
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Jon Awbrey [jawb...@att.net]
Sent: Sunday, November 01, 2015 7:12 AM
To: Jeffrey Brian Downard; biosemiot...@lists.ut.ee; PEIRCE-L
Subject: Re: Peirce's Categories
Jeff, List,
I have to be traveling again and won't be able to get back to this,
except in mobile bits and snatches, until the middle of next week. I
collected a few links and thoughts pertinent to this part of the
discussion in the following blog post:
http://inquiryintoinquiry.com/2015/10/31/peirces-categories-%E2%80%A2-2/
Regards,
Jon
On 10/31/2015 3:47 PM, Jeffrey Brian Downard wrote:
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