Gary F., Gary R., Cathy Legg, John Kaag, Jerry, List,
Jerry says: "My personal feeling about your exposition is that such a view of
material and formal categories leads one into an extra-ordinarily deep philosophical
morass from which you may never emerge."
At the Congress, several people expressed a worry about falling "down the rabbit hole" when studying Peirce. The concern was that spending too much time on the difficult parts of the more challenging essays threatened to pose insurmountable problems in making sense of what Peirce is up to.
Despite your warnings, I will have to trust my own judgment in determining when it makes sense for
me to press on when it comes to the more challenging texts and arguments. My conviction is that
Peirce often is trying to teach us how to employ specific methods in doing philosophy, and that
we'll struggle in our attempts to understand him so long as we lack the experience and skills he
possessed. I dont know about you, but this puts me in a tough position, because I seem to lack
much of his experience and skills. While Peirce tried to put many things in the simplest possible
terms, he often takes it for granted that the reader will "actively think" and draw on
his sentences as "so many blazes to enable him to follow the track of the reader's
thought." (EP, 301)
Reading Peirce presents a challenge. As many scholars have pointed out, he was
a remarkably talented logician, and he possessed an intimate familiarity with
the mathematics of the 19th century and its larger history. What is more, he
was a practicing scientist who had a rich understanding of how to do and not
merely read chemistry, astronomy, classificatory biology, and geodesy. In
addition to being a special scientist working in multiple fields, he had a
synoptic sense of the history philosophy and the conceptual landscapes
represented by different philosophical systemsalong with a rich appreciation
of the different worldviews that philosophers might try to explore. Above all,
he was a student of methodology, and his aim was to develop a systematic method
for improving the methods of inquiry.
Turning from these remarks about the difficulties one faces in trying to understand
Peirce's views--especially the more difficult arguments expressed in the more challenging
texts--to the task of reconstructing some of Peirce's arguments in the text of "New
Elements (Kaina Stoicheia)", let's take a look at the text itself. There are three
main sections. The first contains biographical remarks about the textbook he wrote on
the logic of mathematics--taking topology, projective geometry and metrical geometries as
its subject matter. The second contains a statement of the distinction between
definitions, postulates, axioms, etc. The third, which is the longest section, is
divided into 4 sub-sections. You quote from the fourth and longest of these subsections.
What is Peirce doing in the passage you've quoted? It is possible that we are
reading the text somewhat differently. Let me provide a few of comments about
what he is doing in the pages leading up to the passage you've quoted so that
we might clarify some of the differences in our approaches. I note that you've
quoted the passage, but you've said precious little about what you think is
going on here. You refer to an earlier post by Clark, so perhaps I could turn
to what he says at some later time in an attempt to understand your remarks.
So, in parts I and II, Peirce starts by referring to his own work on the logic
of mathematics. By the fourth part of section III, he has moved from a
discussion of speculative grammar and critical logic to a series of examples
drawn from the theoretical and the practical sciences. You seem to be
particularly interested in his remarks about the various specific uses of the
concepts of cause and effect, including internal and external causes, along
with material, formal, efficient and final causes. He has an exceptionally
long paragraph on the topic starting on page 313 and ending on 316. The point
of this little foray on the different causes is not to argue for big
metaphysical conclusions. He's made those arguments elsewhere. And, he says
as much: Yet I refuse to enter here upon a metaphysical discussion. (EP, )
As he points out in the opening sentence of this paragraph, everything he says here is designed to clarify the distinction between a proposition and an argument. His goal, I think, is to illustrate how we should go about classifying different acts of cognition (e.g., as an act of interrogating, affirming or arguing) and then ascertaining the nature of those acts. So, the question is something like this:
1) If the act is one of affirming an assertion, then what is involved in affirming that the proposition true?
Or this:
2) If the act is one of arguing for a conclusion from a set of premisses,
then what is involved in affirming that the argument is valid?
He is also asking the question: How can we put our questions to nature and get
a reasonable answer? That is, how can we find out what is really the case?
These sound like questions of metaphysics, but he is focusing on a set of
questions that surface in the theory of logic. Namely, what hypotheses
concerning the nature of what is real should we adopt for the sake of
understanding the validity of deductive, inductive and abductive inferences?
He has argued that we need, for the sake of making valid deductive arguments,
to adopt a nominal definition of the real. He sees that induction and
abduction requiring richer hypotheses concerning the real.
Here are some things that he says about the hypotheses that are required for the sake of making valid abductive inferences:
Abduction . . . is the first step of scientific reasoning, as induction is
the concluding step.
In abduction the consideration of the facts suggests the hypothesis. In
induction the study of the hypothesis suggests the experiments which bring to
light the very facts to which the hypothesis had pointed. The mode of
suggestion by which, in abduction, the facts suggest the hypothesis is by
resemblance, -- the resemblance of the facts to the consequences of the
hypothesis. The mode of suggestion by which in induction the hypothesis
suggests the facts is by contiguity, -- familiar knowledge that the conditions
of the hypothesis can be realized in certain experimental ways.
I now proceed to consider what principles should guide us in abduction, or the
process of choosing a hypothesis. Underlying all such principles there is a
fundamental and primary abduction, a hypothesis which we must embrace at the
outset, however destitute of evidentiary support it may be. That hypothesis is
that the facts in hand admit of rationalization, and of rationalization by us.
That we must hope they do, for the same reason that a general who has to
capture a position or see his country ruined, must go on the hypothesis that
there is some way in which he can and shall capture it. We must be animated by
that hope concerning the problem we have in hand, whether we extend it to a
general postulate covering all facts, or not.
We are therefore bound to hope that, although the possible explanations of our
facts may be strictly innumerable, yet our mind will be able, in some finite
number of guesses, to guess the sole true explanation of them. That we are
bound to assume, independently of any evidence that it is true. Animated by
that hope, we are to proceed to the construction of a hypothesis. (CP 7.218-19)
Given the fact that the primary subject matter of the New Elements essay is
the normative science of logic, let us ask: what are the data (i.e., the
observations) for generating hypotheses in logic and then putting them to the
test? As we seek an answer the question, I believe that we need to focus our
attention on the data part of the equation. As he says, the logician has to
be recurring to reexamination of the phenomena all along the course of his
investigations. (EP, 311)
In the paragraphs leading up to his remarks about atomic weights, he considers the following examples: a psychologist studying the experience of déjà vu, a logician studying of the experience of similarity and resemblance, a seamstress buying fabric from a shopkeeper, a homeowner buying a piece of furniture, and a chemist studying the weight of gold. What is the point of these examples? Much of Peirces attention is fastened on the question of how we should arrive at a more scientific understanding of the conditions for making measurements. How should we measure a psychological feeling, or a length of silk, or a the size of a piece of furniture, or the chemical weight of an elementor the degree to which one feeling (or other idea) is, logically speaking, similar to an another.
In some comments on The Basis of Pragmatism in the Normative Sciences, I
forwarded the claim that Peirces phenomenology is, at least in part, an
attempt to answer the following question: what are the formal features in
experience that are necessary for us to draw valid synthetic inferences from
our observations? This is not an easy question to answer. Were looking for
an answer because we want to understand how it is possible to put the qualities
weve observed in a transitive ordering and make comparisons based on the
degree to one resembles or does not resemble another. Id like to add the
following to what Ive said thus far: discovering the formal conditions for
putting things in such a transitive order and comparing them are essential
aspects of what is needed to measure them.
The point he is making about using a yard stick to measure length is analogous
to the point he is making about using a standard for measuring the chemical
weight of gold. In order to make measurements of length, we use something that
is like a rigid bar that can be moved up and down the thing we are measuring
(so that the finite length of the bar does not matter for purposes of making
the measurements). The remark that caught my attention is where he says that
our theory of measurement is based on the idea that we need something that can
serve as a more universal standard. In an effort to make our standard more
universal, scientists have designated one particular bar in Westminster as the
object to which our concept of yard refers. In order to determine whether or
not any other yardstick we might use will lead us into error, we canas a
matter of principlecompare it to the protypical standard in Westminster.
Is this the best way to fix the reference for the concept of a yard? Peirce thinks it is not the best way to remove some of the errors that will crop up in the process of making measurements of length. Instead of relying on a single prototype sitting in a case in Westminster, we should rely on an average taken from a number of different bars made of different materials and kept under different conditions (e.g., at different ranges of temperature). We use the concept of yard in such a way that it refers to the mean length of them all. This is the same kind of thing that a biologist does when she compares a number of different specimens and draws up a conception of a type-specimen as a kind of typical thing that has a normal size and shape.
What is the weight of gold? In saying that it is an elementary chemical substance having a particular atomic weight of about 197 ¼, we are relying upon some kind of standard in making the comparison. The standard, of course, is the atomic weight of hydrogen, which is taken to have a weight of 1. What is it to say that the weight of hydrogen is 1 unit? His answer is that, in comparison to air, it is about 14 ½ times lighter.
In this passage, is Peirce making some kind of metaphysical point about the
deeper logic of the chemical elements? I dont think so. Rather, he is
making a point about what is needed to make comparisons between thingsand then
he is asking what is needed to set up a standard for measuring those things.
The system of measurement set up by Dalton in 1803 was a relative scale that
used the weight of hydrogen as the base unit. Technically speaking, scientists
could say that the mass of hydrogen was exactly one only because it was the
serving as the base unit of measurement in a relative scale. It would not
serve the goals of the scientists to say that the concept of the weight of
hydrogen refers to protypical sample stored in a glass case in Westminster.
Rather, the weight of hydrogen, like the length of a yard, should be taken to
refer to a mean over many observations of the relative weights of gold, carbon,
hydrogen and other elements.
What does this have to do with the normative theory of logic? I believe that
it bears on logic in two ways. First, I believe that an analysis of the things
we observein chemistry, biology, the selling of fabric, etc.requires us to
examine the underlying grounds for making measurements of the various
phenomena. We can draw on mathematics, phenomenology and logic in order to
deepen our understanding of what is necessary to apply one or another kind of
measurement to a given kind of phenomena that has been observed in one or
another of the practical or theoretical sciences. Second, this kind of
question surfaces when we ask what the standards are for analyzing the
phenomena were drawing on in the theory of logic. Peirce says as much in his
discussion of what is needed to make something as simple as a comparison
between two qualities of feeling. Take, for instance, a comparison between two
experiences of the color of blue. In the hospital room where Im sitting with
my daughter, there is a stool and a sheet that have just about the same hue.
From this point on, I will probably refer to this shade of color as hospital
blue. When I compare the intensity of the color I experience when looking at
the stool with the color I experience when looking at the sheet, it seems to me
that the color of the stool is remarkably more intense than the color of the
sheet. The two objects are across the room from each other, so all I can do is
to compare the intensity of the one with my memory of the intensity of the
other. What are my grounds for making such a comparison?
One of the points Peirce is making at this point in subsection 4 is that the
comparison of the intensity of two experiences of the quality of blue is
something that is measured chiefly by aftereffects. (EP, 320) He is laboring
over this point, I believe, because he is keenly interested in set of related
issues. Consider, for instance, the following questions:
1) What is the standard that we can use when comparing the feeling that an argument is a good inference to the feeling that an argument is an invalid inference? Isnt this similar in some respects to comparing the intensity of a one experience of a feeling of blue to another feeling of blue? Isnt it different in other respects?
2) Once we have formed a class of sample arguments that we take to be good
and a class that we take to be bad, what kind of measurements can be made when
comparing these classes? At the very least, we can apply a nominal scale in
saying that they are labeled as different classes. For the sake of the logical
theory, however, we need a stronger standard of measurement, dont we?
3) What is the standard for making the comparison of the goodness or
badness of an argument? Should we take it to be a prototypical argument that
appears to be beyond criticism? Perhaps we should take an argument, such as a
cogito argument, or an ontological argument for Gods reality, or an argument
for the indubitability of the axioms of logic as a prototype, and then place
one or another of these arguments in a glass case in Westminster. I suspect
that this would fail to serve the purpose we have in removing possible errors
from our measurements of the goodness or badness of any given argument.
How can the examples of measuring silk against a yardstick, comparing
biological specimens to a type-specimen , and comparing the weight of carbon
and gold to hydrogen help us think more clearly about the grounds we having for
comparing arguments and saying that one class contains a sample of good
inferences and that another class contains a sample of bad inferences In
making such comparisons, we need something more than just a nominal assignment
of the term good to one class and bad to another. Having said that, dont
we need more than an ordinal scale that enables us to make relative comparisons
of goodness and badness? How might we arrive in our theory of logic at a
standard of measuring the validity of inferences that is richer than a nominal
or ordinal scale? After all, we are relying on our standards for comparing
arguments for the sake of arriving at conclusions about what, really, is true
and false.
These are the kinds of questions that Im particularly interested in trying to answer. My hunch is that, rabbit hole or not, Peirce is pointing us to the resources needed to answer these kinds of questions. As he points us in a specific direction, however, he is assuming that we will "actively think" and draw on his sentences as "so many blazes to enable him to follow the track of the reader's thought." The real danger is not one of following the blazes and heading down the rabbit hole. Rather, it is one of sticking with our personal assumptions and convictions in such a fashion that we make ourselves impervious to the fruitful suggestions that are around us and, in doing so, fail to see that we are sitting in a hole of our own making with no sense of which direction is up and which is down.
That, at least, is my abiding worry. Hopefully, it is one that will spur me to
active inquiry.
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Jerry LR Chandler [[email protected]]
Sent: Friday, August 22, 2014 3:35 PM
To: Peirce List
Cc: Jeffrey Brian Downard
Subject: Re: [PEIRCE-L] Phaneroscopy, iconoscopy, and trichotomic category
theory
Dear Jeff:
Thank you for your exposition on your views on the relations between material
and formal categories.
(From your post below)
First off, if things are sounding mystical to your ears, I hope it is a by
product of the richness of the ideas Peirce is examining--and not a by-product
of the comments I'm offering.
Your hopefulness is partially realized. And partially not.
Your may recall Clarks perceptives postings from Kainia Stoichia on CSP views
on causality. In subsequent sentences, CSP gives a crisp example of his
deductions about relations between gold (as a relative weight) when compared to
hydrogen and then to air.
"What is gold? It is an elementary substance having an atomic weight of about
197¼. In saying that it is elementary, we mean undecomposable in the present state
of chemistry, which can only be recognized by real reactional experience. In saying
that its atomic weight is 197¼, we mean that it is so compared with hydrogen. What,
then, is hydrogen? It is an elementary gas 14¼ times as light as air. And what is
air? Why, it is this with which we have reactional experience about us. The reader
may try instances of his own until no doubt remains in regard to symbols of things
experienced, that they are always denotative through indices; such proof will be far
surer than any apodictic demonstration. From KS.
This crisp example of material and formal categories (and the logical phenomena
inferred by mathematics) about material categories is worthy of careful study.
He presents a logic of relatives. Classification of categories inevitably
brings forth issues of causality, Aristotelian or otherwise, which he
illustrates. You may find it useful to contrast this example with other direct
examples from biology or medicine as you pursue your thinking about these
matters.
My personal feeling about your exposition is that such a view of material and
formal categories leads one into an extra-ordinarily deep philosophical morass
from which you may never emerge. For me, the choice of rhetorical terms in
your exposition leads not to calculations but to a Luciferic network of
semantic entanglements.
Thanks again for clarifying your thoughts.
Cheers
Jerry
On Aug 22, 2014, at 1:39 AM, Jeffrey Brian Downard
<[email protected]<mailto:[email protected]>> wrote:
On Wed, Aug 20, 2014 at 3:05 PM, Jeffrey Brian Downard
<[email protected]<mailto:[email protected]><mailto:[email protected]>>
wrote:
Hi Jerry, List,
First off, if things are sounding mystical to your ears, I hope it is a by
product of the richness of the ideas Peirce is examining--and not a by-product
of the comments I'm offering.
To a large degree, the answers to the questions you are trying to raise are
going to be found in the larger story that is articulated in the theory of
semiotics. At this point, I am trying to offer some comments on some of
Peirce's explanations and definitions as a kind of run up to the
phenomenological categories--and especially the distinction between the formal
and material aspects of those categories. The general suggestion I'm making is
that Peirce is not providing two entirely separate lists of the categories, one
formal and that other material. Rather, there is a close connection between the
two even if they do not, in experience, match perfectly because our experience
of the material categories of quality, brute fact and mediation is always so
richly complex. My general suggestion may seem controversial because some
interpreters seem to be offering a different reading of the relevant texts.
Confining myself to the subject of the phenomenological categories and the role of
mathematics in informing our understanding of the essential formal elements of the monad,
dyad and triad, I do take Peirce to be offering an account of the elements needed for
setting up the frameworks necessary for referring to grounds, objects and interpretants.
One might call them three interrelated "frames of reference."
What do the signs that we use in mathematics refer to? Much depends upon
whether we are using the signs to seeks answer to questions in pure or applied
mathematics. Let's consider the case of pure mathematics. What do the signs
used in topology refer to? In the account he offers in the New Elements, the
key operations for setting up a system of mathematical diagrams are those of
generation and intersection. These are the operations used to generate a line
by moving a particle from a point, or for determining the location of a point
on a line by intersecting it with another line.
As we try to understand the conditions that make it possible for the different representations to
refer, we'll need to be clear in identifying the representations we're talking about. It is one
thing to ask: what does that particle in the diagram that is being moved refer to? It is another
thing to ask, what does the symbol "particle" refer to? I hope it is clear that the
conditions under which the symbol "particle" refers is dependent, in many respects, on
the conditions under which the iconic particle that is draw on the page is able to refer. As a
hypo-icon, the particle we move as we draw the line is remarkably rich as a sign. At any time in
the act of drawing the line on the paper, there are qualisigns, sinsigns and legisigns working
together so that the particle can function as a rich sign complex in a larger process of
interpretation. What is more, the particle embodies the idea of a generator. That is, it embodies
a more general rule that determines how we might generate innumerable other possible lines from the
point This is a more general rule that enables us to interpret the larger mathematical space in
which the line is being constructed. It enables us to understand how one line my be transformed
continuously to give us a line that is homeomorphic with the first, or how various kinds of
discontinuities might be introduced to give us another different line altogether.
I hope you can see that I'm trying to bracket some of the questions you've
raised about the role of real things (i.e., chemical compounds, protein or DNA
molecules, and the like) in serving as the grounds or objects to which one or
another kind of representation might refer. I'm bracketing those questions for
a reason. I'd like to keep the phenomenological analysis of the conditions
under which the signs used in pure mathematics refer free from big metaphysical
assumptions about what is really the case as a positive matter of fact. There
is a long line of philosophers who have tried to import such metaphysical
assumptions into their accounts of the reference and meaning of the signs used
in math and formal logic (e.g., Mill, Quine, etc), but Peirce is resisting this
move--at least until we're ready to address questions in metaphysics. Once we
are ready and we're using the methods appropriate for answering questions in
metaphysics, we'll need to think about the real nature of an ideal system of
mathematical definitions, hypotheses, theorems, etc., and what it is for that
system to be real as a rich and consistent network of possible formal relations.
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Jerry LR Chandler
[[email protected]<mailto:[email protected]><mailto:[email protected]>]
Sent: Tuesday, August 19, 2014 9:28 PM
To: Jeffrey Brian Downard
Cc: Peirce List
Subject: Re: [PEIRCE-L] Phaneroscopy, iconoscopy, and trichotomic category
theory
Jeffrey:
Your posts become increasingly mystical.
This is not a judgement, merely an observation from a philosophy of mathematics
perspective.
At issue is how to you assign meaning to mathematical symbols.
In particular, in light of K.S. and his comments on the meaning of number in
the context of his description of gold?
More to the point, does the meaning of mathematical symbols reside in
mathematics itself or do the meanings refer to the reference systems for the
symbol system, that is the application to a particular material reality, such
as the atomic numbers? Or the sequence numbers for a genetic sequence? Or
protein sequence?
In yet other terms, does the concept of order infer a universal meaning or a
meaning dependent on the nouns of the copulative proposition?
Perhaps you can address these vexing issue?
Cheers
Jerry
On Aug 19, 2014, at 8:28 PM, Jeffrey Brian Downard
<[email protected]<mailto:[email protected]><mailto:[email protected]>>
wrote:
Gary F., Gary R., List,
In an effort to think a bit more about the form/matter distinction as it
applies to the phenomenological categories, let me add few comments about an
explanation that Peirce provides concerning the mathematical form of a state of
things. I'd like to add some remarks about this explanation because I think it
offers us a nice way of responding to a concern Gary F. raised. Here is the
concern:
Gary F. says: "Jeff, Im interested in your question, 'is there any kind of formal
relation between the parts of a figure, image, diagram (i.e., any hypoicon) that does not
have the form of a monad, dyad or triad?' . . . I confess that I have no idea how we
would go about investigating that question."
My initial response was: "The answer to the question involves the whole of Peirce's
semiotic--and not just his account of the iconic function of signs. So Peirce is
bringing quite a lot to bear on the question. For starters, however, I think we should
consider the examples he thinks are most important in formulating an answer. What Peirce
sees is that, in mathematics, the examples we need are as 'plenty as blackberries' in the
late summer. (CP 5.483) What do you know, it is late August. Let's go picking."
As a first stop on our way to the briar patch, let's consider the following definition
from "The Basis of Pragmaticism in the Normative Sciences."
"A mathematical form of a state of things is such a representation of that state of
things as represents only the samenesses and diversities involved in that state of
things, without definitely qualifying the subjects of the samenesses and diversities. It
represents not necessarily all of these; but if it does represent all, it is the complete
mathematical form. Every mathematical form of a state of things is the complete
mathematical form of some state of things. The complete mathematical form of any state of
things, real or fictitious, represents every ingredient of that state of things except
the qualities of feeling connected with it. It represents whatever importance or
significance those qualities may have; but the qualities themselves it does not
represent." (EP, vol. 2, 378)
Peirce suggests that this explanation is "almost self-evident." At this point in his
discussion, however, he merely ventures the explanation as a "private opinion." I cite
this passage because it bears directly on the question of how our understanding of the mathematical
form of something such as a figure or diagram is supposed to inform our understanding of the formal
categories of monad, dyad and triad (or, firstness, secondness, thirdness)--and how we might use
those categories in performing a phenomenological analysis of something that has been observed.
Peirce says that he has introduced this explanation in order to account for the
emphatic dualism we find in the normative sciences. The dualism is especially
marked in logic and ethics (e.g., true and false, valid and invalid, right and
wrong, good and bad), but it is also found in aesthetics. As such, he is
noticing a phenomena that has been widely observed to be a part of our common
experience in thinking about how we ought to act and think, and he is getting
ready to venture a hypothesis to explain what is surprising about the
phenomena. The explanation of the dualism that follows might seem a bit hard to
make out, but I think it is clear that this is what he is trying to do.
That might have seemed a bit opaque, so let me try to restate the point. I
think Peirce is drawing on an understanding of mathematical form for the sake
of performing an analysis of a particular phenomenon that calls out for
explanation. We need to see what it is in the phenomena (i.e., the dualism in
the normative sciences) that really calls out for explanation. Otherwise, we
will not have a clear sense of whether one or another hypothesis is adequate or
inadequate to explain what needs to be explained.
He says the following about his account of the mathematical form of a state of a things:
"Should the reader become convinced that the importance of everything resides
entirely in its mathematical form, he too, will come to regard this dualism as worthy of
close attention?"
Why does Peirce say that the importance of everything resides in its
mathematical form? On my reading of this passage and what follows in the next
several pages of the essay, I think he is developing the claim I asserted
above. That is, every kind of formal relation that might be found between the
parts of a figure, image, diagram and the space in which such things are
constructed must have the form of what we are calling, in our phenomenological
theory, a monad, dyad or triad.
It might sound ridiculous to suggest that the dualism present in our experience of what is
valid or invalid as a reasoning or what is right or wrong as an action can be clarified by
using a mathematical diagram, such as a drawing on a piece of paper of two dots that we might
count by saying "one' and "two," but he says that we shouldn't disregard such a
suggestion. He has argued elsewhere that every observation we might make must involve some
kind of figure or diagram--and the form of such a figure or diagram can be understood in terms
of having the structure of a skeleton set (CP, 7.420-32), or a network figure (CP, 6.211), or
some other kind of really basic mathematical structure. I refer to those particular
mathematical structures because the first can be applied to things in our experience that are
more discrete in character, and the second can placed over things more continuous in character.
Do you buy his claim here? Does the "importance of everything reside in its
mathematical form?" The argument he offers in the rest of section B is worth a look.
--Jeff
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