Re: [PEIRCE-L] Diagrams in mathematics, phaneroscopy, and language (was Modeling
John, List
Thank you for these enriching additions, which show the crucial importance
of diagrams in the discovery sciences. In this respect, I wonder about the
status of diagrams as practiced in Category Theory, well known for its
practice of "diagram chasing," which also diagrammatizes the mental
activity itself.
For example, when we prove a "small lemma of the four," we establish that
when certain hypotheses are gathered in commutative diagrams, a morphism is
a monomorphism.
Let the diagram :
https://upload.wikimedia.org/wikipedia/commons/thumb/c/cc/4_lemma_left.svg/300px-4_lemma_left.svg.png
Hypotheses: it is assumed that
1) that the above diagram is commutative
2) that both lines of the diagram are exact (the image of each morphism is
equal to the kernel of the following morphism)
3) that *m* and *p* are monomorphisms, and that *l* is an epimorphism.
Proof (this is like a game):
Let *c **∈**C* such that *n(c)* = *0; t(n(c)) = 0.*
the square *(C,D,C',D')* is commutative so *t(n(c)) =p(h(c))= 0*
*p* is a monomorphism so *h(c) = 0* (because the kernel of *p *is {0})
The above suite is exact, so there exists an element *b **∈**B* such that *g(b)
= c*.
The square (*B,C,B',C')* is commutative so *s(m(b)) = n(g(b)) = n(c) = 0*.
(because the image of B by *g* is equal to the kernel of *h*)
The lower suite is exact, so there exists an element *a' **∈** A'* such
that *r(a') = m(b*) (because the image of *A'* by r is equal to the kernel
of *s*).
Since *l *is an epimorphism, there exists *a* such that *l(a) = a'.*
The square *(A,B,A',B'*) is commutative so *m(f(a)) = r(l(a) = m(b).*
Since *m* is a monomorphism, *f(a) = b,* then *g(f(a) = g(b) = c.*
but trivially *g(f(a))= 0* so *c = 0*.
It follows that n is an epimorphism (since the kernel of* n* is {0}) *QED*
The semiotic interest of this "*diagram chasing*" is that we can follow it
step by step in the animated gif below in which *b1* and *b2 *distinguish
two instances of *b* :
https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Four_lemma_monic_case.gif/500px-Four_lemma_monic_case.gif
We can see that we start with any element *c *and that at the end of the
sequence of successive movements permitted, it becomes "0". It is thus an
iconic diagram in which the movement of thought is expressed, using the
hypotheses. These diagrams of the homological algebra made up only of
points and arrows indexicalized by letters are of a particular kind. What
semiotic status can we give them?
The use of diagrams do not stop here, and their semiotic interest does not
cease. For more information, see Diagram (category theory) - Wikipedia
.
Best regards,
Robert Marty
Honorary Professor; Ph.D. Mathematics; Ph.D. Philosophy
fr.wikipedia.org/wiki/Robert_Marty
*https://martyrobert.academia.edu/
RE: [PEIRCE-L] Diagrams in mathematics, phaneroscopy, and language (was Modeling
Edwina, Gary F, List, ET: Thank you for this excellent post... I'd like to note that I think a key problem with the arguments over 'where does mathematics or phaneroscopy fit into this process' is the old issue of the differentiation of Mind and Matter and their relations. ET: I feel that De Tienne separates the two in a Cartesian method - and slots each into a separate time and place. But Peirce, with his hylomorphic synechism doesn't separate the two and therefore, the one is always informing and analyzing the other. I think the quotes and outline you have provided show this quite clearly. The other issue is HOW Mind portrays our world to us - and your answer is: diagrams. That's a separate issue...and well argued by your comments. Those are important issues, which Peirce had resolved in a way that is consistent with Aristotle's response to Plato. Mathematical forms are real in the sense that they are independent of anything that anybody may think about them. The forms (or diagrams that represent them) are discovered, not invented by mathematicians, but they exist in actuality only when embodied. That embodiment may be on paper, in a computer, or in our neurons. GF: why [do] you bother to repeat all this, since its all been said before and nobody has questioned any of it. The only question I have is why you insert phaneroscopy in your new subject line, as there is nothing in the entire post about phenomenology/phaneroscopy in particular, because there is nothing in it that differentiates phaneroscopy from Peirce's thought in general. First, I am delighted that we all agree on the central role of diagrams in Peirce's thought. But Gary R did question it by accusing me of putting too much emphasis on diagrams -- because, as he said, diagrams are the foundation of my research on conceptual graphs. I wanted to emphasize that I learned the importance of diagrams from Peirce. GF: Its a good summary of the role of diagrams in Peirces thought, but it does nothing to explain the unique role of phaneroscopy in his classification of sciences or in his philosophy. Second, I wanted to emphasize that the central role of phaneroscopy is the transition from experience to diagrams. Contrary to ADT's slide 25, there is no transition out of mathematics, since diagrams can (a) relate experience to any pattern or structure of any branch of science or common sense, (b) allow mathematics and formal logic to be applied to any and every representation of 1-ness, 2-ness, and 3-ness, and (c) furnish all the data required for the normative sciences to evaluate the truth or relevance of hypotheses (guesses) to other diagrams from memories, reading, or dialogues with other people. GF: What does make [phaneroscopy] unique is precisely the subject of the current slow read of ADTs slides. Third, I have read each of the slides from ADT's original and from each of the transcriptions. I believe that he has made many important points. But as I showed about slide 25, he could have made his presentation more precise and more general if he had recognized the role of diagrams. In short, phaneroscopy is the process of mapping experience to diagrams that can be interpreted by all later sciences. The normative sciences evaluate them by the criteria of esthetics, ethics, and truth. Finally, ADT's phrase "the rest of us" suggested that Peirce's mathematics is inadequate to support common sense. Yet every textbook from elementary school to the most advanced research is illustrated with diagrams, which could be mapped to and from EGs. In particular, the diagrams that linguists use to represent the syntax and semantics of ordinary languages have a direct mapping to and from EGs. I also believe that some kinds of diagrams can even represent the exotic languages that Dan Everett has studied. For reference, I put the nine quotations from my previous note and my nine brief summaries in the file http://jfsowa.com/peirce/diagrams.txt John _ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
RE: [PEIRCE-L] Diagrams in mathematics, phaneroscopy, and language (was Modeling
John, I am puzzled as to why you bother to repeat all this, since it's all been said before and nobody has questioned any of it. The only question I have is why you insert "phaneroscopy" in your new subject line, as there is nothing in the entire post about "phenomenology/phaneroscopy in particular," because there is nothing in it that differentiates phaneroscopy from "Peirce's thought in general." It's a good summary of the role of diagrams in Peirce's thought, but it does nothing to explain the unique role of phaneroscopy in his classification of sciences or in his philosophy. Phaneroscopy is certainly not unique in requiring a previous study of mathematics. What does make it unique is precisely the subject of the current "slow read" of ADT's slides, especially the recently posted ones which you have chosen to ignore - which is your privilege, of course, but why do you claim to be saying something about "phenomenology/phaneroscopy in particular" when you are not? Gary f. From: peirce-l-requ...@list.iupui.edu On Behalf Of John F. Sowa Sent: 26-Aug-21 22:42 To: peirce-l@list.iupui.edu Subject: [PEIRCE-L] Diagrams in mathematics, phaneroscopy, and language (was Modeling Robert M, Gary F, Gary R, Jon AS, List, I changed the subject line to emphasize the role of diagrams in Peirce's thought in general and in phenomenology/phaneroscopy in particular. I cited some of these quotations in previous notes, and I copied others from a note by Robert. All of them are relevant to recent discussions with Gary, Gary, and Jon. At the end of this note, I include seven quotations by Peirce, and two by Cornelis de Waal. The following nine points summarize the issues that Peirce or de Waal make in those quotations. 1. In the first quotation, Peirce explains why "phaneroscopic research requires a previous study of mathematics." 2. "The results of experience have to be simplified, generalized, and severed from fact so as to be perfect ideas before they are suited to mathematical use." 3. A diagram is an "icon, which exhibits a similarity or analogy to the subject of discourse." 4. "we construct an icon of our hypothetical state of things and proceed to observe it... We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features." 5. A diagram may be "a concrete, but possibly changing, mental image of such a thing as it represents." 6. "We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible." 7. "Diagrammatic reasoning is the only really fertile reasoning." 8. According to de Waal, Peirce argues that it is the mathematician who is best equipped to translate the more loosely constructed theories about groups of positive facts generated by empirical research into tight mathematical models. 9. Finally, "The three mental qualities that in Peirce's view, come into play are imagination, concentration, and generalization." I agree with these nine points. And I emphasize that they are not just my opinions. The first eight are by Peirce himself. The ninth is de Waal's summary of quotations by Peirce. And by the way, I mentioned language as the third item in the subject line above. I plan to send another note to P-list to show the role of diagrams in representing the semantics of language. The roots of language are found in phaneroscopy, but they develop into diagrams that can represent the semantics of everything that people think or say. John -- 1. "Phaneroscopy... is the science of the different elementary constituents of all ideas. Its material is, of course, universal experience, -- experience I mean of the fanciful and the abstract, as well as of the concrete and real. Yet to suppose that in such experience the elements were to be found already separate would be to suppose the unimaginable and self-contradictory. They must be separated by a process of thought that cannot be summoned up Hegel-wise on demand. They must be picked out of the fragments that necessary reasonings scatter, and therefore it is that phaneroscopic research requires a previous study of mathematics. (R602, after 1903 but before 1908") 2. The results of experience have to be simplified, generalized, and severed from fact so as to be perfect ideas before they are suited to mathematical use. They have, in short, to be adapted to the powers of mathematics and of the mathematician. It is only the mathematician who knows what these powers are; and consequently the framing of the mathematical hypotheses must be performed by the mathematician.' (R 17:06) 3. there are three kinds of signs which are all indispensable in all reasoning; the first is the diagrammatic sign or icon, which exhibits a similarity or analogy to the subject of discourse. [second is index; third is symbol] (CP 1.369) 4. All necessary
Re: [PEIRCE-L] Diagrams in mathematics, phaneroscopy, and language (was Modeling
BODY { font-family:Arial, Helvetica, sans-serif;font-size:12px;
}John, List
Thank you for this excellent post. I think it clarifies the process
of 'How We Understand Our World'.
I'd like to note that I think a key problem with the arguments over
'where does mathematics or phaneroscopy fit into this process is the
old issue of the differentiation of Mind and Matter and their
relations.
I feel that De Tienne separates the two in a Cartesian method - and
slots each into a separate time and place.
But Peirce, with his hylomorphic synechism doesn't separate the two
and therefore, the one is always informing and analyzing the other.
I think the quotes and outline you have provided show this quite
clearly.
The other issue is HOW Mind portrays our world to us - and your
answer is: diagrams. That's a separate issue...and well argued by
your comments.
Edwina
On Thu 26/08/21 10:42 PM , "John F. Sowa" s...@bestweb.net sent:
Robert M, Gary F, Gary R, Jon AS, List,
I changed the subject line to emphasize the role of diagrams in
Peirce's
thought in general and in phenomenology/phaneroscopy in particular.
I
cited some of these quotations in previous notes, and I copied
others
from a note by Robert. All of them are relevant to recent
discussions
with Gary, Gary, and Jon.
At the end of this note, I include seven quotations by Peirce, and
two
by Cornelis de Waal. The following nine points summarize the issues
that Peirce or de Waal make in those quotations.
1. In the first quotation, Peirce explains why "phaneroscopic
research
requires a previous study of mathematics."
2. "The results of experience have to be simplified, generalized,
and
severed from fact so as to be perfect ideas before they are suited
to
mathematical use."
3. A diagram is an "icon, which exhibits a similarity or analogy to
the
subject of discourse."
4. "we construct an icon of our hypothetical state of things and
proceed
to observe it... We not only have to select the features of the
diagram
which it will be pertinent to pay attention to, but it is also of
great
importance to return again and again to certain features."
5. A diagram may be "a concrete, but possibly changing, mental image
of
such a thing as it represents."
6. "We form in the imagination some sort of diagrammatic, that is,
iconic, representation of the facts, as skeletonized as possible."
7. "Diagrammatic reasoning is the only really fertile reasoning."
8. According to de Waal, Peirce argues that it is the mathematician
who
is best equipped to translate the more loosely constructed theories
about groups of positive facts generated by empirical research into
tight mathematical models.
9. Finally, "The three mental qualities that in Peirce's view, come
into
play are imagination, concentration, and generalization."
I agree with these nine points. And I emphasize that they are not
just my opinions. The first eight are by Peirce himself. The ninth
is
de Waal's summary of quotations by Peirce.
And by the way, I mentioned language as the third item in the
subject
line above. I plan to send another note to P-list to show the role
of
diagrams in representing the semantics of language. The roots of
language are found in phaneroscopy, but they develop into diagrams
that
can represent the semantics of everything that people think or say.
John
--
1. "Phaneroscopy... is the science of the different elementary
constituents of all ideas. Its material is, of course, universal
experience, -- experience I mean of the fanciful and the abstract,
as
well as of the concrete and real. Yet to suppose that in such
experience the elements were to be found already separate would be
to
suppose the unimaginable and self-contradictory. They must be
separated
by a process of thought that cannot be summoned up Hegel-wise on
demand.
They must be picked out of the fragments that necessary reasonings
scatter, and therefore it is that phaneroscopic research requires a
previous study of mathematics. (R602, after 1903 but before 1908")
2. The results of experience have to be simplified, generalized, and
severed from fact so as to be perfect ideas before they are suited
to
mathematical use. They have, in short, to be adapted to the powers
of
mathematics and of the mathematician. It is only the mathematician
who
knows what these powers are; and consequently the framing of the
mathematical hypotheses must be performed by the mathematician.'
(R 17:06)
3. there are three kinds of signs which are all indispensable in all
reasoning; the first is the diagrammatic sign or icon, which
exhibits a
similarity or analogy to the subject of discourse. [second is
index;
third is symbol] (CP 1.369)
4. All necessary reasoning without exception is diagrammatic. That
is,
we construct an icon of our hypothetical state of things and proceed
to
observe it. This observation lea
