Mathematical Method

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Jon Awbrey

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Aug 4, 2020, 11:26:29 AM8/4/20
to Cybernetic Communications, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Mathematical Method • Discussion 1
http://inquiryintoinquiry.com/2020/08/04/mathematical-method-discussion-1/

Re: Peirce List ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/thrd1.html#00001 )
::: John Sowa ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/msg00001.html )

Dear John,

Thanks for the notice of Carolyn Eisele's article — it's always worth reading what she has to say. We've had
discussions of Peirce's distinction between theorematic and corollarial reasoning before and I know there's a
respectable amount of literature out there about it. The subject has curiously enough come up just recently in
discussions on Facebook and Academia.edu, mostly on account of points brought up by John Corcoran. It's also related to
a number of discussions I've had over the years about the difference between “insight” proofs and “routine” proofs,
partly in connection with theorem proving apps and Peirce's logical graphs. Usually these discussions take off into the
stratosphere of high-sounding blue-skying about Gödel incompleteness and all that ... but I'm trying to keep my focus on
more nuts-and-bolts matters at the moment and I'll try to avoid going off on those planes.

Regards,

Jon

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Jon Awbrey

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Aug 5, 2020, 7:40:17 AM8/5/20
to Cybernetic Communications, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Mathematical Method ??? Discussion 2
http://inquiryintoinquiry.com/2020/08/05/mathematical-method-discussion-2/
::: Auke van Breemen ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/msg00002.html )

AvB:
It seems to come down to: never consider the textual production
of a scientist only in itself, but also look at the reality the
text tries to explain.

Dear Auke,

Exactly!

We interpret texts
in relation to
the object in view.

Regards,

Jon

Jon Awbrey

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Aug 7, 2020, 2:14:12 PM8/7/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Mathematical Method • Discussion 3
http://inquiryintoinquiry.com/2020/08/07/mathematical-method-discussion-3/

All,

Here's a revision, hopefully clearer, of a previous comment on
the Peirce List, part of a discussion stemming from John Sowa's
citation of an article by Carolyn Eisele.

Eisele, C. (1982), “Mathematical Methodology in the Thought
of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.
Online: https://www.sciencedirect.com/science/article/pii/0315086082901276/
PDF: https://www.sciencedirect.com/science/article/pii/0315086082901276/pdf

Auke van Breemen wrote:

AvB: It seems to come down to: never consider the textual production
of a scientist only in itself, but also look at the reality the text
tries to explain.

I took this as an admirably succinct statement of the difference
between:
(1) scriptural hermeneutics — I'd call it “corpus hermeneutics”
except for the risk of confusion with Corpus Hermeneticum — and
(2) scientific interpretation, that is, any development of
interpretant texts in relation to an independent object domain
with the aim of forming true descriptions or gaining knowledge
of that domain.

What I wrote in response to Auke was this:

JA:
Exactly!
We interpret texts
in relation to
the object in view.

All I did there was mention the three roles in a sign relation.
We take in texts or whole bodies of work as signs of an object
domain and we form interpretive texts as signs of the same domain.
For my part, I interpret Peirce's work as signs of an object world,
one with respect to which other writers, artists, signifactors of
all sorts have generated signs worthy of our interpretation.

I wouldn't take the words “in view” too literally. I just as
easily could have said “at hand” or “in mind” but I went with
“object in view” on account of the fondness one of my old
teachers had for Dewey's signature “end in view”. As far as
indicial signs are concerned, we're all embroiled in concernful
situations all the time, making our selves the initial signs of
those pragmata, from which we derive all the remainder.

Peirce's distinction between theorematic and corollarial reasoning
has come up before. From what I recall of previous discussions, we
should not read the word “theorematic” in too reductive or purely
deductive a sense. Years ago it was something of a commonplace,
even outside Peircean circles, to call attention to the etymology
of “theorem” as having an observational, even “visionary” sense,
cognate with “theatre”, and some would even point to the sacred
origins of theatre, though maybe that's a bridge too far ...

As far as the iconic aspects of mathematics go, or even our knowledge
representations in general, they are nice when we can get them, but I'm
careful not to stress them too far — it's too easy to “fall victim to a
picture”, in Wittgenstein’s phrase, or succumb to the short-sightedness
of Russell's isomorphism theory of knowledge. Icons are specializations
of symbols and thus fall short of symbols' full potential. There is more
to science than serving as a mirror of nature.

Regards,

Jon

Jon Awbrey

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Aug 8, 2020, 11:40:11 AM8/8/20
to Cybernetic Communications, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Mathematical Method • Discussion 4
http://inquiryintoinquiry.com/2020/08/08/mathematical-method-discussion-4/

Re: Peirce List ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/thrd1.html#00010 )
::: Helmut Raulien ( https://list.iupui.edu/sympa/arc/peirce-l/2020-08/msg00012.html )

Dear Helmut, All ...

It's one of the occupational hazards of the classifying mind
that one can start out consciously characterizing aspects of
real situations and end up unwittingly thinking one's gotten
everything under the sun sorted into mutually exclusive bins.

Once the idols of compartmentality and the illusions of autonomous
abstraction get their hold on our minds it is almost impossible to
reconstitute or synthesize what we've torn asunder, if only in our
own minds. The ounce of prevention here is always keeping in mind
that from which all abstractions are abstracted, living experience.

Regards,

Jon

Jon Awbrey

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Aug 10, 2020, 10:45:13 AM8/10/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Mathematical Method ??? Discussion 5
http://inquiryintoinquiry.com/2020/08/10/mathematical-method-discussion-5/

Re: Ontolog Forum ( https://groups.google.com/d/topic/ontolog-forum/7_sx3_4khCE/overview )
::: Paul Tyson ( https://groups.google.com/d/msg/ontolog-forum/7_sx3_4khCE/GSQ26MC8BQAJ )

All,

This elaborates on a previous reply to the Ontolog Forum.

Dear Paul,

???How We Think??? is a topic for the descriptive science of psychology,
and its ways are legion beyond definitive or exhaustive description.

???How We Ought To Think??? if we wish to succeed at specified purposes
is a topic for the normative science of logic, lumping together for
the moment the evolving varieties of informal, formal, mathematical,
and technologically augmented methods.

They're all good questions and I see no reason not to pursue them all,
aside from the limitations of our brief lives, but we have to keep the
spectrum of different aims sorted.

John Dewey wrote the book How We Think
https://www.gutenberg.org/files/37423/37423-h/37423-h.htm in 1910.
Peirce had earlier summed up his ???non-psychological conception of logic???
https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-%e2%80%a2-on-the-definition-of-logic/
in the pithy motto ???Logic has nothing to do with how we think??? and this led
some scholars to suspect Dewey's title was aimed as a poke in Peirce's ribs.
But the book itself is a How-To guide devoted to improving our capacity for
learning and reasoning, what we'd call today instruction in critical thinking.

All that is prologue to Vannevar Bush's 1945 article, ???As We May Think???
https://www.w3.org/History/1945/vbush/vbush.shtml , projecting the ways
technology may amplify our capacity for inquiry going forward into the future.
I think this is where we came in ...

Reference
=========

??? Eisele, C. (1982), ???Mathematical Methodology in the Thought
of Charles S. Peirce???, Historia Mathematica 9, pp. 333???341.
Online ( https://www.sciencedirect.com/science/article/pii/0315086082901276/ )
PDF ( https://www.sciencedirect.com/science/article/pii/0315086082901276/pdf )

Jon Awbrey

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Aug 13, 2020, 1:30:19 PM8/13/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Mathematical Method • Discussion 6
http://inquiryintoinquiry.com/2020/08/13/mathematical-method-discussion-6/

Re: Ontolog Forum ( https://groups.google.com/d/topic/ontolog-forum/uYPm9lUVSJk/overview )
::: Alex Shkotin ( https://groups.google.com/d/msg/ontolog-forum/uYPm9lUVSJk/eMBxUr14BgAJ )

Dear Alex,

Thanks for the very apt segue from Jon Barwise --

<QUOTE>

Modern mathematics might be described as the science of abstract objects, be they real numbers, functions, surfaces,
algebraic structures or whatever. Mathematical logic adds a new dimension to this science by paying attention to the
language used in mathematics, to the ways abstract objects are defined, and to the laws of logic which govern us as we
reason about these objects. The logician undertakes this study with the hope of understanding the phenomena of
mathematical experience and eventually contributing to mathematics, both in terms of important results that arise out of
the subject itself (Godel's Second Incompleteness Theorem is the most famous example) and in terms of applications to
other branches of mathematics. (Barwise p. 6)

</QUOTE>

When it comes to mathematics as the science of abstract objects I have my
personal favorite classes among its abstract gardens and zoos. One order
of particular interest in the great chain of abstract being descends from
the family of "mathematical relations" to the genus of "triadic relations"
to the species of "triadic sign relations".

By a curious turn, but no real surprise when we stop to think about it,
sign relations, with their object, sign, and interpretant sign domains,
come into being whenever we reflect on the systems of signs we use to
describe any universe of objects, abstract or otherwise, and thus they are
just the tickets we need to enter that "new dimension" of mathematical logic.

References
==========

* Barwise, J. (1977), "An Introduction to First-Order Logic", pp. 5-46
in Barwise, J. (1977, ed.), Handbook of Mathematical Logic, Elsevier
(North Holland), Amsterdam.

* Eisele, C. (1982), "Mathematical Methodology in the Thought
of Charles S. Peirce", Historia Mathematica 9, pp. 333-41.
Resources
=========

* Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )

* Sign Relations ( https://oeis.org/wiki/Sign_relation )

* Triadic Relations ( https://oeis.org/wiki/Triadic_relation )

* Relation Theory ( https://oeis.org/wiki/Relation_theory )

Regards,

Jon

Jon Awbrey

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Aug 15, 2020, 5:05:36 PM8/15/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Mathematical Method ??? Discussion 7
http://inquiryintoinquiry.com/2020/08/15/mathematical-method-discussion-7/

Re: Ontolog Forum
https://groups.google.com/d/topic/ontolog-forum/7_sx3_4khCE/overview
::: Alex Shkotin
https://groups.google.com/d/msg/ontolog-forum/7_sx3_4khCE/Ghv4VGbKBgAJ

Dear Alex,

You raised the following point:

AS: One important usage of a sign is as an element of a language,
especially a formal one, i.e. with a formal grammar.

For context you cited a standard definition of a formal language
with a formal grammar (Aho and Ullman 1972).

Viewed from the standpoint of pragmatic semiotics, where
a sign relation L is a structure of the form L ??? O ?? S ?? I,
we are starting out on pretty much the same page, since I'm
always thinking of a sign s as an element of a sign domain S
and I'm mainly interested in cases where the sign domain S is
a formal language with a formal grammar as defined above or
variants thereof.

That brings us to your question,
"What is the grammar of Peirce's language?",
which I will take up next time.

Regards,

Jon
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