Mathematical Method

[Jon Awbrey]

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

inquiry into inquiry: https://inquiryintoinquiry.com/
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[Paul Tyson]

I've nothing against mathematics, but would like ardent proponents of
them in this forum to first answer Stan Ulam's question: "What makes you
so sure that mathematical logic corresponds to the way we think?"
(Quoted at 1:10:40 in
https://time-issues.org/george-dyson-turings-cathedral-the-origins-of-the-digital-universe-lecture/)

Regards,

--Paul

On 8/4/20 10:26 AM, Jon Awbrey wrote:
> 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 G??del incompleteness

[Jon Awbrey]

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 enhanced methods.

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

Regards,

Jon

[William Frank]

Jon, +2

Had been wracking my brain to imagine a simple reply to Paul Tyson's question,  and couldn't figure out what to say in less than a few thousand words.     Saying just a bit more than you, I would say that very thoughtful and smart people (as you imply) have devoted their lives primarily to this question, so it might just open up a can of worms to begin a discussion of it in an email.  Indeed, a key such person now is John Corcoran, (whose student I have the good fortune to have been) in many papers and current discussions in academia. 

How 'we' do think is mostly unsound -- most people seem to follow the fallacy of denying the antecedent, and even more the post hoc fallacy, and even more, almost always *argue* with each other based on different hidden premises, yet get enraged when these missing premises are pointed out.  Preserving truth is how we ought to think, and mathematical logics are better or worse models of that.  

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Wm

We understand what other people say through empathy—imagining ourselves to be in the situation they were in, including imaging wanting to say what they wanted to say.  

– Zellig Harris    

[Paul Tyson]

On 8/4/20 4:50 PM, William Frank wrote:

Jon, +2

Had been wracking my brain to imagine a simple reply to Paul Tyson's question,  and couldn't figure out what to say in less than a few thousand words.     Saying just a bit more than you, I would say that very thoughtful and smart people (as you imply) have devoted their lives primarily to this question, so it might just open up a can of worms to begin a discussion of it in an email.  Indeed, a key such person now is John Corcoran, (whose student I have the good fortune to have been) in many papers and current discussions in academia.

First, it wasn't my question. It was Stan Ulam's, a supposedly eminent mathematician of recent times.

How 'we' do think is mostly unsound -- most people seem to follow the fallacy of denying the antecedent, and even more the post hoc fallacy, and even more, almost always *argue* with each other based on different hidden premises, yet get enraged when these missing premises are pointed out.  Preserving truth is how we ought to think, and mathematical logics are better or worse models of that. 

Second, the point isn't about whether humans think logically or not, but whether mathematical logic is a useful model for any domain of human experience except, well, those that can be modeled by mathematical logic. Talk of "preserving truth" and "prescribing thought" worries me.

Regards,

--Paul

To view this discussion on the web visit https://groups.google.com/d/msgid/ontolog-forum/CALuUwtCL%3DbJV4NQFBzjTBhy3VKxxaJCA%3DnNN6hPj90164NtgFg%40mail.gmail.com.

[John F. Sowa]

Jon A> Had been wracking my brain to imagine a simple reply to Paul Tyson's...

Short answer:   Nobody thinks in formal proofs, not even professional mathematicians.

But everybody -- including their pet cat or dog -- thinks in diagrammatic patterns that could be represented in mathematics.  Writing down a formal proof is the last step *after* all the hard thinking has been done.

For details, see "Peirce, Polya, and Euclid -- integrating logic,
heuristics, and geometry", http://jfsowa.com/talks/ppe.pdf .

John

[Alex Shkotin]

Paul, 

Is there any chance to make the question more specific: "What makes you 

so sure that mathematical logic corresponds to the way we think in math?"

As math logic brought some great results in math and this way I think it corresponds somehow to the way we think at least in math. 

From the other side math logic is just a part of math:-)

Alex

вт, 4 авг. 2020 г. в 20:03, Paul Tyson <pht...@sbcglobal.net>:

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[Jon Awbrey]

Cf: Mathematical Method ??? Discussion 2
http://inquiryintoinquiry.com/2020/08/05/mathematical-method-discussion-2/

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

::: 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

[jsi...@measures.org]

Paul,

I didn’t listen to the whole piece, but Ulam’s question "What makes you so sure that mathematical logic corresponds to the way we think?" seems an artifact of the days when the assumptions of ‘brain as computer, computer as brain’ provided inspiration.

One can only go so far with those metaphors and then they actually obstruct humans and machines from being appreciated in their own right. Machines and humans do different things well.

Mathematical logic is central to designing the machines. What difference would it make whether it ‘corresponds to the way we think’ or not?

Janet

On Aug 4, 2020, at 3:39 PM, John F. Sowa <so...@bestweb.net> wrote:



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<untitled-[2]>

[Alex Shkotin]

All,

Discussion around John Corcoran's text may be interesting.

FYI https://www.academia.edu/s/184e4e0892/corcoran-shapiro-on-what-mathematical-logic-realy-is-philosophia-1978

He has another one on the same topic but I can't find quickly.

Alex

вт, 4 авг. 2020 г. в 20:03, Paul Tyson <pht...@sbcglobal.net>:

I've nothing against mathematics, but would like ardent proponents of

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[Chris Partridge]

Hi,

I'm not sure I want to get into this discussion, but there is an aspect that could be relevant.

The question sort of seems to assume that thinking is hardware. There are movements in psychology that say we should think about it as software - and so plastic - that is it can be updated/reprogrammed.

There is a community that studies the historical effect of this - arising from (among other things) Ong's book Orality and Literacy. So, for example, how thinking was updated by writing and printing. It is not so far fetched to think that it could be updated by computing - maybe with a mathematical logic module. So the issue is that we have not uploaded the right update yet :)

To view this discussion on the web visit https://groups.google.com/d/msgid/ontolog-forum/81D8296E-11C4-485C-8181-41A562565AD7%40measures.org.

[jsi...@measures.org]

Chris,

As John said, humans only use mathematical logic at the formal proof stage, after the hard thinking is done. It would be nice to access and amplify it at will, and it’s necessary for understanding/designing machine inference. But it’s not a sufficient module for human reasoning. 

Isn’t the goal at this point to support the different capabilities of humans and machines so they work well in teams? At least that’s the discussion in the DoD-adjacent engineering  communities that I follow.

Janet

On Aug 6, 2020, at 3:48 AM, Chris Partridge <partri...@gmail.com> wrote:



To view this discussion on the web visit https://groups.google.com/d/msgid/ontolog-forum/CAMWD8MrPvHUP76i4tfFtEPf0c1Y44Dk293ZkDQ42kPuej7VhRA%40mail.gmail.com.

[Chris Partridge]

Hi Janet,

Yes, you are right.

One of the results of the Orality and Literacy studies is that we have a symbiosis with our current IT.

So the shaping of the way we think is both us thinking more like our IT - to reduce the translation burden - but also offloading the burden where the machine/technology does it better (your point, I think).

I'm not sure where the case of logic sits. There are studies that show that thinking in syllogisms comes with writing, for example. Who can predict what computing will bring? There is also the question whether our current predicate logic is the logic we are going to be using in 100 years or so.

There is an interesting rant from Plato that is often quoted - where he is saying writing and reading will cause us to lose memory skills that sounds much like people talking about (insert your favourite technology here  - e.g. calculators).

I have a view (which I suspect few people share) that we have not yet experienced the full effects of the computing revolution. The classification systems that were enabled by printing (Gray, Linnaeus, etc) took awhile to be established. 

Best,

Chris

To view this discussion on the web visit https://groups.google.com/d/msgid/ontolog-forum/F56F224D-C65F-4217-83DA-856079C3B455%40measures.org.

[jsi...@measures.org]

Hi Chris,

The symbiosis idea is interesting and provides rich opportunities for study and development. 

At a minimum research shows that human teams function best if participants have mental models of each other’s processing. Extending this to bi-directional modeling between humans and tech is part of the AAAI 2020 topic ‘AI Welcomes Systems Engineering: Towards the Science of Interdependence for Autonomous Human-Machine Teams’ (physical meeting postponed to Nov 11-12)

https://sites.google.com/view/scienceofinterdependence

Janet

On Aug 6, 2020, at 8:28 AM, Chris Partridge <partri...@gmail.com> wrote:



To view this discussion on the web visit https://groups.google.com/d/msgid/ontolog-forum/CAMWD8MoqnO54eMKvYUXAaPXAV-%2BXdiLsi9gJYnTkJ8S_8W42qQ%40mail.gmail.com.

[Jon Awbrey]

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]

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]

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]

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.

Online ( https://www.sciencedirect.com/science/article/pii/0315086082901276/ )
PDF ( https://www.sciencedirect.com/science/article/pii/0315086082901276/pdf )

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

[Alex Shkotin]

Dear Jon,

Let me mention that one important usage of sign is as an element of a language, especially formal one i.e. with formal grammar. And we have a huge progress in this special kind of math. 

Let me just refer to another great "handbook": The Theory Of Parsing, Translation, And Compiling, A. Aho, J. Ullman, Published 1972, Computer Science. Volume I:Parsing.

Where Chapter 2 "Elements of language theory" on p.85 has this definition

What is the grammar of Peirce's language? May be an interesting question.

Regards,

Alex

чт, 13 авг. 2020 г. в 20:30, Jon Awbrey <jaw...@att.net>:

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[Jon Awbrey]

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

[Alex Shkotin]

Dear Jon,

I hope the Peirce Language Grammar would be context-free one and I am looking forward to seeing it.
Maybe we got one more language for ontologies.

Regards,

Alex

вс, 16 авг. 2020 г. в 00:05, Jon Awbrey <jaw...@att.net>:

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