Diagrams, Images, pictures, and representations

[John F Sowa]

Alex,

There is a huge difference between a diagram and a picture. 

A diagram has a discrete set of  lines, areas, and structures in two or more dimensions.  Euclid is an excellent example.  But modern mathematics, science, engineering, and architecture follow the same principles and guidelines as Euclid.  Every diagram can be precisely specified in a linear notation that can be exactly translated to and from bit strings in a digital computer.  

A picture is the result of some attempt to represent some aspect of reality (whatever that may be).   A mechanical representation (photograph or sound recording) is usually more precise than a human drawing, painting, sculpture, or other imitation. 

The lines drawn by humans aren't as precise as the lines drawn by a machine.  But both of them are approximations of the same features and relations.   Since a diagram has discrete features, the approximations are irrelevant -- provided that they specify the same formal features.

Relevance to ontology:   Every formal ontology can be translated to and from some kind of diagram.  It is therefore limited to the same kinds of approximations as a diagram.

An informal ontology may be represented by a picture.   That implies that it can be more accurate than a formal  ontology for some aspects of reality.   But no picture is ever sufficiently precise and detailed that it can represent the full content of all things and relations in any part of reality.

Summary:  There is no such thing as a perfect ontology of everything -- or even  a perfect ontology of some limited aspect of reality at every level of detail.  Every ontology is always a work in progress.  There will always be some aspects of reality that will require future revisions and extensions.

Just look at the periodic updates to your computer systems.  It's impossible for any printed version to be a perfect representation of all or even any version.  The same issues are true of any ontology of those systems.

John

 

---

From: "Alex Shkotin" <alex.s...@gmail.com>
Sent: 12/9/25 3:21 AM

John,

You call any drawing, from geometric to engineering, a diagram.  Why?

Alex

пн, 8 дек. 2025 г. в 21:29, John F Sowa <so...@bestweb.net>:

Alex,

Just look at Euclid:  Every definition, theorem, and proof includes a diagram.

In fact, look at the blackboard or whiteboard of any teacher of any branch of science:  It's covered with diagrams.   The algebraic notation is a convenient way to summarize the results, but every step of algebra has an associated operation on a diagram.  For dimensions beyond 2, the diagrams become harder to draw, but the best mathematicians and scientists use their imaginations to "visualize" 2D or 3D projections.

Summary:  Science without diagrams is blind.

John

[jsi...@measures.org]

John,

I see that your chapter in the 2024 Phaneroscopy and Phenomenology book presenting Peirce’s late writings on phaneroscopy and diagrammatic reasoning *is exactly on point* for the issues I have been trying to raise in this thread.

What have been successful lower level ontologies could have (less mysteriously) been called phenomenologies, or theories of specific appearances, while the ULO level has required theory that is different in kind from those. 

ULO convergence has not needed a theory of Being qua Being, but a *meta-theory of the phenomenologies*. As you point out, this has a solid start in Peirce’s phaneroscopy or ‘Science of Phenomenology’ that is perfectly consistent with Von Uexküll’s biosemiotics. Other later work in semiotics and cybernetics, Rosen’s modeling-relational biology, works of Deacon, Friston, Levin, etc. etc., also converge in this same general direction (some with at least partial attribution to Peirce, and others serving as proof of concept by making similar discoveries independently). 

Differences of terminology, field of origin, etc., have prevented convergence across these contributions, though there have been promising hints in recent years. Recognizing this developing meta-theory as the proper domain for ULO convergence could give new life to the initiative begun in 2006.

Janet

On Dec 9, 2025, at 11:46 AM, John F Sowa <so...@bestweb.net> wrote:



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[John F Sowa]

Janet,

The word that I consider problematical is convergence.  Every new discovery opens up more questions than answers.  I very seriously doubt that convergence on a universal upper level ontology is possible or even desirable,

Instead of a goal of convergence, people have developed methods of interoperability despite divergence.

The goal of convergence on a universal ontology requires us to anticipate the innovations in the future and design a solid foundation for all of them.  If we just look back on the discoveries in the past decade, we can see that nobody in 2015 could have anticipated what we see today,

And I'm certain that we cannot anticipate what we will see in 2035.  Any ULO we design for 2025 is certain to be inadequate.  But in looking back at the insights that many brilliant people, such as Peirce, Rosen, and even Aristotle had are still just as sound as they were decades or even centuries ago.

That means we should not expect convergence on a fixed ontology.  Instead, we should look for methods for guiding an open ended journey that can produce unlimited innovation and development.

John

 

---

From: jsi...@measures.org

[Alex Shkotin]

John,

INTRODUCTION

First of all let me introduce an absolutely funny situation. [1]

So, my question was rhetorical. Anyway thank you for your answer.

MAIN TEXT

Regarding the diagrams themselves, a person can only work with a diagram by materializing it. That is, although at its core, according to its description, a diagram is a geometric object defined with precision down to proportionality and mirror image, a person can only work with it by placing it in reality and engaging their fantastic ability to think about observable surfaces and lines.

A most interesting topic!

Alex

[1]

This is my letter as I sent it to you, Janet and Ravi:

This is my letter as you received it:

вт, 9 дек. 2025 г. в 22:45, John F Sowa <so...@bestweb.net>:

Alex,

There is a huge difference between a diagram and a picture. 

A diagram has a discrete set of  lines, areas, and structures in two or more dimensions.  Euclid is an excellent example.  But modern mathematics, science, engineering, and architecture follow the same principles and guidelines as Euclid.  Every diagram can be precisely specified in a linear notation that can be exactly translated to and from bit strings in a digital computer.  

A picture is the result of some attempt to represent some aspect of reality (whatever that may be).   A mechanical representation (photograph or sound recording) is usually more precise than a human drawing, painting, sculpture, or other imitation. 

The lines drawn by humans aren't as precise as the lines drawn by a machine.  But both of them are approximations of the same features and relations.   Since a diagram has discrete features, the approximations are irrelevant -- provided that they specify the same formal features.

Relevance to ontology:   Every formal ontology can be translated to and from some kind of diagram.  It is therefore limited to the same kinds of approximations as a diagram.

An informal ontology may be represented by a picture.   That implies that it can be more accurate than a formal  ontology for some aspects of reality.   But no picture is ever sufficiently precise and detailed that it can represent the full content of all things and relations in any part of reality.

Summary:  There is no such thing as a perfect ontology of everything -- or even  a perfect ontology of some limited aspect of reality at every level of detail.  Every ontology is always a work in progress.  There will always be some aspects of reality that will require future revisions and extensions.

Just look at the periodic updates to your computer systems.  It's impossible for any printed version to be a perfect representation of all or even any version.  The same issues are true of any ontology of those systems.

John

 

---

From: "Alex Shkotin" <alex.s...@gmail.com>
Sent: 12/9/25 3:21 AM

John,

You call any drawing, from geometric to engineering, a diagram.  Why?

Alex

пн, 8 дек. 2025 г. в 21:29, John F Sowa <so...@bestweb.net>:

Alex,

Just look at Euclid:  Every definition, theorem, and proof includes a diagram.

In fact, look at the blackboard or whiteboard of any teacher of any branch of science:  It's covered with diagrams.   The algebraic notation is a convenient way to summarize the results, but every step of algebra has an associated operation on a diagram.  For dimensions beyond 2, the diagrams become harder to draw, but the best mathematicians and scientists use their imaginations to "visualize" 2D or 3D projections.

Summary:  Science without diagrams is blind.

John

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[John F Sowa]

Alex,

The following questions are interesting, but they are about totally different subjects:  (1) What kinds of things  exist or may exist?  (2) Where and how do they exist -- physically, abstractly, imaginatively?  (3) How can they be known -- by observation, by abstract definitions, by vague imaginations?  (4) How can information about them be obtained?  (5) How can they be represented in an ontology?  (6) Is it useful, desirable, or necessary to represent them in a particular ontology? (7) Finally for every one of these six cases, there are two important questions:  How and Why?

For example, there are infinitely many integers.  None of them are physical, but every one of them has a name that can be typed on a keyboard. and one can refer to them by saying "Let x be [some expression that expresses a computation]".   It's also possible to design diagrams that represent equivalent expressions.

And it's also possible to design computer programs that compute the results of those expressions or diagrams.

John  

 

---

From: "Alex Shkotin" <alex.s...@gmail.com>
Sent: 12/10/25 5:44 AM

[Alex Shkotin]

John, 

That's right! These are also interesting topics. "but they are about totally different subjects" and don't touch on the fact that

A diagram is a material object that helps us think.

Alex

чт, 11 дек. 2025 г. в 00:24, John F Sowa <so...@bestweb.net>:

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[Simon Polovina]

Hello all.

I find that TOGAF | www.opengroup.org provides a simple-to-understand, elegant comparison on how we can visualise content. (@John, you may also remember a precursor to this distinction when you worked with John Zachman regarding what became Enterprise Architecture in the 1990s.) See below (NB: The content below was Gen AI-generated, which I checked but didn’t edit, as it conveys the general message.)

Regards,

Simon

 

 

In TOGAF 10, catalogs, matrices, and diagrams are three distinct but complementary artifact types that capture architectural information with varying degrees of organization and visualization, tailored to different stakeholder needs.

1. Catalogs

• Definition: Catalogs are curated lists of building blocks (i.e., the fundamental entities within the TOGAF content metamodel) organized by type or relevance.
• Purpose: Serve as reference or governance tools, providing a clear inventory of architectural elements for querying, reporting, and analysis.
• Characteristics:
• Represent information linearly or hierarchically, like a list or table of contents.
• Include metadata to support stakeholder queries (e.g., ownership, classification, location).
• Example TOGAF 10 catalogs include:
• Principles Catalog
• Organization/Actor Catalog
• Business Service/Function Catalog
• Location Catalog
• Process/Event/Control/Product Catalog

2. Matrices

• Definition: Matrices are two-dimensional grids illustrating relationships among architectural entities listed typically in catalogs.
• Purpose: Highlight interactions, dependencies, or responsibilities that are not obvious from linear lists, making them ideal for analyzing complex associations.
• Characteristics:
• Rows and columns correspond to different entity types.
• Cells indicate relationships, such as “performs,” “owns,” or “consumes.”
• Better suited for tabular presentation of many-to-many relationships rather than graphical abstraction.
• Example matrices include:
• Stakeholder Map Matrix
• Actor/Role Matrix
• Data Entity/Business Function Matrix
• Application/Function Matrix

3. Diagrams

• Definition: Diagrams are graphical representations of building blocks and their relationships within the architecture.
• Purpose: Provide visual context for stakeholders, illustrating patterns, flows, and connections that aid comprehension, communication, and validation.
• Characteristics:
• Visual, spatial layout captures both entities and relationships.
• Can be high-level (“pencil sketch”) or detailed, depending on stakeholder needs.
• Examples include:
• Value Chain Diagram
• Business Footprint Diagram
• Process Flow Diagram
• Functional Decomposition Diagram
• Goal/Objective/Service Diagram
• Integration: Diagrams are often populated using data from catalogs and matrices, thereby bridging list-based and visual architectural representation.

Summary Table

Artifact Type	Primary Function	Structure	Typical Use
Catalog	List building blocks	Linear or hierarchical list with metadata	Reference, governance, querying, completeness checks
Matrix	Show relationships	2D table (rows & columns correspond to entity types)	Analyze dependencies, traceability, relationships
Diagram	Visualize entities and connections	Graphical layout of nodes & edges	Communicate to stakeholders, interpret flows, validate architecture

Key Insight: TOGAF 10 positions catalogs, matrices, and diagrams as complementary artifacts. Catalogs capture "what exists," matrices capture "how things relate," and diagrams show "how elements connect visually," supporting a layered understanding of enterprise architecture.

References: Sources .

Source(s):
1. https://togaf.visual-paradigm.com/2023/10/10/navigating-the-architectural-landscape-unveiling-togafs-building-blocks-catalogs-matrices-and-diagrams/
2. https://coe.qualiware.com/resources/togaf/togaf-artifacts/
3. https://www.archimetric.com/comprehensive-guide-to-togaf-10/
4. https://togaf.visual-paradigm.com/2025/02/18/comprehensive-guide-to-the-modular-structure-of-togaf-10/

[Alex Shkotin]

Hello Simon,

It's important to me to note that a diagram, being a geometric object, has a clear definition in geometry which can be formalized properly. But for a person to be able to work with it, it must be materialized.

So for me Structure is not just "Graphical layout of nodes & edges" but "Some geometrical definition".

Alex

чт, 11 дек. 2025 г. в 13:32, Simon Polovina <si...@similelogics.ltd>:

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[Simon Polovina]

Hi Alex.

Agreed. My message was initially in the TOGAF context, but it can extend to address your remark. My initial purpose was to distinguish between a Catalog (a list), a Matrix (a 2D table), and a Diagram, and which one to choose for which purpose. TOGAF offers this clarity.

Thanks,

Simon

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[Alex Shkotin]

Hi Simon,

Exactly. And for me, the underlying mathematical structure is important. For example, we start with directed graphs, and we have a good mathematical theory for them. But we simply add some labels, and voila—KG. 🏋️
It's the same in geometry: some geometric structures, but when we label their components, we get diagrams.

Alex

чт, 11 дек. 2025 г. в 14:53, Simon Polovina <si...@similelogics.ltd>:

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[Gary Berg-Cross]

Alex

You said:

'So for me Structure is not just "Graphical layout of nodes & edges" but "Some geometrical definition".'

Isn't that (structure) an instance of a (defined/understood) concept versus a concept?

And that seems on the face of it to be 2points of the triangle meaning with the 3rd being the use of a term for the concept in the instance.

Gary Berg-Cross 

Potomac, MD

240-426-0770

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[Alex Shkotin]

Gary,

I referred to the end of Simon's email where we have [1]. And I proposed just to replace "Graphical layout of nodes & edges" by "Some geometrical definition".

This geometrical definition may be like this: Some geometrical figure with labeled components. 

Alex

[1]

Summary Table

Artifact Type	Primary Function	Structure	Typical Use
Catalog	List building blocks	Linear or hierarchical list with metadata	Reference, governance, querying, completeness checks
Matrix	Show relationships	2D table (rows & columns correspond to entity types)	Analyze dependencies, traceability, relationships
Diagram	Visualize entities and connections	Graphical layout of nodes & edges	Communicate to stakeholders, interpret flows, validate architecture

чт, 11 дек. 2025 г. в 17:51, Gary Berg-Cross <gberg...@gmail.com>:

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[Alex Shkotin]

Gary,

To say frankly I do not understand yours 

"Isn't that (structure) an instance of a (defined/understood) concept versus a concept?

And that seems on the face of it to be 2points of the triangle meaning with the 3rd being the use of a term for the concept in the instance.

"

And if we are talking about the meaning of term "structure" usage in this particular table, then it's a good example of verbalization for me. 

When we get sentences represented in the form of a table.

So we have 

"A structure of a catalog is a linear or hierarchical list with metadata." 

"A structure of a Matrix is a 2D table (rows & columns correspond to entity types)." 

"A structure of a diagram is a graphical layout of nodes & edges." 

I think the relationship "X is a structure of Y" points that one type of math object X (list, 2D table, graph(!)) is a base for another type of math objects Y (catalog, matrix, diagram) being more structured, specifically attributed and so on.

We may say that Y is somehow created from X. One type from another.

But we should ask Simon to verify.

In this case if you are talking about that term for example catalog has in its definition term list, I am with you.

Alex

чт, 11 дек. 2025 г. в 17:51, Gary Berg-Cross <gberg...@gmail.com>:

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[Simon Polovina]

Hi all.

In the 2. TOGAF Content Framework and Enterprise Metamodel : TOGAF® Standard — Architecture Content (which I’ve attached in case you don’t want to go through the site’s free registration process), it illustrates how a diagram (in this case, the TOGAF metamodel) can also be depicted as a catalog (list) as you scroll down the page. Hence, how a catalog in this example can be better expressed in a diagram, and the catalog in this case remains useful as it explains the diagram.

At MetroMap (which comes from The SAP Enterprise Architecture Framework derived from TOGAF), you can view some other catalog/map/diagram examples (NB a Map is another name for Matrix). Some of these artifacts look as if they overlap (e.g., is it mainly a catalog or a map?)

Remember, my context is Enterprise Architecture (EA), hence the reference to “graphical layout of nodes & edges”. If you go to, for example, D3 by Observable | The JavaScript library for bespoke data visualization, you’ll get a sense of possible infographic structures. There are many more infographic sites.

For me, almost all diagrams are, under the hood, a “graphical layout of nodes & edges”, but that could reflect my EA focus, so I’m not hard-and-fast about it, and other geometric forms are possible.

Indeed, on another dimension, some diagrams are maps, e.g., Google Maps, alongside more abstract versions such as the London Underground map.

TOGAF offers a simple definition between catalogs (lists), matrices (maps) and diagrams, to support the best way to represent some given content, and the relationships between these three artifact types. It provides the guidance that EA practitioners need (and possibly others could use, too).

Regards,

Simon

 

From: ontolo...@googlegroups.com <ontolo...@googlegroups.com> On Behalf Of Alex Shkotin
Sent: 12 December 2025 09:23
To: ontolo...@googlegroups.com
Cc: CG <c...@lists.iccs-conference.org>
Subject: Re: [ontolog-forum] Diagrams, Images, pictures, and representations

 

Gary,

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[Simon Polovina]

Hi again.

I received a bounce from the CG list, as it didn’t like the attachment without moderator approval. Here’s my message without the attachment (in case you have issues, too), but you’d have to open the first link.

Regards,

Simon

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[John F Sowa]

Simon, Gary, and Alex,

This thread discusses important applications of ontology to software design and development.  But a universal ontology must cover everything that exists or may exist in the universe or in any thoughts (by humans or other beings) in a physical space or in a purely mathematical space -- either formally or informally.

Simon:  I find that TOGAF provides a simple-to-understand, elegant comparison on how we can visualize content. (John, you may also remember a precursor to this distinction when you worked with John Zachman regarding what became Enterprise Architecture in the 1990s.)

I agree that TOGAF has a useful summary of their ontology for the kind of software development that they support.  But the joint article by Zachman and me was more general.  It covered every stage of any project (hardware, software, or whatever).  He also considered the kinds of people involved, each with different backgrounds, views, and requirements.   I credit Zachman for that broad scope, but he did not have any knowledge of formal logic.   In our joint article, I broadened the scope and developed the formal specifications.   See https://jfsowa.com/pubs/sowazach.pdf .

That article was published in the IBM Systems Journal in 1992, and Zachman continued to use it in his lectures and consulting work for his entire career.   For later developments, see https://en.wikipedia.org/wiki/Zachman_Framework .  It includes a link (Ref 5) to a comparison with TOGAF.   I spoke with Zachman a few times after we wrote the article, but my contribution to his project ended in 1992.

Gary:  So for me Structure is not just "Graphical layout of nodes & edges" but "Some geometrical definition".

Yes.    And we must distinguish abstract mathematical structures, which are independent of space and time from physical structures, which exist at some location in time and space.   We also have to consider informal structures, which do not have a precise specification.  The precise mathematical structures usually begin with informal ideas that are formalized after further research and implementation efforts.

Alex:   So we have 

"A structure of a catalog is a linear or hierarchical list with metadata." 

"A structure of a Matrix is a 2D table (rows & columns correspond to entity types)." 

"A structure of a diagram is a graphical layout of nodes & edges." 

This distinction is based on the TOGAF publications.  It is not sufficiently precise and general to apply to all the possible applications of ontology and the systems of formal reasoning.   For example, a catalog or a matrix can be specified in a diagram.  Therefore, diagrams are sufficiently general to include catalogs and matrices as special cases.  

Diagrams can also be generalized to represent anything that can be described in any version of logic.  Conversely, anything described by any version of logic can be represented by a diagram.  For examples, see the ISO standard for Common Logic (CL).  Every CL linear formula can be mapped to and from a diagram drawn as a conceptual graph.  They are 100% equivalent in expressive power.  But other kinds of diagrams are also possible.

The Zachman system has a broader scope than TOGAF, but even Zachman is limited to the kinds of structures that people build.  It does not cover the enormous range of things that exist on earth and even more in the universe outside of earth.

A general ontology must be able to describe every physical, mathematical, or imaginary structure that may exist anywhere, anytime -- either physically or abstractly -- either formally or informally. 

I'll recommend some definitions that satisfy these conditions in my next note in this thread.  For a research project that devoted two years of study by several dozen researchers, see the IKRIS project, which developed the IKL logic: https://jfsowa.com/ikl .

John

 

---

From: "Alex Shkotin" <alex.s...@gmail.com>

[Alex Shkotin]

Hi Simon,

What constitutes a diagram depends on the specific technology in which they are used. And the corresponding definitions are provided there.

For example, in your first letter there is a 

"Definition: Diagrams are graphical representations of building blocks and their relationships within the architecture."

And I'm sure the documentation you linked to has a whole system of definitions for all classes of diagrams used.

I just wanted to emphasize two facts:

a diagram is a material object,

it is necessary to clearly specify the mathematical object that underlies the diagram.

Moreover, in your case, and in most cases, this mathematical object is a directed graph.

But JFS also calls labeled geometric drawings diagrams.

And as I already wrote: why not?

By the way, we discussed a little about the use of undirected graphs here.

Alex

пт, 12 дек. 2025 г. в 18:17, Simon Polovina <si...@similelogics.ltd>:

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[John F Sowa]

Alex,

Euclid's writings were so comprehensive that all later mathematicians accepted Euclid's definitions and conventions as a starting point for their work.  Even today, introductory textbooks begin with a  version of Euclid.  Therefore, the following conventions are accepted everywhere.

1. Every definition, theorem, and proof is specified in terms of an abstract mathematical diagram, which is approximated by a physical diagram that may be drawn, printed, or displayed in some way.

2. In a mathematical diagram, the lines have zero thickness, straight lines are perfectly straight, and they may extend to infinity at both ends, unless they are stopped by labeled points at one or both ends.

3. For every definition or proof, every part of the diagram that is mentioned by any statement has a name that is determined by labels on parts of the diagram.  In Euclid, the name of a point is written near the point.  A straight line is named by the labels of two points on the line

There are many more conventions.  But the important issues begin with the distinction between abstract mathematical drawings and their physical approximations,  Even in ancient times, mathematicians realized that the abstract diagrams could be described by a linear string of words or other symbols.  But those linear notations are harder for people to understand,  Euclid's diagrams are based on drawings that resemble the things they describe.  But it's important to note that the physical drawings are usually approximations.

That word usually is important.   It's possible to design diagrams for which some physical details are irrelevant.  For example, a connection between A and B could be represented by a line of any length or thickness or curvature.  An enclosure could be a circle or an oval or a rather poorly drawn blob.

Peirce's existential graphs have this property.  Therefore, they can be drawn on paper or a blackboard by people who are not good artists without making horrendous mistakes.

Nevertheless, there is still a precise distinction between the abstract mathematical definition and the many acceptable physical approximations.  That abstract definition has a precisely defined mapping of Peirce/s graphs to and from the modern linear notations for logic.

That distinction between the abstract mathematical definition and the physical pattern on some medium is universal.  There are no exceptions in any version of logic or mathematics or programming practice. 

John  

____________

[Chris Partridge]

For more historical detail on how Euclid and the Greeks did mathematics, I have found Reviel Netz useful. Here is a sample of his books.

• The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History, Cambridge: Cambridge University Press, 1999, ISBN 978-0-521-54120-6.
• The Transformation of Mathematics in the Early Mediterranean World: from Problems to Equations, Cambridge: Cambridge University Press, 2007 ISBN 978-0-5210-4174-4.
• A New History of Greek Mathematics, Cambridge: Cambridge University Press, 2022, ISBN 978-1-108-83384-4.

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[Alex Shkotin]

John,

There are no notations in the world of mathematical objects. We introduce them, add them to define, describe, understand, and collectively discuss certain mathematical objects.

When someone tells me, "Imagine an isosceles triangle, but not an equilateral triangle," I might draw something in my mind. But there definitely won't be any letters or other notations.

If you also call some mathematical objects diagrams, I'll write again: why not.

For example, in category theory, there are commutative diagrams. They define what a diagram is and what it means for a diagram to be commutative.

A geometric object has no notations, but a diagram without notations loses its value to the point that even its author wouldn't recognize it.

Consider any geometric figure [1] or a finite system of geometric figures arranged in some way that don't intersect. Precisely because there are no notations on them or their parts, we have to develop a whole non-trivial naming technique.

Alex

[1] framework

rus	Пусть F1 есть совокупность двух или более точек. F1 есть фигура еите для любых двух различных точек p1, p2 из F1 существует разомкнутая линия l1 такая что все точки l1 из F1 и p1, p2 есть концевые точки l1.
eng	Let F1 be a set of two or more points. F1 is a figure iff for any two distinct points p1, p2 from F1, there exists an open line l1 such that all points l1 from F1 and p1, p2 are endpoints of l1.

вт, 16 дек. 2025 г. в 08:42, John F Sowa <so...@bestweb.net>:

--

[Polovina, Simon (BTE)]

Hi everyone.

I’ve probably little to add, but I note John’s remark about Peirce's existential graphs:

Ø    Peirce's existential graphs have this property.  Therefore, they can be drawn on paper or a blackboard by people who are not good artists without making horrendous mistakes.

In 2007, I wrote an Introduction to Conceptual Graphs, in which I referred to Peirce's existential graphs as portrayed in Sowa (1984)’s original work and Heaton’s adaptation (which downloads as a PDF). I further adapted Peirce’s diagram for visuality (and as a non-artist), expressed in my Introduction to Conceptual Graphs paper above (Polovina, 2007). It certainly helped me understand and convey logic in diagrams. I also continue to use simple Conceptual Graphs as diagrams in my work, not least for their human-readability relative to other representations (e.g., knowledge graphs), as in Polovina et al. (2025). That paper also touches on Enterprise Architecture, the subject that prompted my original posting.

Simon

 

References

Polovina, S., Fallon, R. and Saleem, M. 2025. Moregraph: Metadata-Driven Enterprise Architecture Using Conceptual Structures. In: Lecture Notes in Computer Science. Springer Nature Switzerland, 91-106. URL: https://shura.shu.ac.uk/36156/

S. Polovina. 2007. ‘An Introduction to Conceptual Graphs’, in Conceptual Structures: Knowledge Architectures for Smart Applications, U. Priss, S. Polovina, and R. Hill, Eds, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 1–14. URL: https://shura.shu.ac.uk/1175/

J. F. Sowa. 1984. Conceptual structures: information processing in mind and machine. Addison-Wesley Longman Publishing Co., Inc., USA.

[John F Sowa]

Chris,

Thanks for the references.  But can you cite any that are available for a free download?

Since you have found those books useful, can you quote relevant definitions and relate them to Peirce's definitions in my recent copy of the note by Jon Awbrey?   And/or to the recent discussions with Alex?

John

 

---

From: "Chris Partridge" <partri...@gmail.com>
Sent: 12/16/25 2:50 AM
To: ontolo...@googlegroups.com

Subject: Re: [ontolog-forum] Diagrams, Images, pictures, and representations

[John F Sowa]

Alex,

There is a very clear and precise distinction between purely abstract mathematical objects and physical notations:  Mathematical objects and structures do not have any associated matter or energy.

The following symbols written on paper, a computer screen, or bits and bytes inside the computer are associated with physical things that consist of matter and there is an expenditure of energy in writing them or erasing them:  1, 2, 3, A, B, C.

But a line in Euclidean geometry whose endpoints are labeled A and B, and whose length is stated as 1 centimeter is purely abstract.  The letters A and B do not have any weight, and there is no expenditure of energy in assigning those letters to the endpoints of that line.  And that length of 1 cm is not associated with any location in the physical universe.

Mathematical structures and their sizes and labels are purely abstract.  They have no matter or energy of any kind.  But  their representations on paper, a blackboard,,a human voice, a computer screen, or computer storage have very different amounts of matter and energy.

John

 

---

From: "Alex Shkotin" <alex.s...@gmail.com>
Sent: 12/16/25 5:59 AM

[Alex Shkotin]

John,

Among the mathematical objects of Euclidean geometry, there are no letters, much less inscriptions like "1 cm."

Seeing 1 cm in the drawing, in addition to images of geometric figures and geometric symbols (A, B, C), leads me to the conclusion that this is an engineering drawing.

The fact that we all, and especially engineers, successfully apply Euclidean geometry in our lives and work is part of the phenomenology of matter.

Matter appears to us as curved surfaces glowing in different colors, located relative to one another in Euclidean space.

As for the definition of a line segment, see [1]

The definition of 1 cm is 0.01 m.

But the definition of 1 m is so beautiful that it is worth quoting:

"the metre has been defined as the length of the path travelled by light in vacuum during a time interval of ⁠1/299792458⁠ of a second" see

Geometric objects are quite diverse, and several axiomatic theories have been constructed to study them. Hilbert already mentions several constructions. Tarski later developed an axiomatic theory of plane geometry and proved that it was decidable.

A wonderful project GeoCoq is formalizing a theory of geometry.

"Formalization of theoretical knowledge—what could be more sophisticated!", as Leibniz once said, I hope.

Let me add: everybody can formalize facts, try to formalize a theory keeping it in a framework.

Alex

[1] framework lsegment rus:отрезок eng:line segment

rus	Пусть x есть совокупность точек. x есть отрезок еите x состоит из двух точек и всех точек на прямой между ними.
eng	Let x be a set of points. x is a line segment iff x consists of two points and all points on the straight line between them.

ср, 17 дек. 2025 г. в 08:57, John F Sowa <so...@bestweb.net>:

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[Chris Partridge]

Hi John,

I'm afraid I don't know where free downloads are of these books - and it is a pity that they are not just accessible.

Netz was a student, for a while, of https://en.wikipedia.org/wiki/G._E._R._Lloyd - so understandably he takes a historical stance.

What I found interesting was his discussion of Greek mathematics in terms of practices (he explicitly relates this to Kuhn's paradigms) - and his detailed investigation of how diagrams were used. For example, that the practices started as presentations and then migrated to text - and this shaped their structure. That some details can be telling, so, for example, that the diagrams were integral and that they were (probably?) not metrical. This links with current historical and sociological work on mathematical practices.

I copy in the book cover text from The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History - to give you all some idea of scope

The aim of this book is to explain the shape of Greek mathematical thinking. It can be read on three levels: first as a description of the practices of Greek mathematics; second as a theory of the emergence of the deductive method; and third as a case-study for a general view on the history of science. The starting point for the enquiry is geometry and the lettered diagram. Reviel Netz exploits the mathematicians’ practices in the construction and lettering of their diagrams, and the continuing interaction between text and diagram in their proofs, to illuminate the underlying cognitive processes. A close examination of the mathematical use of language follows, especially mathematicans’ use of repeated formulae. Two crucial chapters set out to show how mathematical proofs are structured and explain why Greek mathematical practice manages to be so satisfactory. A final chapter looks into the broader historical setting of Greek mathematical practice.

As this indicates, he thinks the deductive method has a history - and so maybe a future :)

I found Netz from multiple references in Dutilh Novaes, C. (2012). Formal Languages in Logic: A Philosophical and Cognitive Analysis. Cambridge University Press. www.cambridge.org/9781107020917

In this, Caterina says (in relation to Netz) 

"In effect, what follows is a modest attempt at what Netz (1999) has described as ‘cognitive history’: a historical analysis which takes into account the cognitive background, motivations, and implications of the developments in question. More specifically, the underlying idea is that the historical development of formal languages is best understood from the point of view of the extended cognition framework (developed in more detail in Chapter 5). In fact, it would seem that the whole history of notations in mathematics could (should?) be written from the point of view of the concept of extended cognition, but this more ambitious goal falls out of the scope of the present investigation. For reasons of space, the survey here is rather brief, but it emphasizes the role of the development of algorithmic and algebraic techniques for calculation in the Arabic world (against the background of progress in Indian mathematics), which were brought to Europe by the abbaco schools. Without this link, it is impossible to understand the progress in mathematical notation in the sixteenth and seventeenth centuries, initiated by Viète and completed by Descartes."

And:

"Before discussing schematic letters and regimentation specifically, a first observation is in order. Both for logic and for mathematics, a key transformation in ancient Greece was the transition from oral to written contexts. As argued by Netz, the birth of the deductive method takes as its starting point purely oral, dialogical situations, which then become regimented in written forms.  

Greek mathematics reflects the importance of persuasion. It reflects the role of orality, in the use of formulae, in the structure of proofs . . . But this orality is regimented into a written form, where vocabulary is limited, presentations follow a relatively rigid pattern . . . It is at once oral and written. (Netz 1999: 297–8)

The observation applies, mutatis mutandis, to logic as well, which emerged as a codification of dialogical practices in the Academy (Marion and Castelnerac 2009). The transition from the oral to the written medium is epitomized in Plato’s dialogues, which registered and passed on to future generations Socrates’ orally formulated teachings (despite Socrates’ own misgivings vis-à-vis the written medium)."

My intuition is that a close reading of these would shift people's understanding of the questions. I'll leave the hard work of shifting understanding to those who choose to read the texts (where and if they can find them). I suspect there are no easy comments.

Hopefully, the quotes give some idea of the delights in store.

Chris

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[John F Sowa]

Alex,

A diagram of a structure does not require any letters or other symbols.  But a diagram of 

For exam;ple,

 

---

From: "Alex Shkotin" <alex.s...@gmail.com>
Sent: 12/16/25 5:59 AM

[John F Sowa]

I accidentally hit send.   I'll write more tomorrow.

But note the example by Euclid.    Diagrams have labels,

  

John

 

---

From: "John F Sowa" <so...@bestweb.net>
Sent: 12/18/25 11:52 PM
To: ontolo...@googlegroups.com
Cc: CG <c...@lists.iccs-conference.org>, Edward Zalta <edward...@gmail.com>
Subject: [CG] Diagrams, Images, pictures, and representations

[Alex Shkotin]

John,

I am happy we agree that a diagram has labels. There are two more points:

-There is some math object behind the diagram.

-A diagram is a material object: you look at one on your monitor, I look at another on my monitor.

And what is this math object (geometrical one) behind the geometrical picture you call "diagram"?

This is a plane figure consisting of a circle  and 5 line segments. In the picture we have one additional short line segment to visualize the center of diameter.

It's a little bit boring to describe all relationships among these components. And we use a naming technique to keep it more understandable.

Let me just point:

-In this text Euclide used advanced operational naming technique: Let A, B denote some two different points, then AB with default operation denotes line segment between them, not new identifier "AB".

-The short horizontal line segment intersecting the center of the circle we do not need in our math object. Describing our figure we have a definition: Let F denotes a center of line segment CE.

So we should add that a diagram, in addition to labels, sometimes keeps auxiliary images of geometrical objects, usually to connect labels with particular images of parts of a math object.

Let me show a picture of a plane figure itself without any additions.

Notice that now there is no short horizontal line segment at the center.

Is this a diagram?

Alex 

framework

пт, 19 дек. 2025 г. в 08:02, John F Sowa <so...@bestweb.net>:

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[Alican Tüzün]

Dear Alex, 

> There is some math object behind the diagram.

I would argue, not a single object but a structure "behind" the diagram exists. Where the thing behind is metaphysics and the diagram is just about the metaphysics.

E.g., the empty "spaces" within the "diagram" inside the circle is about area. Where each " " term refers to something in the structure (as you mean the thing behind).

İf we accept that metaphysics (the structure behind), diagram (including its parts (e.g., the small horizontal line)) refers to this structure. 

İf we eliminate the "horizontal line" , we are just changing the representation (information), which refers to the metaphysics. However, now the metaphysics still stands but the human cannot make the bridge due to the non-existence of that particular line, which represented the center of a circle.

LG,

AT

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[Alex Shkotin]

Hi Alican,

You touch on many topics. Let me take one.

Of course, we need metaphysics, because a diagram is a physical thing!

In this case, we need the branch of metaphysics called mathematics, and within that, the branch called geometry.

Consider the following geometric figure, i.e., a path-connected set of points.

We will need the following units of knowledge.

Definition of the mathematical object MO1.

MO1 consists of a circle, and two of its chords, such that they are perpendicular, and the first intersects the midpoint of the second.

End of object definition.

Theorem MO1T1.

The midpoint of the first chord of MO1 is the center of the circle.

End of theorem statement.

Proof of Theorem MO1T.

<2bd>

End of proof.

We may be particularly interested in the ontological aspect.

Theorem MO1ot. Figure MO1 is possible, realizable.

Proof of MO1ot. <2bd> End of proof.

Definitions and other units of knowledge are collected in the framework of the theory. Definition MO1 is actually a definition of a class of objects. To solve a problem, for example, an engineering problem, a separate framework is constructed that uses the units of knowledge of the framework of the theory.

Over time, the following unit of knowledge will appear in the framework.

Definition.

Let o1 be a circle and ls1 a straight line segment. ls1 is a chord of o1, iff the endpoints of ls1 are located on o1.

End of definition.

It seems that Kant has a unit of knowledge: mathematics is a part of metaphysics.

The framework of a theory is the ultimate form of structured storage of theoretical knowledge.

Tschüss,

Alex

пт, 19 дек. 2025 г. в 18:10, Alican Tüzün <tuzun...@gmail.com>:

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[Matteo Bianchetti]

Hi. 

Thanks for the very interesting discussion. 

I have read sentences like  "There is some math object behind the diagram" and "a structure behind the diagram exists". These are a bit ambiguous and I would like to point out that, as used in Euclid's Elements, sometimes diagrams show that a mathematical object or configuration of objects does NOT exist. In other words, sometimes Euclidean diagrams represent impossible objects or configurations of objects. Maybe a promising way is to say either (i) that Euclidean diagrams have an epistemological role and not an ontological role or (ii) that there is an ontology of possibly contradictory objects that are used in proofs ad absurdum and an ontology of mathematical objects about which proofs wants to provide a warranted grasp.

Thanks very much,

Matteo

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[John F Sowa]

Alex, Alican, and Matteo,

The subject line mentions four kinds of things:  I believe that we can agree that that a diagram drawn on paper or a blackboard is an approximate physical representation of an abstract mathematical object.   Images and pictures are broader terms that may include drawings of diagrams as well as detailed photographs at various levels of approximation..

In the 19th  and early 20th centuries, mathematicians developed algebraic notations to such a high level of accuracy that they could map the intended content of physical mathematical drawings to linear notations that precisely represented the intended mathematical objects.  Every Euclidean geometrical diagram could be precisely translated to and from an algebraic notation in first order logic (FOL).  Euclid's letters on the diagrams are mapped to and from letters for variables in FOL.  Those letters are essential components of both the Euclidean diagrams and the FOL statements.

In the 20th century, developments in physics and astronomy showed that Euclidean geometry is only approximately true of the physical universe.   For things on earth, the error is so minute that the errors cannot be detected.  But the differences are significant for things in outer space.

Conclusion:  The diagrams drawn by Euclid over 2000 years ago are physical approximations to the abstract mathematical relations.  Those diagrams and the letters that label various parts can be translated to and from formulas in FOL.  All the diagrams that Euclid drew were physical approximations to actual mathematical patterns,for which the lines are perfectly straight and may extend to infinity in both directions.  

No physical drawing can capture those distinctions precisely.  Therefore, the algebraic notation is a less readable, but more precise verrsion of what Euclid intended.  For convenience, we can define the term mathematical diagram as the purely abstract and possibly infinite structure of which the physical drawing is an approximation.  Since the labels on the diagram map to and from variables in the algebraic notation, those labels are an essential part of the mathematical diagram.

John 

 

---

From: "Matteo Bianchetti" <mttb...@gmail.com>

[Alex Shkotin]

Matteo,

The materiality of a drawing makes it perfectly clear that the mathematical object represented in it exists. Claims about its properties may be contradictory or absurd.

For example, the drawing above copypasted by JFS asserts that point G and point F are two distinct points. And this turns out to be absurd.

You point out that drawings and the existing mathematical objects behind them can be used to help prove the absurdity of certain assumptions.

In this sense, yes, not every geometric drawing simply demonstrates the existence of a geometric object; some are used to illustrate more complex situations and reasoning.

Collecting definitions of non-existent objects is entirely possible, but somehow no one gets around to it. It seems unhelpful.

For example, a triangle with two right angles doesn't exist, but it can be defined, formalized, and proven to not exist.

However, in engineering, it is worth remembering that the requirements for a product are demonstrably impossible to fulfill.

But in short, you're certainly right: some geometric drawings play not only an ontological role but also an epistemic one.

I was interested in only two ideas: geometric drawing is material, Sowa calls geometric drawings diagrams (Why not?).

Maybe I should emphasize: "math object behind the diagram" does exists 😂

Alex

вс, 21 дек. 2025 г. в 14:59, Matteo Bianchetti <mttb...@gmail.com>:

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[Alex Shkotin]

John,

You call drawings used to depict Euclidean mathematical objects "diagrams."

It's important to emphasize that a drawing is a material object.

Some, but not all, can draw mental drawings. This ability is tested for aphantasia.

But we can't base our scientific knowledge on the abilities of a particular mind.

Drawing involves geometric thinking, which is Euclidean.

Terminology can always be agreed upon.

A drawing is the materialization of a finite set of straight and curved segments, somehow mutually arranged and intersecting, usually on a plane.

Sometimes certain types of figures (letters, signs) are treated in a radically different way from others, becoming inscriptions, markings, and notations. Working with such drawings requires special training.

Alex

пн, 22 дек. 2025 г. в 00:37, John F Sowa <so...@bestweb.net>:

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[Chris Partridge]

I am not sure that technically Euclid is the best example to take in these discussions. Sure, he is foundational but exactly what he committed to is the subject of much exegesis - quite a bit recent.

For example:  Grattan-Guinness, I.: Numbers, Magnitudes, Ratios, and Proportions in Euclid’s Elements: How Did He Handle Them? Historia Mathematica, 23(4), 355–75, (1996) --- https://www.academia.edu/9477388/ --- "In other words, in Euclid’s geometry the square on the side is not the square of the side, or the side squared; it is a planar region which has this size."

There are real questions as to whether he was as Platonist (i.e. mathematical objects are abstract) as this discussion suggests.
 
Macbeth, D. (2010). Diagrammatic Reasoning in Euclid’s Elements. In Philosophical Perspectives on Mathematical Practice 12 (pp. 235–267). College Publications. (https://www.filosoficas.unam.mx/docs/37/files/Diagrammatic_reasoning_in_Euclids_Elemen.pdf)
"Euclid’s Elements is often described as an axiomatic system in which theorems are proven and problems constructed though a chain of diagram-based reasoning about an instance of the relevant geometrical figure. It will be argued here that this characterization is mistaken along three dimensions. First, the Elements is not best thought of as an axiomatic system but is more like a system of natural deduction; its Common Notions, Postulates, and Definitions function not as premises from which to reason but instead as rules or principles according to which to reason. Secondly, demonstrations in Euclid do not involve reasoning about instances of geometrical figures, particular lines, triangles, and so on; the demonstration is instead general throughout. The chain of reasoning, finally, is not merely diagram-based, its moves, at least some of them, licensed or justified by manifest features of the diagram. It is instead diagrammatic; one reasons in the diagram in Euclid, or so it will be argued."

My gut feel is that some of this exegesis - including whether mathematical objects are abstract - could enrich the discussion.

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[John F Sowa]

Alex,

There is more to say on this topic, and I'll write another note to explain the issues in detail.  But in this note, I must emphasize one important point:  What Euclid wrote in the example below is correct.  

When Euclid let line AB be "bisected at the point D", he based that option on a previous theorem that showed how to bisect any finite line, such as AB. 

When hs wrote "Let DC be drawn at right angles to AB, he also assumed a previous proof for drawing such a line.  He also relied on another theorem for bisecting line CE at point F.

The critical issue:   Euclid wrote "I say that F is the centre of the circle ABC."

He then begins a proof by contradiction:  "For suppose it is not, but if possible, let G be the centre."

Please read the next few sentences carefully.   The last line "therefore, the angle GDB is right" implies that the point G must be identical to the point F.  They are two different names for the same point.  In algebraic notation, G=F.

In the three volumes about Euclid, Heath points out some places where Euclid made minor mistakes and ambiguities.  But this proof is not one of them.

John

 

---

From: "Alex Shkotin" <alex.s...@gmail.com>
Sent: 12/22/25 5:14 AM

Matteo,

The materiality of a drawing makes it perfectly clear that the mathematical object represented in it exists. Claims about its properties may be contradictory or absurd.

For example, the drawing copypasted by JFS asserts that point G and point F are two distinct points. And this turns out to be absurd. . 

 

---

From: "John F Sowa" <so...@bestweb.net>
Sent: 12/18

[Alex Shkotin]

Chris,

Cool links!

About the first:

The Platonists are tricky, because they seem to have assumed that the place where mathematical objects exist is the same for all people.

But we initially have the situation that there are many minds and ideals—at least as many as there are people—and mathematical objects exist there in our minds. The subtlety here may lie in what we call a specific mathematical object, and what we call a concept of a class of specific mathematical objects. I roughly remember the mathematical object picture JFS sent us. It's convenient to consider it a specific mathematical object, but it is defined with an accuracy of proportionality. But all of this exists in my mind. In physics, bodies can take shapes that approximate the shapes of certain mathematical objects.

Regarding the second:

My formalization of geometry is based on Hilbert's axiomatic theory https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf 

What I've done so far is located in the framework. 

I have links somewhere to a formalization of Euclid's theory, partial of course.

And about the ending:

Well, that's a completely different discussion. The main discussion here is about how drawings in geometry are material objects.

For me, mathematical objects are contained in my ideal, in my mind, where I conduct thought experiments with them. And they are abstract precisely in this sense.

Alex

пн, 22 дек. 2025 г. в 14:21, Chris Partridge <partri...@gmail.com>:

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[Alex Shkotin]

John,

I think Matteo was saying that Euclid's drawing depicts an absurd situation: the center of the circle, G, differs from the midpoint of the diameter, F. I simply pointed out that the drawing itself is completely existential and ontological. The assumptions are absurd.

I never even thought to claim that Euclid made a mistake here.

Alex

пн, 22 дек. 2025 г. в 22:00, John F Sowa <so...@bestweb.net>:

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