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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