formalization. types of knowledge processing

[Alex Shkotin]

John,

I'll reply to your reply in a new thread, as the topic is much more important than my forward message.

The set of methods for processing theoretical knowledge on the path to its formalization involves several non-trivial steps.

I'll give an example, which is also a response to Chris's letter, which mentions work on the ontology of coordinate systems.

Let's take Euclidean geometry and Hilbert's theory p.36(41) and ask: how does Hilbert define a coordinate system?

I'll note that some argue that Euclid didn't have the concept of a coordinate system. Descartes is credited with the joy of discovering it.

And so, instead of defining such a remarkable object as a coordinate system, we have Hilbert's description of this concept for a coordinate system in the case where the space is a plane: 

"In a plane α, we now take two straight lines cutting each other in O at right angles as the fixed axes of rectangular co-ordinates, and lay off from O upon these two straight lines the arbitrary segments x and y. We lay off these segments upon the one side or upon the other side of O, according as they are positive or negative. At the extremities of x and y, erect perpendiculars and determine the point P of their intersection. The segments x and y are called the co-ordinates of P. Every point of the plane α is uniquely determined by its co-ordinates x, y, which may be positive, negative, or zero."

What is the first step in "formalization" (aka systematization of theoretical knowledge)? It consists of extracting from a free mathematical text the definition of the mathematical object introduced in this paragraph—a coordinate system on a plane. A definition is one of the fundamental units of knowledge. And yet, it turns out that constructing a definition from Hilbert's text isn't so simple.

Therefore, I will only provide a part of the working definition of what Hilbert presumably intended.

Definition  CoS1: a coordinate system is a set of

two straight lines intersecting at right angles, with each line having a distinct point that does not coincide with the intersection point, and these points are marked with different labels consisting of capital letters (usually "X", "Y").

and two algorithms: put_point: construct a point from s-segments, and get_coordinates: construct s-segments for a point on a plane.

◽

Where s-segment is a segment or point with a possible sign introduced in a previous page. Well, we need a definition for this term too 🏋️

It's important to note that a coordinate system isn't just a mathematical object, like an algebraic system. A coordinate system is an algorithmic system: an algebraic system plus two algorithms. Programmers call this approach to mathematical objects object-oriented.

Such a definition, extracted from the text, must be agreed upon with the author or an expert of the theory. After that, it can be incorporated into the framework of the theory as described here (PDF) Theory framework - knowledge hub message #1, where it can be formalized in one or another formal language.

Alex

[Chris Partridge]

Hi Alex,

I suspect I may be replying to something that is not quite what you intend - apologies in advance.

The reason I gave a thumbs up to John's comment is that he was clearly recognising that computers are formal - and this is important (I hope that is right John).
An interesting case is how we capture the coordinate system, whether Euclid or Hilbert, in code that can run on a computer.

And, insofar as it runs there, it is formal. 

If we are successful, we can deploy on computers.

In my paper (https://www.academia.edu/129382330/Broadening_Ontologization_Design_Embracing_Data_Pipeline_Strategies) I suggest there is an imbalance, that we work too much in the human space, attempting to formalise for humans. 
That it can be helpful to re-balance and work with machines.

Note your comment "Such a definition, extracted from the text, must be agreed upon with the author or an expert of the theory. " and my comments on experts in the paper.

Chris

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[John F Sowa]

Alex and Chris,

I strongly agree with Chris that there are multiple viewpoints or ways of analyzing and formalizing any subject for any particular purpose.

I also agree with Alex that Euclid did not specify any coordinate system.  But in Euclidean space, there are an uncountable infinity of possible coordinate systems.  Any choice is always a special case from some perspective or point of view.  

In the example by Chris, a coordinate system designed for driving a car may be a poor choice for designing a pipeline through a forest.   The new AI systems may use different kinds of coordinate systems based on tensors.

For that reason, Euclid's approach may be considered the most fundamental, and any representation in terms of a coordinate system is just one of an infinity of options.

John

 

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From: "Chris Partridge" <partri...@gmail.com>

Hi Alex,

I suspect I may be replying to something that is not quite what you intend - apologies in advance.

The reason I gave a thumbs up to John's comment is that he was clearly recognising that computers areC formal - and this is important (I hope that is right John).

To view this discussion visit https://groups.google.com/d/msgid/ontolog-forum/CAMWD8MqORuqnbZ-R-KcaLM67T6G5u5_KdDPg65k95%2BrpVHj0Zg%40mail.gmail.com.

[alex.shkotin]

Hi Chris,

I took a look at your Ontologization text; it has many interesting ideas. Here are some reactions. But first let me mention that formalization takes in all knowledge available now and in the future.

1) The "ask an expert" approach includes an important initial phase: "read an expert," i.e., the documentation. In a project, if I start asking an expert, they'll ask if I've read the project documentation.

2) I completely agree: "A good marker of the level of digitalization of an approach is how it handles natural language definitions for humans. The level of maturity of this process is a good guide to the overall level of digitalization."

That's precisely why, when I discover a lack of definitions even in the text of an exposition of a mathematical theory by such an authority as Hilbert, I have to fill this gap. After all, Hilbert didn't provide a definition because everything is clear anyway. 🙂

The art of definition is a subtle matter, even before any attempt at formalization.

3) I'm glad formalization is taking its rightful place. We have the same https://ontologforum.com/index.php/OntologySummit2025 mostly.

Of course there is much more to discuss.

By the way, regarding the computer being formal, I haven't gotten to that point yet. So far, I'm satisfied with the understanding that this is a deterministic system, until it's still working correctly. For now, I'm happy with idea that English is a HOL🕊️

Integrating different theories of the same subject area is one of the tasks of ontologists. They have to identify, formulate, and compare definitions of the same thing in different theories.

Which brings me back to Hilbert's definition of a coordinate system.

Here [1] is the current definition as a unit of knowledge. It's still a bit less polished.

Alex

[1]

Working definition: a coordinate system /SoC/ is

∘ a partially labeled figure consisting of two perpendicular lines: on one line, one of the rays is labeled with the letter X, on the other line, with the letter Y.

The coordinates of a point for the SoC are a pair of o-segments, one of which is labeled with X and the other with Y.▫️

∘ along with two algorithms:

- obtaining the coordinates of a point in the space where the SoC is located.

Let the SoC and the point pt1 be located on plane pl2, and let plane pl1 be empty. On plane pl2:

Construct line pr1 passing through pt1 and perpendicular to the X-axis. Transfer the o-segment from the intersection of pr1 with the X-axis to the origin to pl1 and label it X. Delete pr1 except for pt1 and the intersection with the X-axis.

Construct line pr2 passing through pt1 and perpendicular to the Y-axis. Transfer the o-segment from the intersection of pr2 with the Y-axis to the origin to pl1 and label it Y. Delete pr2 except for pt1 and the intersection with the Y-axis.

STOP.

- construct a point in space using coordinates and label it with an identifier in variable mt1.

Suppose there are two o-segments on plane pl1, one labeled X, the other Y, and SoC is located on plane pl2.

On the plane pl2:

Plot an X-segment on the X-axis. Construct a line pr1 passing through the resulting point and perpendicular to the X-axis.

Plot a Y-segment on the Y-axis. Construct a line pr2 passing through the resulting point and perpendicular to the Y-axis.

Mark the intersection point of pr1 and pr2 with the identifier from mt1.

Delete pr1 except for the intersection points with pr2 and the X-axis.

Delete pr2 except for the intersection points with pr1 and the Y-axis.

STOP.

◽

четверг, 9 октября 2025 г. в 15:35:22 UTC+3, Chris Partridge:

[John F Sowa]

Alex,

That sentence is extremely misleading:  "But first let me mention that formalization takes in all knowledge available now and in the future."  

A vague theory is much more general.    The advantage of a precisely formalized theory is that it is much easier to prove that it is false.

If you start with a vague theory, it's easy to modify parts of it whenever somebody finds a counterexample to one aspect.  But a formal theory is so precise and rigid, that a single false prediction is sufficient to reject everything.

John

 

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From: "alex.shkotin" <alex.s...@gmail.com>
Sent: 10/12/25 6:32 AM
To: ontolog-forum <ontolo...@googlegroups.com>
Subject: Re: [ontolog-forum] formalization. types of knowledge processing

[Alex Shkotin]

John,

Of course, we formalize only verified theoretical knowledge.

Is Euclid's or Hilbert's theory vague? Maybe in some places😁

Can Iwaszkiewicz's formal theory be proven false? No chance.

Let's start with Hilbert's theory. Its processing consists of extracting definitions into separate blocks of knowledge. This is the stage of systematization of the theory's presentation, which precedes formalization.

Our formal ontologies are primarily a structured representation of the theory's terms and their relationships, including definitions accepted by experts. And, of course, they contain many formulas, i.e., formalized knowledge. See OBO Foundry for a lot of examples.

Chris++ describe the ontology of coordinate systems here. For me, this topic begins with looking at how this mathematical structure is defined and applied in geometry. For example, in Hilbert's theory.

And when it turns out that a definition is not given, it has to be carefully extracted from the text, without deviating from the author's meaning.

That's how the definition CoS1 came about here. Incidentally, someone might accuse Hilbert of being vague in his description of the coordinate system cited here. But among mathematicians 125 years ago, this was acceptable.

Other theories will likely have different definitions, and here the ontologist's job is purely logical: analyzing and identifying the relationships between different definitions in different theories. The authors of definitions usually defend them "to the death", and we are forced to either prove their equivalence or provide examples of entities that fall under one definition and not the other.

Systematizing theoretical knowledge is a non-trivial, painstaking task.

Why do it for vague theories?

What do you think about this definition of vague theory?

Definition VTh1. Let th1 is a systematized theory. th1 is vague iff there is a vague unit of knowledge in th1. ◽

Next time we need to define what it means to be vague for axioms, definition, theorem, proof, and algorithm.

Because as we know now (Thanks Chris for pushing) the system of coordinates is an algorithmic one.

Alex

вт, 14 окт. 2025 г. в 01:41, John F Sowa <so...@bestweb.net>:

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[John F Sowa]

Alex,

There are two kinds of knowledge:

1. Pure mathematics, independent of any application of any kind.   This kind of knowledge can and must be formalized with absolute precision.  But any formalization is independent of any application to anything in the physical universe.

2. Knowledge about anything in the physical universe.  This includes any and every branch of pure and applied science:  Physics, Engineering, Philosophy, Psychology, Astronomy, Geology, Biology, Anthropology, Economics, etc, etc, etc.

Euclidean geometry as stated in any version by anybody at any time is purely theoretical.  Every theorem is FALSE about our universe.  But it is sufficiently accurate for most applications on earth; the errors are smaller than most measuring systems can detect.  

For that reason, it is a good approximation for many applications.  But even the very common application of using a cell phone to determine a location on earth depends on non-Euclidean geometry.

Conclusions:  

A. Every branch of pure mathematics has been developed by brilliant mathematicians, whose methods are far superior to 99.9% of anything that subscribers to Ontolog Forum do,  There may be a few such brilliant mathematicians on the Ontolog list, but they have far more important work to do that to participate in this project.  

B. Every application of mathematics is specialized for a particular application or  a specified set of applications.   Any specialization has a limited range of useful applications.  Any version designed for one range of applications is unlikely to be useful for other kinds of applications.

C. The few, if any, pure mathematicians who might subscribe to Ontolog Forum will publish their work in journals that specialize in the branch of mathematics they study.  If they contribute anything useful for ontology, they may send a copy or a link to Ontolog Forum.  But they are not likely to waste their time on an Ontolog project,

D. The applied mathematicians who develop versions that are specialized for various applications may publish their work in journals that are devoted to those applications.  They may also send a copy or a link to Ontolog Forum.  But they are not likely to waste their time on an Ontolog project.

Many subscribers to Ontolog Forum belong to group D.  I consider myself one of them.  I devote my work to (a) theoretical work, which I may publish in books or journals, (b) applications of such theories to practical problems, which I may publish and/or apply to projects of the company Permion Inc., which I and my colleagues are affiliated with.

For these reasons, I send occasional notes about these issues.  But I do not have any spare time to devote to an Onolog project on these topics.  I believe that publications in professional journals are more useful than devoting time to this project.

I also believe that most other people who belong to group D would do something similar.

John

 

---

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

[Alex Shkotin]

John,

Mathematics in ontologies is a great topic for our summit! What kind of mathematics do we use in our ontologies?

Oddly enough, it's usually and almost certainly mathematical logic and some real-number arithmetic, more specifically rational numbers. However, Chris Partridge and colleagues in (PDF) Coordinate Systems: Level Ascending Ontological Options  use a geometric concept called a coordinate system. This makes it particularly interesting!

And this immediately raises a wonderful question: how formalized is the part of the mathematics we use in our ontologies?

Geometry is certainly used extensively and frequently in many ontologies. We could even organize a discussion and even a poll here!

It's clear that mathematical logic is used, although some believe it's finite set theory.

My modest interest in geometry is sparked by the topic "Formalization of Mechanics." Mechanics is a science we all use in everyday life without even thinking about it.

In preparation for formalization, it's interesting to look at the relevant mathematics. She doesn't need to be a professional mathematician to do this, as they create mathematics so that everyone can use it, especially engineers and especially ontologists.

A classic example of the application of mathematics is the Königsberg bridges problem, posed and solved by Euler. Mihai Nadin analyzed it semiotically, and I analyzed it algebraically https://www.linkedin.com/pulse/theory-model-reality-alex-shkotin-dpnue/.

Mathematics is everywhere. Data is a mathematical structure tied to reality, or retaining some remnants of its connection to reality, such as units of measurement. If only numbers somehow interconnected remain, it's abstract data.

I was very surprised that Hilbert didn't define such a nice and useful mathematical object as a coordinate system. Extracting a definition from an ancient mathematical text is like archeology.

Anybody doesn't have to be a professional mathematician to apply mathematics.

Alex

ср, 15 окт. 2025 г. в 01:00, John F Sowa <so...@bestweb.net>:

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[John F Sowa]

Alex,

There is no limit to the amount and the many different kinds of mathematics that are necessary for ontology.

It is true that the hierarchy of types or categories of entities in the ontology can be specified by first-order logic (FOL) -- sometimes called GOFOL (Good Old FOL).

But any kind of mathematics may be necessary for defining the kinds of entities and relations that are represented by simple letters in the GOFOL.

For example, I was just reading an article about the mathematics that may be used to express the relationships that are typically called the unobserved "dark matter" that seems to pervade the universe.  See the article "A new attempt to explain the accelerated expansion of the universe":   https://phys.org/news/2025-10-expansion-universe.html .

By introducing new mathematical definitions, they eliminate the need for the mysterious dark matter.  By introducing the new definitions, they eliminate the category called "dark matter" from their ontology.

Changes like these occur in every branch of science.  A complete specification of any ontology must anticipate such modifications in every branch of science, engineering, business. etc.

There is no such thing as a fixed and fully specified ontology.  Every branch is constantly changing.  Even the kind of mathematics used to specify any branch is constantly changing.

John

__________

 

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

John,

[Alex Shkotin]

John,

FOL isn't sufficient even for mathematics. Barwise, for example, pointed this out; see citations in our forum here. That's why many, including myself, prefer HOL. Especially since English is HOL.

An ontology, like any computer artifact, must be applicable to specific uses, meaning it must indicate where, when, and how to apply it. This could be a great topic for our summit: the practical application of ontologies.

About the note A new attempt to explain the accelerated expansion of the universe.

Even the title of the note suggests that this is an attempt to write yet another theory of the expansion of the universe. Yes, that's how science works: there's a phenomenon, and then there are various theories that more or less explain it. Having multiple theories for the same phenomenon is, in a sense, a headache for ontologists.

Processing a scientific article involves systematization (down to definitions) and formalization.

But a note is just the surface of a scientific article. It's usually better to work with the original source. This is one of the rules of systematization. Popularizers simplify and can distort.

The number of theories in cosmology is significant, and their systematization and construction of frameworks awaits enthusiasts.

The original source is closed to the general public.

I asked Grok to identify [1] the key terms from the article's note. This is the first step in systematizing the theory. Of course, his answer needs to be verified.

The second step is to obtain a definition of each term.

The third is to construct a framework for the theory, analogous to the theory of undirected graphs. 

Alex

[1] https://grok.com/share/bGVnYWN5_d714ff73-3e09-4c43-b590-d7ea943bd774

Q:Highlight the key terms from this short note https://phys.org/news/2025-10-expansion-universe.html

A:

### Key Terms from the Article: "New attempt to explain the accelerated expansion of the universe"

The article discusses a research team's alternative explanation for the universe's accelerating expansion using Finsler gravity, potentially eliminating the need for dark energy. Here are the highlighted key terms and concepts extracted from the main content:

- **Accelerated expansion of the universe**

- **Ever-increasing rate of expansion**

- **Dark energy**

- **Friedmann equations**

- **Dark energy term**

- **General relativity (GR)**

- **Finsler gravity**

- **Finsler-Friedmann equations**

- **Generalized spacetime geometry**

- **Exponentially expanding universe**

- **Vacuum expansion**

чт, 16 окт. 2025 г. в 07:39, John F Sowa <so...@bestweb.net>:

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