The concept of a point from Euclid to Hilbert

[alex.shkotin]

At the session of the second synthesis Terry Bollinger mentioned that Euclid's Elements begins with the definition of a point. A definition that is delightful: "A point is that which has no parts."

I would like to emphasize that 2300 years later, in Hilbert's axiomatic theory for the same entities in the same Euclidean space, the situation with the definition for a point has changed significantly, namely.

For Hilbert, a point is a primary (aka primitive) concept, along with two others: an infinite straight line and a plane.

For these three (there are no other primary) types of objects, 6 primary relations are introduced - all but one are binary, betweenness is ternary.

So what is a point for Hilbert?

This is any class of objects if it is possible to provide two more classes of objects for it, specify the geometry relations on all three, and check that the axioms of geometry are satisfied. The class of objects for which the axioms are satisfied as for a point can be considered points. I think Hilbert once joked "even beer mugs". This example illustrates how we apply our axiomatic theories: specify real or ideal objects for primary classes (sorts), specify primary relations on them, and check that the axioms are satisfied. If the axioms are satisfied (for example, theories of undirected graphs), go ahead, the theory is applicable in all its glory.     

https://www.linkedin.com/pulse/concept-point-from-euclid-hilbert-alex-shkotin-fsg2e/

Alex

[Ravi Sharma]

Alex

Does that address multidimensional at least Euclidean geometry to 3rd Dimension?

We can  Apply Hilbert criteria and verify?

Thanks.

Ravi

(Dr. Ravi Sharma, Ph.D. USA)

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[Alex Shkotin]

Ravi,

Answer to the first question is yes and not "at least" - it's exactly about 3D Euclidean geometry. For planimetry we have Tarski's axiomatic theory.   

And the second question I do not understand. But some thought:

-may be dot in some spaces can have parts.

-is that possible to prove from Hilbert's axioms that dot does not have parts?

Relationship between Hilbert's axiomatic theory and Euclid's propositions is an interesting topic.

And we begin this way: we read, understand, and agree with all Hilbert's axioms from the point of view of our intuition of 3D space around us.

Alex

сб, 17 мая 2025 г. в 15:53, Ravi Sharma <drravi...@gmail.com>:

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[John F Sowa]

Alex,

Re Hilbert:   He was a good mathematician.  But there are many other good mathematicians and scientists and engineers who find it important to adopt different definitions for the same terms for different purposes.  It's not possible or advisable to choose one definition or set of definitions that can or should be adopted as an international standard for all purposes.

That is why Cyc never used single words for their categories.   They could have an open-ended variety of different definitions.  For example, Aristotle-point, Hilbert-point, etc.

Recommendation:  A dictionary is a better guide for a universal ontology than any fixed set of formal definitions.  Special purpose ontologies can be formalized, but different purposes may require very different definitions for the same terms.

John

 

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

[John F Sowa]

Alex,

Those different options and their formalizations in FOL might be useful for differentl ontologies.  There is no reasonn to insist on any one of them as the ONLY choice.  I would recommend the version that defines a point as a locus (a method for specifying a location), NOT as a part of a line.

This is basically what Aristotle stated.  It has many advantages,  For example, if you cut a line in two, what happens to the point at the cut?  Does it go to the left or the right?

But if a line is not composed of points, the locus is just a specification of a location, not an object.  Therefore, there is no "thing" called a point that has to go one way or the other.

This is just one of many reasons why a general ontology should be VAGUE about issues that may be treated in different ways for different applications.

General principle:  If you don't need to make a decision about some detail, it's best to leave it undecided. Different applications may have legitimate reasons for making decisions that you had not considered.

John

 

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From: "alex.shkotin" <alex.s...@gmail.com>

[Alex Shkotin]

John,

IMHO "scientists and engineers" adopts not just definitions but one or another theory to apply it to solve one or another practical problem. Idea of "set of definitions" is much better, but theory is not only a set of agreed definitions (it's a bad idea to take definitions from different theories). Theory has a set of derivation rules. And this set is not just modus ponens, there are many others. In FOL we have approximately six rules we use to construct formal proof. In practice, the number of rules is much larger (see). 

There is no task "to choose one definition or set of definitions that can or should be adopted as an international standard for all purposes."

It may be a task (as I mentioned in our last session) to formalize Hilbert's theory of Euclid's geometry. 

"open-ended variety of different definitions" is a proclamation of a kind of bad infinity. What we have is approximately 10 different axiomatic theories for Euclid's geometry. And this is true that having a problem in mind we choose a theory which best suits our goal and how to solve this problem. Euler invented basic undirected graphs theory trying to solve the problem of Konigsberg bridges.

But today's point is not inventing a new theory. We have a lot. But systematize and formalize existing. Have a look at OBO Foundry ontologies to see how many formulas we have there. What are the theories behind?

Recommendation: Keep in mind which one theoretical knowledge you use to solve a problem. Remember that formal definitions are not for you, it's for computer algorithms to make proper knowledge processing to help you to solve a problem.

Definition is a unit of theory, but theory is more than a system of agreed definitions.

Alex

вс, 18 мая 2025 г. в 21:56, John F Sowa <so...@bestweb.net>:

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[Alex Shkotin]

John,

It's just a proposal to discuss what is a point in different theories. I took Hilbert's theory as it is well known.  It is definitely interesting to take 9 others into account. But this looks like a small research.

If you have a reference to a theory where a point is a locus, please give a reference.

And I think Hilbert does not use the relationship "part_of" in his theory at all. This is a very interesting topic that "part_of" is always an abstraction from real relationships. In Hilbert's theory we have point lies on line, and line lies on plane.

Do we need a definition for a part_of relationship?

General principle: Abstraction is a fundamental mental action. Even what we see with good eyes is an abstraction from real matter. We use abstraction further when we choose a theoretical knowledge to use. We use theoretical knowledge to describe and then solve a problem. Be careful with abstraction. Remember that there may be a child in the water.

Alex

пн, 19 мая 2025 г. в 05:30, John F Sowa <so...@bestweb.net>:

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[John F Sowa]

Alex,

I'll dig up the references, but more discussion is needed to explain the issues.

Most important point:   The general framework for ontology must support an open-ended number of options, because different versions and applications of science, engineering, and ordinary language have many different ways of thinking, talking, and working.  

A universal ontology must support every possible way of thinking.  Since nobody can anticipate what may be invented in the future, a general ontology must be open-ended.  New branches with totally different ways of thinking and acting may be added at any time.

John

 

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

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[Alex Shkotin]

John,

Topics of "general framework for ontology" and "universal ontology" are interesting and classical. I am not sure I am ready to say something new.

Let me continue a little about the research I have mentioned before.

In the Russian version of the article Hilbert's axioms - Wikipedia there is a section about other axiomatic systems by the way for the same kind of space - euclidean and the same kind of objects and relationships in it.

If we truslate this section by Google-trans we get this [1].

In section "More modern axiomatics:" we have only 5 axiomatic theories to look at point appearance. This is a small research, I am sure.

As we see from "Weyl axiomatics  - operates with undefined concepts of a point and a free vector." Here point is a primary (aka primitive) concept.

Axiomatizing theory we find out primary (aka primitive, undefined) concepts and relationships and axiomatize them. But we define all other concepts and relationships. 

Applying theory to an area of interest we must show there our primary concepts and relationships, like this: the stars and planets on the sky are points. Beams of lights are moving along the straight lines, and orbits of planets move over the planes. Well, we can apply Hilbert's axiomatic theory to describe and even solve some astronomical problems. At least their geometrical part.

If we speak more precisely, we must say that we have an axiom: every physical object at any moment of time occupies some place in physical space. Thus, we have a mapping of a physical object into a geometric figure located somewhere in space. This is a mapping of "location". Perhaps you call it a locus.

Alex

[1]

Other axiom systems[ edit | edit code ]

Creators of pre-Hilbert systems:

• Euclid

• Pash

• Shur

• Peano (includes the concept of " movement ")

• Veronese

• M. Pieri (1899)

Related to Gilbert's:

• W. F. Kagan (1902)

• O. Veblen (1904)

• A. Kolmogorov

More modern axiomatics:

• Axiomatics of Alexandrov

• Bachmann's axiomatics

• Tarski's axiomatics

• Birkhoff's axiomatics  - contains the " ruler axiom " and the " protractor axiom ". Its variants are used in most American school textbooks, and Pogorelov's axiomatics is close to it .

• Weyl axiomatics  - operates with undefined concepts of a point and a free vector . A line and a plane are defined as sets of points.

ср, 21 мая 2025 г. в 19:56, John F Sowa <so...@bestweb.net>:

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[Ravi Sharma]

Alex

So are all axioms for Euclidean spaces, including kolmogorov and tarski?

Thanks.

Ravi

(Dr. Ravi Sharma, Ph.D. USA)

NASA Apollo Achievement Award

Former Scientific Secretary iSRO HQ

Ontolog Board of Trustees

Particle and Space Physics

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[Alex Shkotin]

Ravi,

Yes. It is usual to have more than one theory for the same part of reality.

Alex

чт, 22 мая 2025 г. в 15:31, Ravi Sharma <drravi...@gmail.com>:

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[John F Sowa]

Alex,

Your long list of axiom systems show why we should not prioritize any one of them.

Any or all of them should be available for some application.  I'll emphasize the point by the engineer and statistician George Box:  "All models are wrong; some are useful."

For any and every practical application of any kind, there are always error bounds and estimates.

John

 

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

John,

[Alex Shkotin]

John

Exactly! While all theories are mostly beautiful, we need to know how to apply them. And this applicability reasoning is crucial. To open up a little Box's point. It may be an issue that to apply this particular theory to this particular problem is the wrong idea, and in other cases, is not useful. As Landau wrote, we do not apply quantum mechanics where Newtonian is applicable.

One task is to formalize theory itself like the theory of undirected graphs [1]. Another task to formalize one or another problem solving, using this theory [2]. 

But the task #1 is to recognize somehow (by abstraction, reduction, generalization etc.) that this particular theory is applicable to solve this particular practical task properly, taking into account "error bounds and estimates" [3].

Theory itself consists sometimes of a large number of subtheories for different kinds of objects and processes, i.e. situations.

In this case the structure of a theory itself is not trivial.

Alex

[1] (PDF) Theory framework - knowledge hub message #1

[2] Specific tasks of Ugraphia on a particular structure (formulations, solutions, placement in the framework)

[3] https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg

вс, 25 мая 2025 г. в 03:35, John F Sowa <so...@bestweb.net>:

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[Michael DeBellis]

John Sowa said: Most important point:   The general framework for ontology must support an open-ended number of options, because different versions and applications of science, engineering, and ordinary language have many different ways of thinking, talking, and working.  A universal ontology must support every possible way of thinking.  Since nobody can anticipate what may be invented in the future, a general ontology must be open-ended.  New branches with totally different ways of thinking and acting may be added at any time.

I agree with most of what John said but where I disagree is the concept of a universal ontology. I think his arguments show why there can't and shouldn't be one universal ontology at all. If I'm doing quantum physics my ontology isn't going to have the same core definitions as if I'm doing animal behavior. And even more so between doing science and having ontologies that support business systems like data lakes, data catalogs, etc. For those latter ontologies we want a foundational ontology grounded in common sense (Gist for example). For science, common sense is what you are moving beyond, so you do not want your upper model to be based on common sense when doing science. Rather it should be based on whatever core axioms apply to the specific scientific model you are using. 

Michael

https://www.michaeldebellis.com/blog

[Philip Jackson]

Michael,

Thanks for these comments. I guess you would like an "ontology of ontologies", i.e. a meta-ontology. (?)

Phil

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[John F Sowa]

Michael,

Michael,

The point I was making is that you can't have a finite universal ontology since the number of options is infinite.

But an infinite ontology is possible, but you couldn't represent it in any finite universe in a finite time.

So it's theoretically possible in the thought of an infinite God.   But we couldn't store it unless we had infinite Bible.

And if you had an infinite Bible, you wouldn't have the time to find what you're looking for.

John

 

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