Re: subsumption in KR

AI Review based on the Google search string "the difference between
knowledge representation and mathematics "

*"Mathematics is the formal study of quantity, structure, space, and
change, using axioms, proofs, and deductive reasoning. *

*Knowledge representation is a field of artificial intelligence that
focuses on creating frameworks and languages to encode human or real-world
knowledge in a way that a computer can understand and use for reasoning and
decision-making. *

*The key difference is that mathematics provides a specific, universal, and
formal system of knowledge, while knowledge representation is a broader,
more flexible system for encoding various types of knowledge for
computational use, which can include mathematical knowledge.*

*Mathematics*
*Subject: Studies numbers, quantities, shapes, space, and change.*
*Methodology: Uses axioms, theorems, proofs, and deductive reasoning.*
*Nature: A formal, abstract system with universal, timeless truths.*
*Purpose: To establish fundamental truths and provide a tool for
understanding the world through logic and calculation.*

*Knowledge Representation*
*Subject: Aims to represent a wide range of knowledge, including factual,
procedural, and conceptual knowledge, often about the real world.*
*Methodology: Uses a variety of techniques, such as logical formalisms,
semantic networks, and frame-based systems, to structure information for
computers.*
*Nature: A broad field within AI, focused on how to encode knowledge for
machine reasoning, which can be more flexible and less rigid than pure
mathematics.*
*Purpose: To enable intelligent systems to perform tasks like learning,
reasoning, and making decisions by providing them with a structured
understanding of information. *

*Relationship between the two*
*Mathematics is a tool for knowledge representation: Mathematical equations
are one type of knowledge that can be represented in a computer system.
However, this requires carefully choosing how to represent them, and the
computer can use these representations to perform computations.*

*Knowledge representation provides the framework: Knowledge representation
provides the means to structure and use mathematical knowledge, and also to
integrate it with other types of knowledge, like qualitative relationships
or real-world constraints."*

The difference is that mathematics is UNIVERSAL, knowledge representation
presupposes observers and observable processes in THE REAL WORLD.

And all tools used by knowledge representation are MATHEMATICAL.

This is the arrogance of many computer scientists, physicists and academics
who at their own peril want to ignore the Godel-Turing-Tarski-Chaitin
framework of limitations on completeness, consistency and computability and
even informational encoding.

Knowledge implies an observer who communicates information about the real
world.

Some articles to drive the message home.

https://www.quantamagazine.org/a-new-bridge-links-the-strange-math-of-infinity-to-computer-science-20251121/

https://futurism.com/artificial-intelligence/large-language-models-willnever-be-intelligent

https://www.theverge.com/ai-artificial-intelligence/827820/large-language-models-ai-intelligence-neuroscience-problems

https://royalsocietypublishing.org/doi/10.1098/rstb.2024.0314

https://gwern.net/doc/psychology/linguistics/2024-fedorenko.pdf

The bottom line is: knowledge is a domain that cannot be fully captured by
mathematics, knowledge representation,  language or any forms of
communication and formal representation methods.

Natural language fails at representation of knowledge which why LLMs will
fail.

Mathematics that are not necessarily about the real world with or without
observers are bound by Godel-Turing-Tarski-Chaitin.

Computer science and hence computability and hence AI face this DOUBLE
LIMITATION inherent to mathematics and natural language combined.

We can only create AI that rises above the current level if it is natural
language-agnostic and that requires mathematics that are descriptive and
representable set theory based and when we assume V=L, that is the
"universe of all sets" V can be constructively represented in the universe
of representable sets L.

Whereas L implies an observer,  V does not. V=L is the quiet assumption
made by computer scientists in dealing with knowledge representation. And
with profound consequences,  because constructibility implies the
Godel-Tarski-Turing-Chaitin limitations. And when we talk about LLMs it
gets worse.

Now we see why consciousness is such a hard problem, because in rational
thought, which is an aspect of consciousness, natural language and even
mathematics can be sidestepped by the marvelous human brain.

I had the honor of getting my hands on a very inspiring very limited
edition published book from The Cultural Centre of His Holiness the Dalai
Lama,  titled " Quantum Physics, Brain Function in Modern Science and
Buddhist Philosophy ", Tibet House, ISBN 978-81-953361-7-3.

Why bring up this book? Because it sheds some light on why we cannot assume
all knowledge about the real world we observe to be able to be captured
formally.

And I have been posing questions about this through various intermediaries
exactly about how mathematics confirm this.

Mathematics confirms the limitations of knowledge representation in the
real world and thus sits above it.


On Sat, Nov 29, 2025, 11:46 Paola Di Maio <paola.dimaio@gmail.com> wrote:

> I would normally not commen on statements about mathematics because as
> stated in previous posts here we are concerned
> with NL
>
> However, if it helps, a reminder that it is what is generally accepted,
>
>
> 1. maths is  type of KR
> 2. is not NL KR *which is what we use in LLM
>
> Subsumption
> Subsumption is a key concept in knowledge representation, ontology design,
> and logic-based AI. It describes a “is-a” hierarchical relationship where
> one concept is more general and another is more specific.
> mathematics *is* a knowledge representation *although it may be
> understood or defined in other ways because  it provides:
>
>    -
>
>    Formal symbols (numbers, variables, operators)
>    -
>
>    Structured syntax (equations, functions, relations)
>    -
>
>    Precise semantics (well-defined meanings)
>    -
>
>    Inference rules (logical deduction, proof)
>
> and much more not related to what we are discussing here
>
>
>  Other views may also exist, in the vast universe of discourse, that may
> or may not contribute to the discussions in hand.
> .
>
> 9Milton Ponson
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